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Kinetic Boltzmann, Vlasov and Related Equations


Victor Vedenyapin Keldysh Institute of Applied Mathematics (Russian Academy of Sciences) Russia

Alexander Sinitsyn Departamento de Mathemàticas Facultad de Ciencias Universidad Nacional de Colombia Bogotà, Colombia

Eugene Dulov Facultad de Ciencia y Tecnolog´ıa Universidad de Ciencias Aplicadas y Ambientales U.D.C.A

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Elsevier 32 Jamestown Road London NW1 7BY 225 Wyman Street, Waltham, MA 02451, USA First edition 2011 c 2011 Elsevier Inc. All rights reserved. Copyright No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangement with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-123-87779-6

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Preface

The Boltzmann and Vlasov equations played a great role in the past. Their importance can still be seen in modern natural sciences, technique, engineering, and even in the philosophy of science. The classical Boltzmann equation derived in 1872 became a cornerstone for the molecular–kinetic theory, the second law of thermodynamics (increasing of enthropy), and for the derivaion of the basic hydrodynamic equations. Discovering and studying different physical, chemical, and astronomy objects and processes and even popular nanotechnological applications opened new fields for Boltzmann and Vlasov equations. Examples include diluted gas, radiation, neutral particles transportation, atmosphere optics, nuclear reactor modeling, and so on. The Vlasov equation was obtained in 1938 and served as a basis of plasma physics, but it also describes large-scale processes and galaxies in astronomy (the starwind theory). The development of plasma units such as tokomak or plasma engines also are supported by the Vlasov equation. A careful reader who has looked at the Table of Contents of this book will note that it contains not only the basics and common facts, but many of the results discussed in this book were obtained recently. Hence, the first chapter is devoted to the historical introduction and outlines principle details described in other chapters. The second chapter introduces Vlasov-type equations or equations of the self-consistent fields in connection with a problem of multiple body dynamics and the use of the Lagrangian coordinates in the Vlasov equation. It also reminds us about its links with hydrodynamical descriptions. For better understanding, we present several examples with exact solutions. The Vlasov-Maxwell equations are introduced in Chapter 3. To make the derivation technique comprehensible, first we start with the particle shift along the trajectories of an arbitrary system of the ordinary differential equations; we then follow with particle system movement in metric spaces. The next chapter deals with the Vlasov equation for plasma and energetic substitution, given with an analogy with the Bernoulli integral. In Chapter 5 we introduce kinetic equations, the Boltzmann equation, the VlasovPoisson (VP), and the Vlasov-Maxwell (VM) systems, and describe their mathematical structure. Section 5.6 describes several open fundamental problems known for VP and VM systems. Chapter 6 lists references and is devoted to students, engineers, and postgraduate students. Here we introduce an ansatz of distribution function for two-component plasma. Simple problem statements are introduced for nonlinear elliptic equations both for Cauchy and bifurcation cases.

xii

Preface

In Chapter 7, we study special cases of stationary and nonstationary solutions of the VM system. These solutions introduce the systems of nonlocal semilinear elliptic equations with boundary conditions. Applying the lower-upper solution method, we establish the existence theorems for solutions of the semilinear nonlocal elliptic boundary value problem under corresponding restrictions on distribution function. We also give several examples of the solutions at the end of the chapter. The bifurcation problem for the stationary solutions of VM system is considered in Chapter 8. It is translated into the bifurcation problem of the semilinear elliptic system and is studied as an operator equation in Banach space. Using a classical approach by Lyapunov–Schmidt, the branching equation is derived and asymptotics of nontrivial branches of solutions is studied. Here the principal idea is to study a potential BEq, since the system of elliptic equations is potential. Further investigation establishes the existence theorem for the bifurcation points and reveals the asymptotic properties of nontrivial branches of the solutions of VM system. In Chapter 9, we discuss the general and linear Boltzmann equations and correspondence with hydrodynamics and quantum physics. Discrete Boltzmann models are investigated in Chapter 10, paying special attention to the models of interactions between particles in relation with conservation laws and validity of H-theorem. In Chapter 11, we study the Boltzmann equation’s symmetry. Here, commutation of collision operator with rotation group comes first, as it provides us with a solution for momentum system. An appendix to this chapter gives an example for nonlinear equations as convergent series for superposition of running waves. Chapter 12 studies discrete models for gas mixtures with different particle masses and corresponding collision models. This applied problem is extremely important for numerical modeling, as any appropriate model should be checked first to see if it is compliant with conservation laws. Chapter 13 investigates the spectrum of Hamiltonians in application to quantum optics. Here we can use the same “ideology” in applying conservation laws that we used earlier, studying discrete models of Boltzmann equation. This approach to the conservation laws reduces the dimension of the spectrum problem to the finite-dimensional one. It has already been used by physicists in construction of frequency convertors. Chapter 14 studies the stationary self-consistent problem of magnetic insulation under space-charge limitation via the VM system. In a dimensionless form of the VM system, the ratio of the typical particle velocity at the cathode related to the velocity reached at the anode appears as a small parameter. The associated perturbation analysis provides a mathematical framework to the results of Langmuir and Compton. We study the extension of this approach, based on the Child-Langmuir asymptotics to magnetized flows. Chapter 15 shows that when the VM system is written in Hamilton form using nonlocal Poisson bracket, the use of Hamiltonian formal approach for the real kinetic equations still is under discussion. This unique attempt — to approximate the Poisson bracket in VM system by a finite dimensional one — exists similarly between the VM system and the Liouville equation. Hence, the study of the approximate integration methods for analytically integrable and nonintegrable Liouville equations is a cornerstone for development of wavelet solutions for the VM system. Thus, we propose an

Preface

xiii

effective technique of approximate integration for the Cauchy problem of the generalized Liouville equation based on the orthogonal decomposition over Hermite polynomials and Hermite functions. The respective mean convergence theorems are proved. The importance of this approach is related to the possibility of automated analytic computations in modern mathematical packages such as Maple or Mathematica.

About the Authors

Victor Vedenyapin Vedenyapin graduated from Faculty of Mathematics and Mechanics of Lomonosov Moscow State University in 1971 (department of differential geometry). His Ph.D. thesis was defended in 1977 and D.Sci. Thesis in 1989. A list of publications contains more than 100 titles. Since 1992, with M. Maslennicov and V. Dorodnitsyn, Vedenyapin has led the research seminar at the Keldysh Institute of Applied Mathematics on mathematical physics. Professor of Moscow University of Physics and Technology since 1992. USSR State Prize winner (1989) Mathematical Methods in Boltzmann Equation. Fields of interest: kinetic equations; Boltzmann equation; Vlasov equation; entropy; Quantum Hamiltonians, Ergodic theory. Alexandr Sinitsyn Sinitsyn graduated from Irkutsk Polyteknical Institute on 1983. His Ph.D Thesis was defended in 1989 and D.Sci Thesis in 2004. Professor of Colombian National University. He published 60 articles and 2 books. Co-director of INTAS research project “PDE modelling semiconductors.” Worked as visiting professor in Paul Sabatier University, Toulouse, France. Fields of interest: kinetic, Boltzmann, Vlasov equations and their applications. Eugene Dulov Dulov graduated from Faculty of Mathematics and Mechanics of Lomonosov Moscow State University in 1993. His Ph.D. thesis was defended in 1997. He is a Lecturer of Ulyanovsk State University, Professor of Colombian National University. A list of publications contains 38 titles. Fields of interest: numerical methods, kinetic, Vlasov equations and their applications, development of algorithms.

1 Principal Concepts of Kinetic Equations

1.1 Introduction Kinetic equations describe the evolution of distribution function F(t, v, x) of molecules or other objects (like electrons, ions, stars, galaxy, or galactic aggregations) with respect to their velocities v and space coordinates x at moment t. In particular, this means that a number of particles in the element of phase volume dvdx is given by a quantity F(t, v, x) dvdx. The simplest example equation known as an equation of free motion is given below: ∂F ∂F =0 (1.1.1) + v, ∂t ∂x and could be simply resolved—F(t, v, x) = F(0, v, x − vt). The goal of this book is to study the kinetic Boltzmann and Vlasov equations.

1.2 Kinetic Equations of Boltzmann Kind The first kinetic equation to be studied was Boltzmann’s. It considers collision processes via collision integral added into (1.1.1): ∂F ∂F + v, = J[F, F]. (1.2.1) ∂t ∂x Collision integral J[F, F] is a quadratic operator, considering a pairwise collision of particles. Equation (1.2.1) was obtained by Maxwell and Boltzmann for derivation of Maxwellian distribution by velocities. This result has been used for explanation of Clapeyron ideal gas law (see Section 1.4). The already mentioned Maxwellian distribution is connected with one of the first known classic results for the Boltzmann equation (1.2.1)—a proof of the so-called H- theorem. This theorem claims that functional Z H[F] = F ln Fdvdx Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00001-6 c 2011 Elsevier Inc. All rights reserved.

2


does not increase for Boltzmann equation, i.e., dH/dt ≤ 0. Since H equals to entropy with negative sign, this fact was interpreted by Boltzmann as a proof that entropy increases, i.e., a justification of the second thermodynamics law. The inequality of H- theorem holds only sometimes, and we will discuss corresponding conditions later in Chapter 9. When entropy stays unchanged, we obtain a Maxwellian distribution. Hence, the H- theorem justifies not only a stationary state of Maxwellian distribution, but tends to it also, its stability and the second thermodynamics law. However, the Boltzmann equation was developed by Maxwell for more broad aims. The initial Maxwell goal was to obtain the equation of continuos medium (NavierStokes type) derived from the Boltzmann equation; namely, to define transport coefficients (heat and viscosity) and their dependence on intermolecular interaction. He succeeded with intermolecular interaction potential of the form U(r) = r−4 (the socalled Maxwellian molecules), when the collision integral becomes simple. Boltzmann [52] and Hilbert (Hilbert method [67]) tried to state the similar results for another potential, but never succeeded. However, Chempen and Enskog [67, 158] achieved this goal by means of special scheme based on perturbation theory, known today as Chempen–Enskog method. The stakes were high, since a solution could provide qualitative forecasts in molecular-kinetic theory, an issue of hard criticism at that time. Their result predicted a thermal diffusion, and this issue was resolved. The tension was really high and the problem was discussed both by scientists (Mach and Avenarius, for example) and philosophers and polititians. But the solution of Chempen and Enskog was a little bit late, because the Avogadro number was calculated by two different ways and the estimations were quite close. This justified the molecular-kinetic theory at least for scientists and a scientific community calmed down. At the present time, this equation with respective corollaries has several different applications, for example, modeling of the middle atmosphere layers. Tall atmosphere layers are well described by equation of free motion (1.1.1) (also called Knudsen or free gas equation). Lower layers are described by gas dynamics equations, which also are derived from the Boltzmann equation. Derivation and numerical modeling of the two-layer models (see [158]) related with modeling of aircraft motion keeps them important. The other important application deals with chemical kinetics and mixture modeling especially, known as a discrete models of Boltzmann equations (see Chapters 10 and 12). The other widely used corollary of Boltzmann equation is the transport equation, describing the scattering of particles on a fixed background. This is a linear Boltzmann equation. Such equations are used for description of neutrons transport in nuclear reactors and radiation transport in atmosphere when photons are scattered by medium. The limit case of the Boltzmann equation, known as the Landau equation, appears when the main contribution in scattering cross-section is made by strong scattering forward. It is used for plasma description. There also are quantum analogs of the Boltzmann equation, called UehlingUhlenbeck equations. For these equations, Fermi-Dirak or Boze-Einshtain distributions are steady state instead of Maxwell distribution. Therefore, one can represent the hierarchy of Boltzmann-type equations in the scheme seen in Figure 1.1.

Principal Concepts of Kinetic Equations

3

Boltzmann equation

Discrete models

Uehling-Uhlenbeck equation Transport equation

Landau equation ?

Figure 1.1 The hierarchy of Boltzmann type equations illustrates connections among them.

The lines marked with question signs outline the fact that corresponding equations are still undiscovered (Landau approximation for Uehling-Uhlenbeck equations, for example).

1.3 Vlasov’s Type Equations Vlasov-type equations are compared with equations of the Boltzmann type describing short-range interactions. The Vlasov equation or equation of self-consistent field has the form ∂F ∂F ∂F + v, + f (F), = 0. (1.3.1) ∂t ∂x ∂v Here the force f is the functional of a distribution function F, and equation (1.3.1) has the form of shift equation along characteristics. A simplest kind of functional (force f ) describing a dependence from distribution function corresponds to pairwise interaction potential K(x, y): Z f = −∇ K(x, y)F(y, v, t)dvdy. (1.3.2) This kind of interaction introduces the system of Vlasov equations. Generally speaking, mostly we are using a phrase—“Vlasov plus something more” equations—aimed to distinguish between the kinds of interactions. There are VlasovPoisson, Vlasov-Maxwell, Vlasov-Einstein, and Vlasov-Yang-Mills equations (see Chapter 3 for details). The Vlasov-Poisson equation exists for two kinds of problems: for gravitation and plasma. In both cases, we exchange a potential (1.3.2) by Poisson equation applying a Laplace operator. Here K(x, y) assumed to be a fundamental solution [303] of a Laplace operator: 1x K(x, y) = δ(x − y). Therefore, K is a potential of a sin1 |x − y|−1 ) (Coulomb law), of a gle charge in three-dimensional case (K(x, y) = − 4π 1 thread (K(x, y) = 2π ln|x − y|) for a two-dimensional, and plane (K(x, y) = 12 |x − y|) in one-dimensional cases [303]. When we study gravitational case exchanging Newton type interaction by the Einstein’s one, we obtain a so-called Vlasov-Einstein equation. If we exchange electrostatic by electrodynamic interaction required for plasma, we obtain Vlasov-Maxwell equations. If not a charge, but some vector characteristic is

4


Vlasov-Poisson equation

Vlasov-Einstein equation

Vlasov-Maxwell equation

Vlasov-Yang-Mills equation

Figure 1.2 The hierarchy of Vlasov equations.

unchanged (like isotopic charge or color), then we should take matrices instead of electromagnetic 4-potentials, thus, obtaining Yang-Mills equations. These equations represent the modern theory of joint weak, electric and strong interactions. Finally, we can introduce the following hierarchy of Vlasov equations: The given hierarchy (Figure 1.2) gives us an incredible example of interactions between mathematics and several branches of natural sciences. Some of these connections will be presented in the following chapters, when we will study some basic substitutions into Vlasov equation: l

l

l

l

Equation of N bodies dynamics as a corollary of Vlasov equation obtained by substitution of the sum of delta-functions; Substitution in the form of integrals of delta-functions and Lagrangian coordinates is used for oscillators and anti-oscillators and two-Hamiltonian structures; Euler-Lagrangian coordinates and hydrodynamic substitution is used in N-layer and continuum-layer hydrodynamics for modeling expanding universe, overlaping and even applicability of hydrodynamic description (see Chapter 2); Energetic substitution, when distribution function depends only from of energy (see Chapter 4). In this case, equation (1.3.1) is satisfied, and (1.3.2) transforms into nonlinear equation for potential. In applications, there were plasma diode (Lengmure diode), Debay equations for electrolytes and Len-Emden equation in gravitation. In math this equation earlier has been studied in geometry, and it is called Liouville equation. In two-dimensial case it possesses a vast group of symmetries (conformal group).

The last energetic substitution gives equations similar to Bernoulli equations for Euler equations. Their fates are similar—they were initially discovered for particular cases. These equations then were studied before the generalized Vlasov equations had been written. Moreover, they were introduced from the same energetic point of view, expressing the energy conservation law.

1.4 How did the Concept of Distribution Function Explain Molecular-Kinetic and Gas Laws to Maxwell Let F(t, v, x) be a distribution function of molecules by their velocities v ∈ R3 and space x ∈ R3 at moment t. It means that quantity F(t, v, x)dvdx represent a number of particles in the element of phase space dvdx. Let us consider the following macroscopic values:


5

1. The density (a moment of zero order) Z n(x, t) =

(1.4.1)

F(t, v, x)dv. R3

2. The mean velocity (a first moment), as a mean value of velocity v: u(x, y) =

1 n

Z vF(t, v, x)dv.

3. Stress tensor (second centered moment): Z pij = m

(vi − ui )(vj − uj )F(t, v, x)dv,

(1.4.2)

m—is a molecule R mass. R 4. The quantity vϕ(v)Fdv is called the flow of a value ϕ(v)Fdv. For example, nu is a flow of the density n. R 5. An auxiliary relation vi vj Fdv = nui uj + pij /m connects velocity flows and stress tensor. 6. Kinetic energy of moving molecules is represented as a kinetic energy mu2 /2 of molecules motion as whole entity plus kinetic energy of relative motion E(x, t) =

m 2n

Z

v2 Fdv =

mu2 m + 2 2n

Z

(v − u)2 Fdv.

The second term also may be interpreted as internal energy: e=

m 2n

Z

(v − u)2 Fdv.

(1.4.3)

7. Pressure. The idea is to calculate which pressure makes a gas onto the unit ground with a normal n for a given distribution function F(t, v, x). We’ll assume that a ground with area S reflects molecules like a mirror. Then the magnitude of changing molecular momentum with velocity v is 2m(v, n). The number of molecules impacting the ground during 1t equals F(v)dvV. Here V is a volume of a parallelepiped with a bottom S and a side v1t. Namely: V = (v, n)1tS. Hence, we define pressure p as p=

FORCE 2m(v, n) (v, n)SF1tdv = × = 2m(v, n)2 Fdv. AREA S 1t

Integrating an obtained relation in v, we get the formula for pressure Z p(n, x, t) = (v,n)>0

(v, n)2 Fdv.

(1.4.4)

6


Additionally, we would like to outline the relationship between the relation (1.4.4) and two famous physical laws. Pascal law or independence of pressure from direction holds if p does not depend on n. The sufficient condition is isotropy of distribution function, i.e., when F depends only on v2 . Mendeleev-Klapeyron law or state equation: 2 p = en. 3

(1.4.5)

Comparing (1.4.3) and (1.4.2), we see that if one assumes p = (p11 + p22 + p33 )/3, then (1.4.5) fulfills for any distribution function. Comparing (1.4.5) with formula (1.4.4) one deduces when (1.4.4) represent a real, physical pressure: (a) (b)

The mean velocity u is zero. If not, we have to redefine (1.4.4) by exchanging v with v − u; Isotropy of distribution function, when relation (1.4.4) does not depend on n.

1.5 On a Kinetic Approach to the Sixth Hilbert Problem (Axiomatization of Physics) The Sixth Hilbert problem was formulated as a problem on development of axiomatic method in natural sciences. For example, it was partially solved by Kolmogorov for the probability theory. But it hardly could be said to be as a well-studied problem in physics. Better to be named nearly untouched. Nevertheless, its importance is undoubted from the philosophy, natural sciences and even tutorial points of view. Any person assuming to derive one equation from the other gets involved with this problem. For example, kinetic Boltzmann, Vlasov and Landau equations were derived by Bogolyubov [49] using N–body dynamics (see also [67, 95, 227]). Another attempt was made by Godunov [118], who proposed the hyperbolic point of view to classify the fundamental equations found in the famous physical textbook volumes by Lifshitz and Landau. One can obtain hyperbolicity for the Euler-type equation derived from Boltzmann equation as a first approximation in Chempen-Enskog method (see Chapter 9 and [116]). There the special twicely divergent form of Euler type equations is obtained from a simple discrete model of Boltzmann equation. Following the mentioned ideas, we can propose as a basis of physico–mathematical description of the world just two Lagrangians: Vlasov-Einstein and Vlasov-YangMills (see Chapter 3 for details). The first one explains gravitational interactions, while the second one’s weak and strong electric interactions. The dynamics of N bodies can be derived from them by an ordinary substitution, described in Chapter 2. That was already made before for modern introduction in classical and statistical mechanics. Besides, all the main equations describing the aggregate states of a substance, like plasma, gas, fluid or rigid body also should be derived. Here plasma is described by Vlasov-Maxwell equation, which has been “split of” from


7

the Vlasov-Yang-Mills equation. A diluted gas is decribed by Boltzmann equation, which could be obtained using the scheme by Bogolyubov [67, 95, 227] from the dynamics of N bodies. The equations of continuous medium (Euler and Navier-Stokes) are obtained from Boltzmann equation by Chempen-Enskog method. In application to rigid bodies and fluids, quantum kinetic equations [177] are obtained from the correspondence “quantum Hamiltonians—kinetic equations” (see Chapter 13). This correspondence is not well developed yet (see [292] and Chapter 13), but [177] uses it explicitely and implicitely. The quantum Hamiltonians itself are obtained from the second Lagrangian by quantization of connected fields. As a short summary, the above examples already became the cornerstones for the kinetic approach to a construction of modern physics. They were made by a synthesis of Landau and Lifshitz, Bogolyubov, and Godunov approaches. The correspondence “quantum Hamiltonians—kinetic equations” already revealed the possibility to generalize kinetic equations in applications to chemically reacting mixtures and problems with triple and higher number collisions (see Chapter 13). The same correspondence allowed also to obtain a simple formulas for conservation laws (Chapter 13) which led us to the method for constructing discrete models for mixtures with correct number of invariants (see Chapter 12). The other very important research instrument is a careful study of the special conservation laws—linear in the number of particles. Precisely, such conservation laws are the basics for: l

l

l

l

l

l

collision invariants of Boltzmann equation; the uniqueness theorem of the Boltzmann H- function; in conditions of chemical equilibrium; in the studying of the sets of stationary positions of Markov processes and the Pauli Master equation; in the study of spectrums of quantum Hamiltonians; in reasoning of the correspondence “quantum Hamiltonians—kinetic equations.”

1.6 Conclusions 1. The derivation of the Vlasov-type equations is not too complicated. Even for the most advanced Vlasov-Maxwell equation, one can use a more simple method, different from Bogolubov chains technology. 2. The derivation of the Boltzmann-type equations is complicated. So, we introduce only the correspondence principle allowing to solve the problems on interaction cuts (coefficients of Boltzmann-type equations). 3. Axiomatization of physics (Sixth Hilbert problem) is very important from the philosophical, natural sciences and tutorial points of view as justification and classification of different equations.

2 Lagrangian Coordinates 2.1 The Problem of N-Bodies, Continuum of Bodies, and Lagrangian Coordinates in Vlasov Equation Let us consider Vlasov equation (1.3.1)–(1.3.2) when (1.3.2) is substituted into (1.3.1): Z ∂F ∂F ∂F + v, − ∇x K(x, y)F(y, v, t)dvdy, = 0. ∂t ∂x ∂v

(2.1.1)

Considering substitution F(t, v, x) =

N X

ρi δ(v − Vi (t))δ(x − Xi (t))

(2.1.2)

i=1

for δ(x)—Dirac δ-function, Vi (t) and Xi (t) are time-dependent functions (coordinates and velocities of particles), ρi > 0—numbers (weights of particles). When Vlasov started to study this equation he already knew that such substitution could be applied if functions Xi and Vi satisfy the motion equations of N-bodies dynamics [306].  X˙ i =Vi    N X (2.1.3) ˙  V = − ∇1 K(Xi , Xj )ρj ,   i j=1

where ∇1 is gradient vector by the first argument. Consider a similar substitution, if the sum in (2.1.2) is changed by integral Z F(t, v, x) = ρ(q)δ(v − V(q, t))δ(x − X(q, t))dq. (2.1.4) Here, we have to define the right expression as generalized function. This could be done naturally: this is a linear functional defined by the formula Z Z ρ(q)δ(v − V(q))δ(x − X(q))dq, ϕ(v, x) = ϕ(V(q), X(q))ρ(q)dq. Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00002-8 c 2011 Elsevier Inc. All rights reserved.

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This is a quite wide generalization of a “simple layer” notion [303] for parametrically defined surfaces (q → (X(q, t), V(q, t)). Parameter q may run through any domain in the space of arbitrary dimension Rk or through some kind of manifold. Usually one takes k < 6. Let us establish conditions when the substitution (2.1.4) turns (2.1.1) into the equality. Using relation vδ(v − V(q)) = V(q)δ(v − V(q)) and replacing v by V(q, t) according to multiplication rule for δ function [303] we have in (2.1.1) for function (2.1.4): ∂V ρ(q) ∇v δ(v − V(q)), − δ(x − X(q))dq + ∂t Z ˙ + ρ(q)δ(v − V(q)) ∇x δ(x − X(q)), −X(q) dq; Z ∂F v, = v, ρ (q)δ (v − V (q)) ∇x δ (x − X (q)) dq = ∂x Z = V (q, t) , ρ (q)δ (v − V (q)) ∇x δ (x − X (q)) dq . ∂F = ∂t

Z

Finally,

∂F K(x, y)F(y, v, t)dvdy, = ∂v Z = ∇1 K(X(q, t), X(q0 , t))ρ(q0 ), ρ(q)∇v δ(v − V(q))δ(x − X(q))dqdq0 . Z

∇x

Here ∇1 is a gradient over first argument. Here we used the definition of δ-function while integrating over y. Integrating over v we used the property that an intergral of δ-function is equal to one. Also, we changed x by X(q, t) using the multiplication property of δ-function. Comparing all three expressions, we can represent them as  ˙ t) =V(q, t)  X(q, Z ˙ t) = − ∇1 K X(q, t), X(q0 , t) ρ(q0 )dq0 .  V(q,

(2.1.5)

Equations (2.1.5) naturally are called the equations of continuum body dynamics. Consider substitution (2.1.2) for N = 1. Then x˙ = v, v˙ = −ρ∇1 K(x, x). If potential K(x, y) is an even smooth function of the difference x − y, then ∇1 K(x, x) ≡ 0. In other words, it means that a body (particle, etc.) does not have an influence itself (nonselfacting); see [48] for additional details. In the preface of Vlasov’s book [307], Bogolubov said that the Vlasov equation is a cornestone of plasma physics, but it also has an exact solution like function (2.1.2), as in classical mechanics. Functions are exactly such a microscopic solution. Bogolubov also proved that substitution (2.1.2) can be applied for the Boltzmann-Enskog equation [48], using dynamics of hard balls.

Lagrangian Coordinates

11

Conclusions 1. The Vlasov equation accepts the substitution of the form (2.1.2) and contains within itself a description of the motion of N bodies for arbitrary number N. Thus, it proves the equation to be fundamental. 2. The Vlasov equation accepts the substitution of the form (2.1.4) providing equations for continuum of bodies. If q = (X(0), V(0)) the initial coordinates, then q is called Lagrange coordinates and equation (2.1.5) is interpreted as a transformation of it.

2.2 When the Equations for Continuum of Bodies Become Hamiltonian? A useful candidate to be a Hamiltonian for equation (2.1.5) is a functional of the form Z Z 2 1 P (q) dq + K X(q), X(q0 ) ρ(q)ρ(q0 )dqdq0 . (2.2.1) H[P, X] = 2ρ(q) 2 This functional obviously is a generalization of the Hamiltonian of N bodies. Here we exchange the sum by integral. Let us find, when equations (2.1.5) are Hamiltonian, i.e., they comply with the system  δH  ˙ t) =   X(q, δP(q) (2.2.2) δH   ˙  P(q, t) = − . δX(q) On the right side of the expression, we use the variational derivative, analogous to the partial derivative: Let F[g] be some functional. Then its weak differential is d F(g + λh) . F 0 [g]h = dλ λ=0 We call a variation derivative the following equality: Z δF 0 F [g]h = h(x)dx. δg(x)

δF of the functional F as a function, defined by δg(x)

This equality can be interpreted in the sense of generalized functions [303]. In this case, we will call variational derivative—a linear functional acting according to the formula δF d , h(x) = F(g + λh) . δg(x) dλ λ=0

12


Hence, equation (2.2.2) also should be considered in the sense of generalized functions (i.e., the equility of the functionals). Exercise 2.1. Derive (2.2.2) from (2.2.1). Solution.

P(q) δH = ; δP(q) ρ(q)

Z 1 δH =− ∇1 K X(q), X(q0 ) ρ(q)ρ(q0 )dq0 − δX(q) 2 Z 1 − ∇2 K X(q0 ), X(q) ρ(q)ρ(q0 )dq0 . 2

(2.2.3)

We can see, that this equation coincides with (2.1.5) if (a) (b)

P(q) ; ρ(q) Integrals in expression (2.2.3) are equal. V(q) =

The first condition represents the natural condition that velocity V and impulse P are related; the second one is fulfilled if potential K is symmetric with respect to its arguments K(x, y) = K(y, x). Conclusion. If potential K is symmetric, then the equations for the continuum of bodies (2.1.5) are Hamiltonian with a multiparticle Hamiltonian, defined by (2.2.1). They lead us to the Vlasov equation (2.1.5) in Lagrangean coordinates (2.2.2). Another characteristic Hamiltonian is given by expression H=

v2 + U(x, t), where U = 2

Z K(x, y)F(y, v, t)dvdy,

∂F + {H, f } = 0 in Euler coordinates. ∂t ∂H ∂f ∂f ∂H Here {H, f } is a Poisson bracket {H, f } = − . ∂v ∂x ∂v ∂x and defines Vlasov equation (2.1.1) of the form

2.3 Oscillatory Potential Example ω2 Let us investigate a special case in which K(x, y) = (x − y)2 . 2 The problem has an exact solution in Lagrange coordinates. Equations (2.2.2) are simplified to ¨ t) + ω2 ρ0 X(q, t) = ω2 X(q, R Here, ρ0 = ρ(q)dq.

Z

X(q0 , t)ρ(q0 )dq0 .


13

V

V

X

(a)

X

(b)

Figure 2.1 Single dimensional (a) elliptic and (b) hyperbolic phase portraits.

Multiplying by ρ(q) and integrating over dq, we obtain an equation for middle R ¨ = 0, where Q(t) = X(q, t)ρ(q, t)dq. coordinate Q Then ˙ Q(0) Q(0) √ √ X(q) = A(q, t) cos(ω ρ0 t) + B(q) sin(ω ρ0 t) + t+ . ρ0 ρ0 So, one can see that particle system with oscillatory potential oscillate around com√ ˙ mon center with ω ρ0 frequency. This center, in its turn, moves with a velocity Q(0) ρ0 . Coordinate q enumerates particles and defines an amplitude. ˙ Assume Q(0) = Q(0) = 0. Drawing the phase portrait picture for single dimension, we will see that all particles have an elliptic trayectory, moving clockwise; see Figure 2.1a. X √ The rotation formula in is given below: V/(ω ρ0 ) X(q, t) cos(χt) sin(χ t) X(q, 0) √ √ = . V(q, t)/ω ρ0 − sin(χt) cos(χ t) V(q, 0)/ω ρ0 √ Here, χ = ω ρ0 .

2.4 Antioscillatory Potential Example The antioscillatory potential is the same the thing as before, just taking the different sign: K(x, y) = −

ω2 (x − y)2 . 2

14


Obviously, the solution looks similar if we exchange trigonometric functions with hyperbolic ones: ch(χt) sh(χ t) X(q, t) X(q, 0) √ √ = . −sh(χt) ch(χ t) V(q, 0)/ω ρ0 V(q, t)/ω ρ0 √ Here, χ = ω ρ0 . The corresponding phase portrait (see Figure 2.1b) represents the divergent hyper√ bolic movement with asimptotes X = ±V/ω ρ0 . Later we’ll study this solution again, comparing this trajectory with real ones that appear in plasma and gravitational modeling problems.

2.5 Hydrodynamical Substitution: Multiflow Hydrodynamics and Euler-Lagrange Description We call as hydrodynamical substitution the substitution introduced in [259] F(t, v, x) = ρ(x, t)δ (v − V(x, t)) . This analytical expression means one simple thing: at each point x there exists only one velocity value V(x, t). In particular (singleflow hydrodynamics), it gives the system of equations for ρ and V:

∂V ∂V + Vi + ∂t ∂xi

∂ρ + div (Vρ(x)) = 0, ∂t Z

(2.5.1)

∇x K(x, y)ρ(y, t)dy = 0.

The mentioned substitution could be generalized. Namely, N-flow hydrodynamics is defined when [259] F(t, v, x) =

N X

ρl (x, t)δ (v − Vl (x, t)) .

l=1

Also, we can regard continuum-layered, or continuum-flow hydrodynamics, when the sum is replaced by integral Z F(t, v, x) = ρ(x, s, t)δ (v − V(x, s, t)) ds. The right part of the expression denotes generalized function providing the value Z ρ(x, s)ϕ (x, V(x, s)) dsdv for the testing function ϕ(x, v); s—Lagrange coordinate.


15

Now we can derive the equations over ρ and V. First, we need to evaluate all terms in (2.1.1): Z Z ∂ρ ∂V ∂F = δ(v − V)ds + ρ ∇v δ(v − V), − ds, ∂t ∂t ∂t Z ∂F v, = v, (∇x ρ)δ(v − V)ds + ∂x Z ∂Vi (α) + v, ρ∇i δ(v − V) − ds = ∂x Z Z Z ∂Vi = v, (∇x ρ)δ(v − V)ds + V, ρ∇i δv (v − V) − ds− ∂x Z ∂Vi − δij δ (v − V)ρ(s)ds, ∂xj Z Z ∂F = ∇x K(x, y)F(y, v, t)dvdy, ∂v Z Z = ∇x K(x, y)ρ(y, s)dyds, ρ(x, q)∇v Fdq . In the second equility (α), we changed v by V using the multiplication rule for gradient of the δ-function vj ∇i δ(v − V) = Vj ∇i δ(v − V) − δij δ(v − V). Collecting δ-function members, we obtain the system of equations of variable ρ(x, q, t) ∂ρ + (V, ∇x ρ) + ρ divV = 0. ∂t

(2.5.2)

The second equation of variable V is obtained by collection of δ-function gradient coefficients: Z ∂V ∂V + Vi + ∇x K(x, y)ρ(y, q, t)dydq. (2.5.3) ∂t ∂xi Equations (2.5.2) and (2.5.3) are the equations of continuum-layered hydrodynamics. One should note that this equation is not valid, due to ambiguity of the functions when the number of layers differs from point to point and is time-dependent (so-called overlapping). The last case corresponds to a free motion, when K = 0. Nevertheless, we can use a Lagrange descriptive technique to investigate it. Example 2.1. Free motion. Assuming K = 0 in (2.1.1), we get (1.1.1). A. B.

There is no overlapping for any moment of time (Figure 2.2a). The overlapping occurs when fast-moving particles leave the slower ones behind (Figure 2.2b).

16


V

V

X

(a)

X

(b)

Figure 2.2 (a) Non overlapping and (b) Overlapping cases.

Equations (2.5.1) and (2.5.2), (2.5.3) cannot describe the situation after the overlapping, but the initial ones (2.1.1), (2.1.5) still are good. Exercise 2.2. Find overlapping moments using different types of coordinate X(s, 0) and velocity V(s, 0) dependencies in Lagrange coordinates s. Example 2.2. Let F(x, v, 0) = ρ(x, 0)δ(v − V(x, 0)) at the initial moment is defined by conditions 1 |x| ≤ 1, ρ(x, 0) = V(x, 0) = x2 a branch of parabola. 0 |x| > 1, Using Lagrange coordinates, we get X(s, 0) = s, V(s, 0) = s2 and the movement equa¨ t) = 0. It can be solved explicitly: V(s, t) = s2 , X(s, t) = s2 t + s. To tion becomes X(s, ∂X turns to zero for the first time find the overlapping moment, we have to study when ∂V ∂X or when the gradient catastrophe occurs when ∂V turns to infinity. Reducing s, we √ ∂X get X(V, t) = Vt − V for the left branch. Differentiating, we obtain ∂V = t − √1 . 2 V

∂X Hence, function ∂V turns to zero for the first time on the segment 0 ≤ V ≤ 1 for V = 1, 1 t = 2 . After the moment t = 12 function V(x, t) is not uniquely defined, and the hydrodynamic description is of no use while the solution in Lagrange coordinates is kept. Overlappings also are the basics for the disk theory of a large-scale universe. Making a projection on the x coordinate, one gets a density singularity, approaching the basics of the Lagrange singularities theory [15].

Exercise 2.3. Derive from equation (2.1.1) a hydrodynamic-type equation assuming f (x, v) to be an arbitrary force instead of a self-consistent field.

2.6 Expanding Universe Paradigm The term expanding universe first appeared after the Friedman’s solutions of Einstein equations and Hubble’s discovery of red shift. There exists a simple nonrelativistic


17

analogous solution, known as a “self–gravitating” ball, known also as the MilnMcKree model [319]. In addition this is an exact solutions of Vlasov-Poisson equation in gravitational field of the form ∂F ∂F ∂F + v, − ∇u, = 0, ∂t ∂x ∂v Z 1u = 4πγ F(x, v, t)dv. Here the quantity γ is a well-known gravitational constant. Here solutions can be written as Z F(t, v, x) = δ(v − V(q, t)) δ(x − X(q, t))dq while X(q, t) satisfies the equation Z dq0 ¨ t) = −γ . X(q, |X(q) − X (q0 )| q The considered solutions are spherically symmetric, since X(q, t) = |q| R(r, t), r = |q|. The particle velocities are directed along radius vectors and depend only on radius. Hence, the movement equation takes the form

¨ t) = −γ R(r,

M(r) . R2 (r, t)

Here M(r) is a mass of the ball with a Lagrange coordinate less than r. Thus, r could be interpreted as a spherical layers enumerator. Here we used the known fact, that a uniform spherical layer applies zero force to any point inside it. The force of attraction of outside points exactly equals to the force of attraction by the same mass put in center of the layer. Integrating this equation, we obtain 1 2 M(r) R˙ − γ = C. 2 R(r, t)

(2.6.1)

If we take C > 0 (see Figure 2.3), then we have an unbounded expansion, also called an open model. Assuming C < 0, an expansion is exchanged by contraction, providing us with closed or oscillating Universe model (compare with Sections 2.3 and 2.4). Dividing equation (2.6.1) by R2 , we get 2 R˙ M(r) 2C = H 2 (t) = 2γ 3 + , R R (r, t) R2 where quantity H(t) is called Hubble constant. One should know that definition of this quantity was made initially in 1929 assuming the expansion rate of Universe to be constant. Advanced models are time-dependent, but they also mean to use the same historical name Hubble constant.

18


V C>0 X U(R) = −γ M R

C 0, d > 2, = 4π e2α 0 mα  1 gα (eα V) ≥ 0, d = 2. = −4π

e2α Bd

All of these calculations are valid if gα (E) ≥ 0 are continuously differentiable and g0α

mα V 2 C + eα S V d−1 < 1+ε 2 V

when V > R for some constants C > 0, R > 0 and ε > 0. We see that the expression 9 0 (S) is nonnegative for d ≥ 2 when gα (E) ≥ 0. Assuming d = 1, we cannot use the same reasoning, because the lower limit involves the infinite value while integrating by parts. To be specific, the inequality gα (E) ≥ 0 is violated for some nonmonotonic functions gα (E). The most important are the monoenergetic functions, when gα (E) = Aα δ(E − Eα ). That allows the possibility of obtaining nontrivial periodic solutions, usually called Bernshtein-Green-Kruskal waves. Exercise 4.2. Prove the nonmonotonicity of function 9 when gα (E) = Aα δ(E − Eα ). Obtain such periodical solutions [8, 286]. Exercise 4.3. Obtain a solution of the Lengmuir diode problem. Hint: How are the electron density and catode–anode potential related? When will the electric current flow? (The solution can be found in [210, 319]).

4.5 Conclusions 1. Taking monotonically decreasing distribution functions, one has monotonic dependence of the density distribution function 9(U) from the potential: 9 0 (U) ≥ 0. Hence, the periodical solutions are absent, and any boundary-value problem has unique solution for any velocity dimension space. The main example is obtained for gα (E) = Aα e−βα E , i.e., the BoltzmannMaxwell distribution function. In the case of gravitation, a sign of inequality is inverse and the boundary-value problem is ill-posed and has no physical sense. 2. Since the inequality 9 0 (U) ≥ 0 becomes valid for d ≥ 2, then the boundary-value problem (4.4.1) is also correct, and periodical solutions are absent.

Energetic Substitution

33

3. For d = 1, the monotonicity of function 9(U) is violated for nonmonotonic functions gα (E). In particular, for monoenergetic distributions gα (E) = Aα δ(E − Eα ), there exist periodical solutions.

Remark 4.1. In case of energetic substitution, the trajectories of particles depend only from the initial point and thus define dynamic system. Regarding nonstationary Vlasov equations, the trajectories of particles depend from overall configuration and do not define a dynamic system, because a self-consistent potential U(x, t) is time dependent. As a final note, we should say that paper [16] proved the nontrivial existence theorems; paper [189] investigated several quantizations of the Vlasov equation. For the further theory development, we recommend paper [247] in which the generalizations of energetic substitution have been investigated for the Vlasov-Maxwell equation.

5 Introduction to the Mathematical Theory of Kinetic Equations

5.1 Characteristics of the System In this subsection, we briefly turn our attention to the application of classical characteristic theory (Bouchut [54], Hartman [129]) to transport an equation of the general form ∂t u + a(t, x) · ∇x u = 0,

0 < t < T,

x ∈ RN

(5.1.1)

with a smooth coefficient a : (0, T) × RN → RN and also to transport equation of divergence type ∂t f + divx [a(t, x)f ] = 0,

0 < t < T,

x ∈ RN .

(5.1.2)

Definition 5.1. A function X(s) ∈ C1 in an interval R with values in RN , satisfying equation dX = a(s, X(s)), ds

(5.1.3)

is said to be characteristic for (5.1.1) or (5.1.2). Theorem 5.1 (see [129] for details). Let a ∈ C([0, T] × RN ), ∇x a ∈ C([0, T] × RN ) and ∀t ∈ [0, T] ∀x ∈ RN

|a(t, x)| ≤ k(1 + |x|).

Then for all t ∈ [0, T] and x ∈ RN there exists unique characteristic on [0, T], satisfying X(t) = x and denoting by X(s, t, x). Here X ∈ C1 ([0, T]s × [0, T]t × RN x ), ∂s ∇x X, ∇x ∂s X exist and coincide in C([0, T]s × [0, T]t × RN x ). If a, ∇x a ∈ Ck−1 ([0, T] × RN ), k ≥ 1, then X ∈ Ck ([0, T]s × [0, T]t × RN x ). Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00005-3 c 2011 Elsevier Inc. All rights reserved.

36


Proposition 5.1. Assume conditions of Theorem 5.1 are fulfilled. Then (i) ∀t1 , t2 , t3 ∈ [0, T], ∀x ∈ RN , X(t3 , t2 , X(t2 , t1 , x)) = X(t3 , t1 , x).

(5.1.4)

(ii) For all s, t ∈ [0, T] mapping x → X(s, t, x) is C1 diffeomorphism in RN with inverse mapping X(t, s, ·). (iii) Jacobian J(s, t, x) = det(∇x X(s, t, x)) satisfies equation ∂J = (divx a)(s, X(s, t, x))J, ∂s

J > 0.

Proof. A property (i) follows from the definition of mapping X and the uniqueness of the Cauchy problem; (ii) follows from (i) for t1 = t3 . Denoting by I(A) conjugate comatrix A, one obtains ∂J = tr(I(∇x X(s, t, x))∂s ∇x X(s, t, x)) = ∂s = tr(I(∇x X(s, t, x))∇x [a(s, X(s, t, x))]) = = tr(I(∇x X(s, t, x))∇x a(s, X(s, t, x))∇x X(s, t, x)) = = tr(∇x a(s, X(s, t, x))∇x X(s, t, x)I(∇x X(s, t, x)) = = det ∇x X(s, t, x)tr(∇x a(s, X(s, t, x))) = = J(divx a)(s, X(s, t, x)). Since J(t, t, x) = 1, we obtain J > 0, which proves (iii). Remark 5.1. If divx a = 0, then J ≡ 1 and mapping X(s, t, ·) preserves the measure (Liouville theorem). In that case, equations (5.1.1) and (5.1.2) coincide, and X induces “incompressible” flow. One of the properties of the flow X(s, t, x) states that it can described as a differential equation in variable s by means of (5.1.3) written as ∂s X(s, t, x) = a(s, X(s, t, x))

(5.1.5)

and as a partial-differential equation in terms of variables (t, x). Proposition 5.2. A flow X is described by equation ∂t X(s, t, x) + (a(t, x) · ∇x )X(s, t, x) = 0.

(5.1.6)

Proof. Differentiating (5.1.4) in t2 , we obtain ∂t X(t3 , t2 , X(t2 , t1 , x)) + ∇x X(t3 , t2 , X(t2 , t1 , x))∂s X(t2 , t1 , x) = 0. Using (5.1.5) and the property (ii) of the proposition 5.1, we obtain (5.1.6) after an evident change of variables.

Introduction to the Mathematical Theory of Kinetic Equations

37

Theorem 5.2. (i) If u0 ∈ C1 (RN ), then there exists a unique solution u ∈ C1 ([0, T] × RN ) of (5.1.1) with initial value u(0, x) = u0 (x) of the form u(t, x) = u0 (X(0, t, x)). (ii) Let a, ∇x a ∈ C1 . If f 0 ∈ C1 (RN ), then there exists a unique solution f ∈ C1 ([0, T] × RN ) of (5.1.2) with initial value f (0, x) = f 0 (x) of the type f (t, x) = f 0 (X(0, t, x))J(0, t, x).

(5.1.7)

Proof. To prove the first part of the theorem, we multiply (5.1.6) by ∇x u0 (X(0, t, x)). Therefore, if u ∈ C1 satisfies (5.1.1), then for any (t0 , x0 ) one has d [u(s, X(s, t0 , x0 ))] = ∂t u(s, X(s, t0 , x0 )) + ds + ∇x u(s, X(s, X(s, t0 , x0 )))a(s, X(s, t0 , x0 )) = 0 and u(s, X(s, t0 , x0 )) = u0 (X(0, t0 , x0 )). Putting X(s, t0 , x0 ) = x, one obtains u(s, x) = u0 (X(0, t0 , X(t0 , s, x))) = u0 (X(0, s, x)). Assuming f to be a solution of (5.1.2), we can prove the second statement. Namely, d [f (s, X(s, t0 , x0 ))] = ∂t f (s, t0 , x0 ) + ∇x f (s, X(s, t0 , x0 ))a(s, X(s, t0 , x0 )) = ds = −(divx a)(s, X(s, X(s, X(s, t0 , x0 ))))f (s, X(s, t0 , x0 )) = = −f (s, X(s, t0 , x0 ))

∂s J(s, t0 , x0 ) . J(s, t0 , x0 )

Hence, f (s, X(s, t0 , x0 )) = f 0 (X(0, t0 , x0 ))

J(0, t0 , x0 ) . J(s, t0 , x0 )

Differentiating (5.1.4) in x and opening determinant, one obtains J(t3 , t2 , X(t2 , t1 , x))J(t2 , t1 , x) = J(t3 , t1 , x). Taking t3 = 0, t2 = s, t1 = t0 , x = x0 , we have J(0, t0 , x0 ) = J(0, s, X(s, t0 , x0 )), J(s, t0 , x0 ) which induces (5.1.7). Or, vice versa, function f defined by (5.1.7) satisfies the equation (∂t f + divx (a(t, x)f ))(s, X(s, t0 , x0 )) = 0, for all (t0 , x0 ), which implies (5.1.2).

38


Remark 5.2. Taking f 0 = 1 in the preceeding equation, we have ∂t J + divx (a(t, x)J) = 0. Proposition 5.3. We assume that a, ∇x a ∈ C1 . If f 0 ∈ L1 (RN ), then there exists a unique solution f ∈ C([0, T], L1 (RN )) of (5.1.2) with f (0, x) = f 0 (x) such that f is defined by (5.1.7). Proof. If f 0 ∈ C0∞ (RN ), then it follows from (5.1.7) that f ∈ C([0, T], L1 (RN )). Then for f 0 ∈ L1 (RN ) there exists a sequence fn0 ∈ C0∞ such that fn0 → f 0 in L1 . Since fn (t, ·) → f (t, ·) is uniform in a time scale for L1 , then f ∈ C([0, T], L1 ) and the limit in sense of distributions (weak limit) shows that f satisfies (5.1.2). To prove this uniqueness, we assume that f ∈ C([0, T], L1 (RN )) satisfies (5.1.2) in terms of distributions. We define the function g(t, x) = f (t, X(t, 0, x))J(t, 0, x).

(5.1.8)

Then g ∈ C([0, T], L1 (RN )) ( f is approximated in C([0, T], L1 (R)) by a sequence of smooth functions fn ) and, in terms of distributions, one has for g ∂t g = [∂t f (t, X(t, 0, x)) + ∇x f (t, X(t, 0, x))∂s X(t, 0, x)]J(t, 0, x) + + f ∂s J(t, 0, x) = [∂t f + divx (af )](t, X(t, X(t, 0, x)))J(t, 0, x) = 0. Thus g(t, x) = f 0 (x). Inverting the relation (5.1.8), we see that f is defined by formula (5.1.7). Proposition 5.4. Let X1 (s), . . . , Xp (s) be the function satisfying the system dXj = a(s, Xj (s)), ds λ1 , . . . , λp ∈ R and we define the function f (t, x) =

X

λj δ(x − Xj (t)).

j

Then f satisfies equation in terms of distributions ∂t f + divx (af ) = 0

and

[0, t] × RN .

Proof. Let φ ∈ C0∞ ([0, T] × RN ). Then we obtain the following chain of derivations, from which follows the result of the proposition:


< ∂t f , φ > = − < f , ∂t φ >= −

X j

=−

X

λj

ZT

j

=

X

λj

j

λj

ZT

39

∂t φ(t, Xj (t))dt =

0

d φ(t, Xj (t)) − ∇x φ(t, Xj (t))˙xj (t) dt = dt

0

ZT

∇x φ(t, Xj (t))a(t, Xj (t))dt =< af , ∇x φ >=

0

= − < divx (af ), φ >.

5.2 Vlasov-Maxwell and Vlasov-Poisson Systems The Vlasov equation describes the evolution of the system of particles in the force field F(t, x, p), which depends on time t, position x, and momentum p. For every particle with index j, we can write motion equations x˙ j = v(Pj ),

p˙ j = F(t, Xj , Pj ),

where Xj denotes a position j-th particle and Pj its momentum. In general, v is a function of momentum; in a classical case, one obtains v(p) =

p . m

(5.2.1)

Defining the density of particles in the form f (t, x, p)dxdp in the phase space (x, p) ∈ RN × RN for any fixed t as f (t, x, p)dxdp =

X

δ(x − Xj (t), p − Pj (t)).

j

It follows from Proposition 5.4 that the function f must satisfy the Vlasov equation: ∂t f + divx (v(p)f ) + divp (F(t, x, p)f ) = 0

and

N Rt × RN x × Rp .

(5.2.2)

Between the infinite number of particles we look for “smooth” solutions f of (5.2.2); N therefore, we may claim that function at least f ∈ L1 (RN x × Rp ). In this case, when the particles are subjected to collisions, we should add a nonlinear term into the Vlasov equation (5.2.2). Moreover, it is transformed to a nonlinear kinetic equation with collision operator Q( f ) in the right part ∂t f + divx (v(p)f ) + divp (F(t, x, p)f ) = Q( f ).

40


Depending on the structure of the collision operator Q( f ), this equation is said to be a Boltzmann or Fokker-Plank-Landau equation. Let us consider a collisionless case in which the model describes the evolution of particles in a selfconsistent electromagnetic field. In this case, systems simulating the described process are constructed in the following way: we define particle density ρ and current density j by means of Z Z ρ(t, x) = q f (t, x, p)dp, j(t, x) = q v(p)f (t, x, p)dp, where q—charge of the particle. Force F is given as F(t, x, p) = qE(t, x),

E(t, x) = −∇x φ(t, x)

for a Vlasov-Poisson (VP) system (5.2.2) −4x φ = ρ,

(5.2.3)

or in the form (Lorentz force) F(t, x, p) = q(E(t, x) + v(p) × B(t, x)) for a Vlasov-Maxwell (VM) system (5.2.2) ∂t E − c2 rot B = −j, ∂t B + rot E = 0, divx E = ρ,

(5.2.4)

divx B = 0.

In a relativistic case, the velosity v in VM system is given by the formula v(p) = p

p/m 1 + |p|2 /m2 c2

.

In a VP system, which is formally the classical limit (c → ∞) of a VM system, velocity is defined by (5.2.1).

5.3 Weak Solutions of Vlasov-Poisson and Vlasov-Maxwell Systems By a weak solution, we mean a solution in sense of distributions. In that case, the field of force is not smooth enough to apply a classical characteristic theory described in subsection 5.1. The proof of existence of weak solutions for VP and VM systems is very complicated and a group of outstanding mathematicians (see Arsen’ev [17], DiPerna, P. Lions [92], and Illner, Neunzert [148]) have tried to resolve it. Here we


41

present (without proof) the principle existence theorems of weak solutions of VP and VM systems obtained by Arsen’ev, DiPerna and P. Lions, and Horst. We note that the existence of global (week) solutions for a three-dimensional VM system has been proven by DiPerna and P. Lions. Proposition 5.5. For a VP system (5.2.2), (5.2.3) one has the following relation: ∂t

Z

|E|2 |p|2 fdp + 2m 2

Z + divx

|p|2 v(p)f dp + φ( j + ∂t E) = 0 2m

and conservation principle of total energy Z Z E(t) ≡

|p|2 f dxdp + 2m

Z

|E|2 dx. 2

Proposition 5.6. For a VM system (5.2.2), (5.2.4) the following integral relation holds: ∂t

|E|2 + c2 |B|2 mc2 (γ (p) − 1)f dp + + 2 Z + divx mc2 (γ (p) − 1)v(p)f dp + c2 E × B = 0

Z

with γ (p) =

q

1 + |p|2 /m2 c2 ,

moreover, one has a conservation law of total energy Z Z E(t) ≡

mc2 (γ (p) − 1)f dxdp +

Z

|E|2 + c2 |B|2 dx. 2

Theorem 5.3. We suppose that N ≥ 3 and let f 0 ∈ L1 N |p|2 f 0 ∈ L1 (RN x × Rp ),

T

N 0 L∞ (RN x × ×Rp ), f ≥ 0,

x ∗ f 0 (x, p)dp ∈ L2 (RN x ). |x|N

Then there is a solution of the VP system (5.2.2), (5.2.3) f ∈ C([0, ∞], N 0 L∞ (RN x × Rp ) − w∗) with initial value f , satisfying for any t ≥ 0 0 p ≤ || f || p , || f (t, ·)||Lxp Lxp

E(t) ≤ E(0).

1 ≤ p ≤ ∞,

42


Theorem 5.4. Let f 0 ∈ L1 conditions divB0 = 0,

T

divE0 = q

N 0 0 0 L∞ (RN x × Rp ), f ≥ 0, E , B satisfy the agreement

Z

f 0 dp,

and we assume that N |p| f 0 ∈ L1 (RN p × Rp ),

E0 , B0 ∈ L2 (RN x ).

N Then there is a solution of VM system (5.2.2), (5.2.4) f ∈ C([0, ∞]), L∞ (RN p × Rp ) − N 0 0 0 w∗) and E, B ∈ C([0, ∞], (Rx ) − w) with initial value f , E , B and satisfying for any t ≥ 0 0 p ≤ ||f || p , || f (t, ·)||Lxp Lxp

1 ≤ p ≤ ∞,

E ≤ E(0).

5.4 Classical Solutions of VP and VM Systems The existence of classical (smooth) solutions of the VP system (5.2.2), (5.2.3) can be established via two methods. The first method (see Batt [22], Horst [143], Rein [235]) consists of transformation of problem and getting decay estimation of function f (t, x, p) at |p| → ∞. The second one implies direct obtaining of that decay starting from a priori estimations. To realize this goal, there are two main methods: e.g., the P. Lions and B. Perthame method based on dispersion estimation and the characteristic method developed by Pfaffelmoser [229]. For the first time, Pfaffelmoser proved global solvability of Cauchy problem for a three-dimensional VP system with an arbitrary initial value. These results were simplified and improved by Horst and Schaeffer. For a VM system, at present, the situation is the following: the first step of reduction has been obtained by Glassey and Strauss [111], but decay estimations in velocity p remain an open problem. The existence of global solutions of VM system for dimension 2.5 (2 dimensions in x and 3 in p) was proved by Glassey and Schaeffer [115]. Existence of global classical solutions of the VM system for 3 dimensions still remains an open problem.

5.5 Kinetic Equations Modeling Semiconductors A distinctive peculiarity of mathematical analysis of semiconductors is its connection with hierarchy of models of transport of charged particles in various mediums. It is explained by that transport processes have a practical interest for various scales of length and time generated by various physical effects. In particular on the very small scales using in microelectronic technology, quantum and/or kinetic effects are important.


43

The Boltzmann equation for semiconductors simulates the flow of charged electrons in semiconductor crystals. It describes the evolution of distribution function f = f (x, k, t) in the phase space. Here x ∈ R3 —state variable, k ∈ B—wave vector, B denotes Briullen zone, connected with corresponding crystal chain. A semiclassical Boltzmann equation is written in the form q ∂t f + v(k) · ∇x f + ∇x V · ∇k f = Q(f ), x ∈ R3 , k ∈ B, t > 0, h¯ f (k, k, 0) = fi (x, k), x ∈ R3 , k ∈ B. Physical constants q and h¯ denote elementary charge and reduced Plank con¯ k (k)—velocity of electron, (k)—energy and V = stant respectively, v(k) = (1/h)∇ V(x, t)—electrical potential. Collision operator Q(f ) simulates a short (by scale of length) interaction of electrons with crystal inclusions, phonons and electrons. Electrical potential V is the given function or selfconsistently closed via Poisson equation 4V = q(n − C(x)),

(5.5.1)

where denotes conductivity of semiconductor and C = C(x) simulates given an ion background. The density of electron is defined by Z f dk.

n= B

For modeling super-small electronic devices, we need to consider quantum effects. The quantum Boltzmann equation is a kinetic equation describing evolution the Vigner function w = w(x, v, t) q ∂w + v · ∇x w + 2h [V]w = Qh (w), ∂t m∗ w(x, v, 0) = w0 (x, v),

x, v ∈ R3 ,

t > 0,

x, v ∈ R3 .

Here Qh (w)—collision operator, V—effective potential, satisfying the Poisson equation (5.5.1). Electronic density and current density are defined by formulas Z n=

Z w(x, v, t)dv,

R3

jn = −q

vw(x, v, t)dv.

R3

Operator 2h [V]w is pseudodifferential operator of special form. In the semiclassical limit h¯ → 0, a quantum Boltzmann equation is transformed to a quantum Vlasov equation ∂w q + v · ∇x w + ∇x V · ∇v w = 0, ∂t m∗

(5.5.2)

44


which is closed by a Poisson equation (5.5.1) for potential V. The system (5.5.2), (5.5.1) is a nonlinear, nonlocal, integro-differential system. In the last 2–3 years, many leading mathematicians both with point of view of existence problem of solutions and from an applied point of view (semiconductor modeling) have considered this problem. Transport modeling of electrons in nanostructure, for example, resonance tunnel diodes, superchains leads to quantum models: Schro¨ dinger and Vigner equations. Generally quantum effects exhibit in the local domains of devices, and in the other part (global domain) are subjected to classical equations. In this case, there arises the question: how to correctly choose the boundary conditions for quantum models and adjust them with classical kinetic models—Vlasov or Boltzmann equations? The answer is contained in construction of new composed classico-quantum models with transition boundary conditions between quantum and classical domains. As an example of such models, consider a one-dimensional nonstationary Vlasov-Vigner model (Ben Abdallah, Degond, Gamba [37]). Let a < b—two real numbers, V(x)—smooth stationary potential on R. Then, in the classical domain, the motion of electrons is described by a Vlasov equation ∂f dV ∂f ∂f +p − = 0; ∂t ∂x dx ∂p

x ∈ [a, b],

t ∈ R,

t ∈ R,

and, in a quantum zone, the equation of matrix density ihρt = (Hx − Hx0 )ρ,

(x, x0 , t) ∈ R3 ,

∂ where ρ = ρ(x, x0 , t)—matrix density; H = − h2 ∂x 2 + V(x). Boundary interaction of classical domain and open quantum system is realized by means of particles incoming from exterior (classical domain) in the previous time interval and outcoming from S quantum domain in the next interval. We denote by 2− = ({a} × R∗+ ) ({b} × R∗− ) the part of boundary corresponding to incoming particles and g(x, p, t)—distribution S function on the boundary. Here electrons are quantum in [a, b] and, in [0, a] [b, L], behave as classical particles. Let fC (x, p, t)—distribution function in the classical zone and fQ —Vigner function in quantum zone. Then the classical distribution function is calculated by means of solution of boundary value problem for Vlasov or Boltzmann equation in classical zone with additional conditions of surface (boundary) “interchange” at x = a or x = b 2

2

fC (a, p, t) = Rh−p fC (a, −p, t − τRh (−p)) + Tph fC (b, p, t − τth (p)),

p < 0,

fC (b, p, t) = Rh−p fC (b, −p, t − τRh (−p)) + Tph fC (a, p, t − τth (p)),

p > 0.

If fC is found, then fQ is calculated as Vigner transformation for matrix of density ρ = ρ(x, x0 , t).


45

5.6 Open Problems for Vlasov-Poisson and Vlasov-Maxwell Systems In this section is given the list of the open problems for the VP and VM equations, kindly sent to the authors by Professor Ju¨ rgen Batt. Problem 1. One way to construct a stationary spherically symmetric solution of the VP system is described in [23]. A second method is the so-called inversion method: given a function h(u, r) such that the equation 1 2 0 (r u (r))0 = h(u(r), r) r2 is solvable, that is, has a solution u = u(r), one defines ρ(r) = h(u(r), r) and has to solve the problem to find a function ϕ = ϕ(E, F) with ϕ(E, F) = 0 for E < a := u(∞) such that 2π h(u, r) = r

Za

2r2Z(E−u)

u

0

ϕ(E, F) p d(E, F). 2r2 (E − u) − F

(5.6.1)

This equation then guarantees ρ(|x|) = 4π

Z

! v2 2 2 2 ϕ + u(|x|), x v − (xv) dv, 2

R3

v2 2

+ u(|x|) =: E,

x2 v2 − (xv)2 =: F,

that is, the above defined ρ is the local density of the distribution function f := ϕ(E(x, v), F(x, v)). The substitution ξ = a − u, η = 2r2 (one-dimensional variables) gives (5.6.1) in the form r Zξ 1 η = √ h a − ξ, 2 π η 0

η(ξ Z−s1 ) 0

√ 2 2π 2 ϕ(a − s1 , s2 ) ds2 ds1 √ η(ξ − s1 ) − s2

or g(ξ, η) =

1 √

π η

Zξ 0

η(ξ Z−s1 ) 0

f (s1 , s2 ) ds2 ds1 √ η(ξ − s1 ) − s2

in R+ × R+ with r η h a − ξ, =: g(ξ, η), 2

√ 2 2π 2 ϕ(a − s1 , s2 ) =: f (s1 , s2 ).

And, the problem is, when g is given, to find distribution f .

46


The substitution s = s1 and t = s1 + 1 g(x, y) = π

Zx Z t 0 0

s2 η

gives

f (s, y(t − s)) ds ds dt, √ x−t

ξ =: x,

η =: y.

Abel’s integral equation 1 g(x, y) = π

Zx 0

ψ(t) dt, √ x−t

where ψ(t) =

Zt 0

f (s, y(t − s)) dt √ x−t

has the solution 1 d ψ(x) = √ π dx

Zx 0

g(s, y) ds. √ x−s

The problem remains, given ψ, to solve (for f ) the equation ψ(x) =

Zx

f (s, y(x − s))ds

0

or ψ(x) =

Z

2(y)a

1 1 = 0. J. Batt and G. Rein proved hη (t) = O(t2 ) in the x-periodic case. For cylindrically symmetric solutions the estimate is not known. Sharp estimates found in [321] gave an example of a uniquely determined weak solution of the VP system whose (x, v)—support is uniformly bounded for all times. t

1 y

x

s

In general, the uniqueness of weak solutions to an initial value problem is open, for the total energy one only knows E(t) ≤ E(0) (instead of equality). The paper by Robert R. [239] seems to be a further attempt in this direction. A.J. Majda, G. Majda and Y. Zheng [182] have proven the nonuniqueness of weak (rather singular) solutions. The exact border between nonuniqueness and uniqueness in the field of weak and strong solutions is not known. E. Horst and R. Hunze [141] have developed a concept to get weak solutions for the VP system, which has become a guideline to handle other cases (Vlasov-Poisson-Fokker-Plank, flat case of the VP system). Problem 8. In article [28], an elementary proof for the approximability of the solutions of the unmodified VP system by the mass-point solutions of modified N-body problems when N → ∞ and the modification parameter δ → 0 in an appropiate way was given. However, this is not what the physicists call the “mean field limit”. For the modified VP system, the mean field limit has been proven in the now classical paper of W. Braun and K. Hepp [56] “The Vlasov dynamics and its fluctuations in the N1 limit of interating classical particles.”

48


Problem 9. The same as for problem 8, but for the Vlasov-Maxwell system. E. Horst has shown the global existence for classical solution of the modified VM system, where modification means that the current density j is replaced by the convolution j(·, t) ∗ δ, where δ is a member of a canonical sequence of δ-functions. Problem 10. The “flat” VP system is obtained, if x, v ∈ R3 are replaced by x, v ∈ R2 and in the integral form of Poisson‘s equation, i.e., R in the formula for the potential u and R in the definition of the local density ρ, the integrals R2 over R3 are replaced by integrals R2 over R2 (over all x or all v, respectively). The question is: Can one obtain the solutions of the flat VP system from the solutions of the VP system (in 3D) by a suitable limit process? The “jump relations” (see R. B. Guenther, J. W. Lee [124], Chapters 6–8) might play an important role. S. Dietz in her dissertation [93], (unpublished) has proven the existence of classical solutions locally, the existence of weak solutions globally. Problem 11. The unicity of classical solutions of the VP system (or of related systems) is not fully understood. Are two C1 —solutions f1 , f2 of the initial value problem with the same initial value f 0 ∈ Cc1 (R6 ) equal? Problem 12. Generalize the approach of [25] introduced for Emden-Fowler equation to another equation.

6 On the Family of the Steady-State Solutions of Vlasov-Maxwell System

6.1 Ansatz of the Distribution Function and Reduction of Stationary Vlasov-Maxwell Equations to Elliptic System Let us consider stationary VM system qi 1 ∂ ∂ E+ v×B fi = 0, v fi + ∂r mi c ∂v rotE = 0 divB = 0 divE = 4π

i = 1, 2

(6.1.2) (6.1.3) Z 2 X qj fj (v, x)dv j=1

rotB = Z Z

4π c

(6.1.1)

2 X j=1

(6.1.4)

R3

Z qj

vfj (v, x)dv

(6.1.5)

R3

fi (v, x)dxdv = 1.

(6.1.6)

R3 1

We will assume that q1 > 0 and q2 < 0, which means that f1 (v, x) and f2 (v, x) respectively are the ion and electron distribution functions. Taking into account condition (6.1.6), we also will find the stationary distribution functions of the form fi (v, x) = fi (−αi v2 + ϕi , vdi + ψi ) = fˆ (R, G),

(6.1.7)

where R = −αi v2 + ϕi , G = vdi + ψi and the corresponding electromagnetic fields E(x), B(x) satisfying the system of equations (6.1.2)–(6.1.5). Here we assume ϕi : R3 → R, ψi : R3 → R, x ∈ 1 ⊆ R3 , v ∈ R3 , αi ∈ R+ = [0, ∞), di ∈ R3 .

Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00006-5 c 2011 Elsevier Inc. All rights reserved.

50


Let us build the system of equations determining the functions ϕi , ψi . After substitution (6.1.7) into (6.1.1), we come to the relationships mi ∇ϕi 2αi qi mi c B(x) × di = − ∇ψi qi (E(x), di ) = 0. E(x) =

(6.1.8) (6.1.9) (6.1.10)

Moreover, it follows from (6.1.8), (6.1.10) that (∇ϕi , di ) = 0.

(6.1.11)

By contrast, from (6.1.9) we obtain (∇ψi , di ) = 0.

(6.1.12)

Let us note that (B(x), di ) = λi , where λi (x) is an arbitrary function at this moment. Thus, for definition of B(x) we need the joint solution of equations (6.1.9), (6.1.12) taking into account a fact that (B(x), di ) = λi (x). Vector B(x) takes the form B(x) =

λi (x) mi c di − di × ∇ψi , 2 di qi di2

(6.1.13)

2 + d2 + d2 . where di2 = d1i 2i 3i The most outstanding fact is that fields E(x) and B(x), determined by means of formulas (6.1.8), (6.1.9) does not depend on index i. Hence, the functions ϕi , ψi can be searched in the form

ϕi = ϕî + ϕi (x),

ψi = ψˆ i + ψi (x),

(6.1.14)

where ψˆ i , ϕî are constants; functions ϕi (x), ψi (x) satisfy the relations ϕ2 (x) = l1 ϕ1 (x), ψ2 (x) = l2 ψ1 (x). Parameters l1 , l2 , with respect to (6.1.8), (6.1.9) are connected with the relations m2 m1 = l1 α1 q1 α 2 q2 q1 q2 l2 d 1 = d2 . m1 m2 m1 α2 q2 m2 α1 q1

(6.1.15) (6.1.16)

< 0, mi > 0, αi > 0, q1 q2 < 0. As it follows from (6.1.16), vectors d1 , d2 are linearly dependent. Since ϕî and ψˆ i defined in (6.1.14) are constants, then ∇ϕi = ∇ϕi (x), ∇ψi = ∇ψi (x). In this case, l1 =

On the Family of the Steady-State Solutions of Vlasov-Maxwell System

51

Substituting (6.1.7) and (6.1.8) into (6.1.4), we will obtain the equations 2

X mi qj 4ϕi (x) = 4π 2αi qi j=1

Z

fj (v, x)dv.

(6.1.17)

R3

Since div[di × ∇ψi ] = 0 and taking into account (6.1.12), substituting (6.1.13) into (6.1.3) gives (∇λi (x), di ) = 0.

(6.1.18)

Substituting (6.1.13) into (6.1.5), one obtains the system of equations 2

∇λi × di =

4π 2 X mi c di 4ψi (x) + d qj qi c i j=1

Z vfj dv.

R3

The above system is solved if and only if functions ψi (x) satisfy the equations

−

Z 2 mi c 4π X 4ψi (x) = qj (v, di )fj dv. qi c j=1

(6.1.19)

R3

In this case   Z 2 X 4π γi   qj di × vfj dv ∇λi (x) = 2 di + c di j=1

(6.1.20)

R3

because of [di × dj ] = 0, (i, j) = 1, 2. Finally, combining (6.1.20), (6.1.18) and di 6= 0, we obtain   Z 2 X 4π   qj di × vfj dv . (6.1.21) ∇λi (x) = c j=1

R3

Relations (6.1.17), (6.1.19) are the desired system of elliptic equations related to the functions ϕi (x), ψi (x). Thus, the problem of finding the steady-state solutions of the VM system (6.1.1)–(6.1.5) came to a joint study of equations (6.1.17), (6.1.19) with conditions of orthogonality (6.1.11), (6.1.12) and normalization condition (6.1.6). Systems (6.1.17), (6.1.19) will be studied assuming 2 +ϕ

fi (v, x) = e−αi v

i (x)+di v+ψi (x)

.

(6.1.22)

52


In this case Z

fj (v, x)dv =

R3

Z

π αj

3/2

(di , dj ) (v, di )fi (v, x)dv = 2αj

R3

e Z

ϕi +ψi

e

dj2 4αj

(6.1.23)

fj (v, x)dv

(6.1.24)

R3

Z di ×

dj vfj dv = di × 2αj

R3

Z

fj (v, x)dv,

(6.1.25)

R3

where i, j = 1, 2. Since vectors d1 , d2 are linearly dependent, then the right parts in (6.1.25) are equal to zero. Therefore, if fi (v, x) is defined by (6.1.22), then, due to (6.1.21), ∇λi (x) = 0. Hence, λi (x) = βi is constant in (6.1.13). The right parts of (6.1.17), (6.1.19) are completely defined, due to (6.1.23), (6.1.24), and the last one takes the form 3/2 dj2 2 X mi π ϕj +ψj 4αj e e 4ϕi = 4π qj 2αi qi αj

(6.1.26)

j=1

2

d 2 mi c 4π X (di , dj ) π 3/2 ϕj +ψj 4αj j − e e , 4ψi = qj qi c 2αj αj

(6.1.27)

j=1

where i = 1, 2; ϕi = ϕî + ϕi (x); ψi = ψˆ i + ψi (x), (d1 , d2 ) = kd12 ;

d22 = k2 d12 ;

k=

α2 l2 . α1 l1

Rewriting (6.1.26), (6.1.27) in the vector form 4ϕ = Af (ϕ + ψ) 4ψ = Bf (ϕ + ψ),

(6.1.28) (6.1.29)

where a b bk a x1 x1 2α1 cy1 2α2 cy1 ; B = A = a b a b x x 2α cy k 2α cy 1 2 2 2 2 2 ψ ϕ 1 1 ϕ = ; ψ = ϕ2 ψ2 ϕ +ψ e 1 1 f (ϕ + ψ) = ϕ2 +ψ2 e


a = 4π q1 xi =

π α1

mi ; 2αi qi

d12

3/2

e 4α1 ;

yi = −

mi c ; qi di2

and introducing notations 81 G = A + B, 8 = , 82

b = 4π q2

π α2

3/2

53

d22

e 4α2

i = 1, 2

8i = ϕi + ψi ,

after summation one obtains the equation 48 = G f (8)

(6.1.30)

instead of (6.1.28), (6.1.29). For studying (6.1.26) and (6.1.27), we will use the following results. Lemma 6.1. Let 8 satisfy system of equations (6.1.30). Then system (6.1.28), (6.1.29) has a solution ϕ = ϕ0 + u1 , ψ = −ϕ0 + u2 , where ϕ0 is an arbitrary harmonic vectorfunction; u1 , u2 are vector functions satisfying the linear Poisson equations 4u1 = Af (8),

4u2 = Bf (8).

Moreover, if detG 6= 0, then u1 = AG −1 8,

u2 = BG −1 8.

Starting a study of (6.1.30) let us consider two cases: 1. detG 6= 0; 2. detG = 0.

Lemma 6.2. If detG 6= 0, then solution of the system (6.1.30) of the form 8 ˆ +u 8 = ˆ 1 82 + u corresponds the solution of algebraic system ˆ = I, I = 1 G f (8) 1 ˆ = −I , G f (8) where function u satisfies Liouville equation 4u = eu (4u = −eu ).

(6.1.31) (6.1.32)

54


Remark 6.1. An analogous result occurs in the case when G is the nonsingular matriz of the n–th order. c Lemma 6.3. If detG = 0, then a general solution λ 1 , where λ—an arbitrary conc2 ˆ stant of the uniform system G f (8) = 0 has a correspondence to the solution family of the system (6.1.30) of the form 8 ˆ 1 +u . 8= ˆ (6.1.33) 82 + lu Here u satisfies the equation of the type 4u = λc3 (eu − elu ), l, c1 , c2 , c3 —constants; ˆ i = ln λci . 8 Proof. Since detG = 0, then there exists a constant l such that a a G = 11 12 . la11 la12 Therefore, substituting (6.1.33) into (6.1.30), one obtains the equation 4u = a11 e81 +u + a12 e82 +lu , ˆ

ˆ

where 1 8ˆ 1 e 8ˆ = λ a11 e 2 −a 12

if a12 6= 0. Since a11 e81 + a12 e82 = 0, then ˆ

ˆ

a11 e81 +u + a12 e82 +lu = λa11 [eu − elu ]. ˆ

ˆ

6.2 Boundary Value Problem In this section we aim to the construction of the distribution functions fi (v, x), electromagnetic fields E(x), B(x) and setting the adecuate boundary-value problems. We will consider the distribution functions fi (v, x) and the fields E(x), B(x) corresponding to equations (6.1.22), (6.1.8), (6.1.13), where functions ϕi , ψi satisfy (6.1.26), (6.1.27), (6.1.14)–(6.1.16) and conditions (6.1.11), (6.1.12). Let us consider two cases: m1 α2 q2 , i.e., l2 = l1 ; m2 α1 q1 m1 α2 q2 2. l2 = 6 , i.e., l2 6= l1 . m2 α1 q1 1. l2 =


55

The first calls to Lemma 6.3. Actually, since detG = −

ab (l2 − l1 )2 , 2cα1 l1 x1 y1

then detG < 0 for l2 6= l1 and detG = 0 for l2 = l1 . Thus, in the first case detG = 0. Therefore, it is possible to use Lemma 6.3. We note, due to (6.1.15), (6.1.16), l2 = 1 α2 q2 l1 = l = m m2 α1 q1 < 0, α2 d1 = α1 d2 . In this case, matrix G can be transformed as 1 1 a b G= − . x1 2α1 cy1 la lb 8 ˆ 1 +u , then (6.1.30) degenerates into one equation Since 8 = ˆ 82 + lu 4u =

1 1 ˆ ˆ ae81 +u + be82 +lu , − x1 2α1 cy1

(6.2.1)

ˆ 1, 8 ˆ 2 are defined from the system moreover 8 8ˆ 1 e G = 8ˆ = 0, e 2 due to Lemma 6.3. Since we are interested in real solutions, sign ab = sign q1 q2 < 0, ˆ 1 = ln λ, 8 ˆ 2 = ln | − λ a |, where λ ∈ R+ is an arbitrary parameter. then 8 b Equation (6.2.1) takes the form 4u = λa11 (eu − elu ),

(6.2.2)

where a11 =

3/2 2 2πq21 4αd1 π 1 (4α 2 c2 − d 2 ) e , 1 1 2 α1 α1 m1 c

l ∈ R− .

Let x ∈ 2 ⊂ R3 , 0 = ∂. We will search for a nontrivial solution of (6.2.2) satisfying boundary condition u|0 = 0

(6.2.3)

and relation (∇u, d1 ) = 0.

(6.2.4)

With respect to (6.2.4), one needs u = u dx11 − dy12 , dy12 − dz13 , if d1k 6= 0, (k = 1, 2, 3). In addition, three-dimensional problem (6.2.2)–(6.2.4) is transformed into

56


a two-dimensional one. Dirichlet problem (6.2.2)–(6.2.4) has a trivial solution u = 0. Nonlinear Dirichlet problems (6.2.2)–(6.2.4) can have a small nontrivial real solution u → 0 for λ → λ0 in the neighborhood of eigenvalue λ0 of linearized problem 4u = λa11 (1 − l)u, x ∈ 2 ⊂ R3 u|0 = 0, (∇u, d1 ) = 0. Assuming without loss of generality that λ0 is smallest eigenvalue, then if it is unique, in the semineighborhood of the point λ0 , there exists a small nontrivial solution u → 0 at λ → λ0 (λ → −λ0 ). We note that Lyapunov-Schmidt branching equation of nonlinear problem has the form L(µξ ) = ∇ξ U(ξ, µ), µ = λ − λ0 . This equation is potential for any eigenvalue of linearized problem. Therefore, any eigenvalue λ0 will be a bifurcation point. If λ0 is odd-multiple, then real solution exists at least in semi-neighborhood of the point λ0 . Detailed description of the domain 2 allows us to build the asymptotic behavior of the corresponding branches of nontrivial solutions u of (6.2.2). For determining the functions ϕi , ψi , we use Lemma 6.1. Knowing vector-function 8, we construct functions ϕ1 + ψ1 = 8ˆ1 + u, ϕ2 + ψ2 = 8ˆ2 + lu, substituting them in the right part of the first equation of (6.1.26). As a result, we obtain linear equation m1 4ϕ1 (x) = λa(eu − elu ) 2α1 q1 ϕ1 (x)|0 = 0, (∇ϕ1 , d1 ) = 0. We will find ϕ1 (x) in the form ϕ1 (x) = 2u, where 2—constant. Then we come to the equation 2

m1 4u = λa(eu − elu ), 2α1 q1 u|0 = 0, (∇u, d1 ) = 0.

Assuming 2 =

2α1 q1 a m1 a11 ,

we obtain identity, because u is a solution of (6.2.2) under

ˆ 1 + 2u + ϕ0 (x), where ϕ0 (x) an arbiconditions (6.2.3), (6.2.4). Hence, ϕ1 (x) = 8 trary harmonic function. Since E(x) = 2αm11q1 ∇ϕ1 (x), then 2αm11q1 ϕ1 (x) is a potential of required electric field. Assume that the value of this potential P0 is given on the boundary 0, then function ϕ0 (x) is defined concretely. For its determination we will obtain linear Dirichlet boundary value problem 4ϕ0 (x) = 0, ϕ0 (x)|0 = P0 −

m1 ˆ 8. 2α1 q1

ˆ 1 + u, then, having obtained ϕ1 , we construct ψ1 by formula ψ1 = Since ϕ1 + ψ1 = 8 ˆ u, ϕ1 , ψ1 and using (6.1.8), (6.1.13), (6.1.22), we find (1 − 2)u − ϕ0 (x). Knowing 8,


57

required E(x), B(x), fi : fi (v, x) = e−αv +vdi +8i (x) , m1 E(x) = ∇ (2u + ϕ0 (x)) , 2α1 q1 m1 c β1 [d1 × ∇ [(1 − 2)u − ϕ0 (x)]] , B(x) = 2 d1 − d1 q1 d12 2

where 81 (x) = ln λ + u, 82 (x) = ln | − λ ab | + lu; d1 ∈ R3 ,

d2 =

α2 d1 , α1

αi ∈ (0, ∞);

l=

m1 α2 q2 < 0. m2 α1 q1

Further, we consider system of Vlasov-Maxwell equations (6.1.1)–(6.1.5) with condition (E, n0 )|0 = e0 .

(6.2.5)

Finding stationary distributions in the form (6.1.7), as it is was made above, we derived to (6.1.17), (6.1.19) and (6.1.20). Let us assume that the folowing condition holds: A). fi (v, x) = fi (−αi v2 + vdi + ϕi + ψi ) Z Z fi (v, x)dv < +∞, vfi (v, x)dv < +∞. R3

R3

Systems (6.1.17), (6.1.19) possess the specific symmetry expressed by the following property: Property I. The second equations in systems (6.1.17), (6.1.19) coincide with the first ones. For the proof of Property I it suffices to take into consideration that ϕ2 (x) = l1 ϕ1 (x),

ψ2 (x) = l2 ψ1 (x)

and taking into account (6.1.15), (6.1.16). Lemma 6.4. If for some function 8(x), (α ∈ R+ , d ∈ R3 ) the conditions Z Z f (−αv2 + vd + 8(x))dv < +∞, vf (−αv2 + vd + 8(x))dv < +∞, R3

R3

are satisfied, then Z Z d 2 f (−αv2 + vd + 8(x))dv. vf (−αv + vd + 8(x))dv = 2α R3

R3

(6.2.6)

58


d we have identity Proof. Since under change of variables v = ξ + 2α Z Z d2 vf (−αv2 + vd + 8(x))dv = ξ f −αξ 2 + + 8 dξ 4α R3

R3

d2 f −αξ + + 8(x) dξ, 4α

Z

d + 2α

2

R3

then Z

vi f (−αv2 + vd + 8(x))dv =

R3

Z =

Z d2 di d2 2 2 ξi f −αξ + + 8 dξ + f −αξ + + 8(x) dξ, 4α 2α 4α

R3

R3

ξ ∈ R3 , (i = 1, 2, 3). Introducing the spherical coordinates % ≥ 0, 0 ≤ ϕ ≤ π , 0 ≤ 2 ≤ 2π , ξ1 = % sin ϕ cos 2, ξ2 = %2 = % sin ϕ sin 2, ξ3 = % cos ϕ, it is easy to see that Z d2 + 8(x) dξ = 0, (i = 1, 2, 3). ξi f −αξ 2 + 4α R3

Thus, Z

Z d2 di vi f −αv + + 8(x) dv = f dξ. 4α 2α

2

R3

R3

d Since ξ = v − 2α , then (6.2.6) is satisfied.

Lemma 6.5. Let the distribution functions f1 , f2 satisfy condition A . Then system (6.1.17), (6.1.19) and (6.1.20) is transformed to the form p4ϕ(x) = q1 A1 (λ1 + ϕ(x) + ψ(x)) + q2 A2 (λ2 + l1 ϕ(x) + l2 ψ(x)), ℘4ψ(x) = q1 A1 (λ1 + ϕ(x) + ψ(x))

+ q2 A2 × 2α1 (d1 , d2 ) × (λ + l1 ϕ(x) + l2 ψ(x)) , 2α2

where ϕ(x) = ϕ1 (x),

ψ(x) = ψ1 (x),

λ1 = c11 + c21 ,

p=

λ2 = c12 + c22 ,

Functions A1 , A2 are defined in (6.2.9).

(6.2.7)

d12

m1 8π α1 q1 m 1 c2 ℘=− . 4πq1

(6.2.8)


59

Proof. Due to Property I, the second equations in (6.1.17), (6.1.19) can be omitted. Since we have identities Z

fi (−αi v2 + vdi + ϕi + ψi )dv = Ai ,

(i = 1, 2)

(6.2.9)

R3

from Lemma 6.4, where 4

A1 = A1 (λ1 + ϕ + ψ),

4

A2 = A2 (λ2 + l1 ϕ + l2 ψ),

then the first equations in (6.1.17), (6.1.19) can be written in the form of (6.2.7), (6.2.8). Since Z

dk vfk dv = − 2αk

Z fk dv,

di × dk = 0 (i, k = 1, 2),

R3

R3

then Lemma 6.5 is proven. As already mentioned, during the study of systems (6.1.17), (6.1.19) two cases were detected. Case 1 l2 = α2 q2 m1 /α1 q1 m2 . Then l2 = l1 , α2 d1 = α1 d2 . System (6.2.7), (6.2.8) takes a form 4ϕ(x) = q1 A1 (λ1 + ϕ + ψ) + q2 A2 (λ2 + l(ϕ + ψ)), µ4ψ(x) = q1 A1 (λ1 + ϕ + ψ) + q2 A(λ2 + l(ϕ + ψ)) 4

l1 = l2 = l,

=

m1 , 8π α1 q1

µ=−

(6.2.10) (6.2.11)

α1 c2 m1 . 2πq1 d12

Let us consider nonlinear equation 4u = a(q1 A1 (λ1 + u) + q2 A2 (λ2 + lu)) a=

2πq1 (4α12 c2 − d12 ), α 1 c2 m 1

(6.2.12)

u = ϕ(x) + ψ(x).

If u∗ satisfies (6.2.12), then (6.2.10), (6.2.11) has a solution ϕ(x) = 2u∗ (x) + ϕ0 (x),

ψ(x) = (1 − 2)u∗ (x) − ϕ0 (x),

where 2 = 4α12 c2 /(4α12 c2 − d12 ); ϕ0 an arbitrary harmonic function. Since solution of (6.2.10), (6.2.11) is expressed via the solution of equation (6.2.12) and harmonic

60


function, then in the Case 1 VM system is reduced to a study of “resolving” equation (6.2.12). Hence, it follows: Theorem 6.1. Let distribution functions f1 , f2 satisfy condition A . Then the corresponding solution of system (6.1.1)–(6.1.5) can be written in the form f1 (v, x) = f1 (−αv2 + vd1 + λ1 + u∗ (x)),

d1 ∈ R 3 ,

f2 (v, x) = f2 (−αv2 + vd2 + λ2 + lu∗ (x)), d2 ∈ R3 , m1 ∇ 2u∗ (x) + ϕ0 (x) , E(x) = 2α1 q1 γ1 m1 c B(x) = 2 d1 − d1 × ∇((1 − 2)u∗ (x) − ϕ0 ) , 2 d1 q1 d1

(6.2.13)

where u∗ (x) is defined from (6.2.12); γ1 , λ1 , λ2 are arbitrary constants; ϕ0 (x) an arbitrary harmonic function; (∇u∗ , d1 ) = 0, (∇ϕ0 (x), d1 ) = 0. Let us consider solutions of “resolving” equation (6.2.12). If arbitrary constants λ1 , λ2 are connected by means of equation q1 A1 (λ1 ) + q2 A2 (λ2 ) = 0,

(6.2.14)

then (6.2.12) has a trivial solution u∗ (x) = 0. In this case, construction of nontrivial solutions of (6.2.12) is a well-known problem about bifurcation point for nonlinear equations. For the solution of this problem we need use boundary condition (6.2.5). Due to (6.2.5), one has ∂ 2α1 q1 e0 (x). 2u∗ (x) + ϕ0 (x) |0 = ∂n m1 Assuming ∂ ∗ u (x)|0 = 0, ∂n

∂ 2α1 q1 ϕ0 (x)|0 = e0 (x), ∂n m1

we obtain a linear boundary value problem 4ϕ0 (x) = 0, ∂ 2α1 q1 ϕ0 (x)|0 = e0 (x), ∂n m1

(∇ϕ0 (x), d1 ) = 0

(6.2.15)

and a nonlinear boundary value problem 4u(x) = a(q1 A1 (λ1 + u) + q2 A2 (λ2 + lu)), ∂ u(x)|0 = 0, (∇u(x), d1 ) = 0, ∂n a = 2πq1 (4α12 c2 − d12 )/α12 c2 m1 ,

(6.2.16)


61

where arbitrary constants λ1 , λ2 satisfy (6.2.14). So (6.2.16) would have the nontrivial solution u → 0, λ → λ∗ , where λ = (λ1 , λ2 ) satisfies (6.2.14), it is necessarily that linearized problem 4ϕ(x) = a q1 A01 (λ∗1 ) + q2 A02 (λ∗2 ) ϕ(x) ∂ ϕ(x)|0 = 0, ∂n

(∇ϕ(x), d1 ) = 0

should have the nontrivial solution u → 0, λ → λ∗ . Case 2 l2 6= α2 q2 m1 /α1 q1 m2 . In this case, we restrict to the construction of solutions ϕ(x), ψ(x) of (6.2.7), (6.2.8) satisfying the condition ϕ(x) + ψ(x) = l1 ϕ(x) + l2 ψ(x).

(6.2.17)

Let us assume that the condition is satisfied. B). There are constants λ1 , λ2 , which satisfy the identity "

# d12 q1 A1 (λ1 + u) 4α1 (1 − l1 ) − (1 − l2 ) = α 1 c2 (d1 , d2 ) (l2 − 1) = q2 A2 (λ2 + u) 4α1 (l1 − 1) − α2 c2 for functions f1 , f2 together with condition A and some u ∈ R1 . Lemma 6.6. Let condition B be satisfied. Then solution of (6.2.7), (6.2.8) with (6.2.17) has the form ϕ(x) =

l2 − 1 ∗ u (x), l2 − l1

ψ(x) =

1 − l1 ∗ u (x), l2 − l1

where u∗ (x) is a solution of equation 4u = bA1 (λ1 + u), b=

(6.2.18)

8πq21 (l2 − l1 )(d12 α2 − (d1 , d2 )α1 ) . m1 (4α1 α2 c2 (l2 − 1) − (d1 , d2 )(l2 − 1))

Proof. After nondegenerate variable change u1 = ϕ + ψ, u2 = l1 ϕ + l2 ψ, system (6.2.7), (6.2.8) becomes 4u1 = (a1 − a2 )q1 A1 + (a1 − a3 )q2 A2 , 4u2 = (l1 a1 − l2 a2 )q1 A1 + (l1 a1 − l2 a3 )q2 A2 ,

62


where a1 =

8πα1 q1 , m1

a2 =

2π q1 d12 , α1 c2 m1

a3 =

2πq1 (d1 , d2 ) . α2 c2 m1

The right sides of the last system are equal to each other in view of condition B. Due to the same condition, equations over 4u1 , 4u2 are reduced to (6.2.18). α2 q2 m1 hold. Then system (6.1.1)–(6.1.5), Theorem 6.2. Let conditions A, B, l2 6= α1 q1 m2 (6.2.5) has a solution fi (v, x) = fi (−αi v2 + vdi + λ1 + u∗ (x)) (i = 1, 2), m1 (l2 − 1) E(x) = ∇u∗ (x), 2α1 q1 (l2 − l1 ) m1 c(1 − l1 ) γ1 B(x) = 2 d1 − d1 × ∇u∗ (x), d1 q1 d12 (l2 − l1 ) where γ1 is an arbitrary constant. Function u∗ (x) satisfies (6.2.18) with the conditions ∂ ∗ 2α1 q1 (l2 − l1 ) u |0 = e0 (x). ∂n m1 (l2 − 1)

(∇u∗ , d1 ) = 0,

Let us show the application of general Theorem 6.1 and Theorem 6.2. Assume, it is necessary to determine distributions 2 +vd +8 (x) 1 1

,

d1 ∈ R 3 ,

2 +vd +8 (x) 2 2

,

d2 ∈ R3 ,

f1 (v, x) = e−α1 v f2 (v, x) = e−α2 v

(6.2.19)

v ∈ R3 , x ∈ 2 ∈ R3 and the corresponding fields E, B satisfying VM system (6.1.1)–(6.1.5) with boundary condition (6.2.5). Due to Lemma 1.5, unknown functions 81 (x), 82 (x) can be occured as 81 (x) = λ1 + ϕ(x) + ψ(x),

82 (x) = λ2 + l1 ϕ(x) + l2 ψ(x).

We suppose that parameters α1 , α2 , d1 , d2 , l1 , l2 satisfy conditions (6.1.15), (6.1.16). Consider two cases. Case 1 l2 = α2 q2 m1 /α1 q1 m2 . Then for finding of parameters λ1 , λ2 and functions ϕ(x), ψ(x), the equation (6.2.12) should be resolved, and, therefore, boundary value problems (6.2.15), (6.2.16) are solved regarding Theorem 6.1. Since 3/2 d2 π λ + 1 e 1 4α1 eu , A1 (λ1 + u) = α1 3/2 d2 π λ + 2 A2 (λ2 + lu) = e 2 4α2 elu , α2


63

then Condition A is satisfied. Equation (6.2.12) will have the nontrivial solution for some λ1 , if 

3/2 q1 λ1 + 14 α2 e λ2 = ln− α2 q2

d22 d12 α1 − α2



.

In this case, boundary value problem (6.2.16) is Neumann problem for equation of the type 4u = σ g(α1 , d1 )(eu − elu ), ∂ u|0 = 0, ∂n

(6.2.20)

(∇u, d1 ) = 0,

where σ = eλ1 ,

g(α1 , d1 ) = ag1

π α1

3/2

d1 2

e 4α1 .

Problem (6.2.20) can have nontrivial solutions u(σ ) → 0 for σ → σ ∗ , only in neighborhood of eigenvalues σ ∗ of the problem 4u = σ ∗ g(α1 , d1 )(1 − l2 )u, ∂ u|0 = 0, (∇u, d1 ) = 0. ∂n After obtaining eigenvalues σ ∗ (bifurcation points of (6.2.20)) and having built solutions u(σ ) → 0 for σ → σ ∗ , we find solutions of VM system (6.1.1)–(6.1.5), (6.2.5) by formulas (6.2.13). Case 2 l2 6= α2 q2 m1 /α1 q1 m2 . Then l2 6= l1 and to find λ1 , λ2 , ϕ, ψ, we need use Theorem 6.2. Moreover, Ai (λi + u) =

π αi

d2

3/2 e

λi + 4αi

i

eu .

Therefore, condition A is satisfied, moreover, condition B is reduced to B0 : ! d12 (1 − l2 ) 1 q1 e 4α1 (1 − l1 ) − = α1 c2 3/2 d2 π (d1 , d2 )(l2 − 1) λ2 + 4α2 2 4α2 (l1 − 1) − = q2 . e α2 α2 c2

π α1

3/2

d2

λ1 + 4α1

64


Since q1 q2 < 0, then from B0 , one can define eλ2 , if parameters α1 , α2 , d1 , d2 , l1 , l2 satisfy inequility

τ

− d12

ω α1

ω τ − (d1 , d2 ) α2

where τ = 4α1 (1 − l1 ), ω = Liouville equation:

1−l2 . c2

> 0,

The solution of equation (6.2.18) is well-known

4u = σ h(α1 , α2 , d1 , d2 )eu , ∂ 2α1 q1 (l2 − l1 ) |0 = e0 , (∇u, d) = 0, ∂n m1 (l2 − 1) 3/2 d2 1 π σ = eλ , h(α1 , α2 , d1 , d2 ) = b e 4α1 . α1

(6.2.21)

Thus, in the case of distributions of exponential form (6.2.19), the solution of (6.1.1)–(6.1.5), (6.2.5) is reduced to linear Neumann problem (6.2.15) for l2 = l1 and to the problem of bifurcation point for equation of sh-Gordon type or to Neumann problem for Liouville equation (6.2.21), where σ ∈ R+ an arbitrary constant for l2 6= l1 and to additional restriction on parameters α1 , α2 , d1 , d2 , l1 , l2 included in (6.2.19).

6.3 Solutions with Norm Let us examine constructed solutions with respect to the norm definition. With respect to normalization condition (6.1.6) and taking into account relation (6.1.23) free parameters α1 , α2 , d1 , λ, l must satisfy the conditions Z a 8ˆ 1 e eu(x) dx =1, 4πq1 2 Z (6.3.1) b 8ˆ 2 lu(x) e e dx =1, 4πq2 2

ˆ 1 = ln λ, 8 ˆ 2 = ln | − λ a |. Therefore, where 8 b aλ 4πq1

Z

eu(x) dx = 1,

2

−

aλ 4πq2

Z

elu(x) dx = 1,

2

moreover, function u(x) satisfies equation (6.2.1), which takes the form

1 1 4u = λa eu − elu . − x1 2α1 cy1

(6.3.2)


65

Hence, one needs to solve (6.3.2) with conditions (6.3.1). Thus, we obtain solutions with norm. Excluding aλ from (6.3.1), we obtain equation

1 1 − 4u = 4π x1 2α1 c1 y1

"

q elu(x) q eu(x) R 1 R 2 + u(x) dx lu(x) dx 2 e 2 e

# (6.3.3)

with condition Z q1 elu(x) + q2 eu(x) dx = 0.

(6.3.4)

2

Thus, normed solution u(x) leads to the problem (6.3.1), (6.3.2), or to the problem (6.3.3), (6.3.4). If one finds solution u∗ (x) satisfying (6.3.3) and equation 4u = 4π

1 1 q1 − eu − elu R , u(x) dx x1 2α1 cy1 e 1

then condition (6.3.4) also will be satisfied for this solution, giving us a normalized solution. α q m 2 2 1 Let l2 6= . Then l2 6= l1 and detG 6= 0. α 1 q1 m 2 In this case we study (6.1.26), (6.1.27) using Lemma 6.2 and searching for real valˆ of (6.1.31), (6.1.32). So, we have to define conditions when systems ued solutions 8 x 1 G = 1 = x2 1 x1 1 G = = − x2 1

(6.3.5) (6.3.6)

have the positive solutions x1 > 0, x2 > 0. If aij elements of matrix G , then G −1 =

1 a22 −a12 ; detG −a21 a11

moreover, detG < 0. Therefore, one of the systems has the positive solution, if D = (a22 − a12 )(a11 − a21 ) > 0. Since l1 − 1 k(1 − l2 ) + , x1 2α2 cy1 1 − l1 l2 − 1 a11 − a21 = a + , x1 2α1 cy1

a22 − a12 = b

66


where ab < 0, l1 < 0, then 1 + |l1 | k2 (l2 − 1) 1 + |l1 | l2 − 1 D = |ab| + + . x1 2α1 cy1 x1 2α1 cy1 Moreover, D > 0, if 1 + |l1 | 1 − l2 1 k > max , , x1 2cy1 α1 α2 or 1 + |l1 | 1 − l2 1 k < min , . x1 2cy1 α1 α2 Substituting values l1 , x1 , y1 into the inequilities given above, we obtain 4α1 c2 l2 , (α q m + α |q |m ) > max 1 − l , (1 − l ) 1 1 2 2 2 1 2 2 l1 q1 d12 m2 4α1 c2 l2 (α1 q1 m2 + α2 |q2 |m1 ) < min 1 − l2 , (1 − l2 ) . l1 q1 d12 m2

(6.3.7)

Introducing notation L(αi , qi , mi , d1 ) =

4α1 c2 (α1 q1 m2 + α2 |q2 |m1 ), q1 d12 m2

we rewrite (6.3.7) in the form l2 , L(αi , qi mi , d1 ) > max 1 − l2 , (1 − l2 ) l1 l2 L(αi , qi , mi , d1 ) < min 1 − l2 , (1 − l2 ) . l1

(6.3.8) (6.3.9)

We construct straight line 1 − l2 on plane yOl2 and parabola y = (1 − l2 ) ll21 , where l1 is a given constant (see figure). 3

y

2

1

0 −1 −2

γ

αβ −1

0

1

l2 2


67

If we fix αi , qi , mi , d1 , then l1 and value L(αi , qi mi , d1 ) in (6.3.8), (6.3.9) will become the specific constants. Let us draw straight line y = 1 and let us denote as α, β, γ abscissas of intersection points with parabola. Then for l2 ∈ (−∞, α), inequality (6.3.9) will be satisfied, a22 − a12 < 0, a11 − a21 < 0, and for l2 (β, γ ), inequality (6.3.8) with a22 − a12 > 0, a11 − a21 > 0. Since detG < 0, then system (6.3.5) has the positive solution for l2 ∈ (−∞, α), and for l2 ∈ (β, γ ), sistema (6.3.6) has the positive solution. ˆ 1, 8 ˆ 2 , we build vector Having defined 8 8 ˆ 1 +u 8= , ˆ 2 +u 8 ˆ 1, 8 ˆ 2 correspond to the solution where u satisfies a Liouville equation 4u = eu , if 8 ˆ 1, 8 ˆ 2 correspond to the solution of of system (6.3.5) or of equation 4u = −eu , if 8 (6.3.6). For determining the functions ϕ1 , ψ1 , we will use Lemma 6.1. Knowing 8, we substitute it in the right side of the first equation of system (6.1.26). One obtains linear equation m1 ˆ ˆ 4ϕ1 (x) = (ae81 + be82 )eu . 2α1 q1

(6.3.10)

Let, for the definition, function u(x) satisfy equation 4u(x) = eu(x) , then for 2=

2(ae81 + be82 )α1 q1 , m1 ˆ

ˆ

ˆ 1 + 2u(x). Since 81 = function ϕ1 (x) = 2u(x) satisfies (6.3.10). Thus, ϕ1 (x) = 8 ϕ1 + ψ1 , then ψ1 (x) = (1 − 2)u(x). We demonstrate that, here, functions fi (v, x) do not satisfy normed condition (6.1.6), not with what values of the free parameters, if q2 = −q1 . Indeed it is necessary for norming that b ˆ 1 a 8ˆ 1 e = e82 = R u(x) , q1 q2 e dx 2

where 8ˆ 1 e 1 G 8ˆ = . 1 e 2 Therefore, 1 1 k 1 ˆ ˆ − e81 + b − e82 = 1, a x1 2α1 cy1 x1 2α2 cy1 l1 l2 l1 kl2 ˆ ˆ a − e81 + b − e82 = 1, x1 2α1 cy1 x1 2α2 cy1

68


or the same q1 a 8ˆ 1 q2 kq2 b 8ˆ 2 q1 − e + − e = 1, x1 2α1 cy1 q1 x1 2α2 cy1 q2 q1 l1 q 1 l2 a 8ˆ 1 q2 l1 q2 kl2 b 8ˆ 2 − e + − e = 1. x1 2α1 cy1 q1 x1 2α2 cy1 q2 Since

ˆ1 a 8 q1 e

q1

=

ˆ2 b 8 q2 e ,

then it is necessary

1 − l1 1 − l2 − x1 2α2 cy1

+ q2

1 − l1 1 − l2 −k x1 2α2 cy1

= 0.

If q2 = −q1 , then it is necessary that α11 = αk2 . Since k = αα21 ll21 , then α11 = αl12l1 , i.e., l1 = l2 . However, in our case, l1 6= l2 , providing us with contradiction. So, while l1 6= l2 , the normalyzing condition (6.1.6) is fulfilled for q2 6= −q1 .

7 Boundary Value Problems for the Vlasov-Maxwell System

7.1 Introduction At present the investigation of the Vlasov equation goes in two different directions. The first direction is related to the existence theorems for Cauchy problem and uses an a priori estimation technique as the basis for research. The second one implements the reduction of the initial problem to a simplified one introducing a set of distribution function (ansatz), followed by reconstruction of the characteristics for electromagnetic fields in an evident form. This is a rather restrictive approach, since distribution function has a special form. By contrast, it allows us to solve a problem in an explicit form, which is important for applications. The statement and investigation of the boundary value problem for the Vlasov equation are very difficult and have only been considered in simplified cases (see Abdallach [30], Guo [125], Degond [89]). Reducing it to the boundary value problem for a system of nonlinear elliptic equations allows us to show a solvability in some cases. Doing the same for the initial statement of problem is not that simple. Nevertheless, both directions are related in terms of special structures used for studying kinetic equations. For example: l

l

Energy integral is applied in both cases for obtaining energy estimations in existence theorems and for construction of Lyapunov functionals; Virtual identities in stability and instability analysis in special classes of solutions of Vlasov equation.

It is known that the solution of Vlasov equation (see Vlasov [305, 306]) are arbitrary functions of first integrals of the characteristic system (till now smoothness of the solutions remains a complicated unsolved problem), defining the trajectory of a particle motion in electromagnetic field qi 1 ˙ r˙ = V, V = E(r, t) + V × B(r, t) , (7.1.1) mi c 4

4

where r = (x, y, z) ∈ 2 ⊆ R3 , V = (Vx , Vy , Vz ) ∈ 1 ⊂ R3 —position and velocity of 4

4

a particle, E = (Ex , Ey , Ez )—a tension of electrical field, B = (Bx , By , Bz )—magnetic Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00007-7 c 2011 Elsevier Inc. All rights reserved.

70


induction and mi , qi —mass and charge of a particle of i-th kind. For N-component distribution function, the classical Vlasov-Maxwell system has the form 1 qi E + V × B ·∇V fi = 0, i = 1, . . . , N, (7.1.2) ∂t fi + V · ∇r fi + mi c ∂t E = c curlB − j, (7.1.3) divE = ρ, (7.1.4) ∂t B = −c curlE, (7.1.5) divB = 0. (7.1.6) The charge and current densities are defined by formulas Z n X qi fi dV, ρ(r, t) = 4π i=1

1

(7.1.7)

Z n X j(r, t) = 4π qi fi VdV. i=1

1

We impose the specular reflection condition on the boundary for distribution function fi (t, r, v) = fi (t, r, v − 2(vN (r))N (r)),

t ≥ 0, r ∈ ∂,

v ∈ ,

where N (r) is a normal vector to the boundary surface. In applied problems, the impact of magnetic field is often neglected. This limit system is known as the Vlasov-Poisson (VP) one, where the Maxwell equations degenerates to the Poisson equation 4ϕ = 4π

Z n X qi fi dV, i=1

(7.1.8)

1

where ϕ(r, t)—a scalar potential of the electrical field. In a general case, the distribution function may be represented in the form fi = fi (Hi1 , Hi2 , . . . , Hil ),

i = 1, . . . , N,

where Hil is the first integral (is constant along the characteristics of the equation) for (7.1.1). In reality, it is not easy to select a structure of the distribution function which is connected with electromagnetic potentials aiming to transform the initial system into simplified form. Hence, in practice, one is usually restricted to the integrals of energy Hi = −ci |V|2 + ϕ(r, t) or Hi0 = −di |V|2 + ϕ(r) as in the stationary problem case (see Vlasov [305, 306]). At the same time, an introduction of the following ansatz X j Hil = ϕil + (V, dil ) + (Ail V, V) + ailmkj V1m V2k V3 m+k+j=3

Boundary Value Problems for the Vlasov-Maxwell System

71 4

generalizes the form of the distribution function. Here V = (V1 , V2 , V3 ) and (Ail V, V) is a quadratic form; the following are the forms of higher degrees. In this case, matrices Ail and coefficients ailmkj should be connected with the system (7.1.2)–(7.1.6) converting the first integrals for the characteristic system (7.1.1) into Hil . The first statement of existence problem of classical solutions for the onedimensional Vlasov equation has been given by Iordanskii [149], and the existence of generalized (weak) solutions for the two-dimensional problem has been proven by Arsen’ev [17]. The results of Neunzert [216], Horst [140], Batt [22], Illner, Neunzert [148], Ukai, Okabe [278], DiPerna, Lions P. [92], Wollman [310, 311], Batt, Rein [26], and Pfaffelmoser [229] are devoted to existence of solutions for (7.1.2) and (7.1.8). Degond [86], Glassey, Strauss [111], Glassey, Schaeffer [112]–[115], Horst [142, 143], Cooper, Klimas [76], Schaeffer [253], Guo [125], and Rein [235] concern its generalization to the VM system (7.1.2)–(7.1.6). Some rigorous results obtained recently (see Guo [125], Abdallach [30], Degond [89], Abdallach, Degond, Mehats [32], Vedenyapin [284], [286], Batt, Fabian [27], Braasch [55], Guo, Ragazzo [126], Dolbeault [96], Poupaud [232], Caffarelli, Dolbeault, Markowich, Schmeiser [59], Ambroso [4]) relate to analysis of (7.1.2)–(7.1.6), (7.1.2)–(7.1.8) in the bounded domains with boundary conditions. We have to mention that techniques used to prove the existence of solutions of Cauchy problem for the VM and VP systems for (x ∈ R3 , v ∈ R3 ) have limited applicability in bounded domains. Hence, a necessity to study VM and VP systems with boundary conditions is valid. That is why, before presenting our own results, we have to outline some already published results on VM and VP systems in bounded domains.

7.2 Existence and Properties of the Solutions of the Vlasov-Maxwell and Vlasov-Poisson Systems in the Bounded Domains In the case of spherical symmetry rather complete results were obtained by Batt, Faltenbacher, and Horst [23]. In the next paper by Batt, Berestycki, Degond, Perthame [24], a family of “local isotropic” solutions of nonstationary problem of the VP system for distribution function (U − Ar)2 , f (t, r, V) = 8 W(t, r) + 2 t ∈ R,

r ∈ D ⊂ R3 ,

U(t, r) = W(t, r) +

v ∈ R3 ,

(Ar)2 , 2

8 : R → [0, ∞),

W : R3 → R,

was constructed. Here U—potential and A—antisymmetric 3 × 3- matrix. Under this assumption, the VP system is reduced to the Dirichlet boundary value problem for the

72


nonlinear elliptic equation Z 1 4W + 2|w| = 4π 8 W + |v|2 dv, 2

w = (w1 , w2 , w3 ) ∈ R.

R3

The existence of the solution for the named problem is proved using the lower—upper solution method. The stationary solutions of n-component VP system for distribution function depending on integral of energy fi (E) were studied by Vedenyapin [284], [286]. He proved the existence of solution for Dirichlet problem −4u(r) = ψ(u), ψ(u) = 4π

n X k=1

u(r)|∂D = u0 (r), Z 1 2 mk |v| + qk u dv, qk gk 2

(7.2.1)

R3

d where an arbitrary function ψ satisfies the condition (i), du ψ(u) ≥ 0. Here u(r)— scalar potential, gk (·)—nonnegative continuously differentiable functions, D ⊂ R3 — domain with a smooth boundary enough, u0 (r)—potential given on the boundary. If r ∈ D ⊂ Rp , v ∈ Rp , then the boundary value problem (7.2.1) has a unique solution for arbitrary nonnegative functions gk (Vedenyapin [284]). Rein [236] proved the existence of solution of (7.2.1) by variational method under condition (i). In their paper [27], Batt and Fabian studied a transformation of the stationary VP system into (7.2.1) in general case, considering distribution functions depending on energy fi (E) and on the sum of energy and momentum fi (E + P). Using a lower— upper solution method (Pao [224]), they proved the existence of the solutions (7.2.1) under condition (i). Therefore, the condition (i) became a primary condition to prove the existence theorems for the problem (7.2.1). The general weak global solution of the VP system has been presented by Weckler in [309]. Dolbeault [96] proved the existence and uniqueness of Maxwellian solutions

f (t, x, v) =

−|v|2 1 ρ(x)e 2T , (2π T)N/2

(x, v) ∈ × RN

using variational methods. A new direction in the study of the VP system is connected with a free boundary problem for semiconductor modeling. Caffarelli, Dolbeault, Markowich, and Schmaiser [59] considered a semilinear elliptic integro-differential equation with Nuemann boundary condition 4φ = q(n − p − C) – , ∂φ = 0 – ∂, ∂ν

(7.2.2)


73

where local densities of electrons n(x) and holes p(x) in insulated semiconductor are given by Boltzmann—Maxwellian statistics P exp(−qφ(x)/(kT)) p(x) = R . exp(−qφ/(kT))dx

N exp(qφ(x)/(kT)) , n(x) = R exp(qφ/(kT))dx

C(x)—is given background, x ∈ . ⊂ Rd is a bounded domain. Using a variational problem statement they proved the existence and uniqueness of the solutions and showed that the limit potential is a solution of the free boundary problem. Concerning a study of nonlocal problem (7.2.2), we recommend the paper by Maslov [192].

7.3 Existence and Properties of Solutions of the VM System in the Bounded Domains p Changing velocity v by its relativistic analogue vˆ = v/ 1 + |v|2 we have to face another complicated problem, since the classical VM system is not invariant in the sense of Galilei and Lorentz. Adding boundary conditions E(t, x) × N (x) = 0,

B(t, x)N (x) = 0,

t ≥ 0,

x ∈ ∂

to the system (7.1.2)–(7.1.7) we obtain a different problem statement. Here N — vector of the unit normal to ∂ and reflection condition fk (t, x, v) = fk (t, x, v˜ (x, v)),

t ≥ 0,

x ∈ ∂,

v ∈ R3 ,

(7.3.1)

where v˜ : R3 → R3 —bijective mapping for x ∈ ∂. One of the most known reflection mechanisms is a specular reflection condition of the form v˜ (x, v) = v − 2(vN (x))N (x),

x ∈ ∂,

v ∈ R3

or reflection condition v˜ (x, v) = −v,

x ∈ ∂,

v ∈ R3 .

At the present, only a few number of papers study the VM system in bounded domains. For the first time, the boundary value problem for the single dimensional VM system has been considered by Cooper, Klimas [76].

74


In the paper [248] Rudykh, Sidorov, and Sinitsyn constructed stationary classical solutions (f1 , . . . , fn , E, B) for the VM system of the special form (Rudykh—Sidorov— Sinitsyn ansatz) fk (x, v) = ψk (−αk v2 + µ1k U1 (x), vd + µ2k U2 (x)), 1 ∇U1 (x), E(x) = α1 q1 1 B(x) = − (d × ∇U2 (x)). q1 d 2 Here functions ψk : R2 → [0, ∞) and parameters d ∈ R3 \{0}, αk > 0, µik 6= 0 (k ∈ {1, . . . , n}, i ∈ {1, 2})—are given; functions U1 , U2 have to be defined. This approach (RSS ansatz) is closely connected with the paper of Degond [86]. Batt and Fabian [27] applied RSS ansatz technique for the VM system with distribution functions ψ(E), ψ(E, F), ψ(E, F, P), where functions E(x, v), F(x, v) and P(x, v)—are the first integrals of Vlasov equation (7.1.2). Braasch in his own thesis [55] extended RSS results to the relativistic VM system.

7.4 Collisionless Kinetic Models (Classical and Relativistic Vlasov-Maxwell Systems) In this area, existence theorems (and global stability) of renormalized solutions in the bounded domains (when trace is defined in the boundary) were proven by Mischler [205, 206]. Abdallah and Dolbeault [36] also developed the entropic methods in bounded domains for qualitative study of behavior of global solutions of the VP system. Regularity theorems of weak solutions on the basis of scalar conservation laws and averaging lemmas were proved by Jabin and Perthame [151]. Jabin [150] also obtained the local existence theorems of weak solutions of the VP system in the bounded domains. For modeling of ionic beams, Ambroso, Fleury, Lucquin-Desreux, and Raviart [5] proposed some new kinetic models with a source. Existence theorems of global solutions of the Vlasov-Einstein system in the case of hyperbolic symmetry were proved by Andreasson, Rein, and Rendall [6].

7.4.1 Quantum Models: Vigner-Poisson and Schrödinger-Poisson Systems In their paper, Abdallah, Degond, and Markowich [33] considered the Child-Langmuir regime for stationary Schro¨ dinger equation. The Authors developed a semiclassical analysis for quantum kinetic equations with passage in limit h → 0 to classical Vlasov equation with special boundary “transition” conditions from quantum zone to classical. New results were obtained for Boltzmann-Poisson, Euler-Poisson, VignerPoisson-Fokker-Plank systems (like existence and uniqueness of the solutions, hydrodynamic limits, solutions with a minimum energy and dispersion properties).


75

7.4.2 Mixed Quantum–Classical Kinetic Systems Abdallah [37] considered the Vlasov-Schro¨ dinger (VS) and Boltzmann-Schro¨ dinger systems for one-dimensional stationary case. Nonstationary problems for VS system with boundary “transition” conditions from classical zone (Vlasov equation) to quantum (Schro¨ dinger equation) are studied in the paper by Abdallah, Degond, and Gamba [37]. We study the special classes of stationary and nonstationary solutions of VM systems. Being constructed, such solutions lead us to the systems of nonlocal semilinear elliptic equations with boundary conditions. Applying the lower-upper solution method, the existence theorems for solutions of the semilinear nonlocal elliptic boundary value problem under corresponding restrictions upon a distribution function are obtained. It was shown that, under certain conditions upon electromagnetic field, the boundary conditions and specular reflection condition for distribution function are satisfied.

7.5 Stationary Solutions of Vlasov-Maxwell System In this section we consider the system qi 1 ∂ ∂ E+ V ×B · fi (r, V) = 0, V · fi (r, V) + ∂r mi c ∂V rot E = 0, divB = 0, Z n X divE = 4π qk fk (r, V)dV, k=1

(7.5.1) (7.5.2) (7.5.3) (7.5.4)

1

Z n 4π X rot B = qk V fk (r, V)dV. c k=1

(7.5.5)

1

4

Here fi (r, V)—distribution function of the particles of i-th kind; r = (x, y, z) ∈ 4

∈ 2 , V = (Vx , Vy , Vz ) ∈ 1 ⊂ R3 —coordinate and velocity of particle respectively; E, B—electric field strength and magnetic induction; mi , qi —mass and charge of particle of i-th kind. We shall seek stationary distributions of the form 4

fi (r, V) = fi (−αi |V|2 + ϕi , V · di + ψi ) = fî (R, G)

(7.5.6)

and corresponding selfconsistent electromagnetic fields E and B satisfying (7.5.2)– (7.5.5). We assume that i) fî (R, G)—fixed differentiable functions of own arguments; αi ∈ R+ , di ∈ R3 are free parameters, |di | 6= 0; ϕi = c1i + li ϕ, ψi = c2i + ki ψ, where c1i , c2i —constant; for all ϕi , ψi

76


the integrals Z fi dv, R3

Z V fi dv R3

are converged. Unknown functions ϕi (r), ψi (r) have to be defined in such manner that system (7.5.1)–(7.5.5) will satisfy the relation (E(r), di ) = 0, i = 1, . . . , N. The last condition is necessary for solvability of (7.5.1) in a class (7.5.6) for ∂ fî /∂R|v=0 6= 0.

7.5.1 Reduction of the Problem (7.5.1)–(7.5.5) to the System of Nonlinear Elliptic Equations We construct the system of equations to define the set of functions ϕi , ψi . Substituting (7.5.6) into (7.5.1) and equating to zero the coefficients at ∂ fî /∂R and ∂ fî /∂G, we obtain mi ∇ϕi , 2αi qi mi c B(r) × di = − ∇ψi , qi (E(r), di ) = 0.

E(r) =

(7.5.7) (7.5.8)

Here ϕi , ψi —arbitrary functions satisfying conditions (∇ϕi , di ) = 0, (∇ψi , di ) = 0.

i = 1, . . . , N,

(7.5.9) (7.5.10)

Vector B is B(r) =

λi (r) mi c di − [di × ∇ψi ] 2 , di2 qi di

(7.5.11)

where λi (r) = (B, di )—function has to be defined. Having defined ϕi , ψi such that system (7.5.2)–(7.5.5) to be satisfied, one can find unknown functions fi , E, B by formulas (7.5.6), (7.5.7), (7.5.11). Unknown vectors ∇ϕi , ∇ψi are linear dependent by virtue of (7.5.7), (7.5.8). Then we shall seek ϕi , ψi in the form ϕi = c1i + li ϕ,

ψi = c2i + ki ψ,

(7.5.12)

where c1i , c2i —constants. Because of (7.5.7), (7.5.8) parameters li , ki are connected by the following relations m1 αi qi , i = 1, . . . , N, α1 q1 mi q1 qi ki d1 = di . m1 mi li =

(7.5.13) (7.5.14)


77

From (7.5.4) with (7.5.7), one obtains the system n

4ϕi =

Z

8παi qi X qk mi k=1

fk (r, V)dV.

1

Since div[di × ∇ψi ] = 0, then substituting (7.5.11) into (7.5.3), one has (∇λi (r), di ) = 0.

(7.5.15)

Taking into account (7.5.11), from (7.5.5) we obtain the system of linear algebraic equations for ∇λi n

mi c 4π 2 X d ∇λi × di = di 4ψi + qk qi c i k=1

Z V fk dV.

(7.5.16)

1

To solve (7.5.16), it is necessary and sufficient, due to Fredholm theorem (see [274]), that ψi is satisfied the equation n

4ψi = −

4π qi X qk m i c2 k=1

Z

(V, di )fk dV.

1

Furthermore, vector Ci (r)di + di × J(r)

(7.5.17)

is a general solution of (7.5.16) with Z n 4π X qk V fk dV, J= c 4

k=1

1

Ci —arbitrary function. Taking into account (7.5.12)–(7.5.14), it is easy to show that functions ϕ, ψ satisfy the system n

4ϕ =

8παq X qk m k=1

4ψ = −

4

4π q mc2 4

n X k=1

Z fk dV,

(7.5.18)

(V, d)fk dV,

(7.5.19)

1

Z qk 1 4

4

with α = a1 , q = q1 , m = m1 , d = d1 .

78


Lemma 7.1. Vector di × J(r) is a potential and unique solution of (7.5.16) satisfying condition (7.5.15). Proof. Since ψ satisfies (7.5.19), then (7.5.17) is a general solution of (7.5.16). Due to (7.5.15), one can put Ci ≡ 0. Therefore, di × J—unique solution of (7.5.15), (7.5.16). We show that di × J—potential. In fact rot [di × J] = −(di , ∇)J + d(∇, J), where (∇, J) ≡ 0,

(di , ∇)J = (di , ∇)rot B = rot (di , ∇)B.

Due to (7.5.11), di mi c mi c λi × (di , ∇)B = (di , ∇) 2 di − [di × ∇ψi ] 2 = 2 (∇λi , di ) − di qi di di qi di2

× [di × ∇(di , ∇ψi )],

(∇λi , di ) = 0,

(∇ψi , di ) = 0.

Hence, rot [di × J] ≡ 0, di × J = ∇λi , and Lemma is proved. Corollary 7.1. ∇λi (r) = [di × J(r)]. Lemma 7.2. Let b(x) = (b1 (x), b2 (x), b3 (x)), x ∈ R3 , ∂bj ∂bi = , ∂xj ∂xi

i, j = 1, 2, 3.

Then b(x) = ∇λ(x), where λ(x) =

Z1

(b(τ x), x)dτ + Const.

0

The proof is developed by straight calculation. Corollary 7.2.   Z1 d  di λi = 2 β + d × J(τ x), x dτ  , d di2

i = 1, . . . , N,

β − Const.

0

The result follows from Lemma 7.2, Corollary 7.1, and (7.5.14).

(7.5.20)


79

We are searching for the solutions (7.5.18), (7.5.19) satisfying orthogonality conditions (7.5.9), (7.5.10). Assuming d1i 6= 0, i = 1, 2, 3, we shall seek solutions in the form ϕ = ϕ(ξ, η), ψ = ψ(ξ, η) d2 z x y y − + 2 11 2 − , d12 d13 d11 + d12 d11 d12 x y |d1 |d11 d12 4 − , d1 = (d11 , d12 , d13 ). η= 2 2 d13 (d11 + d12 ) d11 d12 ξ=

(7.5.21)

Moreover, the problem is reduced to the study of nonlinear (semilinear) elliptic equations 4ϕ = µ

n X

Z qk

k=1

4ψ = ν

n X

fk dV,

(7.5.22)

(V, d)fk dV,

(7.5.23)

1

Z qk

k=1

1

where 4

d = d1 , µ=

∂ 2· ∂ 2· + ; ∂ξ 2 ∂η2 4πq ν=− 2 ; mc w(d)

4· =

8παq ; mw(d)

w(d) =

d2 . 2 + d2 ) d13 (d11 12

We note that every solution (7.5.22), (7.5.23), due to (7.5.21), satisfies orthogonality conditions (7.5.9), (7.5.10). From preceeding it follows Theorem 7.1. Let distribution function have the form (7.5.6). Then electromagnetic field {E, B} is defined by formulas m ∇ϕ, 2αq   Z1  d  mc B(ˆr) = 2 β + (d × J(τ rˆ ), rˆ )dτ − [d × ∇ψ(ˆr)] 2 ,   d qd

E(ˆr) =

0

4

where rˆ = (ξ, η); β −Const; functions ϕ(ˆr), ψ(ˆr) satisfy system (7.5.22), (7.5.23). Let us introduce a scalar and vector potentials U(r), A(r) E(r) = −∇U(r),

B(r) = rot A.

80


Then, due to (7.5.7), (7.5.11), and (7.5.20), field {E, B} is defined via potentials {U, A} by formulas U=−

m ϕ, 2αq

A=

mc ψd + A1 (r), qd2

where (A1 , d) = 0. Unknown potentials U, A can be defined in a subspace D of enough smooth functions on the set ⊂ R3 with a smooth boundary ∂ and moreover to satisfy conditions (∇U, d) = 0,

(∇(A, d), d) = 0

and on the boundary U|∂2 = u0 (r),

(A, d)|∂2 = u1 (r).

(7.5.24)

Corollary 7.3. Let distribution function be (7.5.6). Then the VM system (7.5.1)–(7.5.5) with boundary conditions (7.5.24) has a solution fi = fi (−αi |V|2 + c1i + li ϕ ∗ (r), di V + c2i + ki ψ ∗ (r)), where li , ki satisfy (7.5.13) and (7.5.14), m ∇ϕ ∗ (r), 2αq   Z1  mc d  B = 2 β + (d × J ∗ (τ r), r)dτ − [d × ∇ψ ∗ (r)] 2 ,   d qd

E=

0

Z n 4π X J ∗ (r) = qk Vf dV. c k=1

1

Functions ϕ ∗ , ψ ∗ belong to D and are defined from system (7.5.22), (7.5.23) with boundary conditions 2αq u0 (r), m q = u(r). mc

ϕ|∂2 = −

(7.5.25)

ψ|∂2

(7.5.26)

7.5.2 Reduction of System (7.5.22), (7.5.23) to Single Equation Lemma 7.3. If f (V + d, r) = f (−V + d, r),

d ∈ R3 ,


81

then the following inequality holds j = d · ρ,

(7.5.27)

R is vector of current density and ρ = 1 f dV—charge density. R Proof. Making changes of variables in integral 1 V f dV of the form Vi = ξi + di (i = 1, 2, 3), one obtains

where j =

Z

R

1 V f dV

Vi f (V, r)dV = J1 + J2 + J3 ,

where Z

f (ξ + d, r)dξ,

J1 = di 1

Z0 Z0 Z0 Z∞Z∞Z∞ ξi f (ξi + d, r)dξ + ξi f (ξi + d, r)dξ. J2 + J3 = −∞ −∞ −∞

0 0 0

It is easy to show that J3 = −J2 and (7.5.27) follows. Taking into account Lemma 7.3, (7.5.22), (7.5.23) can be transformed to the form 4ϕ = µ

n X

qi Ai ,

(7.5.28)

i=1 n

4ψ =

νd2 X ki qi Ai , 2α li

(7.5.29)

i=1

R where Ai = 1 fi dV, i = 1, . . . , N. Let (ξ, η) ∈ , where is bounded domain in R2 with a smooth boundary ∂. We set a value of scalar potential on the boundary ∂: ϕ(ξ, η)|∂ = A(ξ, η).

(7.5.30)

Consider when (7.5.28), (7.5.29) is reduced to one equation.

Case 1. li = ki , i = 1, . . . , N. Lemma 7.4. If li = ki and u∗ satisfies equation 4u = a(d, α)

n X k=1

qi Ai (γi + li u)

(7.5.31)

82


with γi = c1i + c2i , a(d, α) = 2πq

i = 1, . . . , N,

4α 2 c2 − d2 mc2 αw(d)

u = ϕ +ψ

,

then system (7.5.28), (7.5.29) possesses a solution ϕ = 2(d, α)u∗ + ϕ0 , ψ = (1 − 2(d, α))u∗ − ϕ0 , where 2(d, α) = 4α 2 c2 /(4α 2 c2 − d2 ),

4α 2 c2 6= d2 .

Knowing some solution u∗ of the equation (7.5.31) being solved under the conditions of Lemma 7.4 and the value of potential on the boundary ϕ|∂ = A(ξ, η), one finds ϕ0 by means of solution of linear problem 4ϕ0 = 0, ϕ0 | = A(ξ, η) − 2u∗ |∂ .

(7.5.32)

Hence, in the first case, we transformed the problem to a solution of “solving” equation (7.5.31) and the linear Dirichlet problem (7.5.32). This has the following result: Theorem 7.2. Let ki = li , i = 1, . . . , N. Then the VM system (7.5.1)–(7.5.5) with boundary condition (7.5.30) has a solution fi = fi (−αi |V|2 + Vdi + γi + li u∗ (ξ, η)), m E= (2(d, α)∇u∗ (ξ, η) + ∇ϕ0 ), 2αq   Z1  d  B = 2 β + (d × J(τ rˆ ), rˆ )dτ −  d  0

− [d × (∇(1 − 2(d, α))u∗ (ξ, η) − ϕ0 )]

mc . qd2 4

u∗ (ξ, η)—function satisfying “solving” equation (7.5.31); γi , βi —Const; rˆ = (ξ, η) and ϕ0 (ξ, η) is a harmonic function defined from linear problem (7.5.32). 4

4

Case 2. l2 = · · · = ln = l, k2 = · · · = kn = k, l 6= k. We note that for N = 2, Cases 1 and 2 exhaust all possible connections between parameters li and ki . We construct solution ϕ, ψ of (7.5.28), (7.5.29) satisfying condition ϕ + ψ = lϕ + kψ.


83

4

Let functions fi = fi (−αi |V|2 + Vdi + ϕi + ψi ) such that the following condition holds. (A). There are constants γi , i = 1, . . . , N such that 2qA1 (γ1 + u) + τ

n X

qi Ai (γi + u) = 0

i=2

for 2 = 4α 2 c2 (1 − l) + d2 (k − 1),

k τ = 4α 2 c2 (1 − l) + d2 (k − 1) . l

We remark that the corresponding distribution function satisfies the condition of Lemma 7.3. 4

4

Lemma 7.5. Let l2 = l3 = · · · = ln = l, k2 = k3 = · · · = kn = k, l 6= k. We assume that condition (A) holds. Then (7.5.28), (7.5.29) possesses a solution ϕ=

k−1 ∗ u , k−l

ψ=

1−l ∗ u , k−l

where u∗ satisfies equation h A1 (γ1 + u), a(α, l) + b(d, k, l) 1 d2 (k − l)2 8π αq2 = 2, h = , a = 4α 2 (1 − l)l, mw(d) c

4u =

(7.5.33) b = d2 (k − 1)k.

Proof. By change ϕ = lu, ψ = ku system is reduced to (7.5.33), due to (A). Since ϕ=

k−1 u, k−l

ψ=

1−l u. k−l

From Lemma 7.5 one obtains: Theorem 7.3. Let α2 q2 /m2 = · · · = αn qn /mn , k2 = · · · = kn = k. Let k ∈ / { αmn qnn , 1} and condition (A) holds. Then the VM system (7.5.1)–(7.5.5) with boundary condition (7.5.30) on scalar potential ϕ has a solution 4

fi = fi (−αi |V|2 + Vdi + γi + u∗ ), m(k − 1) E= ∇u∗ , 2αq(k − l)   Z1  d  cm(1 − l) B = 2 β + (d × J(τ rˆ ), rˆ )dτ − [d × ∇u∗ ] 2 .  d  qd (k − l) 0

84


Here u∗ satisfies (7.5.33) with condition u∗ |∂ =

k−1 m A(ξ, η), k − l 2αq

(7.5.34)

4

β, γi —constants, rˆ = (ξ, η). The problem (7.5.33), (7.5.34) at → 0 possesses solution u∗ = u0 + O(), where u0 is a harmonic function satisfying condition (7.5.34). Existence of another solutions for equations (7.5.33), (7.5.34) can be proved using parameter continuation and branching theory methods.

7.6 Existence of Solutions for the Boundary Value Problem (7.5.28)–(7.5.30) We realize the form of distribution function. Let fi = exp(−αi |V|2 + Vdi + γi + li ϕ + ki ψ).

(7.6.1)

Distributions (7.6.1) have meaning in applications. Substituting (7.6.1) into (7.5.28) and (7.5.29), taking into account (7.5.12)–(7.5.14) and (7.5.27), we come to the system 3/2 n X di2 π 4ϕ = µ qi exp γi + exp(li ϕ + ki ψ), ai 4αi k=1 3/2 n di2 ki d2 ν X π exp γi + exp(li ϕ + ki ψ) . 4ψ = qi 2α αi 4αi li

(7.6.2)

i=1

Introducing normalization condition Z Z

fi dVdx = 1, i = 1, . . . , N;

∈ R2 ;

4

x = (ξ, η),

R3

we transform (7.6.2) to the form

4ϕ = µ

n X

 qi exp(li ϕ + ki ψ) 

i=1

Z

−1 exp(li ϕ + ki ψ)dx

,

 −1 Z n d 2 ν X ki qi exp(li ϕ + ki ψ)  exp(li ϕ + ki ψ)dx . 4ψ = 2α li i=1

(7.6.3)


85

Consider when it is not possible to transform (7.6.3) to one equation. Without loss 4

of generality, we can consider that l2 6= k2 ; q = q1 . Let q1 < 0, qi > 0, i = 2, . . . , N. Introducing new variables u1 = ϕ + ψ,

ui = −(li ϕ + ki ψ),

i = 2, . . . , N.

And using them with boundary conditions (7.5.25)–(7.5.26), one obtains system −4ui =

n X

Cij Aj ,

i = 1, . . . , N,

(7.6.4)

j=1

where  A1 = eu1 

Z

−1 eu1 dx

 ,

Aj = e−uj 

Z

−1 e−uj dx

j = 2, . . . , N,

ai 1 8π · |qi |qj 1 − 2 Zi Zj , Cij = w(d1 ) mi 2d1 c2 ui = u0i ,

,

Zi =

x ∈ ∂,

(d1 , di ) , αi

i = 1, . . . , N. (7.6.5)

It is easy to check that (7.6.3) and (7.6.4) are equivalent in the sense that solutions (7.6.4) define completely solutions of (7.6.3). In fact ϕ, ψ are defined via u1 , u2 , because l2 , k2 and ui are linear dependent for i = 3, . . . , N. Here we assume that u0i ∈ C2+α , ∂ ∈ C2+α , α ∈ (0, 1). We give auxiliary results. Lemma 7.6. Let n X

 Cij > 0,

j=1

n X



 Cij < 0 .

j=1

Then Fi (u) =

n X

Cij Aj (u) ≥ 0,

j=1

 Fi (u) =

n X

ui ≥ min u0i , ∂

 Cij Aj (u) ≤ 0,

ui ≤ max u0i  .

j=1

Proof. It is easy to see that

R

Fi (u)dx =

∂

Pn

j=1 Cij

> 0. Moreover, the set + = {x ∈

: Fi (u(x)) > 0} is nonempty. We denote connected components in − , i.e., maximum (by inclusion) connected subspace − = {x ∈ : Fi (u(x)) < 0}, and we show

86


that − = ∅. Hence, on the one hand, Fi (u(x)) = 0, where x ∈ ∂, and, on the other hand, −4ui (x) = Fi (u(x)) < 0,

x ∈ .

¯ Thus, ui is bounded in , and it reaches it’s maximum on ∂ = \, i.e., max u(x) = u(x ), x ∈ ∂. However, since function F (u) decreases for fixed ¯ 0 0 i x∈ R ¯ that contradicts def( e−uj dx)−1 , then one obtains Fi (u(x)) > Fi (u(x0 )) = 0, x ∈ P inition of the set − . By analogy, case nj=1 Cij < 0 is considered (see [165]). Lemma 7.7 (Gogny, Lions [120]). Let max(u − v)(x) = (u − v)(x0 ) > 0.

Then  e

u(x0 ) 

−1

Z e

u(x)

 >e

dx

v(x0 ) 

−1

Z

v(x)

e

dx

,

 e−u(x0 ) 

Z

−1 e−u(x) dx

< e−v(x0 )

Z

e−v(x) dx

−1

.

¯ n as a lower and an We define the vector-function v(x), w(x) ∈ C2 ()n ∩ C1 () upper solution of (7.6.4), (7.6.5) in the following sense −4vi ≤

−4wi ≥

n X j=2 n X j=2

vi ≤ u0i ,

e−wj ew 1 R + C ≤ Fi (v), i1 −vj dx v1 e e dx

Cij R

ev1 e−vj ≥ Fi (w), Cij R −wj + Ci1 R w 1 dx e e dx wi ≥ u0i ,

x ∈ ∂

x ∈ , (7.6.6) x ∈ , (7.6.7)

with v = (v1 , . . . , vn )0 , w = (w1 , . . . , wn )0 . It is easy to show that Aj (u) is invariant under translation on the constant vector, therefore, one can change on (7.6.7) vi ≤ 0,

wi ≥ 0,

x ∈ ∂.

(7.6.8)

¯ and an upper wi (x) ∈ Theorem 7.4. Let there exist a lower vi (x) ∈ C2 () ∩ C1 () ¯ solution satisfying inequalities (7.6.6), (7.6.8), such that vi (x) ≤ wi (x) C2 () ∩ C1 () ¯ Let u0i ∈ C2+α (∂). Then (7.6.4), (7.6.5) has a unique classical solution ui (x) ∈ in . 2+α ¯ and, moreover, vi (x) ≤ ui (x) ≤ wi (x) in , ¯ i = 1, . . . , N. C ()


87

¯ be given, vi ≤ zi ≤ wi . We define operator Proof. Let functions zi (x) ∈ C() ¯ n → C() ¯ n by formulas u = Tz, z(x) ∈ ∈ C() ¯ n , where u = (u1 , . . . , un )0 is T : C() a unique solution of the problem −4ui =

n X

4

Cij Aj (p(z)) + q(zi ) = Fî (z),

ui = u0i ,

x ∈ ∂,

(7.6.9)

j=1

where p(z) = max{v, min{z, w}},  w −z i i , zi ≥ w i ,    1+z2i vi ≤ zi ≤ w i , q(zi ) = 0,  v −z  i i  vi ≤ zi . 2, 1+zi

ˆ It is evident that function F(z) is continuous and bounded. Then, due to smoothness ¯ n , i.e., u(x) ∈ of ∂ and boundary conditions, (7.6.9) is only solvable in C1+α () ¯ n . Here we used Theorem 8.34 from [110]. Due to compactness of embedding C1+α () ¯ ⊂ C() ¯ and continuity of F(z), ˆ C1+α () it follows that operator T is a completely continuous (compact) operator. Then by Schauder theorem (see [146]), operator T pos¯ n . On the other hand, since u ∈ C1+α () ¯ n, sesses a fixed point u = Tu with u ∈ C() α n 2+α n ¯ and from classical theory follows that u ∈ C ¯ . ˆ then F(u) ∈ C () () Next we show that vi ≤ ui ≤ wi . We suppose that there exist a number k ∈ ¯ such that {1, . . . , N} and the point x0 ∈ (vk − uk )(x0 ) = max(vk − uk ) = > 0. ¯

Evidently, x0 , due to (7.6.7), can not belong to the boundary ∂. Then due to maximum principle, one has contradiction X ew1 (x0 ) e−wj (x0 ) 0 ≤ −4(vk − uk )(x0 ) ≤ Ck1 R v + Ckj R −vj − 1 dx e dx e n

j=2

ep(u1 )(x0 )

− Ck1 R

ep(u1 ) dx

−

n X j=2

e−p(uj )(x0 )

Ckj R

e−p(uj ) dx

+

(uk − vk )(x0 ) < 0. 1 + u2k (x0 )

Thus, vi ≤ ui . By analogy, the proof of inequality ui ≤ wi is given. ¯ such We assume that there exists a number l ∈ {1, 2, . . . , N} and the point y0 ∈ 1 2 1 1 2 that there are two solutions u , u of (7.6.4), (7.6.5), ui ≡ ui , i 6= l, ul (y0 ) > u2l (y0 ). Using Lemma 7.7, we come again to contradiction: 0 ≤ −4(u1l − u2l )(y0 ) < 0, which proves uniqueness. P We construct upper and a lower solutions of (7.6.4), (7.6.5). Let nj=1 Cij > 0, i = 1, . . . , N. Then from Lemma 7.6 it follows ui ≥ 0. At first, we construct an upper

88


solution of the form: vi ≡ 0, − 4wi =

n X

Cij |C | R i1 , −wj dx − w1 e e dx

R j=2

wi |∂ = max u0i ≡ w0

(7.6.10) (7.6.11)

i,∂

with x = (ξ, η) ∈ ⊂ R2 . From (7.6.6) follows that wi must be satisfied inequalities n X

Cij e−wj − |Ci1 |ew1 ≥ 0,

i = 1, . . . , N.

(7.6.12)

j=2

Consider auxiliary problem −4g = 1,

g|∂ = w0 .

We assume that domain is contained in a strip 0 < x1 < r, and one introduces the function q(x) = w0 + er − ex1 . It is easy to show that 4(q − g) = −ex1 + 1 < 0 in , q − g = er − ex1 ≥ 0 on ∂. Therefore, according to maximum principle (see [110]) ¯ and q − g ≥ 0, if x ∈ 4

w0 ≤ g(x) ≤ w0 + er − 1 = M.

(7.6.13)

We denote write part in (7.6.10) by zi = const ≥ 0. Then from (7.6.10) and (7.6.13) we obtain wi ≤ Mzi , wi = zi g(x), and (7.6.10), (7.6.11) is equivalent the following finitedimensional algebraic system zi =

n X

R j=2

Cij |Ci1 | 4 − R z g = Li (z). e−zj g dx e 1 dx

Let us introduce the norm |z| = max1≤i≤N |zi |. Then, due to (7.6.13), we obtain the following chain of inequalities n X Cij |Ci1 | R R |L(z)| ≤ max − ≤ z1 g 1≤i≤n e−zj g dx e dx j=2   n   X 1 max Cij eMzj − |Ci1 |e−Mz1 ≤ ≤  || 1≤i≤N  j=2   n   X 1 ≤ max Cij eM|z| − |Ci1 |e−M|z| ,  || 1≤i≤N  j=2

(7.6.14)


89

where || = mes , ⊂ R2 . P Lemma 7.8. Let nj=1 Cij > 0. Introduce notations n X

4

Cij = ai ,

|Ci1 | = bi ,

j=2

min

1≤i≤N

ai = α 2 > 1. bi

Let the inequalities αai −

|| 1 bi ≤ ln α, α M

i = 1, . . . , N

(7.6.15)

hold. Then equation Lz = z has a solution zi ≤ M1 ln α and functions vi ≡ 0, wi = zi g(x) are a lower and an upper solutions of the problem (7.6.4), (7.6.5). Proof. Let |z| = R. From (7.6.12) follows ai e−MR − bi eMR ≥ 0 with R ≤ M1 ln α. Substituting a maximum value R = M1 ln α in (7.6.14), it is easy to check that (7.6.15) gives estimation |L(z)| ≤ |z| and existence of the fixed point Lz = z follows from Brayer theorem (see [146]). P Let now nj=1 Cij ≤ 0, i = 1, . . . , N. By analogy with preceeding we obtain the following result. P Lemma 7.9. Let nj=1 Cij < 0, β 2 = min1≤i≤N (bi /ai ) > 1 and the inequalities bi || − βai ≤ ln β, β M

i = 1, . . . , N

(7.6.16)

hold. Then functions vi = −zi g(x), wi ≡ 0 are a lower and an upper solutions of (7.6.4), (7.6.5) with zi = −Li (−z). It follows from Theorem R 7.4 and smoothness of the function Fi (u) under the fixed functional coefficients ( e−uj dx)−1 that there exists a constant M(v, w) > 0 such that ∂ ∂uj Fi

¯ n → C() ¯ n defined ≥ −M with i, j = 1, . . . , N. Moreover the mapping G : C() by formulas Gi u = Fi + Mui will be monotonic, increasing in ui because of monotonicity of coefficients. We set operator T1 : z = T1 z, −4zi + Mzi = Gi u > 0,

zi |∂ = u0i .

(7.6.17)

Due to maximum principle, zi > 0 (u0i > 0). Thus, operator T1 is positive and monotonic. Moreover, T1 is completely continuous, being proven in the same way that for operator T also. It is evident, v ≤ T1 v, T1 w ≤ w. We note that a cone of nonnegative ¯ Therefore, due to uniqueness Theorem (7.4), we can functions is normal in C(). apply the classical theory of monotone operator (see [160]) for problem (7.6.17) and obtain the following result:

90


Theorem 7.5. Operator T1 has a unique fixed point u = T1 u, vi ≤ ui ≤ wi , where for any y0 : vi ≤ y0i ≤ wi , successive approximations yn+1 = T1 yn are uniformly converged to u. Corollary 7.4. We define successive approximations by u0i = 0, − 4un+1 + Mun+1 = Fi (un ) + Muni , i i un+1 |∂=u0i , i

i = 1, 2, n = 0, 1, . . . ; m1 m2 q α k k (z2 − zk )un1 + (zk − z1 )un2 , unk = mk (z2 − z1 ) |q1 |α1 q2 α2

k = 3, . . . , n.

Then {uni }, i = 1, . . . , n are monotone and uniformly converged to solution (7.6.4), (7.6.5). Remark 7.1. In the case n = 1, boundary value problem (7.6.4), (7.6.5) was considered in Gogny, Lions [120] and Krzywicki, Nadzieja [165].

7.7 Existence of Solution for Nonlocal Boundary Value Problem Here we consider the problems (7.5.28), (7.5.29), (7.5.25), (7.5.26). Assume plasma in domain ⊂ R2 with a smooth boundary ∂ ∈ C1 consisting of N kinds of charged particles. It is assumed that particles interact among themselves only by means of owns charges q1 , . . . , qn ∈ R\{0}. Every particle of i-th kind is described by distribution function fi = fi (x, v, t) ≥ 0, where t ≥ 0—time, x ∈ —position and v ∈ R3 —velocity. Plasma motion is described by the classical VM system (7.5.1)–(7.5.5) with boundary conditions (7.5.24). We impose reflection condition (7.3.1) for distribution function. In this section we study stationary solutions ( f1 , . . . , fn , E, B) of the VM system of special form fi = fî (−αi v2 + c1i + li ϕ(x), vdi + c2i + ki ψ(x)), m ∇ϕ, E(x) = 2αq cm B(x) = − 2 (d × ∇ψ), qd

(7.7.1) (7.7.2) (7.7.3)

where functions fî : R2 → [0, ∞) and parameters d ∈ R3 \{0}, αi > 0, c1i , c2i , li , ki (see formulas of connection (7.5.12)–(7.5.14)) are given, and functions ϕ, ψ have to be defined. Earlier, using the lower-upper solutions method, existence theorem of classical solutions of boundary value problem (7.5.28)–(7.5.30) is proved for distribution function fî = eϕ+ψ . Under proof existence Theorem 7.4, we essentially apply monotonic property of the right parts of (7.6.4). In general case of distribution function


91

(7.7.1), system (7.5.28), (7.5.29) does not possesses good monotonic properties and, therefore, we can not apply techniques of lower and upper solutions for nonlinear elliptic system in a cone developed by Amann [3]. Therefore, we show existence of solutions of the boundary value problem (7.5.28), (7.5.29), (7.5.25), (7.5.26) by the method of lower-upper solutions without monotonic conditions. We note that approach (7.7.1)–(7.7.3) is connected with papers of P. Degond [86] and J. Batt, K. Fabian [27]. In these papers they are introduced integrals f (E), F(x, v) and P(x, v) of the Vlasov equation and solutions of the VM system for distribution function (i = 1—particles of single kind) of the form fˆ (E), fˆ (E, F) or fˆ (E, F, P) are considered. The case of distribution function of fˆ (E, P) and particles of various kinds (species i = 1, . . . , N) in these papers are not considered. Thus, we consider the boundary value problem (7.5.28), (7.5.29), (7.5.25), (7.5.26). Let q < 0 (electrons), qi > 0 (positive ions), i = 2, . . . , N. Then (7.5.28), (7.5.29) takes the form ! n X 8παq qA1 − |qi |Ai = h1 , (7.7.4) 4ϕ = mw(d) i=2 ! n X ki 4πq d2 qA1 − |qi |Ai = h2 , (7.7.5) −4ψ = li mc2 w(d) 2α i=2

where Ai =

R

fi dv,

i = 1, . . . , N, and fi is ansatz (7.7.1).

Remark 7.2. In case ki = li , system (7.7.4), (7.7.5) is transformed to one equation, and we may use Theorem 7.4. Theorem 7.6 (McKenna-Walter [194]). Let ⊂ Rn —bounded domain with boundary ¯ × Rn → Rn satisfy the following smoothness ∂ ∈ C2,µ for some µ ∈ [0, 1]. Let h : ¯ and ∀y, y1 , y2 ≤ r: conditions: ∀r > 0 there exist Cr > 0 such that ∀x, x1 , x2 ∈ I. There hold the inequalities |h(x1 , y) − h(x2 , y)| ≤ Cr |x1 − x2 |µ , |h(x, y1 ) − h(x, y2 )| ≤ Cr |y1 − y2 |; II. There exists ordered couple (v, w) of lower v and upper w solutions, i.e., v, w ∈ T ¯ n , v ≤ w in , ¯ v ≤ 0 ≤ w on ∂, C2 ()n C1 () ∀x ∈ : ∀z ∈ Rn , v(x) ≤ z ≤ w(x), zk = vk (x) : 4vk (x) ≥ hk (x, z) and ∀x ∈ : ∀z ∈ Rn , v(x) ≤ z ≤ w(x) : zk = wk : 4wk (x) ≤ hk (x, z) for all k ∈ {1, . . . , N} (Here the vector inequality v(x) ≤ z ≤ w(x) means a component wise comparison).

92


¯ n of the problem Then there is a solution u ∈ C2,µ () 4u = h(·, u(·))

in

∂

u=0

¯ such that v ≤ u ≤ w in .

Because the right parts in (7.7.4), (7.7.5) are nonlocal, we give sufficient conditions to function fî to make it possible to apply a McKenna-Walter theorem. Lemma 7.10. Let α > 0 and fˆ : R2 → [0, ∞) satisfy the following conditions: 1. 2.

fˆ ∈ C1 (R2 ); fˆ and fˆ 0 are bounded there exists R0 ∈ R such that supp(fˆ ) ⊂ [R0 , ∞) × R. Then function hα,fˆ : R2 → R2 , given via hα,fˆ (u) =

4π q mw(d)

2αq − c12 klii

Z R3

! fˆ (−αv2 + u1 , vd + u2 )dv

is continuously differentiable, and there are R, C1 , C2 such that ! 3/2 0 C1 (u1 + R)+ ≤ hα,fˆ (u) ≤ −C2 (u1 + R)2+ C2 (u1 + R)2+ for any function u ∈ R2 .

Proof. Passing on to a spherical system of coordinates v1 = ρ sin 2 cos ϕ,

v2 = ρ sin 2 sin ϕ,

v3 = ρ cos 2,

we obtain h1 ˆ (u) = α, f

8παq2 mc2 w(d)

Z

fˆ (−αv2 + u1 , vd + u2 )dv =

R3

Z∞Zπ Z2π = P(q, α, d, m) fˆ (−αϕ 2 + u1 , ϕk(ρ, 2) + u2 ) sin(2)ϕ 2 dρd2dϕ =

=

P(q, α, d, m) α2

0 0 0 Zu1 Zπ Z2π

p fˆ (s, α −1 k(ρ, 2) (u1 − s) + u2 )×

−∞ 0 0

p × sin(2)(u1 − s) (u1 − s)dρd2ds = Zu1 p P(q, α, d, m) = K (s, u − s, u )(u − s) (u1 − s)ds, 1 1 2 1 α2 −∞


93

where k(ρ, 2) = d1 cos(ρ) sin(2) + d2 sin(ρ) sin(2) + d3 cos(2) and ZπZ2π √ fˆ (s, α −1 k(ρ, 2) t + ϕ) sin(2)dρd2. K1 (s, t, ϕ) = 0 0

Similar expressions are satisfied for h2 ˆ and K2 (s, t, ϕ). Due to condition (2) kernels α,f

K1 , K2 are bounded and applying Lebesgue theorem on dominant convergence, it is easy to prove that hα,fˆ ∈ C1 (R2 )2 .

Theorem 7.7. Let ⊂ R2 —two-dimensional domain with boundary ∂ ∈ ∈ C2,µ , µ ∈ [0, 1]. Let fˆ1 , . . . , fˆn : R2 → [0, ∞) satisfy conditions (1), (2) of Lemma 7.10. Then the problem (7.5.28), (7.5.29), (7.5.25), (7.5.26) possesses a smooth ¯ ψ ∈ C2 (). ¯ Moreover distribution function fn ∈ C1 ( ¯ × R3 ) solution ϕ ∈ C2 (), generates the classical stationary solution ( f1 , . . . , fn , E, B) of the VM system of the form (7.7.1)–(7.7.3) in . Proof. Consider the system (7.7.4), (7.7.5). The right parts of these expressions may have different signs, depending on relations P A1. qA1 − ni=2 |qi |Ai = G(q, A) > 0. Hence, qA1 >

n X

|qi |Ai >

i=2

A2. qA1 −

Pn

qA1
0 c2i |G|(li u1 + R)2+ G1 0},

l− = min{|li | |

li < 0}

and l = min(l+ , l− ). We define a lower and an upper solution in   −l+ 2  v =  −1 Pn |li | R2 c |G| 1 + 4 2i i=1 l and l−

 w=

−4−1



2  |li | c |G| 1 + R2 i=1 2i l

Pn

and on the boundary vi ≤ u0i ,

wi ≥ u10i ,

x ∈ ∂

with v = (v1 , v2 )0 , w = (w1 , w2 )0 . Assuming that the right parts h1 (·), h2 (·) of (7.7.4), (7.7.5) are invariant under the transition on the constant vector, we can change last conditions on the following ones vi ≤ 0,

wi ≥ 0,

x ∈ ∂.

Moreover operator 4−1 is defined with respect to zero boundary conditions and v ≤ ¯ 0 ≤ w in . Due to the above given estimation for hf and conditions (A1), (A2), we obtain 4v1 = 0 ≥ h1f (v1 , z2 ),

z2 ∈ R,

4w1 = 0 ≤ h1f (w1 , z2 ), n X

z2 ∈ R,

c2i |G|(li z1 + R)2+ ≥ h2f (z1 , v2 ),

4v2 ≥

z1 ∈ [v1 , w1 ]

i=1

and 4w2 ≤ −

n X

c2i |G|(li z1 + R)2+ ≤ h2f (z1 , w2 ),

z1 ∈ [v1 , w1 ].

i=1

¯ U = (ϕ, ψ)0 of (2.54), (2.55) (respectively Thus, existence of solutions U ∈ C2,µ (), (2.40, (7.5.29)) (7.5.25), (7.5.26) follows from McKenna-Walter theorem.


95

Remark 7.3. Existence of stationary solutions for the relativistic VM system has been proven in the dissertation of P. Braasch [55] with using RSS [183] ansatz.

7.8 Nonstationary Solutions of the Vlasov-Maxwell System 7.8.1 Reduction of the Vlasov-Maxwell System to Nonlinear Wave Equation Let us consider the nonstationary VM system (7.1.2)–(7.1.6) for N-component distribution function with additional condition Z n X q2 i

i=1

mi

1 E + [V × B] ·∇V fi dV = 0. c

(7.8.1)

R3

We shall seek distribution functions of the form fi = fi (−αi |V|2 + Vdi + Fi (r, t)),

di ∈ R 3 ,

αi ∈ [0, ∞)

(7.8.2)

and corresponding fields E(r, t), B(r, t) satisfying equations (7.1.2)–(7.1.6), (7.8.1). If functions Fi (r, t), vectors di and vector-functions E, B are connected among themselves by relations ∂Fi qi + (E, di ) = 0, ∂t mi 2αi qi qi ∇Fi − E+ [B × di ] = 0, mi mi c

(7.8.3) i = 1, . . . , N,

(7.8.4)

then functions (7.8.2) satisfy (7.1.2) and one has the following equations ∂Fi 1 + (∇Fi , di ) = 0, ∂t 2αi ∂fi 1 + (∇fi , di ) = 0. ∂t 2αi

(7.8.5)

Introducing auxiliary vectors Ki = (Kix (r, t), Kiy (r, t), Kiz (r, t)), we transform (7.8.4) to the system 2αi qi E = Ki , mi

(7.8.6)

qi [B × di ] = −Ki . mi c

(7.8.7)

∇Fi −

We note that equation (7.8.7) is solvable with respect to vector B, if (Ki , di ) = 0.

(7.8.8)

96


We define functions Fi (r, t) and vectors Ki (r, t) in the form Fi = λi + li U(r, t), Ki = ki K(r, t), where λi , ki , li —constants, l1 = k1 = 1. Then from (7.8.6) and (7.8.7) follows that mi (li ∇U − ki K), 2αi qi γ ki m i c B(r, t) = 2 di + [K × di ] , di qi di2

E(r, t) =

where γi (r, t) = (B, di ) are remained arbitrary functions. Let m1 αi qi , α 1 q1 m i αi d1 = α1 di , αi γ1 = α1 γi , li = k i =

i = 1, . . . , N.

Then m (∇U − K), 2αq γ mc B(r, t) = 2 d + [K × d] 2 , d qd

(7.8.9)

E(r, t) =

(7.8.10)

where the following notations are introduced 4

m = m1 ,

4

α = α1 ,

4

d = d1 ,

4

γ = γ1 .

Moreover, K ⊥ d. Due to (7.8.3), (7.8.8), function U(r, t) satisfies linear equation 2α

∂U + (∇U, d) = 0. ∂t

(7.8.11)

Having defined U, K such that the Maxwell equations (7.1.2)–(7.1.5) to be satisfied for distribution function fi = fi (−αi |V|2 + Vdi + λi + li U(r, t)),

(7.8.12)

we can find unknown functions fi , E, B using (7.8.9), (7.8.10) and (7.8.12). Lemma 7.11. Densities of charge ρ and current j defined by formulas ρ(r, t) = 4π

Z X n R3

i=1

qi fi dV,

j(r, t) = 4π

Z X n R3

i=1

qi Vfi dV,


97

are connected among themselves by the following relation j=

1 dρ + rot Q(r) + ∇ϕ 0 (r), 2α

4ϕ 0 (r) = 0.

(7.8.13)

An equality (7.8.13) follows directly from continuity equation ∂ρ +∇ ×j = 0 ∂t and ∂ρ 1 + (d, ∇ρ) = 0, ∂t 2α which is corollary of (7.8.5). Substituting (7.8.9), (7.8.10) into (7.1.3), (7.1.5), one obtains n

4U = divK +

8π αq X qi m

Z fi dV,

(7.8.14)

mc (d, rot K) = 0. (d, ∇γ ) + q

(7.8.15)

i=1

R3

Due to Lemma 7.11 and taking into account rot Q(r) + ∇ϕ 0 = 0 (that always can be assured by calibrating), Z Z d V fi dV = fi dV. 2α R3

R3

Thus, after substitution (7.8.9), (7.8.10) into (7.1.2), we obtain the relation n

2π d2 X md2 ∂ (∇U − K) + d qi ∇γ × d = 2αcq ∂t αc i=1

Z fi dV −

mc rot [K × d]. (7.8.16) q

R3

Having used the Fredholm alternative, we set the function U(r, t), and from condition that its solution ∇γ is a gradient of function γ (r, t), we find K(r, t) as function of U. Thus, from solvability condition of (7.8.16) with respect to (7.6.16), one obtains n

∂ 2U 2πqd2 X = qi αm ∂t2 i=1

Z

fi dV + c2 divK.

R3

Due to (7.8.14), this equility is transformed into n

Z

i=1

R3

X 2π q 2 ∂ 2U = c2 4U + (d − 4α 2 c2 ) qi 2 αm ∂t

fi dV.

(7.8.17)

98


Below we apply (7.8.17) for solvability (7.8.16). If function U satisfies (7.8.17), then (7.8.16) is satisfied and, moreover, mc md2 ∂ 1 4 ν = F, (∇U − K) ∇γ = 2 d + d × − rot [K × d] + q 2αcq ∂t d d2

(7.8.18)

where ν(r, t) = (∇γ , d) is kept arbitrary. It follows from (7.8.18) that vector field F(r, t) must be irrotational. Since U satisfies (7.8.11), we define K in a class of vectors satisfying condition 2α

∂K + (d · ∇)K = 0. ∂t

(7.8.19)

Then d × rot [K × d] = −2α[d × ∂K/∂t] and (7.8.18) transforms ν ∂K ∂ m ∇γ = 2 d + d × (4α 2 c2 − d2 ) . + d2 ∇U ∂t ∂t d 2αcqd2 Up to arbitrary function b(U) and arbitrary vector-function a(r), one can put K(r, t) =

d2 (∇U + b(U)d + a(r)). d2 − 4α 2 c2

(7.8.20)

Then ∇γ =

ν . d2

(7.8.21)

If b(U) = −

1 (∇U, d), d2

a(r) = ∇ϕ0 (r),

where ∇ϕ ⊥ d, then (7.8.20) satisfies (7.8.19). Proof is developed by direct substitution of (7.8.20) into (7.8.19) with account of (7.8.11). Thus, vector K(r, t) =

d2 1 (∇U, d)d + ∇ϕ (r) , ∇U − 0 d2 − 4α 2 c2 d2

(7.8.22)

where ∇ϕ ⊥ d satisfies condition (7.8.19). Moreover, it is evident that K ⊥ d. If 4ϕ0 (r) = 0, then for any U(r, t) satisfying (7.8.17) vector-function (7.8.22) satisfies (7.8.14) that can be showed by substitution (7.8.22) into (7.8.14). We show that in (7.8.21) ν ≡ 0. In fact, (d, rot ((∇U, d)) = (d, ∇(∇U, d) × d) ≡ 0


99

for arbitrary U, (d, rot K) = 0 and, due to (7.8.15), d ⊥ ∇γ . But then from (7.8.21), ν ≡ 0. Therefore, ∇γ = 0, and γ is a constant. It remains to show that functions (7.8.9), (7.8.10), where U(r, t) satisfies (7.8.17) and K(r, t) are expressed via U and ϕ0 by formula (7.8.22), satisfy (7.1.4). From substitution (7.8.9) and (7.8.10) in (7.1.4), we obtain the chain of equalities 1 m 1 ∂K × d − rot K = q d2 ∂t 2α m ∂ 1 = [∇U × d] + rot ((∇U, d)d) = 2α q(d2 − 4α 2 c2 ) ∂t ∂U 1 m ∇ + rot (∇U, d) ×d = 0. = ∂t 2α q(d2 − 4α 2 c2 ) Remark 7.4. If (7.8.13) holds, then functions γ 6= Const, ∇γ = d × rot Q. Hence, it follows. Theorem 7.8. Let fi (S)—an arbitrary differentiable functions, moreover Z

fi (−|V|2 + T)dV < ∞,

T ∈ (−∞, +∞),

αi ∈ [0, ∞),

di ∈ R 3 ,

R3

αi d = αdi ,

4

α = α1 ,

4

d = d1 ,

then every solution U(r, t) of hyperbolic equation (7.8.17) with condition (7.8.11) corresponds solution of the system (7.1.1)–(7.1.5) of the form fi = fi (−αi |V|2 + Vdi + λi + li U(r, t)),

(7.8.23)

γ mc d+ [∇(U + ϕ0 (r)) × d], 2 2 d q(d − 4α 2 c2 ) m E= {∇(4α 2 c2 U + d2 ϕ0 (r) − (∇U, d)d)}, 2αq(4α 2 c2 − d2 )

(7.8.24)

B=

where ϕ0 (r)—arbitrary function satisfying 4ϕ0 = 0, ∇ϕ0 ⊥ d. Corollary 7.5. In the stationary case, (7.8.17) is transformed to the form n

Z

i=1

R3

X 2π q 2 2 2 4U(r) = (4α c − d ) qi αmc2

fi dV

(7.8.25)

with condition (∇U, d) = 0.

(7.8.26)

100


Remark 7.5. If fi = e s ,

S = −αi |V|2 + Vdi + λi + li U,

li =

αi mqi , αmi q

then Z

fi dV = (

R3

π 3/2 ) exp{di2 /4αi + λi + li U}. αi

In that case, “solving” equation (7.8.17) is written n

X ∂ 2U 2π q 2 (d − 4α 2 c2 )π 3/2 = c2 4U + qi (αi )−3/2 exp{di2 /4αi + λi + li U}. 2 αm ∂t i=1

As seen in paper [184], for N = 2 (two-component system), this equation is transformed to: ∂ 2U = c2 4U + λb(eU − elU ), l ∈ R− , λ ∈ R+ , ∂t2 2πq2 π 3/2 2 2 b= (d − 4α 2 c2 )ed /4α . αm α

(7.8.27)

Due to l = −1, (7.8.27) is a wave sh-Gordon equation ∂ 2U = c2 4U + 2λbsinhU. ∂t2

(7.8.28)

Remark 7.6. Due to conditions of Theorem 7.8, a scalar 8 and vector A potentials are defined by formulas m {4α 2 c2 U(r, t) + d2 ϕ0 }, 2αq(d2 − 4α 2 c2 ) mc A= d{U(r, t) + ϕ0 } + 42(r), q(d2 − 4α 2 c2 )

8=

(7.8.29)

where 42(r) =

γ (d2 z, d3 x, d1 y)0 + ∇p(r), d2

4

d = (d1 , d2 , d3 )

(7.8.30)

and p(r) is arbitrary harmonic function. Since function U(r, t) satisfies (7.8.11), then potentials 8, A are connected among themselves by Lorentz calibration. 1 ∂8 + divA = 0. c ∂t


101

For analysis (7.8.28), we direct a constant vector d ∈ R3 along axis Z, i.e., we 4

assume that d = d1 (0, 0, dz ). Moreover, solution U(x, y, z, t) for (7.8.11) has the form U = U(x, y, z −

d t). 2α

(7.8.31)

Solution (7.8.31) describes the wave spreading velocity running in positive direction along axis Z with a constant velocity d/2α, where d/2α < c. By substitution ξ = z − (d/2α)t, we reduce (7.8.28) to ∂ 2 U ∂ 2 U (4α 2 c2 − d2 ) ∂ 2 U + 2 + = 2λp sinhU, ∂x2 ∂y 4α 2 c2 ∂ξ 2

(7.8.32)

where 2π q2 π 3/2 (4α 2 c2 − d2 )exp(d2 /4α) > 0; p= αmc2 α 4

λ ∈ R+ .

Moreover, introducing a new variable η = (4α 2 c2 /(4α 2 c2 − d2 ))1/2 ξ , we transform (7.8.32) ∂ 2U ∂ 2U ∂ 2U + 2 + 2 = 2λp sinhU, ∂x2 ∂y ∂η

4

U = U(x, y, η).

(7.8.33)

Using formulas (2.69), it is easy to reconstruct some solutions of (7.8.33) by Hirota method [134].

7.8.2 Existence of Nonstationary Solutions of the Vlasov-Maxwell System in the Bounded Domain Here we consider the classical solutions ( f1 , . . . , fn , E, B) of the VM system of special form (7.8.23)–(7.8.24), which we write in the following form: fi (x, v, t) = fî (−αi v2 + vdi + li U(x, t)), m 2 2 E(x, t) = 4α c ∇U(x, t) + ∂ U(x, t)d , t 2αq(4α 2 c2 − d2 ) mc B(x, t) = − ∇U(x, t) × d, q(4α 2 c2 − d2 )

(7.8.34) (7.8.35) (7.8.36)

where functions fî : R → [0, ∞) and vector d ∈ R3 \{0} are given, and function U : ¯ → R has to be defined. Assuming that ∂ ∈ C1 , we add the VM system [0, ∞) × by the boundary conditions for electromagnetic field E(x, t) × n (x) = 0,

B(x, t)n (x) = 0,

t ≥ 0, x ∈ ∂,

(7.8.37)

102


and specular reflection condition for distribution function on the boundary fi (t, x, v) = fi (t, x, v − 2(vn (x))n (x)),

t ≥ 0, x ∈ ∂, v ∈ R3 ,

(7.8.38)

where n —unit vector of normal to ∂. To prove existence of classical solutions of (7.1.2)–(7.1.6), (7.8.34)–(7.8.38), we apply the method of lower-upper solutions developed for nonlinear elliptic systems. In contrast to stationary problem, nonstationary is more complicated, because we need to add equation of first order (7.8.11) to nonlinear wave equation (7.8.17). Hence, the problem is not “strongly” elliptic, and one needs to develop further the method of lower-upper solutions. Lemma 7.12. Let ⊂ Rn —bounded domain with boundary ∂ ∈ ∈ C2,α , α ∈]0, 1[. ¯ and h ∈ C0,1 ( ¯ × R) such that h(x, ·)—monotonic increasing funcLet u0 ∈ C2,α () loc tion for every x ∈ . Then boundary value problem 4u = h(·, u(·)) in , u = u0 on ∂

(7.8.39)

¯ possesses a unique solution u ∈ C2,α (). Proof. Due to monotonicity of h, it is easy to check that there exist p1 , p2 ∈ C0,α () such that p2 (x) ≤ 0 ≤ p1 (x) and ≤ p1 (x) for s ≤ 0, h(x, s) ≥ p2 (x) for s ≥ 0 ¯ Let u01 = min(u0 , 0) and u02 = max(u0 , 0). Let uk ∈ C2,α ()—solution ¯ for all x ∈ . of linear boundary value problem for k ∈ (1, 2) 4uk = pk in , uk = u0k on ∂. ¯ From the last one follows that u1 — Due to the maximum principle, u1 ≤ 0 ≤ u2 in . a lower solution and u2 —an upper solution for (7.8.39). Then from the theorem of existence (see Pao [224], Theorem 7.1]) follows, (7.8.39) has a unique solution u ∈ ¯ C2,α (). Remark 7.7. Lemma (7.12) is a well-known statement and does not require additional comments. We remark only, on the condition of monotonicity of function h(x, ·) for the VP system as applied first by Vedenyapin [284], [286]. Introduce the following conditions on function fˆ : R → [0, ∞) : (f1) (f2) (f3) (f4)

fˆ ∈ C1 (R); ∀u ∈ R : f ∈ L1 (u, ∞); f is measurable function and f (s) ≤ Ce−s for a.e. s ∈ R; f is decreasing, f (0) = 0 and ∃µ ≥ 0 : ∀s ≤ 0 : f (s) ≤ C|s|µ .


103

Lemma 7.13 (Braasch [55]). Let function f : R → [0, ∞) is given and hf (u) = c

Z

f (v2 + vd + u),

u ∈ R.

R3

Then the following claims hold: 1. Assume conditions ( f2),( f3). Then hf : R → R is continuous and nonnegative, 2

c1 hf (u) = |d|

Z∞ |d|s Z 1

sf (s + t + u)dtds

−|d|s2

for all u ∈ R. 2. Assume condition ( f3). Let ψ : →[0, ∞)—measurable function and ψ ≤ f (a.e.). Then hψ ≤ hf . 3. Assume condition ( f4) and |d| < 1. Then the following conditions ( f2),( f3), hf —continuously differentiable and hf (u) ≤ Ce−u for all u ∈ R are satisfied. 4. Assume ( f4) and |d| < 1. Then from ( f4) it follows that hf —is decreasing function and |hf (u)| ≤ C|u|µ for all u ∈ R, where C = C(µ, |d|).

Lemma 7.14. Let ∈ R2 with a smooth boundary ∂ ∈ C1 . Let fˆ1 , . . . , fˆn : R → ¯ × R → R is given by [0, ∞) satisfy conditions ( f1)—( f3) and |d| < 1. Let hf : n

hf (x, U) = −

X 2πq (4α 2 c2 − d2 ) qi αm i=1

Z

fî (−αv2 + vdi + li U(x, t))dv,

R3

¯ and we assume U ∈ C2 ()—solution of boundary problem (

4

LU = ∂∂tU2 − c2 4U = hf (·, U) in U = 0 on ∂. 2

,

(7.8.40)

We define ˜ + td), U(x, t) = U(x 4

K(x, t) = −

d2 4α 2 c2 − d

¯ t ≥ 0, x ∈ , −2 ∇U(x, t) − |d| ∂ U(x, t)d , t 2

t ≥ 0,

¯ x ∈ ,

K ∈ C1 ([0, ∞[×)3 and E, B by means of (7.8.35), (7.8.36). Then (f1 , . . . , fn , E, B, ) is classical solution of the VM system in and, it satisfies boundary conditions (7.8.37), (7.8.38).

104


Proof. Due to Lemma 7.13, hf —continuous and continuously differentiable function. The function U satisfies equation (7.8.11). Therefore, it follows from Theorem 7.8 that f1 . . . , fn is a solution of the Vlasov equation, and E, B—solution of the Maxwell system. Since U vanish on ∂, then from definition U and translation invariance in d we obtain that U and ∂t U vanish on [0, ∞) × ∂. Hence, ∇U × n = K × n = 0 on [0, ∞) × ∂. From the last one, one obtains E(x, t) × n (x) = (K(x, t) − ∇U(x, t)) × n (x) = 0 and B(x, t) × n (x) = |d|−2 (n (x) × K(x, t))d = 0 at t ≥ 0 and x ∈ ∂. Therefore, the boundary conditions (7.8.37) are satisfied. Theorem 7.9. Let ⊂ R3 . Let f1 , . . . , fn : R → [0, ∞) satisfy condition (f1) and are (pointwisely) less than corresponding functions ψ1 , . . . , ψn : R → [0, ∞) satisfying condition (f4) with µ > 0. We suppose that |d| < 1 and there exists function ˜ ∈ C2 () such that U C ˜ + td), U(x, t) = U(x

t ≥ 0,

x ∈ .

Then (7.8.35) in Lemma 7.14 possesses a smooth solution and f1 , . . . , fn generates the classical solution ( f1 , . . . , fn , E, B) of the VM system in of the form (7.8.34)–(7.8.36). Proof. Since elliptic operator L in (7.8.40) has constant coefficients, then by linear change of coordinates, it is possible to transform to Laplace operator L = 4. Introduce 4

notations F = ( f1 , . . . , fn ), and we write the right part hF of (7.8.40) as hF (x, U) = −c1 (c2 − d2 )

n X

qi hfi (li U(x)),

i=1

where functions hf1 , . . . , hfn are defined in Lemma 7.14. From Lemmas 7.12 and 7.13 we obtain  4  ≥ −c (c − d ) P ˜ 1 2 2 qi >0 Ci |qi |hψi (|li |U(x)) = h1 (x, U), hF (x, U) 4  ≤ c1 (c2 − d2 ) P ˜ Ci |qi |(−|li |U(x)) = h2 (x, U), qi 0 and ∀W0 ∈ S, ∃δ = δ(, T) such that when the norm ||W then the norm ||W(r, V, t) − W0 || < for 0 < t < T, where 0 < T < ∞. The equilibrium configuration, which we tested on stability, represents the charged electron-ion bundle with nonrelativistic movement of particles confined in a cylinder with a finite radius and retained by the magnetic field (see Davidson [83]). Moreover, (d, n) = 0.

(7.9.3)

106


In cylindrical geometry (ρ, 2, Z) the boundary conditions (7.9.1), (7.9.2), along with (7.8.24) and (7.9.3), are concretized ∂U |∂ = 0, ∂2 2

∂U |∂2 = 0. ∂Z

Let Gi (fi ) be smooth functions and Z

Gi (fi )dV < ∞,

1

Z

fi dV < ∞,

1

then (7.1.2)–(7.1.6), describing the behavior of the electron-ion bundle in the cylinder, have the following first integrals by (7.9.1) and (7.9.2):

T=

F1 =

1 8π

Z

{E2 + B2 }dr +

(7.9.4)

i=1 1 2

2

N Z X

N Z Z X 1 mi V 2 fi drdV, 2

Z

Gi ( fi )drdV,

i=1 1 2

F2 = (d, P), Z N Z Z X 1 E × Bdr, P= Pi fi drdV + 4π c i=1 1 2

(7.9.5) Pi = mi V.

2

Definition 7.2. (Holm, Marsden, Ratiu, Weinstein [137]) The stationary solution 4

W0 = ( f0i , E0 , B0 ) for the system of equations (7.1.2)–(7.1.6) is called formally stable if there exists the Lyapunov functional L, which possesses an isolated minimum in a stationary point W0 . 4

If the second variation of functional L is strongly positive, then W0 = ( f0i , E0 , B0 ) is an isolated minimum. Following Chetaev’s method, we introduce the Lyapunov functional in the form of a bundle of the first integrals L − L0 = T + F1 + λF2 ,

(7.9.6)

where L0 is a functional value that is calculated along the nonperturbed (stationary W0 ) state of the system and λ is an auxialiary parameter (Lagrange’s coefficient). Calculate the first variation of functional (7.9.6) on variables fi , E, B. In addition, we restrict our consideration to the two-component (N = 2) system of (7.1.2)–(7.1.6).


107

The first variation of energy integral (7.9.4), in the point of equilibrium, has the type 1 δT = 4π

Z 2

2 Z Z X mi 2 {E0 δE + B0 δB}dr + V δfi drdV. 2

(7.9.7)

i=1 1 2

Reduce the first subintegral expression in a functional (7.9.7) using the connection of fields E, B with potentials ϕ and A E = −∇ϕ − B = rot A,

1 ∂A , E0 = −∇ϕ0 , c ∂t B0 = rot A0

(7.9.8) (7.9.9)

and gauge condition (7.8.30). After transformations, we have I Z 1 1 ∂2 [ ϕ0 (δE, n)dS + δT = ϕ0 2 2 δϕ − 4δϕ −4A0 δA dr − 4π c ∂t 2

I

∂2

(δA × B0 )ndS ] +

+

2 Z X

Z

i=1 1 2

∂2

1 mi V 2 δfi drdV. 2

Similar calculations of the first variation for the momentum integral (7.9.5), taking into account (7.9.8), (7.9.9), and (7.8.30), give  Z d  1∂ (d, δP) = − ∇ϕ0 ∇δA + ϕ0 4δA + 4A0 δϕ + [d × δA]rot A0 dr +  4πc c ∂t 2

 I

2

X  {ϕ0 [n × δB] − ϕ0 ∇δA + [n × B0 ]δϕ}dS + d

+ ∂2

Z Z

Pi δfi drdV

i=1 1 2

and the expression Z 1∂ (d, ∇ϕ0 )∇δA + [d × δA]rot A0 dr c ∂t 2

vanishes through the use of (7.8.26), (7.8.29). In addition, introduce the first integrals of the type Z F3 = 8(ϕ)dr,

(7.9.10)

2

F4 =

2 Z Z X i=1 1 2

9i (x, y, z, t)fi drdV,

(7.9.11)

108


with twice differentiable function 8(ϕ) and function 9i satisfying the equation 1 ∂9i + (∇9i , d) = 0, ∂t 2α

(7.9.12)

where, by the terminology of Moisseev, Sagdeev, Tur, Yanovskii [208], the functions 9i are Lagrangian invariants. We show that the functionals (7.9.10) and (7.9.11) are really first integrals of (7.1.2)–(7.1.6) in view of the symmetry problem along d. Differentiate (7.9.10), (7.9.11) with time. Since the function 8(ϕ) satisfies the equation ∂8 1 + (d, ∇8) = 0, ∂t 2α then, due to (7.8.11) and (7.8.29), we obtain Z Z Z I d ∂8 1 1 8dr = dr = − (d, ∇8)dr = − 8(n, d)dS = 0. dt ∂t 2α 2α 2

2

2

∂2

Further 2

dX dt

Z Z

9i fi drdV =

i=1 1 2



=

2  Z X

Z



i=1  1 2



=

2  Z X

Z



i=1  1 2

   qi 1 ∂ 9i fi − 9i ∇r (Vfi ) − ∇V · 9i E + [V × B] drdV =  ∂t mi c    I Z ∂9i 1   + (∇9i , d ) fi drdV −  9i V fi dV  dS − ∂t 2α ∂2

 qi − mi

 Z

I   ∂1

2

1

  

1  9i E + [V × B] fi dr dS1 = 0.  c 

Remark 7.8. The integral (7.9.10) admits a generalization of the type Z F3 = 8(ϕ, 91 , . . . , 9n )dr, 2

where the functions 9n are Lagrangian invariants of (7.9.12). Consider a final structure of the Lyapunov functional by (7.9.10) and (7.9.11) Lˆ = L − L0 = T + F1 + λF2 + F3 + F4 .

(7.9.13)


109

After preliminary calculations, the first variation of functional (7.9.13) has the form ˆ i , δE, δB) = δ L(δf

2 Z Z X mi 2 V + G0i ( f0i ) + dλPi + 90i + qi ϕ0 δfi drdV− 2 i=1 1 2

Z

d 1 4 A0 δA + λ (ϕ0 4δA + 4A0 δϕ)−80 (ϕ0 )δϕ dr− 4π 4πc 2 I 1 − {ϕ0 (δE, n) − [δA × B]n}dS− 4π ∂2 I d −λ {ϕ0 [n × δB] − ϕ0 ∇δA + [n × B0 ]δϕ}dS, 4πc −

∂2

where λ = −1/2α. Assuming ϕ0 |∂2 = 0

(7.9.14)

and taking into account (7.8.24), (7.8.29) we have δ Lˆ =

2 Z Z X d 1 2 0 mi V + Gi ( f0i ) − Pi + 90i + qi ϕ0 δfi drdV + 2 2α i=1 1 2

Z + 80 (ϕ0 ) + 2

d2 4ϕ0 δϕdr 16π α 2 c2

(7.9.15)

with the condition δϕ0 |∂2 = 0. From equality to zero of the first variation (7.9.15) of functional (7.9.13), we have the equations of the equilibrium, which states d 1 Pi + 90i + qi ϕ0 = 0, G0i ( f0i ) + mi V 2 − 2 2α d2 80 (ϕ0 ) + 4ϕ0 = 0 16πα 2 c2

(7.9.16) (7.9.17)

with condition (7.9.14). The correlation (7.9.16) permits us to concretize a structure of the distribution functions for which we can show stability. Introduce notation 1 d G0i (f0i ) = −H, H = mi V 2 − Pi + 90i + qi ϕ0 . (7.9.18) 2 2α

110


A choice of the functions G0i ( f0i ) of type (7.9.18) implies 4

f0i (r, V) = f0i (H). Then by putting G0i ( f0i ) =

1 f0i ln βi , βi γ

γ >0

(7.9.19)

we concretize the distribution functions f0i = γ exp(−βi H).

(7.9.20)

From (7.9.19) we have Gi ( f0i ) =

1 { f0i ln f0i − f0i − f0i lnγ }. βi

(7.9.21)

Analysis of the equations of equilibrium states (7.9.16), (7.9.17) and stationary equations (7.8.25), (7.8.26) define the values of parameter βi in (7.9.20) and functions 90i βi =

2αi , mi

90i = −

qi (d, A0 ). 2αc

(7.9.22)

Moreover, on the basis of (7.8.29) and (7.9.17), we obtain d2 mc 4U0 + 80 (U0 ) = 0, 8π αq(d2 − 4α 2 c2 ) U0 |∂2 = −

d2 ϕ 0 |∂2 ; 4α 2 c2

4ϕ 0 = 0;

ϕ 0 |∂2 = b(r),

(7.9.23)

where b(r) is a given function. The harmonic function ϕ 0 in (7.9.23) may be discussed as a given external field on the boundary ∂2 . Further, it is easy to write the sufficient conditions of positive definiteness of the initial Lyapunov functional (7.9.13) taking into account (7.9.21) and (7.9.22) |d| < c; 2α 2 X 1 i=1

ln f0i ≥ 1 + ln|γ |,

d Pi +9i (r) > 0, 2 2α R 4 1 a(V)f (r, V)dV = R , 1 f (r, V)dV mi V 2 −

(7.9.24) (7.9.25)


111

or 2 X 1 i=1

2

mi V 2 −

X 2 m 0 d qi d 2 ϕ − Pi > ϕ . 0 2α 2αq 4α 2 c2 i=1

Since d/2α =< V >= V − VT , where VT is a mean of random heat velocity, then (7.9.25) becomes 2 X 1 i=1

2

mi VT2

X 2 qi d 2 1 m 0 2 > − mi d ϕ0 − ϕ . 2 2αq 4α 2 c2

(7.9.26)

i=1

The condition of (7.9.26) places a restriction on the value of fields in the system, moreover, d2 /c2 1. The second variation of the functional (7.9.13) has the form 1 δ L= 4π 2ˆ

+

Z 2

1 {(δE) + (δB) }dr + 4απc

2 Z Z X

2

Z

2

G00i ( f0i )(δfi )2 dVdr +

i=1 2 1

(d, δB × δE)dr +

2

Z

800 (ϕ0 )(δϕ)2 dr.

(7.9.27)

2

It is easy to show that, taking into account (7.9.1)–(7.9.3), the second variation of the Lyapunov functional (7.9.27) is an integral of a linearized VM system. Sufficient conditions for the stability of equilibrium solutions (7.9.16), (7.9.17) can be obtained from the condition of the positive definiteness of the subintegral expression in formula (7.9.27). For the positive definiteness of δ 2 Lˆ in the neighborhood of a stationary state, it is sufficient that we have the following conditions: |d| < c; 2α

G00i (f0i ) > 0;

800 (ϕ0 ) > 0,

∂f0i (H) < 0; ∂H

800 (ϕ0 ) > 0,

or by (7.9.16) |d| < c; 2α

i = 1, 2.

Using the stability theorem [262], we obtain the sufficient conditions of stability for the stationary (equilibrium) solutions (7.9.16) and (7.9.17) by measure ρ. As a measure by which the stability is studied, we choose the quantity ρ = ||δE||2L2 (2 ) + ||δB||2L2 (2 ) +

2 X i=1

||δfi ||2L2 (1 ×2 ) + ||δϕ||2L2 (2 ) .

(7.9.28)

112


Let the potential ϕ0 and a solution ϕ 0 of the linear boundary—value problem (7.9.23) satisfy the inequalities (7.9.24) and (7.9.26), then, for the stability of stationary (equilibrium) solutions (7.9.16) and (7.9.17) by measure (7.9.28), it is sufficient that the following conditions are satisfied |d| < c, 0 < Ci ≤ G00i ( f0i ) ≤ bi , bi > 0, Ci < bi , 2α 0 < l1 ≤ 800 (ϕ0 ) ≤ l2 , l2 > 0, l1 < l2 ; Ci , bi , l1 , l2 − Const, or the conditions |d| < c, 2α 1 ∂f0i (H) 1 − ≤ ≤ − < 0, Ci ∂H bi 0 < l1 ≤ 800 (ϕ0 ) ≤ l2 .

7.10 Application Examples with Exact Solutions Taking into account distribution (7.6.1), the solution of the equation (7.5.31) reduces to the expression 3/2 N X di2 li u π 4u(ξ, η) = a(d, α) exp γi + e . qi αi 4αi

(7.10.1)

i=1

We assume N = 2; q1 < 0, q2 > 0, i.e., f1 (r, v), f2 (r, v) distribution functions of ions and electrons respectively, defined by formulas (7.6.1). Taking arbitrary constant γ2 in the form 1 d12 d22 |q1 | α2 3/2 γ2 = γ1 + − +ln , 4 α1 α2 q2 α1 we will obtain an equation of the sh-Gordon type 4u = ω(eu − elu ),

l ∈ R− ,

(7.10.2)

where w(d1 , α1 , γ1 ) = |q1 |

π α1

3/2

d2 a(d, α)exp γ1 + 1 . 4α1

Let us consider the construction of some exact solutions of equation (7.10.2). It follows from (7.6.1) that a value αi /mi is proportional to temperature of i component of plasma. Thus, for the concrete definition, we assume that temperatures of the


113

components of plasma are equal α1 /m1 = α2 /m2 . We use coonection of the charges q1 and q2 : q2 = −Zq1 , where Z = 1, . . . , N. If Z = 1 (the case of the completely ionized hydrogen plasma), then l = −1, and (7.10.2) takes the form 4u(ξ, η) = 2w sinh u(ξ, η). Assuming (see [218]) u(ξ, η) = 2 ln (X(ξ ) + Y(η))/(X(ξ ) − Y(η)) ,

(7.10.3)

(7.10.4)

we transform (7.10.3) into the system of ODEs (X 0 )2 = m2 X 4 − (n2 − w)X 2 + k2 , (Y 0 )2 = −m2 Y 4 + n2 Y 2 − k2 .

(7.10.5)

Here m2 , n2 , and k2 are arbitrary parameters. For m2 = 0, n2 6= 0, k2 6= 0, n2 −w > 0 there is a partial solution X(ξ ) = sin[ξ(n2 − w)1/2 ]k(n2 − w)−1/2 , Y(η) = cosh(nη)kn−1 .

(7.10.6)

As an example, Figures 7.1–7.5 are several 3D graphs of solution (7.10.6). From these graphs it is possible to note that the solution possesses the periodic structure, connected with the phenomenon of magnetic islands for the infinite plasma.

Figure 7.1 Partial solution for (7.10.6) for n = 2, ω = 2.77, k = 34 , ξ = −π . . . π, µ = −0.6 . . . 0.6.

114


Figure 7.2 Partial solution for (7.10.6) for n = 2, ω = 2.77, k = 34 , ξ = −π . . . π, µ = −0.6 . . . 0.6.

Figure 7.3 Partial solution for (7.10.6) for n = 2, ω = 2.77, k = 34 , ξ = 0.695π . . . 0.71π, µ = 0.11 . . . 0.175.


Figure 7.4 Partial solution for 0.69862π, µ = 0.173 . . . 0.174.

(7.10.6)

for

115

n = 2, ω = 2.77, k = 34 , ξ = 0.69727π . . .

Substituting (7.10.6) into (7.10.4), inverting variables by the formulas (7.5.21) and solving linear Dirichlet problem (7.5.32), it is easy to reconstruct fields E, B in domain and distribution functions f1 , f2 by Theorem 7.2. We construct some exact solutions of sh-Gordon equation (7.10.3), using Xirota method [134]. According to this method, we will find solution in the form F + G , u = 2ln F − G

(7.10.7)

where F and G are the functions of ξ and η. Substituting (7.10.7) into (7.10.3), we obtain equations for definition F and G (F 2 + G2 )D24 F ◦ G − FGD24 (F ◦ F + G ◦ G) = FG(F 2 + G2 )2w.

(7.10.8)

Here let D24 be bilinear Hirota operator, acting in the following way: 4

D24 F ◦ G = (D2ξ + D2η )F ◦ G = ∂ ∂ 2 ∂ ∂ 2 4 = − 0 + − 0 F(ξ, η)G(ξ 0 , η0 ) ∂ξ ∂ξ ∂η ∂η

ξ =ξ 0 , η=η0

116


Figure 7.5 Partial solution for (7.10.6) for n = 2, ω = 2.77, k = 34 , ξ = 0.697711π . . . 0.697712π, µ = 0.17329892 . . . 0.173299.

or D24 F ◦ G = F4G + G4F − 2∇F∇G and hence, D24 F ◦ F = 2(F4F − (∇F)2 ). To solve (7.10.8) let D24 F ◦ G = FG(2w),

D24 (F ◦ F + G ◦ G) = 0.

We reduce the last system to one equation by change of variables F = (f + fˆ )/2, G = (f − fˆ )/2 1 D24 f ◦ f = ( f 2 − fˆ 2 )(2w). 2


117

One can solve this equation by means of the choice of functions f and fˆ in the form     N N X X X √ X µi π/2 + π/4 , µi µj Aij  sin  µi ηi + exp  fN = 2 µ=0,1

i=1

1≤i≤j

i=1







 N N X X X √ X fˆN = − 2 exp  µi ηj + µi µj Aij  sin  µi π/2 − π/4 , µ=0,1

i=1

1≤i≤j

i=1

(7.10.9)

4

N = 1, 2, 3. Here sum in µ passes on all sets, Aij = ln aij , aij = (ki − kj )2 /(ki + kj )2 , 4

ki = (kiξ , kiη ) arbitrary vectors, normed by conditions ki2 = 2w, ηi = ki (r − r0 ), r0 — constant vector, i, j = 1, . . . , N. Proof of (7.10.9) is carried out by the standard reasonings. As example, let us consider solutions of Sinh-Gordon equation for N = 1, 2. For N = 1 we obtain u(ξ, η) = ln cotanh2 [k(r − r0 )/2]. For N = 2 from (7.10.9) follows that

u = 2 ln

1/2

n

1 2 [(k1 + k2 )(r − r0 ) + A12 ]

o

− cosh

n

1 2 (k1 − k2 )(r − r0 )

1/2

n

1 2 [(k1 + k2 )(r − r0 ) + A12 ]

o

+ cosh

n

1 2 (k1 − k2 )(r − r0 )

a12 sinh a12 sinh

o o.

(7.10.10)

Solution (7.10.10) is illustrated by graphs 7.6–7.10 with the following values of the vectors k1 = [0, 2], k2 = [2, 0]. Vectors k1 and k2 can be complex values. If, moreover, k1 = k¯2 , then u(ξ, η) is real. If we assume k1 = a + ib, then a2 − b2 = 2w, a · b = 0, then a12 sinh[a(r − r0 ) + A12 /2] − e(1/2)a(r−r0 ) cos b(r − r0 ) 1/2

u(ξ, η) = 2ln

1/2

a12 sinh[a(r − r0 ] + A12 /2] + e(1/2)a(r−r0 ) cos b(r − r0 )

. (7.10.11)

Figures 7.11–7.14 demonstrate graphical solution of (7.10.11). Here orthogonal vectors a, b are a = [−1.2; 1.2], b = [0.3; −4.2].

Figure 7.6 Partial solution for (7.10.10) for x = −2 . . . 2, y = −2 . . . 2.

Figure 7.7 Partial solution for (7.10.10) for x = −2 . . . 0, y = −2 . . . 0.


119

Figure 7.8 Partial solution for (7.10.10) for x = −1.75 . . . − 0.15, y = −1.25 . . . − 0.15.

When constructing other real solutions of (7.10.3), it is necessary that the vectors ki would satisfy some additional relationships. Let us note that solution (7.10.3) given above is also valid in the 3D- case. Consider equation (7.10.2) in case Z = 2 (ionized helium); l = −2 and (7.10.2) becomes 4u = w(d, α, γ )(eu − e−2u ) with the corresponding solution u(ξ, η) = ln 1 −

−2

3 2cosh2 ( 12 w1/2 kr)

,

which is illustrated by Figures 7.15a, 7.15b, and 7.16. Further, we consider application of Theorem 7.3 to the system (7.6.2). In this case condition (A) will be satisfied, if 2|q1 |α −3/2 ed

2 /4α+γ

=τ

N X i=2

−1/2 di2 /4αi +γi

qi αi

e

.

(7.10.12)

120





121

Figure 7.11 Partial solution for (7.10.11) in complex case for x = −2 . . . 20, y = −2 . . . 20.

Taking into account (7.10.12), the “resolving” equation (7.5.33) takes the form 3/2 b π 2 u 4u = me , m = ed /4α+γ . (7.10.13) a + b α As an example of a partial solution, we can take n h p p io u = ln α/4 1/sinh2 α/2x + c + 1/cos2 α/2y + c shown on Figure 7.17. Remark 7.9. If it is necessary to find the solution of system (7.5.1)–(7.5.5) with additional condition of normalization (6.1.6), then we take  −1 3/2 Z αi 2 eγi = e−di /4αi  eli u dx , π

in formula (7.10.1) for Case 1 Theorem (7.2), and we obtain integro-differential equation. Equation (7.10.1) has a constant solution u = lnC, where C is defined from algebraic equation 3/2 N X di2 π qi exp γi + Cli = 0. αi 4αi i=1

122


Figure 7.12 Partial solution of for (7.10.11) in complex case for x = −2.5 . . . 2.5, y = −2.5 . . . 2.5.

Figure 7.13 Partial solution for (7.10.11) in complex case for x = 10 . . . 20, y = 10 . . . 20.


123

Figure 7.14 Partial solution of for (7.10.11) in complex case for x = 10 . . . 20, y = 10 . . . 20.

(a)

(b)

Figure 7.15 Partial solution for k1 = k2 = 0.05; a) ω = 1.0, x = −20 . . . 10, y = −20 . . . 10; b) ω = 0.1, x = −12 . . . 10, y = −12 . . . 10.

124


Figure 7.16 Partial solution for k1 = k2 = 0.005, ω = 0.5, x = −75 . . . 75, y = −75 . . . 75.

Figure 7.17 Partial solution for α = 4, m = 1, c = 1, x = −10 . . . 10, y = −3 . . . 4.

If parameters γi are assigned so that N X i=1

qi α

−3/2

di2 exp γi + = 0, 4αi

then we can take C = 1 that corresponds a trivial solution u = 0.


125

For case 2 Theorem (7.3) under condition (7.10.12) and coefficient m in equation (7.10.13),

eγi =

αi π

3/2

 −di2 /4αi

e

Z



−1 eu dξ dη

,

i = 1, . . . , N.

Therefore, we have condition 2q1 + τ

N X

qi = 0,

i=1

and the related Liouville equation becomes 4u =

eu b R . u a + b e dξ dη

Equations of such type have been studied by Dancer [82] using methods of branching theory. We note that the general approach to find exact solutions of equations considered in this section is given in monography [7].

7.11 Normalized Solutions for a One-Component Distribution Function Consider the distribution function of type (7.6.1) for N = 1. In this case, the system (7.6.2) takes the form 8π αq2 p(d, α)eϕ+ψ , m 2πq2 d2 −4ψ(r) = p(d, α)eϕ+ψ , αmc2 4ϕ(r) =

(7.11.1) (7.11.2)

where p(d, α) =

3/2 π 2 ed /4α α

with the conditions (∇ϕ, d) = 0,

(∇ψ, d) = 0.

(7.11.3)

126


If we substract the respective sides of (7.11.1) from both sides of (7.11.2), we then obtain the Liouville equation 48(r) = a(d, α)

d2 − 4α 2 c2 −8 e , αc2

(7.11.4)

with 8 = −ϕ − ψ;

a(d, α) =

2πq2 p(d, α). m

By determining the function 8 from (7.11.4) and substituting it into (7.6.1) for N = 1, we obtain the desired distribution function f . By knowing 8, we are able to determine ∇ϕ and ∇ψ from (7.11.1) and (7.11.2) and, hence, to construct the desired fields E and B by means of (7.5.7) and (7.5.11) by N = 1. By virtue of (7.11.3), it is necessary to find only the solutions of (7.11.4) of the form x y y z 4 8(r) = 8 − , − . (7.11.5) d1 d2 d2 d3 Let us consider the solution of (7.11.4) of the form 8(S) with x y y z S= − +k − , k = Const. d1 d2 d2 d3 This class of solutions presents a most comprehensive investigation, because it is easy to construct E, B, f in an explicit form. Indeed, the corresponding solution 8(S) of (7.11.4) satisfied the ordinary differential equation 800 (S) = b(d, α)e−8 ,

(7.11.6)

where d2 − 4α 2 c2 b(d, α) = a(d, α) αc2

1 (k − 1)2 k2 + + 2 d12 d22 d3

!−1 .

Consider two cases: 1. d2 − 4α 2 c2 > 0 and 2. d2 − 4α 2 c2 < 0.

Note, the case when d2 − 4α 2 c2 = 0 is trivial, because b(d, α) = 0, 8(S) = S0 S + S1 , where S0 , S1 —Const. Therefore, f = exp(−αV 2 + Vd − S0 S − S1 ) and the constants S0 , S1 , K and vector d are chosen from the normalization condition (6.1.6) by N = 1 3/2 Z π 2 exp(d /4α) exp(−S0 S − S1 )dxdydz = 1. α 2


127

Let d2 − 4α 2 c2 > 0. In this case, the general solution of (7.11.6) is determined from the relation e−8 =

2S12 b(d, α)cosh2 S1 (S − S0 )

.

(7.11.7)

Substituting (7.11.7) into (7.6.1) for N = 1, we have f=

2S12 b(d, α)cosh2 S1 (S − S0 )

exp(−αV 2 + Vd)U.

(7.11.8)

On the basis of (6.1.6), for the distribution function (7.11.8), the normalization condition has the form 3/2 Z 2S12 π dxdydz = 1. exp(d2 /4α) b(d, α) α cosh2 S1 (S − S0 )

(7.11.9)

2

Since 2 = R3 , then at k = 1, k = 0 and the integral in (7.11.9) diverges. If k 6= 1, k 6= 0, then this integral is calculated, and the relationship (7.11.9) can be transformed |d1 d2 d3 |πmαc2 K, |S1 | = 12|k(k − 1)|q2 (d2 − 4α 2 c2 )

K=

! 1 (k − 1)2 k2 + + 2 . d12 d22 d3 (7.11.10)

The relationship (7.11.10) is a condition relating the parameters α, d and the integration constant S1 . Note, when 1 = R3 , 2 = R3 the normalization condition (6.1.6) for the distribution function (7.11.8) can also be ensured on account of the parameter α = (−l + (l2 + 16h2 c2 d2 )1/2 )/8hc2 > 0 with l = |d1 d2 d3 |πmc2 K;

h = 12|k(k − 1)||S1 |.

For determining the corresponding fields E, B from (7.5.7) and (7.5.11), it is enough to know ∇ϕ and ∇ψ. In our case ∇ϕ = ϕ 0 (S)K,

∇ψ = ψ 0 (S)K,

1 k−1 k 0 K = , , − d1 d2 d3

(7.11.11)

and, due to (7.11.1), a function ϕ(S) satisfies the equation ϕ 0 (S) =

8α 2 c2 S1 tanhS1 (S − S0 ) + S2 , (d2 − 4α 2 c2 )

(7.11.12)

128


where S2 - const and a function ψ(S) by (7.11.7) ψ 0 (S) = −

2S1 d2 tanhS1 (S − S0 ) − S2 . 2 (d − 4α 2 c2 )

(7.11.13)

Using (7.11.11)–(7.11.13) from (7.5.7), (7.5.11), we determine the fields m 8α 2 c2 S1 E= tanhS1 (S − S0 ) + S2 K, 2αq (d2 − 4α 2 c2 ) mc 2S1 d2 γ tanhS1 (S − S0 ) + S2 K. B = 2d+ 2 d× d qd (d2 − 4α 2 c2 ) Let d2 − 4α 2 c2 < 0, then (7.11.6) takes the form F 00 (S) = |b(d, α)|eF ,

(7.11.14)

where F = −8. From (7.11.14), it follows that Z

dF + c0 = S. (2|b(d, α)|eF + 2c1 )1/2

In this case, it is easy to see that the general solution of (7.11.14) is determined depending on the sign of the constant c1 , by either eF = −

2|b(d, α)|cos2

c1 , √ |c1 |/2(S − c0 )

c1 < 0,

(7.11.15)

or eF =

|b(d, α)|sinh

c1 2√

c1 /2(S − c0 )

,

c1 > 0.

(7.11.16)

By determining R from (7.11.15) the function F and taking into account (7.6.1), it is easy to verify that R3 ×R3 f drdV = ∞. From (7.6.1), subject to (7.11.16), we have d1 2 d2 2 d3 2 f = expF · exp −α Vx − + Vy − + Vz − . 2α 2α 2α

(7.11.17)

From the norming condition (6.1.6) for (7.11.17) when 1 = R3 , 2 = R3 , it follows that the parameters α > 0, k 6= 0, k 6= 1, d ∈ R3 and the integration constants c1 > 0, c0 must satisfy the relation 2π|d1 d2 d3 |p(d, α) = 1. √ 3 2c1 |k(k − 1)||b(d, α)|


129

In this case, the electric E and magnetic B fields are determined in a similar fashion as above. When seeking the solution of the Liouville equation (7.11.4) in the form of (7.11.5), we have supposed that d1 6= 0, d2 6= 0 and d3 6= 0. In this case, the function 8(x, u), u = y/d2 − z/d3 satisfies the elliptic equation 8xx +

1 d2 − 4α 2 c2 −8 1 + 8 = a(d, α) e . uu αc2 d22 d32

By solving this, it is possible to use previous results. If d1 = d2 = 0 and d3 6= 0, then according to (7.11.5), the function 8(x, y) satisfies the Liouville equation 48(x, y) = a(d, α)

d2 − 4α 2 c2 −8 e . αc2

Note, in the case when 1 = R3 and 2 = R3 , the distribution function is not normalized, because the integral in (6.1.6) will diverge. Let d1 = d2 = d3 = 0 (this case corresponds to a cold plasma). Then from (7.5.8), (7.5.9) and (7.6.1) for N = 1, it accordingly follows ψ(r) = 0 − const,

E=

m ∇ϕ, 2αq

f = exp{−αV 2 + D(r)}

(7.11.18)

and from (7.11.1) we obtain 4D = (α)eD

(7.11.19)

with D = 42 + ϕ,

(α) =

8π αq2 π 3/2 . m α

On the other hand, from (7.5.3), (7.5.5) N = 1, we have divB = 0,

rot B =

4πq D e c

Z

2

Ve−αV dV.

(7.11.20)

1

If 1 ⊆ R3 is a finite or infinite symmetric domain, then from (7.11.20) it follows that B = −∇3, 43 = 0, i.e., 3 is a scalar harmonic function. If D and 3 are sought in the form D(S), 3(S) with S = a1 x + a2 y + a3 z, then by similar reasoning as above, from (7.11.18)–(7.11.20) we have f=

2S12 a21 + a22 + a23

(α)sinh2 S1 (S − c0 )

exp(−αV 2 ),

130


a1 mS1 cothS1 (S − c0 ) a2 , E= αq a 3

a1 B = −h0 a2 , a3

and, in this case, 3 = h0 S + h1 , where c0 , S1 , h0 , h1 , a1 , a2 , a3 are arbitrary constants. If ⊆ R3 is an arbitrary domain, then, through a choice of the parameters involved in the distribution function f , it is possible to fulfill (6.1.6). Consider the case when the domain 1 ⊂ R3 is not a symmetric one. From (7.11.20) subject to (7.11.19), we have divB = 0,

rot B = U(r),

(7.11.21)

where U=

mS12 (a21 + a22 + a23 ) αcq(π/α)3/2 sinh2 S1 (S − c0 )

Z

Vexp(−αV 2 )dV

1

is the given vector. Relations (7.11.21) lead to the equalities B = rot A,

(7.11.22)

4A = −U,

if the vector potential A satisfies the Lorentz condition divA = 0. Subsequently, we shall suppose 1 = R+ × R+ × R+ . If the components of the vector potential A are sought in the form Ax (S), Ay (S), Az (S), where S = a1 x + a2 y + a3 z, then using (7.11.22) we obtain Ax = Ay = Az = µ(α)ln|sinhS1 (S − c0 )| + u0 S + u1 , a2 − a3 B = (S1 µ(α)cothS1 (S − c0 ) + u0 ) a3 − a1 a1 − a2

(7.11.23)

under the condition that the coefficients a1 , a2 , a3 satisfy the relation a1 + a2 + a3 = 0, where µ(α) = (π)−1/2 (α)−3/2 m/(8cq); u0 , u1 —Const. Therefore, we have a1 mS1 , a2 E= cothS1 (a1 x + a2 y − (a1 + a2 )z − c0 ) αq −(a1 + a2 )

f=

2S12 (a21 + a22 + (a1 + a2 )2 ) (α)sinh S1 (a1 x + a2 y − (a1 + a2 )z − c0 ) 2

2

e−αV .

(7.11.24)


131

Note, by the choice of the parameters involved in the f , it is easy to achieve for it the fulfillment of the norming condition (6.1.6). For example, if 2 = R3 , 1 = R+ × R+ × R+ , S1 > 0, a1 > 0 and a2 > 0, then the distribution function has the form (7.11.24) S1 =

(a21 + a1 a2 + a22 )π m . 96 αq2 a1 a2 |a1 + a2 |

Remark 7.10. Formula (7.11.23) defines all solutions of system (7.11.21) of the form B = B(S). If a1 + a2 + a3 6= 0, then the system of equations (7.11.21) does not have a solution of such a form. Indeed, by substituting B(S) into (7.11.21), we get the system of ordinary differential equations ˙ AB(S) = g(S),

(7.11.25)

where 0 a A = 3 −a2 a1

−a3 0 a1 a2

a2 −a1 , 0 a3

Ux U g(S) = y . Uz 0

The system (7.11.25) has the solution if and only if the solvability condition (g, l) = 0, where A∗ l = 0 is satisfied. If a1 6= 0, then l = (1, a2 /a1 , a3 /a1 , 0)0 , if a2 6= 0, then l = (a1 /a2 , 1, a3 /a2 , 0)0 and finally if a3 6= 0, then l = (a1 /a3 , a2 /a3 , 1, 0). Thus, let a1 + a2 + a3 = 0, then the general solution of (7.11.25) is determined by the formula Z B(S) =

A∗ g(S)dS + c,

c = (c1 , c2 , c3 ).

8 Bifurcation of Stationary Solutions of the Vlasov-Maxwell System

8.1 Introduction The problem of the bifurcation analysis of a VM system, formulated for the first time by Vlasov, proved to be very complicated against the general background of the progress of the bifurcation theory in other directions, and it remains open at the present moment. There exist only separate results. One simple theorem about the point of bifurcation is covered by Sidorov and Sinitsyn [257], and another is proven in paper [256] for the stationary VM system. The goal of this chapter is to prove the general existence theorems for the potentials of electromagnetic field and for the density of charge and current; to find bifurcation points of stationary VM system with the given boundary conditions. For studying the bifurcation points of VM system are used the results of the branching theory of Trenogin, Sidorov [272], Vajenberg, Trenogin [279] and the index theory of Conley [75], Kronecker [163]. Let us note that the methods, which were adapted in [256, 257], were not sufficient for examination of the situation in general, which we study below. Let us consider many component plasma consisting of electrons and positively charged ions of various kinds, described by many particle distribution functions of the form fi = fi (r, v), i = 1, . . . , N. Plasma is located in domain D ⊂ R3 with a smooth boundary. Particles interact among themselves only by means of their own charges; we neglect collisions between particles. The behavior of plasma is described by the following (classical) version of stationary VM system [305]: 1 v · ∂r fi + qi /mi E + v × B ·∂v fi = 0, c r ∈ D ⊂ R3 ,

i = 1, . . . , N, curlE = 0, divB = 0,


(8.1.1)

134


divE = 4π

N X k=1

Z qk

4

fk (r, v)dv = ρ,

(8.1.2)

R3

Z N 4π X 4 qk vfk (r, v)dv = j. curlB = c k=1

R3

Here ρ(r), j(r) are the densities of charge and current, and E(r), B(r) are electric and magnetic fields, respectively. We look for the solution E, B, f of the VM system (8.1.1), (8.1.2) for r ∈ D ⊂ R3 with boundary conditions on potentials and densities U |∂D = u01 , (A, d) |∂D = u02 ; ρ |∂D = 0, j |∂D = 0,

(8.1.3) (8.1.4)

where E = −∂r U, B = curlA, and U, A scalar and vector potentials. We call a trivial solution E 0 , B 0 , f 0 if ρ 0 = 0 and j 0 = 0 inside domain D. Here we study the case of the distribution functions of the special form [248] fi (r, v) = λfî (−αi v2 + ϕi (r), ϕi : R3 → R, λ ∈ R+ ,

4 v · di + ψi (r)) = λfî (R, G),

ψi : R3 → R, 4

αi ∈ R+ = [0, ∞),

r ∈ D ⊆ R3 , di ∈ R 3 ,

(8.1.5)

v ∈ R3 ,

i = 1, . . . , N,

where functions ϕi , ψi , which generate appropriate electromagnetic field (E, B), should be found. We are interested in the dependence of unknown functions ϕi , ψi on parameter λ in distribution function (8.1.5). First we consider λ, which does not depend on physical parameters αi , di used in (8.1.5). For example, in case αi = αi (λ), di = di (λ) the distribution function 3/2 −m|v|2 m · exp + d · v + ϕ(r) f (r, v) = 2π kT 2kT gives a dependence α = α(λ), where λ = (m/(2π kT))3/2 , α = −m/(2kT), k— Boltzmann constant, and T—temperature of electrons. In this case parameter, λ has a dimension of the temperature. Definition 8.1. A point λ0 is called the bifurcation point of VM system with conditions (8.1.3), (8.1.4), if in any neighborhood of vector (λ0 , E 0 , B 0 , f 0 ) correspond to trivial solution with ρ 0 = 0, j0 = 0 in domain D: there exists a vector (λ, E, B, f ), satisfying (8.1.1)–(8.1.4) for which k E − E 0 k + k B − B 0 k + k f − f 0 k> 0.

Bifurcation of Stationary Solutions of the Vlasov-Maxwell System

135

Let ϕi0 , ψi0 be constants with corresponding densities ρ 0 and j0 , introduced in medium by distributions fi which are equal to zero in D for ϕi0 , ψi0 . Then N Z X k=1 3 R

qk fˆk dv = 0,

N Z X

qk vfˆk dv = 0

k=1 3 R

for ϕi = ϕi0 and ψi = ψi0 , k = 1, . . . , N. Let di = σi d1 , i = 1, . . . , n and let σi be a constant. Vector di ∈ R3 and parameter αi ∈ R+ characterize the chaotic heat motion of particles of the kind i. We examine the case, when di and αi are different, i.e., nonisothermic plasma as most frequently being encountered in the applications. Then VM system possesses a trivial solution fi 0 = λfî (−αi v2 + ϕi0 ,

v · di + ψi0 ),

E 0 = 0,

B 0 = βd1 ,

β − Const.

for any λ. Our aim is to construct nontrivial solutions of the stationary VM system. We obtain conditions implying the existence of points λ∗ ∈ R+ (bifurcation points), processing a neighborhood, where a VM system has nontrivial solutions in domain D ⊂ R3 . For these solutions we have ρ|D 6= 0, j|D 6= 0 but ρ|∂D = 0, j|∂D = 0. It is assumed that the scalar and vector potentials of the desired electromagnetic field are given at the boundary of the domain. The branching equation (BEq) was derived by Vainberg and Trenogin in [274, 279]. We proved that for sufficiently general case of fi = fî (a(−αi v2 + ϕi ) + b(v · di + ψi )), where a, b are constants, BEq be the potential equation. On this basis the asymptotics are constructed for nontrivial branches of solutions in a neighborhood of the bifurcation point. Let us note that the problem of bifurcation points in the theory of collisionless plasma without allowance for magnetic field was studied in Holloway [135], Holloway, Dorning [136], and Hesse and Schindler [131]. Apparently the problem of bifurcation points for the general VM system was not considered earlier. The chapter is organized as follows: In Section 8.2, two existence theorems of the bifurcation points for the nonlinear operator equation in Banach space, generalizing results on bifurcation points in [255, 272], are proven. The method of proof of these theorems uses index theory of vector fields [75, 163] and makes it possible to study not only points, but also bifurcational surfaces with the minimum limitations to the equation. In Section 8.4, we reduced the problem about a bifurcation point of a VM system to the problem on bifurcation point of semilinear elliptic system, considered as operator equation in Banach space. The boundary value problem and the problem on bifurcation points was formulated, and a spectrum of the problem for linearized system is studied. In Section 8.5, the BEq is constructed. In Section 8.6, the existence theorem for bifurcation points is proven on the basis of the analysis of the BEq, and the asymptotics of nontrivial branches is constructed for the solutions of the VM system.

136


8.2 Bifurcation of Solutions of Nonlinear Equations in Banach Spaces Let E1 , E2 be Banach spaces; ϒ be normed space. Let us consider equation Bx = R(x, ε).

(8.2.1)

Here B : D ⊂ E1 → E2 be a closed linear operator with dense domain of definition in E1 . Operator R(x, ε) with the values into E2 is defined, continuous and continuously differentiable in Frechet sense over x in the neighborhood = {x ∈ E1 ,

ε∈ϒ :

k x k< r,

k ε k< %}.

Assume R(0, ε) = 0, Rx (0, 0) = 0. Let operator B be Fredholm operator. Let us introN N ∗ ∗ duce basis {ϕi }N 1 in subspace N(B), basis {ψi }1 in N(B ), and also systems {γi }1 ∈ E1 , N {zi }1 ∈ E2 , which are biorthogonal to these bases. Definition 8.2. A point ε0 is called the bifurcation point of (8.2.1), if in any neighborhood of point x = 0, ε0 there is a couple (x, ε) with x 6= 0, satisfying equation (8.2.1). It is known [279] that the problem about bifurcation point of equation (8.2.1) is equivalent to the problem on bifurcation point of finite-dimensional system L(ξ, ε) = 0,

(8.2.2)

where ξ ∈ RN , L : RN × ϒ → RN . We will call (8.2.2) the branching equation (BEq). We write 8.2.1 as the system ˜ = R(x, ε) + Bx

N X

ξ s zs ,

(8.2.3)

s=1

ξs =< x, γs >,

s = 1, . . . , n,

(8.2.4)

P defn where B˜ = B + N s=1 < ·, γs > zs has it inverse bounded. Equation (8.2.3) has a unique small solution

x=

N X

ξs ϕs + U(ξ, ε)

(8.2.5)

s=1

for ξ → 0, ε → 0. Substitution (8.2.5) into (8.2.4) gives formulas for coordinates of vector-function L : RN × ϒ → RN ! X N Lk (ξ, ε) = R ξs ϕs + U(ξ, ε), ε , ψk . s=1

(8.2.6)


137

Here the derivatives ∂Lk defn |ξ =0 = Rx (0, ε)(I − 0Rx (0, ε))−1 ϕi , ψk = aik (ε) ∂ξi are continuous in neighborhood of point ε = 0, k 0Rx (0, ε) k< 1. Introducing the set = {ε | det[aik (ε)] = 0} containing the point ε = 0, we are able to formulate the following condition: Condition (A) We assume that there exists the S set S, in a neighborhood of T point ε0 ∈ , which posesses Jordan continuum S = S+ S− , ε0 ∈ ∂S+ ∂S− . Moreover, there is continuous mapping ε(t), t ∈ [−1, 1] such that ε : [−1, 0) → S− , ε : 1 (0, 1] → S+ , ε(0) = ε0 , det[aik (ε(t))]N i,k=1 = α(t), where α(t) : [−1, 1] → R is continuous function becoming zero only in t = 0. Theorem 8.1. Let condition A) hold and α(t) is a monotonic increasing function. Then ε0 is bifurcation point of bifurcation equation (8.2.1). Proof. Let us take arbitrary small r > 0 and δ > 0. Let us consider continuous vector field defn

H(ξ, 2) = L(ξ, ε((22 − 1)δ)) : RN × R → RN , given for ξ, 2 ∈ M, where M = {ξ, 2 | k ξ k= r, 0 ≤ 2 ≤ 1}. Case 1. Case 2.

If there exists a pair (ξ ∗ , 2∗ ) ∈ M with H(ξ ∗ , 2∗ ) = 0, then by definition 8.2, ε0 is bifurcation point. We assume that H(ξ, 2) 6= 0 for ∀(ξ, 2) ∈ M and, hence, ε0 is not the bifurcation point. Then vector fields H(ξ, 0) and H(ξ, 1) are homotopic on the sphere k ξ k= r. Hence, their rotations [161] coincide J(H(ξ, 0), k ξ k= r) = J(H(ξ, 1), k ξ k= r).

(8.2.7)

Since vector fields H(ξ, 0), H(ξ, 1) and their linerization N defn X

L1− (ξ ) =

aik (ε(−δ))ξk |N i=1 ,

k=1 N defn X

L1+ (ξ ) =

aik (ε(+δ))ξk |N i=1

k=1

are nondegenerated on the sphere k ξ k= r, then, due to smallness of r > 0, fields (H(ξ, 0), H(ξ, 1)) are homotopic in their linear parts L1− (ξ ) and L1+ (ξ ). Thus, J(H(ξ, 0), k ξ k= r) = J(L1− (ξ ), k ξ k= r), J(H(ξ, 1), k ξ

k= r) = J(L1+ (ξ ), k ξ

L1± (ξ )

k= r).

(8.2.8) (8.2.9)

Since linear fields are nondegenerate according to the theorem of Kronecker index, the following equalities J(L1− (ξ ), k ξ k= r) = sign α(−δ), J(L1+ (ξ ), k ξ k= r) = sign α(+δ)

138


holds. Since α(−δ) < 0, α(+δ) > 0 then, due to (8.2.8), (8.2.9) relation, (8.2.7) is not valid. Hence, we found a couple (ξ ∗ , 2∗ ) ∈ M, for which H(ξ ∗ , 2∗ ) = 0 and ε0 is bifurcation point.

Remark 8.1. If conditions of the Theorem 8.1 are satisfied for ∀ε ∈ 0 ⊂ , then 0 is a bifurcation set of equation (8.2.1). Moreover, if 0 is a connected set and every point is contained in their own neighborhood are homeomorphous with some domain from RN , then 0 is called n-dimensional bifurcation manifold. For example, taking ϒ = Rn+1 , n ≥ 1 we have 0 —a bifurcation set of (8.2.1) and it contains the point ε = 0, while ∇ε det[aik (ε)] |ε=0 6= 0. The generalization of this result (see [272]) follows from Theorem 8.1 with ϒ = R, and also other known strengthenings of Krasnosel’s kij theorem about bifurcation point of the odd multiplicity [161]. Stronger results in the theory of bifurcation points are obtained for (8.2.1) with the potential BEq in ξ , when L(ξ, ε) = gradξ U(ξ, ε).

(8.2.10)

∂Lk N is symmetrical. By means of the dif∂ξi i,k=1 ferentiation of superposition, we find from (8.2.6) that * ! + N X ∂Lk ∂U = Rx ϕi + , ψk , ξs ϕs + U(ξ, ε), ε (8.2.11) ∂ξi ∂ξi

This condition is satisfied if matrix

s=1

where, according to (8.2.3), (8.2.5), ϕi +

∂U = (I − 0Rx )−1 ϕi . ∂ξi

(8.2.12)

Operator I − 0Rx is continuously invertible, because k 0Rx k< 1 for the sufficiently small by norm ξ and ε. Substituting (8.2.12) into (8.2.11) we obtain equalities ∂Lk −1 = Rx (I − 0Rx ) ϕi , ψk , i, k = 1, . . . , n. ∂ξi The following assertion occurs: Lemma 8.1. In order for BEq (8.2.2) to be potential, it is sufficient that matrix

N 4 = Rx (0Rx )m ϕi , ψk i,k=1 would be symmetrical ∀(x, ε) in the neighborhood of point (0, 0). Corollary 8.1. Let all matrices

N Rx (0Rx )m ϕi , ψk i,k=1 , m = 0, 1, 2, . . . be symmetrical in any neighborhood of point (0, 0). Then BEq (8.2.2) is potential.


139

Corollary 8.2. Let E1 = E2 = H, H—Hilbert space. If operator B is symmetrical in D, and operator Rx (x, ε) is symmetrical for ∀(x, ε) in neighborhood of point (0, 0) in D, then BEq is potential. Work [255] gives more precise conditions for potentiality of BEq. We assume that BEq (8.2.2) is potential. Then it follows from the proof of Lemma 8.1 that corresponding potential U in (8.2.10) takes the form U(ξ, ε) =

N 1 X ai,k (ε)ξi ξk + ω(ξ, ε), 2 i,k=1

where k ω(ξ, ε) k= 0(| ξ |2 ) for ξ → 0. Theorem 8.2. Let BEq (8.2.2) be potential. Let condition A) hold. Moreover, let symmetrical matrix [aik (ε(t))] possess ν1 positive eigenvalues for t > 0 and ν2 positive eigenvalues for t < 0, ν1 6= ν2 . Then ε0 will be the bifurcation point of equation (8.2.1). Proof. Let us take arbitrary small δ > 0 and consider function U(ξ, ε((22 − 1)δ)), defined into 2 ∈ [0, 1] in neighborhood of critical point ξ = 0. Case 1. Case 2.

If there exists 2∗ ∈ [0, 1] such that ξ = 0 is nonisolated critical point of function U(ξ, ε((22∗ − 1)δ)), then, due to definition 8.2, ε0 is the bifurcation point. We assume that point ξ = 0 is the isolated critical point of function U(ξ, ε((22−1)δ)) for ∀2 ∈ [0, 1], where ε(t) is continuous function from condition A). Then with ∀2 ∈ [0, 1] for this function, the Conley index [75] K2 of critical point ξ = 0 is determined. Let us recall that det k

∂ 2 U(ξ, ε((22 − 1)δ)) kξ =0 = α((22 − 1)δ). ∂ξi ∂ξk

Since α((22 − 1)δ) 6= 0 for 2 6= 21 , then critical point ξ = 0 for 2 6= 21 is not degenerated. Thus, Conley index K2 with certain 2 6= 12 necessarily is equal to the number of positive eigenvalues of the corresponding Hessian according to definition (Conley [75], p. 6). Therefore, K2 = ν1 , K1 = ν2 , where ν1 6= ν2 , due to conditions of Theorem 8.2. Hence, K2 6= K1 . We assume that ε0 is not the bifurcation point. Then ∇ξ U(ξ, ε((22 − 1)δ)) 6= 0 with 0 0 sufficiently small number, 2 ∈ [0, 1]. In view of the homotopic invariance of Conley index ([75], p. 52, Teorem 4), K2 is a constant for 2 ∈ [0, 1] and K0 = K1 . Hence, in the second case, we will always find the pair (ξ ∗ , 2∗ ) satisfying equation ∇ξ U(ξ, ((22 − 1)δ)) = 0 for arbitrary small r > 0, δ > 0, where 0 = δij ;

< ϕ i , ψj

>= δij ,

i, j = 1, . . . , n;

N X

Pi = k.

i=1

Let us recall that (Pi )

(1) 4

ϕi

= ϕi = 0ϕi

,

(1) 4

ψi

(Pi )

= ψi = 0 ∗ ψi

,

−1 N X (Pi ) (Pi ) , 0 = B+ < ·, ψi > ϕi

(8.2.14)

1

where k = l1 + · · · + ln is the root number of operator B − Rx (0, ε). Small eigenvalues ν(ε) of operator B − Rx (0, ε) satisfy the following BEq [279] 4

L(ν, ε) = det |< (Rx (0, ε) + νI)(I − 0Rx (0, ε) − ν0)−1 ϕi , ψj >|ni,j=1 = 0. (8.2.15) Due to a Veyerstrass theorem [279] and relations (8.2.13), (8.2.14), equation (8.2.15) can be transformed into L(ν, ε) ≡ (ν k + Hk−1 (ε)ν k−1 + · · · + H0 (ε))(ε, ν) = 0


141

in the neighborhood of zero, where Hk−1 (ε), . . . , H0 (ε) = 4(ε) continuous functions ε, (0, 0) 6= 0, H0 (0) = 0. Hence, operator B − Rx (0, ε) has k ≥ n small eigenvalues νi (ε), i = 1, . . . , n, which we can find from equation ν k + Hk−1 (ε)ν k−1 + · · · + 4(ε) = 0. Then

Qk i

νi (ε) = 4(ε)(−1)k .

Let ε ∈ R. Let us consider computation of asymptotics of eigenvalues µ(ε) and ν(ε). Let us introduce the block presentation of matrix [aik ]N i,k=1 satisfying the following: l rik 0 l Condition (B) Let [aik (ε)]N i,k=1 = [Aik (ε)]i,k=1 ∼ [ε Aik ]i,k=1 for ε → 0, where 4

[Aik ] blocks of dimension [ni × nk ], n1 + · · · + nl = n, min(ri1 , . . . , ril ) = rii = ri and rik > riQ for k > i (or for k < i), i = 1, . . . , l. Let l1 det[A0ii ] 6= 0. Condition B denotes that matrix [aik (ε)]N i,k=1 admits the block presentation, which is “asymptotically triangular” for ε → 0. Lemma 8.3. Let condition B hold. Then Y l 0 N n1 r1 +···+nl rl det | Aii | +0(1) , det[aik (ε)]i,k=1 = ε 1

formulas µi = εri (Ci + 0(1)),

i = 1, . . . , l,

(8.2.16)

define the dominant terms all n eigenvalues of matrix | aik (ε) |N i,k=1 , where µi , Ci ∈ 0 n i R , Ci vector of eigenvalues of matrix Aii . Proof. Due to condition B and property of linearity of determinant, we obtain 0 A11 + 0(1) 0(1) . . . . . . . . . . . . 0(1) 0 0 det[aik (ε)] = εn1 r1 +···+nl rl det A21 + 0(1) A22 + 0(1) 0(1) . . . 0(1) = ......... ......... ......... A0 + 0(1) . . . . . . . . . 0 + 0(1) A l1 ll ! l Y n1 r1 +···+nl rl 0 =ε det | Aii | + 0(1) . i

Substituting µ = εri c(ε), i = 1, . . . , l, into equation det | aik (ε) − µδik |N i,k=1 = 0 and using the property of linearity of determinant, we obtain the following equation ε

n1 r1 +···+ni−1 ri−1 +(ni +···+nl )ri

Y i−1

det | A0jj | ×

j=1

× det(A0ii − c(ε)E)c(ε)ni+1 +···+nl

+ ai (ε) = 0,

i = 1, . . . , l,

(8.2.17)

142


where ai (ε) → 0 for ε → 0. Hence, the coordinates of the dominant terms Ci in asymptotics (8.2.16) satisfy equations det | A0ii − cE |= 0, i = 1, . . . , l. If k = n, then operator B − Rx (0, ε), just as matrix [aik (ε)]N i,k=1 has n small eigenvalues. In this case the following result holds. Corollary 8.4. Let operator B not have I adjoint elements and assume condition B. Then formula νi = −ε ri (Ci + 0(1)),

i = 1, . . . , l,

(8.2.18)

defines all n small eigenvalues of operator B − Rx (0, ε), where Ci ∈ Rni is vector of eigenvalues of matrix A0ii , i = 1, . . . , l, n1 + · · · + nl = n. P n (root number k = n) and Proof. In this case, due to Lemma 8.2, we have N 1 Pi = P operator B − Rx (0, ε) possesses n small eigenvalues. Since l1 ni = n, A0ii is the square matrix, then (8.2.18) gives n eigenvalues, where dominant terms coincide with dominant terms in (8.2.16) with an accuracy to the sign. For computing eigenvalues ν of operator B − Rx (0, ε), we transform (8.2.15) into L(ν, ε) ≡ det[aik (ε) +

∞ X

(j)

bik ν j ]N i,k=1 = 0,

(8.2.19)

j=1

where (j)

bik =< [(I − 0Rx (0, ε))−1 0] j−1 (I − 0Rx (0, ε))−1 ϕi , γk >. Substituting ν = −εri c(ε) into (8.2.19) and taking into account the property of linearity of determinant, we obtain equation which differs from (8.2.17) only in terms of error of calculation ai (ε). Then in conditions of Corollary 8.4 the dominant terms define all small eigenvalues of operator B − Rx (0, ε) and matrices −[aik (ε)] are found from the same equations and, hence, are identical.

8.3 Conclusions 1. Due to Lemma 8.3, we can change condition A in Theorem 8.1: Condition (A∗ ) Let E1 = E2 = E, ν = 0 be isolated Fredholm point of operator-function B − νI. Let in neighborhood of the S point ε0 ∈ there be the set S, containing point ε0 , which presents the continuum S = S+ S− . Furthermore, let us assume that ε0 ∈ ∂S+

\

∂S− ,

Y i

νi (ε)

ε∈S+

·

Y i

νi (ε)

< 0,

ε∈S−

where {νi (ε)} are small eigenvalues of operator B − Rx (0, ε).


143

2. If the dominant terms in asymptotics of small eigenvalues of operator B − Rx (0, ε) and matrix [aik (ε)]N i,k=1 coincide, then Theorem 8.2 is applicable. Due to Corollary 8.4, it is possible, if E1 = E2 = H. Then operators B and Rx (0, ε) are symmetrical and condition B is satisfied. We note that condition B is also realized in papers of Sidorov and Trenogin [255, 272] on the bifurcation point with potential BEq, moreover r1 = . . . = rn = 1.

8.4 Statement of Boundary Value Problem and the Problem on Point of Bifurcation of System (8.4.7), (8.4.13) Let us present one preliminary result about the reduction of VM system (8.1.1)–(8.1.2) with boundary conditions (8.1.3), (8.1.4) to the quasilinear system of elliptic equations for the distribution function (8.1.5). Assume that the following condition is satisfied. Condition (C) fî (R, G) is given, differentiable functions in distribution (8.1.5); αi , di constant parameters; |di | 6= 0, ϕi = c1i + li ϕ(r), ψi = c2i + ki ψ(r), c1i , c2i —Const, parameters li , ki are connected by relations li =

m1 αi qi , α1 q1 mi

and integrals

ki

fî dv,

R R3

R

qi q1 d1 = di , m1 mi

k1 = l1 = 1,

(8.4.1)

fî vdv converge for ∀ϕi , ψi .

R3 4

4

4

Let us introduce the following notations m1 = m, α1 = α, q1 = q. Theorem 8.3. Let fi be defined in (8.1.5) and condition C hold. Also assume that vector-function (ϕ, ψ) is a solution of the system of equations N X

4ϕ = µ

4ψ = ν

Z fk dv,

qk

k=1

R3

N X

Z qk

k=1

ϕ |∂D = −

µ=

(v, d)fk dv,

8π αq , m 4πq , mc2

(8.4.2)

i = 1, . . . , N.

(8.4.3)

ν=−

R3

2αq u01 , m

ψ |∂D =

q u02 mc

in subspace (∂r ϕi , di ) = 0,

(∂r ψi , di ) = 0,

Then VM system (8.1.1)–(8.1.4) has a solution   Z1 m d  mc E= ∂r ϕ, B = 2 β + (d × J(tr), r)dt − [d × ∂r ψ] 2 , 2αq d qd 0

(8.4.4)

144


where Z N 4π X qk vfk dv, c

4

J=

k=1

β − Const.

R3

Potentials U=−

m ϕ, 2αq

A=

mc ψd + A1 (r), qd2

(A1 , d) = 0,

(8.4.5)

satisfying condition (8.1.3), are defined over this solution. We introduce the notations Z Z ji = vfi dv, ρi = fi dv, R3

i = 1, . . . , N

R3

and introduce the following condition: Condition (D) there exist vectors βi ∈ R3 such that ji = βi ρi , i = 1, . . . , N. For example condition D is satisfied for distribution fi = fi (a(−αi v2 + ϕi ) + b((di , v) + ψi ))

(8.4.6)

b di , a, b—Const. 2αi a We assume that condition D holds. Then system (8.4.2) is transformed into

for βi =

4ϕ = λµ

N X

qi Ai ,

4ψ = λν

i=1

N X

qi (βi , d)Ai ,

(8.4.7)

i=1

where 4

Ai (li ϕ, ki ψ, αi , di ) =

Z

fî dv,

(8.4.8)

R3

βi =

b di , 2αi a

(βi , d) =

b d 2 ki , a 2α li

(di , d) d2 ki = . αi α li

In the case of normalized distribution functions, this system admits the following generalization. Let Z Z fi dvdr = Ni , D R3


145

where fi = f1i (ϕ, ψ, v) + f2i (ϕ, ψ) · M, R and the integral R3 M(v)dv converges. Then one has the integral identity R R Ni − D R3 f1i dvdr Z Z R . fi dv = f1i (ϕ, ψ, v)dv + f2i (ϕ, ψ) · D f2i dr R3

(8.4.9)

R3

Therefore, for the functions ϕ and ψ occuring in the distribution fi in the case of normalized distributions functions fi , we obtain the following system of quasilinear integro-differential equations 4ϕ = λµ

N X

qi A˜ i ,

4ψ = λν

i=1

Z

qi (βi , d)A˜ i ,

(8.4.10)

i=1

A˜ i =

N X

f1i dv + λf2i ·

R3

ξi − λ

R R 1

D R3 f1i dvdr

R

D f2i dr

,

4

where λ = ||N||, N = (N1 , . . . , Nn ), ξi = Ni /||N||. Remark 8.4. Apparently distribution functions of the form (8.4.9) can be useful for the analysis of stationary solutions of the Boltzmann equation, since they permit one to simplify the collision integral by separating the variables r and v. If, in addition, we normalize the function f1i by Z Z f1i dvdr = Ki , D R3

then we can study distribution functions with different numbers Ni and Ki . From now on, for simplicity, we consider the auxiliary vector d in (8.1.5) directed along the axis Z. Due to (8.4.3), we can take ϕ = ϕ(x, y), ψ = ψ(x, y), x, y ∈ D ⊂ R2 in system (8.4.7). Moreover let N ≥ 3 and klii 6= Const. Let D—be a bounded domain in R2 with boundary ∂D of class C2,α , α ∈ (0, 1). Boundary conditions (8.1.3), (8.1.4) for local densities of charge and current provide equalities: N X

qk Ak (lk ϕ 0 , kk ψ 0 , αi , di ) = 0;

k=1 N X k=1

qk (βk , d)Ak (lk ϕ 0 , kk ψ 0 , αi , di ) = 0

(8.4.11)

146


for ∀ε ∈ ι, where ι is a neighborhood of the point ε = 0 and ϕ0 = −

2αq u01 , m

ψ0 =

q u02 . mc

(8.4.12) 2

di d ki , then, due to (8.4.11), (8.4.12) and (βi , d) = 2α Remark 8.5. If N = 2 and βi = 2α li , i we obtain altarnative: either in (8.4.11) A1 = A2 = 0 or ki = li , i = 1, 2. In this case, and also for klii = Const, system (8.4.7) is reduced to one equation and the bifurcation of the solutions is impossible.

Using (8.4.11), (8.4.12) for the system (8.4.7) with boundary conditions ϕ |∂D = ϕ 0 ,

ψ |∂D = ψ 0

(8.4.13)

one obtains a trivial solution ϕ = ϕ 0 , ψ = ψ 0 , ∀λ ∈ R+ . Then, due to Theorem 8.3 for any λ, the VM system with boundary conditions (8.1.3), (8.1.4) has the trivial solution m E0 = ∂r ϕ 0 = 0, B 0 = βd1 , r ∈ D ⊂ R2 , 2αq f 0 = λfî (−αi v2 + c1i + li ϕ 0 , (v, di ) + c2i + ki ψ 0 ). Under this condition, ρ and j vanish in domain D. Thus, our aim is to find λ0 providing nontrivial solution for neighborhood systems (8.4.7), (8.4.13). Then the corresponding densities ρ and j vanish in domain D, and point λ0 is the bifurcation point of VM system (8.4.1), (8.4.2), (8.1.3), (8.1.4). Let functions fi —analytical in (8.1.5). Using the Taylor series expansion i ∞ X 1 0 ∂ 0 ∂ A(x, y) = (x − x ) + (y − y ) A(x0 , y0 ) i! ∂x ∂y i≥0

and expressing the linear terms, we can rewrite system (8.4.7) in operator form (L0 − λL1 )u − λr(u) = 0.

(8.4.14)

Here

4 0 , u = (ϕ − ϕ 0 , ψ − ψ 0 )0 ; 0 4 " # N s s X µls ∂A µks ∂A 4 ∂x ∂y L1 = qs = ∂As s νls (βs , d) ∂A νk (β , d) s s ∂x ∂y x=ls ϕ 0 ,y=ks ψ 0 s=1 4 µT1 µT2 = , νT3 νT4 L0 =

r(u) =

∞ X N X i≥l s=1

%is (u)bs ,

(8.4.15)

(8.4.16) (8.4.17)


147

where ∂ ∂ i qs ls u1 + ks u2 As (ls ϕ 0 , ks ψ 0 ) %is (u) = i! ∂x ∂y 4

are ith-order homogeneous forms in u; ∂ i1 +i2 As (x, y) ∂xi1 ∂yi2 x=ls ϕ 0 ,

y=ks ψ 0

=0

with 2 ≤ i1 + i2 ≤ l − 1,

s = 1, . . . , N,

l ≥ 2,

4

bs = (µ, ν(βs , d))0 .

The existence problem for a bifurcation point, λ0 , of the system (8.4.7), (8.4.13) also can be stated as the existence problem for a bifurcation point for the operator equation (8.4.14). ¯ and C 0,α (D) ¯ with the norms k · k2,α , Let us introduce the Banach spaces C2,α (D) 2,2 k · k0,α , respectively, and let W (D) be the ordinary Sobolev L2 space in D. 4

Let us introduce the Banach space E of vectors u = (u1 , u2 )0 , where ui ∈ L2 (D), L2 is the real Hilbert space with inner product (·, ·) and the corresponding norm k · kL2 (D). 4

◦

We define the domain D(L0 ) as the set of vectors u = (u1 , u2 ) with ui ∈W 2,2 (D). Here ◦

W 2,2 (D) consists of W 2,2 functions with zero trace on ∂D. Hence, L0 : D ⊂ E → E is a linear self-adjoint operator. By virtue of the embedding, ¯ W 2,2 (D) ⊂ C0,α (D),

0 < α < 1,

(8.4.18)

the operator r : W 2,2 ⊂ E → E—is analytic in a neighborhood of the origin. The operator L1 ∈ L (E → E) is linear, bounded. We keep same notations for matrix corresponding to operator L1 . By the embedding (8.4.18) any solution of equation (8.4.14) is a Ho¨ lder function in D(L0 ). Moreover, since the coefficients of system (8.4.14) are constant, the vector r(u) is analytic, and ∂D ∈ C2,α ; it follows from well known results of regularity theory for weak solutions of elliptic equations (see Ladyzhenskaya and ◦

Ural’zeva [168]) that the generalized solutions of equation (8.4.14) in W 2,2 (D) actually belong to C2,α . Definition 8.3. (See [274]). A point λ0 is called a bifurcation point of the problem (8.4.7), (8.4.13) if every neighborhood of the point (ϕ 0 , ψ 0 , λ0 ) contains a point (ϕ, ψ, λ) satisfying system (8.4.7), (8.4.13) such that ||ϕ − ϕ 0 || ◦ 2,2 + ||ψ − ψ 0 || ◦ 2,2 > 0. W

W

Here || · || ◦ 2,2 —is the norm in the space W 2,2 (D). W

148


According to the Theorem 8.3 on the reduction of the VM system, the bifurcation points of problem (8.4.8), (8.4.13) will be reffered to as the bifurcation points of the stationary VM system (8.4.1), (8.4.2), (8.1.3), (8.1.4). Under the above assumptions about L0 and L1 all singular points of the operator 4

L(λ) = L0 − λL1 are Fredholm points. If N(L(λ0 )) = {0} then by the implicit operator theorem [274], for any δ > 0 there exists a neighborhood S of the point λ0 such that for all λ ∈ S the ball ||u||E < δ contains only the trivial solution u = 0, so that λ0 is not a bifurcation point. Therefore, to find the bifurcation points it is necessary (but not sufficient) to find number λ0 such that N(L0 − λ0 L1 ) 6= {0}. The bifurcation points of the nonlinear equation (8.4.14) are necessarily spectral points of the linearized system (L0 − λL1 )u = 0.

(8.4.19)

To analyze the spectral problem (8.4.19) for physically acceptable parameter values, first we search for eigenvalues and eigenvectors of the matrix L1 in (8.4.19). To achieve it, we need several auxiliary assertions to be made. Let us introduce the conditions: (i) (ii)

T1 < 0, (T1 T4 − T2 T3 ) > 0.

Lemma 8.4. If ∂ fˆk | 0 > 0, ∂x x=lk ϕ then condition (i) is satisfied. 4

Proof. We can assume that q = q1 < 0, qi > 0, i = 2, . . . , N, and signqi li = signq. By the assumption of the lemma we have ∂Ai = ∂x

Z R3

∂ fî dv > 0. ∂x

Therefore T1 < 0. Let us introduce matrix 2 = ||2ij ||i,j=1,...,n 6= [0], 2ij = qi qj (lj ki − kj li )(βi − βj , d). Lemma 8.5. If ∂Ai ∂Ai = , ∂x ∂y

i = 1, . . . , N,

N ≥ 3,

∂Ai >0 ∂x

and the matrix 2—is positive, then conditions (i) and (ii) are satisfied.


149

Proof. For any ∂Ai /∂x, we have the identity X X X X T1 T4 − T2 T3 = li a i ki (βi , d)ai − ki ai li (βi , d)ai = =

N X i−1 X

ai aj (lj ki − kj li )(βi − βj , d),

4

where ai = qi

i=2 j=1

∂Ai . ∂x

By virtue of this identity, for the expression T1 T4 − T2 T3 to be positive it is sufficient that 2ij = qi qj (lj ki − kj li )(βi − βj , d) > 0,

i, j = 1, . . . , N,

N ≥ 3.

(8.4.20)

Since the matrix 2—is positive we have (8.4.20). Remark 8.6. If βi = di /(2αi ), then (βi , d) =

d 2 ki 2α li

(8.4.21)

and 2ij =

d 2 qi qj (lj ki − li kj )2 > 0, 2α li lj

i, j = 1, . . . , n,

because sign(qi /li ) = signq. Remark 8.7. If N = 2 and βi = di /(2αi ) then by conditions i), ii) and (8.4.21), the following alternative takes place: either A1 = A2 = 0 under conditions (8.4.11), (8.4.12) or ki = li , i = 1, 2. For example if βi = N X i−1 X

di 2αi ,

then (βi , d) =

= ai aj (lj ki − li kj )2 ·

i=2 j=1

d2 ki 2α li

and

d2 > 0. 2αli lj

Lemma 8.6. Let distribution function have the form (8.1.5) and fi0 > 0. Then condidi , and system (8.4.7) is transformed to tions D and (i), (ii) are satisfied for βi = ab 2α i potential type " ∂V # ϕ a1 0 ∂ϕ 4 =λ (8.4.22) ∂V , ψ 0 a2 ∂ψ

where V=

N X qk k=1

alk ϕ+bk Z kψ

Ak (s)ds,

lk

a1 = µ/a,

a2 =

0

To prove it we just substitute (8.4.23) into (8.4.22).

νd2 . 2ab

(8.4.23)

150

Kinetic Boltzmann, Vlasov and Related Equations 4

4

Lemma 8.7. Let r = x ∈ R, v ∈ R2 , d = d2 . Then system (8.4.22) with potential (8.4.23) can be written as Hamiltonian system ϕ˙ = ∂pϕ H, ψ˙ = ∂pψ H

p˙ ϕ = −∂ϕ H, p˙ ψ = −∂ψ H, with Hamiltonian H=−

p2ϕ 2

−

p2ψ 2

+ V(ϕ(x), ψ(x)).

Here V(ϕ, ψ) = λa1

Zalk ϕZ N X qk k=1

lk 0

A(s, ψ)ds + λa2

bk Z k ψZ N X qk k=1

R2

lk 0

A(ϕ, s)ds.

R2

Proof follows from Lemma 2.2 (see Guo, Ragazzo [126], p. 1152). Lemma 8.8. Let conditions (i), (ii) be satisfied. Then the matrix L1 in (8.4.16) has one positive eigenvalue χ+ = µT1 + 0(1) and one negative eigenvalue χ− = η

T1 T4 − T2 T3 + O(), T1

η=

4π | q | >0 m

4

as = c12 → 0. Eigenvalue χ− generate eigenvectors of matrices L1 and L10 , respectively, ∗ T2 − T1 c1 c1 0 + O(), = ∗ = 1 + O(). c2 c 0 2 Proof. The characteristic equation of the matrix µT1 − χ µT2 +ηT3 +ηT4 − χ has the form χ 2 − χ(µT1 + ηT4 ) + ηµ(T1 T4 − T2 T3 ) = 0. Since q 1 2 χ1,2 = µT1 + ηT4 ± (µT1 + ηT4 ) − 4ηµ(T1 T4 − T2 T3 ) , 2

(8.4.24)


151

we obtain χ+ = µT1 + O(1),

χ− = η

T1 T4 − T2 T3 + O() T1

as → 0. Since µ < 0 and T1 < 0 it follows that χ+ > 0. It can be checked in a similar way that χ− < 0 by virtue of the inequalities η > 0, T1 T4 − T2 T3 > 0, T1 < 0. Solving the homogeneous systems (4 − χ− )c = 0,

(40 − χ− )c∗ = 0,

we can find the eigenvectors corresponding to χ− . Let us proceed to the computation of the bifurcation point λ0 . Assuming that λ = λ0 + in (8.4.14), consider the system (L0 − (λ0 + )L1 )u − (λ0 + )r(u) = 0

(8.4.25)

in a neighborhood of the point λ0 . Assume that either T2 6= 0 and T3 6= 0 or T2 = T3 = 0. To symmetrize the system for T2 6= 0, T3 6= 0, we multiply both sides of equation (8.4.25) by matrix 4 µT2 10 , where a˜ = M= 6= 0. 0 a˜ νT3 We rewrite (8.4.25) in the form Bu = B1 u + (λ0 + )R(u).

(8.4.26)

Here B = M(L0 − λ0 L1 ),

4

4

R(u) = Mr(u) = (r1 (u), r2 (u)),

B1 ∈ L(E → E)—is a self-adjoint operator, since it is generated by the symmetric matrix B1 = ML1 , and B : D(L0 ) ⊂ E → E—is a self-adjoint Fredholm operator. Remark 8.8. If As (als ϕ + bks ψ) then ∂As = A0s b, ∂y

∂A = A0s a, ∂x

a˜ =

µb d2 a, ν 2α

βs =

b ds . a 2αs

In decomposition (8.4.17) %is =

qs (i) A (als ϕ 0 + bks ψ 0 )(als u1 + bks u2 )i . i! s

Therefore, ∂r1 /∂u2 = ∂r2 /∂u1 in this case, the matrix Ru (u) is symmetric for any u, and the operator Ru : E → E—is self-adjoint for any u.

152


Remark 8.9. If T2 = T3 = 0 then one can set a˜ = 1. If either T2 = 0 and T3 6= 0 or T3 = 0 and T2 6= 0 then the problem cannot be symmetrized and the derivation of the BEq in Subsection 8.5 must be performed directly for equation (8.4.25). Let us introduce the following condition: (iii) Let µ—be an eigenvalue of the Dirichlet problem −4e = µe,

e|∂D = 0,

and {e1 , . . . , en }—be an orthonormal basis in the subspace of eigenfunctions. Let c− = (c1 , c2 )0 —be the eigenvector of the matrix L1 corresponding to the eigenvalue χ− < 0. Lemma 8.9. Let condition (iii) hold, and λ0 = −µ/χ− . Then dim N(B) = n and the system {ei }N i=1 , where ei = c− ei is a basis in the subspace N(B). Proof. Consider the matrix 0 whose columns are the eigenvectors of L1 corresponding to the eigenvalues χ− , χ+ . Then 0−1 L1 0 =

χ− 0 , 0 χ+

L 0 0 = 0L 0 ,

and substitution u = 0U reduces the equation Bu = 0 into M[L0 0U − λ0 L1 0U] = M[0(L0 U − λ0 0−1 L1 0U)] = 0. It follows that the linearized system (8.4.19) splits into the two linear elliptic equations 4U1 − λ0 χ− U1 = 0, 4U2 − λ0 χ+ U2 = 0,

u1 |∂D = 0, u2 |∂D = 0,

(8.4.27) (8.4.28)

where λ0 χ− = −µ, λ0 χ+ > 0. By condition (iii) we have µ ∈ σ (−4). Therefore, U1 =

N X

αi ei ,

αi = Const,

U2 = 0,

i=1

and, hence, N X u1 = 0U = c1− c1+ · u1 = c1− α i ei . u2 c2− c2+ 0 c2− i=1

Lemma 8.10. The operator B does not have B1 -adjoint elements.


153

Proof. Since L1 c− = χ− c− , we obtain an equility 1 0 c1− c1− e = L c e , c e = χ− < B 1 ei , ek > = e, 0 a˜ 1 − i − k c2− i c2− k = χ− (c21− + a˜ c22 )δik ,

i, k = 1, . . . , n.

N 2 Therefore, det|| < B1 ei , ek > ||N ˜ c22− )N 6= 0, because i,k=1 = χ− (c1− + a

χ− 6= 0,

c21i + a˜ c22− ≈ −2α

T2 2 c , T3

and, hence, according to the definition of generalized Jordan sequences [279], the operator B does not have B1 -adjoint elements. Without loss of generality we can assume that the eigenvector c1− of the matrix L1 is chosen so that χ− (c21− + a˜ c22− ) = 1. Then the system of vectors {B1 ei }N i=1 is biorthogonal to {ei }N . Hence, by Schmidt’s Lemma [279] the operator i=1 B˘ = B +

N X

< ·, γi > γi ,

i=1 4

with γi = B1 ei , has a bounded inverse 0 ∈ L (E → E). Thus, 0 = 0∗,

0γi = ei .

(8.4.29)

Remark 8.10. It follows from the proof of Lemma 8.7 that to construct the operator 0 one can use the equation 01 0 −1 −1 0 M , 0 = 0 0 02 where 01 =

Z

G1 (x, s)[·]ds, 02 =

D

Z

G2 (x, s)[·]ds,

D

G1 (x, s)—is the modified Green’s function of the Dirichlet problem (8.4.27), and G2 (x, s)—is the Green’s function of the Dirichlet problem (8.4.28).

8.5 Resolving Branching Equation Let us rewrite equation (8.4.26) in the form of the system X (B˘ − B1 )u = (λ0 + )R(u) + ξi γi ,

(8.5.1)

i

ξi =< u, γi >,

i = 1, . . . , n.

(8.5.2)

154


From (8.5.1), by the inverse operator theorem, we have N

u = (λ0 + )(I − 0B1 )−1 0R(u) +

1 X ξi ei . 1−

(8.5.3)

i=1

Moreover, in virtue of (8.5.2) and (8.4.29) we must have λ0 + ξi + < R(u), ei >= 0, 1− 1−

(8.5.4)

where R(u) = Rl (u) + Rl+1 (u) + · · · . According to the implicit operator theorem equation (8.5.3) has the unique solution u = u1 (ξ e, ) + (λ0 + )(I − 0B1 )−1 0{ul (ξ e, ) + ul+1 (ξ e, ) + . . .}

(8.5.5)

for sufficiently small and |ξ |, where N

u1 (ξ e, ) =

1 X ξ i ei , 1−

ul (ξ e, ) = Rl (u1 (ξ e, )),

i=1

ul+1 (ξ e, ) = Rl+1 (u1 (ξ e, ))+ 0, + 0 0R2 (u1 (ξ e, ))(λ0 + )(I − 0B1 )−1 0u2 (ξ e, ),

l>2 l = 2,

and so on. By substituting the solution (8.5.5) into (8.5.2), we obtain the desired bifurcation system (BEq) ξ + L(ξ, ) = 0, 1−

(8.5.6)

where L = (L1 , . . . , LN ), λ0 + 1 Li = Rl (ξ e), ei + Rl+1 (ξ e), ei + 1− (1 − )l+1    0, l > 2 + λ0 + + ri (ξ, ),  (1−)4 R02 (ξ e)(I − 0B1 )−1 0R2 (ξ e), ei , l = 2  ri = o(|ξ |l+1 ),

i = 1, . . . , n.

If L(ξ, ) = grad U(ξ, ), then the BEq (8.5.6) is said to be potential. In the potential case matrix Lξ (ξ, ) is symmetric. Let fi = fi (ali ϕ + bki ψ), i = 1, . . . , N in (8.1.5). Then by Remark 8.8 the matrices Ru (u) are symmetric for any u.


155

Let us show that the BEq (8.5.6) is potential if the matrix u(u)—is symmetric for any u. Indeed, by (8.5.4) Li (ξ, ) =

λ0 + R(u(ξ e, )), ei , (1 − )

i = 1, . . . , n,

where u(ξ e, ) is defined by the series (8.5.5). Therefore, the vector field L(ξ, ) is potential if and only if the matrix N Ru ∂u(η, ) ej , ei ∂η

4

(η = ξ e)

(8.5.7)

i,j=1

is symmetric. By virtue of (8.5.3) and the inverse operator theorem we have the operator identity −1 ∂u(η, ) 1 −1 = I − (λ0 + )(I − 0B1 ) 0Ru (u(η, )) ∂η 1− in a sufficiently small neighborhood of the point ξ = 0, = 0. Since B1 , 0 and Ru — are self-adjoint operators it follows that

∂u ∂η

∗

= [I − (λ0 + )Ru (u(η, ))]−1

1 . 1−

Therefore, ∂u ∂u ∗ 1 = Ru Ru = [I − (λ0 + )Ru 0(I − B1 0)−1 ]−1 Ru ∂η 1− ∂η by virtue of the operator identity [I − (λ0 + )Ru 0(I − B1 0)−1 ]−1 Ru = Ru [I − (λ0 + )(I − 0B1 )−1 0Ru ]−1 . Hence, the operator Ru

∂u :E→E ∂η

in the matrix (8.5.7) is self-adjoint. Therefore, the matrix (8.5.7) is symmetric and L(ξ, ) = grad U(ξ, ).

156


The foregoing implies the following assertion: Lemma 8.11. Let conditions (8.4.11), (8.4.12), (i)–(iii) be satisfied and λ0 = −µ/χ− . Then the number of solutions of Eq. (8.4.25) such that u → 0 as for λ → λ0 coincides with the number of small solutions ξ → 0 as → 0 of the BEq (8.5.6). If Ai = Ai (ali ϕ + bki ψ), i = 1, . . . , N in system (8.4.2), (8.4.3), (8.4.7), (8.4.13), and a, b—are constants, then the BEq is potential, that is, L(ξ, ) = ∇ξ U(ξ, ), U : RN × R → R, where N X λ0 + Rl (ξ e), ei ξi − u(ξ, ) = − (l + 1)(1 − )l+1 i=1

N X λ0 + Rl+1 (ξ e), ei ξi − − (l + 2)(1 − )l+2 i=1   0, N − (λ0 + ) P 0 −1 R2 (ξ e)(I − 0B1 ) 0R2 (ξ e), ei ξi ,  (1−)4 i=1

+ o(|ξ |

l+1

 l > 2 l = 2

).

+

(8.5.8)

8.6 The Existence Theorem for Bifurcation Points and the Construction of Asymptotic Solutions The BEq (8.5.6) is the desired Lyapunov-Schmidt BEq (see Vainberg and Trenogin [279]) for the bifurcation point of the boundary value problem (8.4.7), (8.4.13). In the sequel we need some properties of the real solutions and the structure of the BEq 4

L(ξ, ) =

ξ + Ll (ξ, ) + o(|ξ |l ) = 0. 1−

(8.6.1)

We state these results from [255, 272] in the form of two lemmas. Lemma 8.12. Let 1. n be odd; or P j 2. l be even and N j=1 |Ll (ξ, 0)| 6= 0 for ξ 6= 0 or 3. L(ξ, ) = ∇W(ξ, ).

Then in every neighborhood of the point ξ = 0, = 0 there exists a pair (ξ ∗ , ∗ ), ξ ∗ 6= 0 satisfying (8.6.1). Proof. If the point ξ = 0 is a nonisolated singular point of the vector field L(ξ, 0), then in every neighborhood of the point ξ = 0, = 0 there exists a pair (ξ ∗ , 0) such that ξ ∗ 6= 0 satisfies (8.6.1), and the lemma is true. Assume ξ = 0—is an isolated point of the vector field L(ξ, 0). Let us consider three cases:


157

1. Let n be odd. Let us take the neighborhood |ξ | ≤ r, || ≤ % and introduce the vector field 8(ξ, t) =

(2t − 1)% ξ + L(ξ, (2t − 1)%). 1 − (2t − 1)%

If 8(ξ, t) 6= 0 for t ∈ [0, 1] and |ξ | = r then the degree of the map 8 Jt = J , S(0, r) ||8|| of the boundary of the sphere |ξ | = r into the unit sphere is well defined [12], and, hence, Jt —is the same integer for each t ∈ [0, 1]. But J0 = (−1)N , J1 = 1N . Hence, Jt 6= Const. Therefore, for all r > 0 and % > 0 there exist t∗ ∈ [0, 1] and ξ ∗ , |ξ ∗ | = r such that 8(ξ ∗ , t∗ ) = 0. The corresponding pair (ξ ∗ , (2t∗ − 1)%) satisfies system (8.6.1). 2. Let l be even and let N X

j

|Ll (ξ, 0)| 6= 0.

j=1

In this case, the field L(ξ, ) is homotopic to Ll (ξ, 0) on the sphere S(0, r) for || < δ with δ sufficiently small. Hence, L(ξ, ) Ll (ξ, 0) J , S(0, r) = J , S(0, r) ||L|| ||Ll || is an even number, because l is even. Let us fix an ∗ ∈ (−δ, δ) and introduce the Kronecker index [163] γ0 of the isolated singular point ξ = 0 of the field L(ξ, ∗ ), γ0 = (sign ∗ )N . By Kronecker theorem [163] X L(ξ, ∗ ) J , S(0, r) = γi . (8.6.2) ||L|| i

Since the left-hand side of equation (8.6.2) is even, and γ0 —is odd, it follows that along with the point ξ = 0 the sphere S(0, r) contains another singular point ξ ∗ 6= 0 of the field L(ξ, ∗ ). The pair (ξ ∗ , ∗ ) satisfies system (8.6.1). 3. Let L(ξ, ) = grad W(ξ, ), where X ξi2 + U(ξ, ), |ξ | < δ, δ > 0. W(ξ, ) = 2(1 − ) i

If the point ξ = 0 is a nonisolated critical point of the potential W(ξ, 0) (of the potential W(ξ, ) for 0 < || < δ) then the lemma is true. Let the point ξ = 0—be an isolated critical point of the potential W(ξ, 0) and of the potential W(ξ, ) for 0 < || < δ. Then the Morse-Conley indices (see Conley [75]) are defined for the critical point ξ = 0 of the potential W(ξ, 0) and of the potential W(ξ, ) for 0 < || < δ, where δ is sufficiently small. By the homotopic invariance property of the Morse-Conley index (see Conley [75], p. 67, Theorem 1.4), these indices are equal. But for 0 < || < δ the point ξ = 0 is a nondegenerate critical point, because 2 ∂ N det w(ξ, ) = 6= 0. ∂ξi ∂ξj (1 − )N =0

158


Therefore, the Morse-Conley index is equal to the number of negative eigenvalues of the corresponding Hessian . 1 − δik i,k=1,...,n But then this index is zero for > 0, and for < 0 it is equal to n. Therefore, the point ξ = 0 cannot be an isolated critical point of the potentials W(ξ, 0) and W(ξ, ) for 0 < || < δ. Hence, ξ = 0 is a bifurcation point of the BEq L(ξ, ) = 0.

Lemma 8.13. 1. Let l—be even, and let the system ξ + Ll (ξ, 0) = 0

(8.6.3)

have a simple real solution ξ 0 6= 0, then in a neighborhood of the point = 0, system (8.6.1) has a real valued solution of the form ξ = (ξ 0 + o(1)) 1/(l−1) .

(8.6.4)

2. Let l—be odd, and let system (8.6.3) or the system −ξ + Ll (ξ, 0) = 0

(8.6.5)

have a simple real solution ξ 0 6= 0, then in the half-neighborhood > 0 ( < 0) there exist two solutions of the form ξ = (±ξ 0 + o(1))||1/(l−1) .

(8.6.6)

3. Let Ll (ξ, 0) = grad U(ξ ) and let ξ 0 —be an isolated extremum of the function U(ξ ) on the sphere |ξ | = 1, U(ξ 0 ) 6= 0, then there exists a solution of the form ξ = (c + o(1))||1/(l−1) ,

(8.6.7)

where 1/(l−1) sign ξ 0, 0) (l + 1)U(ξ c= 1/(l−1)  1  ± ξ 0, (l + 1)U(ξ 0 )    

(8.6.8) U(ξ 0 ) > 0.

Proof. 1), 2). We seek the solutions of equation (8.6.3) in the form ξ = η() 1/(l−1) . To define η(0) we obtain two systems: one for > 0 and the other for < 0, that, are systems (8.6.3), (8.6.5). If l—is even then the substitution ξ = −ξ transforms equation (8.6.3) into equation (8.6.5). If l—is odd this substitution does not change equations. Therefore, in the case of simple real solutions ξ 0 the existence of solutions of the form (8.6.4), (8.6.6) follows from the implicit function theorem.


159

3). Let Ll (ξ, 0) = grad U(ξ ). Then we seek the solutions of (8.6.7) in the form ξ=

2q

1/(l−1)

,

where |η| = 1 and q()—is a scalar parameter satisfying the condition q(0) > 0 for odd l. For q and η we obtain the system 2qη + Ll (η, 0) + 2(η, q, ) = 0,

|η| = 1,

where ||2|| = o(1) as → 0. Therefore, 2q(0)η(0) + Ll (η(0), 0) = 0 and |η(0)| = 1. Since, by assumption, ξ 0 —is an isolated extremum of the function U(ξ ) on the sphere |ξ | = 1 and U(ξ 0 ) 6= 0, we set q(0) = (l + 1)U(ξ 0 ),

η(0) = ξ 0 .

Consider the perturbed vector field ∂U 8 (η, q) = 2qηi + + 2i , ∂ηi

|η| = 1 .

Let S—be the sphere of radius % > 0 centered at the point (q(0), η0 ) in Rn+1 . Let us introduce the degree of the map (see Rothe [258]): 8 J ,S = ||8 || +1, if (q(0), ξ 0 )—is an arg of the min q(0)|ξ |2 + U(ξ, 0), = n+1 (−1) , if (q(0), ξ 0 )—is an arg of the max q(0)|ξ |2 + U(ξ, 0). Since this degree is nonzero, it follows that the vector field 8 (η, q) = 0 has a singular point in a neighborhood of the point (q(0), ξ 0 ) for || < δ with δ sufficiently small. With the help of Lemmas 8.11, 8.12, and 8.13, it is now possible to prove the following results on the bifurcation point for problems (8.4.7), (8.4.13). Theorem 8.4. Let conditions (8.4.11), (8.4.12), (i)–(iii) with λ0 = −µ/χ− , as well as one of the following three conditions, be satisfied: 1. n is odd; P 2. l is even and N i=1 | < Rl (ξ e), ei > | 6= 0 for ξ 6= 0; 3. fi = fi (a(−αi v2 + ϕi ) + b(vdi + ψi )), i = 1, . . . , N, and a, b—are constants.

Then λ0 —is a bifurcation point of the boundary value problem (8.4.7), (8.4.13). Proof. By assumptions (1)–(3) of the theorem the assumptions of Lemmas 8.11 and 8.12 are satisfied for the BEq (8.5.6) of the boundary value problem (8.4.7), (8.4.13). Equation (8.5.5) establishes a one-to-one correspondence between the desired solutions of the boundary value problem and small solutions of the BEq (8.5.6). Therefore, the validity of Theorem 8.4 follows from these lemmas.

160


Corollary 8.5. Let the potentials of the electromagnetic field satisfy conditions (8.1.3), and let the assumptions of Theorem 8.4 be satisfied. Then λ0 —is a bifurcation point of the VM system (8.1.1)–(8.1.4). Example 8.1. Let the distribution functions of the VM system have the form [248] fi = λexp(−αi v2 + (di , v) + γi + li ϕ(r) + ki ψ(r)), and N X

−3/2 qi αi exp

i=1

di2 = 0, γi + 4αi

N X

−3/2 qi αi exp

i=1

di2 ki γi + = 0. 4αi li

Then conditions (8.4.11), (8.4.12) for βi = di /(2αi ) and assumptions (1)–(3) of Theorem 8.4 are satisfied. Thus, the BEq (8.5.6) is potential. If µ—is an eigenvalue of the Dirichlet problem (iii) then, by Corollary 8.5, λ0 = −µ/χ− is a bifurcation point of the VM system with conditions (8.1.3), (8.1.4), where u10 = u20 = 0. Thus, 4 X 4 X 4 X T1 = a i li , T 2 = ai ki , T 3 = a i li , 2 di2 π 3/2 4 4 X ki exp γi + , T4 = ai , ai = qi li αi 4αi q 1 2 νT4 − µT1 − (νT4 − µT1 ) + 4νµ[T1 T4 − T2 T3 ] . χ− = 2 Theorem 8.5. Let conditions (8.4.11), (8.4.12), (i)–(iii) be satisfied with λ0 = −µ/χ− as well as the conditions of one of the three statements of Lemma 8.13. If 1) is satisfied then the boundary value problem (8.4.7), (8.4.13) has the solution ! N X 0 u= ξi ei + o(1) (λ − λ0 )1/(l−1) ; i=1

2) is satisfied then there exist two solutions ! N X 0 u± = ± ξi ei + o(1) |λ − λ0 |1/(l−1) , i=1

defined in the half-neighborhood λ > λ0 (λ < λ0 ) provided that ξ 0 satisfies system (8.6.3) (ξ 0 satisfies system (8.6.5)); 3) is satisfied then ! N X u= ci ei + o(1) |λ − λ0 |1/(l−1) , (8.6.9) i=1

where the vector c is defined by equation (8.6.8).

The proof follows from Lemmas 8.11, 8.13, and equation (8.5.5).


161

Corollary 8.6. Let the potentials of the electromagnetic field satisfy conditions (8.1.3), and let the assumptions of Theorem 8.5 be satisfied. Then the VM system (8.1.1)– (8.1.2) with conditions (8.1.3), (8.1.4) has the solution (8.4.4), where ϕ=−

2αq u10 + u1 (r, λ), m

ψ=

q u02 + u2 (r, λ), mc

(8.6.10)

the functions u1 , u2 → 0 are defined by Theorem 8.5 as λ → λ0 . Let us consider more detailed distribution functions of the form fi = fi (a(−αi v2 + c1i + li ϕ) + b(vdi + c2i + ki ψ)),

(8.6.11)

where li , ki are connected by the linear relations (8.4.1), the integrals Z fi dv = Ai (ali ϕ + bki ψ) R3

converge, and ∂Ai (s)/∂s < 0 for all s. In this case the conditions (8.4.11), (8.4.12), (i), (ii) and the assumptions of Theorem 8.5 are satisfied by Lemmas 8.12, 8.13; hence, according to Lemma 8.12 the BEq (8.5.6) is potential. In Theorem 8.5, case (3) occurs. Therefore, the form of the functions u1 (r, λ), u2 (r, λ) in (8.6.10) can be specified; namely, in the case of the distribution (8.6.11) the vector u = (u1 , u2 ) in equation (8.6.10) can be given by equations (8.6.9), (8.6.8). Thus, if it will be found that vector c in equation (8.6.8) corresponds to a nonisolated extremum of the corresponding potential, then some of its coordinates may be arbitrary points of some sphere S ⊂ Rk , where k ≤ n (see [255, 272]). Then problem equation (8.4.25) will have a solution depending on free parameters. This case is possible if the domain D— is symmetric and problem equation (8.4.25) has a spherical symmetry. Thus, the free parameters remaining in the solution have a group meaning. Let us show that this is just the situation that arises in our problem in the case of a circular cylinder. Let us introduce the condition (iv) D = {x ∈ R2 |x12 + x22 = 1}, and the matrix R0 (u)—is symmetric for any u. Let us pass to the polar coordinates x1 ρ cos 2, x2 = ρ sin 2 in system (8.4.25). Then 4=

∂2 1 ∂ 1 ∂2 + + , ∂ρ 2 ρ ∂ρ ρ 2 ∂22

u|ρ=1 = 0.

Condition (iii) now makes sense: µ ∈ {µ(s) σ , 2

s = 0, 1, . . . ,

σ = 1, 2, . . .}, (s )2

(s)

where µσ —are the zeros of the Bessel function Js (µ). If µ = µσ00 , s0 ≥ 1 then dim N(B) = 2; the vectors 0) c− Js0 (µ(s σ0 ρ) cos s0 2,

0) c− Js0 (µ(s σ0 ρ) sin s0 2

162


form a basis in the subspace N(B). By Lemma 8.11, the BEq (8.5.6) is potential on the whole. Moreover, the BEq (8.5.6) admits the group O(2) and by [179], Theorem 1, has the form ξi |ξ |−1 L(|ξ |, ) = 0,

(8.6.12)

where L(|ξ |, ) =

∞ X

L2i+1 ()|ξ |2i+1 ,

l1 () =

i=0

1−

is analytical. Let us note that in this case the forms (8.6.10), which are even in ξ , must be lacking on the left-hand side of the BEq (8.5.6), since L(|ξ |, ) is odd in ξ . Remark 8.11. In view of (8.6.12) the potential U of the BEq in (8.5.8) has the form Z|ξ | U = − L(s, )ds + 0

1 ξ12 + ξ22 . 1− 2

Therefore, Z1 1 . U |ξ |=1 = − L(s, )ds + 2(1 − ) 0

Let L2i+1 () ≡ 0,

i = 1, 2, . . . , m − 1,

l2m+1 () 6= 0

in (8.6.12). Assuming that ξ1 = r cos α, ξ2 = r sin α, let us reduce the system (8.6.12) to the single equation + L2m+1 ()r2m + O(r2m+2 ) = 0. 1−

(8.6.13)

Note that L2m+1 (0) = λ0 < R2m+1 (ξ e), ej > ξj−1 |ξ |−2m ,

j = 1, 2,

for all ξ if R2 u = · · · = R2m (u) = 0. Remark 8.12. If R2 (u) 6= 0, then for all ξ we have L3 (0) = λ0 ξj−1 |ξ |−2 < R3 (ξ e) + R02 (ξ e)0R2 (ξ e), ej >,

j = 1, 2.


163

From equation (8.6.13) we find two solutions s − 2m + O(||1/2m ), r1,2 = ± L2m+1 (0) which are real for L2m+1 (0) < 0. The two solutions s ξ1 − 1/2 cos α = ± 2m + O(|| ) ξ2 sin α L2m+1 (0)

(8.6.14)

of the BEq correspond to these solutions, where the parameter α ∈ R corresponding to the group O(2) remains arbitrary. Substituting (8.6.14) into (8.5.5) and the vector (8.5.5) into (8.6.10), we obtain two solutions   2αq ± u01  ϕ − =  qm ± ± ψ u02 mc s λ − λ0 −T2 /T1 2m (s0 ) + 0(|λ − λ0 |1/(2m) ). ± js0 (µσ0 ρ) cos s0 (2 − α) − 1 L2m+1 (0) For (λ − λ0 )L2m+1 (0) < 0, to these solutions there correspond two real solutions fi± , E± , B± of the stationary VM system (8.1.1)–(8.1.2) with boundary conditions (8.1.3), (8.1.4) determined by equations (8.4.4). In conclusion, we note that instead of condition (8.4.11) it sufficies to impose the following condition: (v) the potentials U, A of the electromagnetic field (E, B) satisfy conditions (8.1.3), and, hence, N X

qi Ai (li ϕ 0 , ki ψ 0 ) = 0,

i=1

N X

qi (βi , d)Ai (li ϕ 0 , ki ψ 0 ) = 0,

i=1

where ϕ 0 , ψ 0 —are harmonic functions with the boundary conditions ϕ 0 |∂D = −

2αq u01 (r), m

ψ 0 |∂D =

q u02 (r). mc

Moreover, if distribution functions has the form fî = fi (a(−αv2 + ϕi ) + b(vdi + ψi )), then results are obtained by analogy and in case of inconstant values u01 , u02 . The procedure presented can be used for constructing the nontrivial solutions of an integro-differential system (8.4.10). Therefore, analogous results occur, also, in the problem about the point of bifurcation of VM system with normalized distribution functions.

9 Boltzmann Equation 9.1 Collision Integral Describing the difference between Vlasov and Boltzmann equations, one should remember that Vlasov’s equation describes long-range action, while Boltzmann’s describes the short-range action, i.e., how the particle collision processes behave. Vlasov equation describes a shift of distribution function along trajectories of dynamic system, while trajectories also depend from the distribution function. The Boltzmann equation takes into account the number of collisions in a cell of the typical size. This is an evolutionary equation for distribution function f (t, x, v) of the particles over velocities v and space coordinates x describing in-flow and out-flow of colliding particles ∂f ∂f + v, = I[ f , f ]. (9.1.1) ∂t ∂x Assuming the right part of the equation (9.1.1) to be zero, one obtains the equation of free motion. The function I[ f , f ] describes collisions and is called collision integral or collision operator. The word operator is used to outline the fact that we establish the correspondence between functions f and I[ f , f ]. The last represents quadratic operator, which acts as distribution function via variable v only, since x and t are assumed to be parameters [52, 53, 64, 67]: Z (u, n) 0 0 I[g1 , g2 ](v) = g1 (v )g2 (w ) − g1 (v)g2 (w) × B |u|, dndw. |u| Here n ∈ S2 —the unit sphere vector, indicating a direction of relative particle velocity after a collision u = v − w—relative velocity; v0 and w0 —particles velocities after collisions: v0 =

v + w |u|n + , 2 2

w0 =

v + w |u|n − . 2 2

(9.1.2)

In order to represent or describe the process of collisions, the diagram of collisions u+w is drawn. First, we draw the sphere of radius R = |u| 2 with center coordinates 2 . (see Figure 9.1). For the given precollision velocities v and w, all possible opposite points of this sphere v0 and w0 are the particle velocities after collisions. We note that relation Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00009-0 c 2011 Elsevier Inc. All rights reserved.

166


v′

w

v

w′

Figure 9.1 Collision diagramm.

v + w = v0 + w0 as conservation of impulse and v2 + w2 = v0 2 + w0 2 as conservation of energy are the basis for deriving (9.1.2). The similar expression will be obtained below for the particles of different masses. Function B(|u|, µ) is a product relative velocity module |u| and differential scattering cross-section σ (|u|, µ): B(|u|, µ) = |u|σ (|u|, µ). For power potentials U(r) = r1α , one has B = |u|1−4/α gα (µ). Potential U(r) = r−4 , α = 4 is independent from |u| function B and is called a Maxwellian molecular potential. Under this assumption the expression for collision integral is simplified and Maxwell was able to derive the equations on macroscopic characteristic variables: density, velocity, and temperature. This successful result was quite unexpected, so he tried to prove that all natural forces act as minus fifth degree of distance. This fact was not justified later in experiments: simplest solutions do not always describe the real natural phenomena behavior. Here n = 5 defines a relation between force and distance and corresponds to α = 4 for the potential.

9.2 Conservation Laws and H- Theorem R Let us study the evolution of integral characteristics 8ϕ = ϕ (v, f (t, v, x)) dvdx for the Boltzmann equation, regarding some function ϕ(v, f ). Using (9.1.1), one obtains Z Z d8ϕ ∂f ∂f = ϕf0 dvdx = − v, ϕ 0 dvdx + dt ∂t ∂x f Z + ϕf0 I[ f , f ]dvdx = S1 + S2 . (9.2.1) Denote ϕf0 = ψ. Now we can transform the second term Z S2 =

(u, n) ψ(v)[ f (w )f (v ) − f (w)f (v)]B |u|, dwdvdndx. |u| 0

0

Boltzmann Equation

167

We transform the above expression identically, exchanging v ↔ w and obtaining symmetric expression on S2 , adding two equal terms with coefficient 1/2. Next, we exchange (v, w) ↔ v0 , w0 . Therefore, the element dwdvdn becomes dw0 dv0 dn0 . Taking into account that v0 and w0 are linearly dependent on v and w, 0 v v = A , w0 w due to (9.1.2), one finds the Jacobian J(A) of transformation A equal to the determinant A: J(A) = det A, dw0 dv0 = | det A|dwdv. Here A2 = I, hence (det A)2 = 1, and, therefore, dw0 dv0 = dwdv. Thus, we obtain two more terms and coefficient 1/4: Z 1 ψ(v) + ψ(w) − ψ(v0 ) − ψ(w0 ) × f (w0 )f (v0 ) − f (w)f (v) × S2 = 4 (u, n) dwdvdndx. (9.2.2) × B |u|, |u| Example 9.1. Calculate det A. Hint: Writedown matrix A in the form E−p p A= , p E−p with p = pu = (u, m)m—one-dimensional projector onto vector m: v0 = v + (u, m)m, w0 = w − (u, m)m, u = v − w. This expression is useful as an additional form, representing collisions (9.1.2). Answer: det A = −1. Using formulae (9.2.2), we can introduce some corollaries.

9.2.1 Conservation Laws If we assume ψ(v) + ψ(w) = ψ(v0 ) + ψ(w0 ), then S2 ≡ 0. The given equation posesses five-dimensional solution space over continuous functions ψ(v) = φ05 (v) = av2 +

3 X i=1

bi vi + c

168


(see [52, 64, 67] for details). Here a, bi (i = 1, 2, 3), c are arbitrary constants. These types of solutions are called summator invariants or additive R invariants of collisions. The solution basisRcorresponds to the number of particles f dvdx, three components R of velocity vector vi f (v, x)dvdx and kinetic energy v2 f (v, x)dvdx. The expression φ05 (v) denotes 5 as a summator invariants space dimension, and 0 means conservation.

9.2.2 Boltzmann H- Theorem Assuming ψ(v) = ln f , we obtain S2 =

1 4

Z ln

f (v)f (w) [ f (v0 )f (w0 ) − f (v)f (w)]Bdvdwdndx. f (v0 )f (w0 )

In this integral function, B is nonnegative and strongly positive almost everywhere. Since the first two factors have different signs, one obtains S2 ≤ 0, that corresponds to the first part of H- theorem [52]. The second part consists of condition, providing equility S2 = 0. Observing nonnegativity of integrand function and using inequility B > 0 strongly almost everywhere, one obtains an equality condition f (v0 )f (w0 ) = f (v)f (w). Taking the logarithm of this equality, we get ln f (v) as a summator invariant: ln f (v) = φ05 (v). Hence, 2 f = f0 = Ae−β(v−v0 ) . Using the physical sense, we treat A > 0, because a distribution function is a nonnegative one; we also take β > 0, since negative β means an increasing of distribution function in infinity and gives divergent expressions for local velocity, density 1 of particles and temperature; β = 2kT is proportional to inverse temperature, k—is a Boltzmann constant, v0 —mean velocity. Parameter A is convenient to write in the 3 form A = ρ[2π kT]− 2 , treating quantity ρ as density. All these parameter quantities ρ, T and v0 could be coordinate x and time t dependent. Such distribution is called local-Maxwellian. R H- theorem means that the functional H = f ln f dvdx decreases in time due to collision: S2 ≤ 0. And S2 = 0 if the distribution function is local Maxwellian.

9.2.3 Beams Generally speaking, the local-Maxwellian distribution is not an exact solution of Boltzmann equation: free motion “spoils” it. Boltzmann carefully investigated such cases of exact solutions [52, 67]. We give one special example for T = 0, producing beams. Please note that, while temperature T tends to zero and density ρ(x, t) kept unchanged, the local-Maxwellian distribution transforms into f (t, x, v) = ρ(x, t)δ(v − v0 (x, t)). We are already studied such distributions (see Chapter 2, Section 2.5, called hydrodynamic substitution). They describe the beams of particles. Now we see that collision integral vanishes for such distributions and also gives exact solutions of Boltzmann

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equation if ρ and v0 satisfy the system of equations of free motion (see Sections 2.5.1, 2.5.2 with K = 0). If the number of beams is greater than 1 (see N-layer hydrodynamics section of Chapter 2), then collision integral do not vanish.

9.2.4 Validity Conditions for Conservation Laws and H- Theorem Let us consider now the first term S1 in equality (9.2.1). Transforming it to the integral from divergence operator Z ∂ (vi ϕ)dvdx, S1 = − ∂xi we found it trivial in many cases. Let us consider three of them:

Infinite space Here we transform the above expression integrating every term over its derivative. We obtain Z S1 = − v1 [φ(x1 = +∞, x2 , x3 ) − φ(x1 = −∞, x2 , x3 )]dx2 dx3 dv + + two similar terms. Hence, if it is seen that f (t, v, x) tends to the same function g(t, v), independent of variable x, then S1 = 0. Vise versa, if f tends to different functions, then there appears to be the flows of the corresponding values. Thus, both inequality of H- theorem and conservation laws could be violated. A well known typical example is the shock wave, when the distribution function tends to different Maxwellian distribution at x → ±∞.

Periodical problem Let f (t, v, x + (n, T)) = f (t, v, x) for some linear independent vector set T = (T1 , T2 , T3 ) and for all vectors n = (n1 , n2 , n3 ) with integer components. Then, integrating over periods, one obtains conservation laws and H- theorem on the basis of equality S1 = 0. Therefore, the conservation laws and H- theorem are fulfilled in periodical problem.

Boundary-value problem If x ∈ D, where D is domain with a smooth boundary ∂D, then, integrating over that domain and using the Ostrogradsky formulas, we find Z Z ∂ S1 = − (vi , ϕ) dvdx = − (v, n) ϕdvdS. ∂xi D

∂D

Hence, the equality to zero of the flows through the boundary of corresponding values is necessary for global conservation laws.

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The most popular boundary conditions [67] are Maxwell conditions: 1. specular reflection: f (x, v, t) = f (x, v − 2n(n, v), t), x ∈ ∂D, (v, n) > 0, where n—vector of normal to surface in the point x; 2. diffuse reflection: when we obtained locally-Maxwellian distribution under reflection.

One may find details in [64, 67].

9.3 Boltzmann Equation for Mixtures Consider mixture of two gases with masses of molecules M and m. Let F(t, p, x) and f (t, q, x) distribution functions of molecules of heavy and light kinds, respectively, defined by impulses ( p ∈ Rd , q ∈ Rd , d = 1, 2, 3), space x ∈ Rd in moment t. Because of impulse conservation for particle collisions, it is convenient to consider impulses instead velocities. The Boltzmann equation is reduced to the following system: ∂f q ∂f + , = I[F, f ] + I[ f , f ], ∂t m ∂x ∂F p ∂F + , = I[ f , F] + I[F, F], ∂t M ∂x

(9.3.1)

where I[F, f ]—has the form Z I[F, f ] =

(un) dpdn. [F(p )f (q ) − F(p)f (q)] |u|σ |u| , |u| 0

0

Here u = (p/M) − (q/m)—relative velocity, p0 , q0 —impulses after collisions: p+q 1 (1 + δ) + |u| µn, 2 2 p+q 1 q0 = (1 − δ) − |u| µn, 2 2 p0 =

(9.3.2)

where δ = (M − m)/(M + m), µ = (2Mm)/(M + m)—mean harmonic of masses (2/µ = 1/M + 1/m); n—vector of unit sphere: |n| = 1, n ∈ Rd . If we define p0 and q0 in this way, they satisfy conservation law for impulse 0 p + q0 = p + q and energy (p2 /2M) + (q2 /2m) = ((p0 )2 /2M) + ((q0 )2 /2m). Let us show that parametrization (9.3.2) is obtained from conservation laws. Putting p + q = Q and expressing impulses of heavy particles p0 = Q − q0 and p = Q − q from conservation law of impulse, we substitute them into conservation law of energy. After

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171

p′ q O′

A

O p

q′

Figure 9.2 Collision diagramm.

transformations, we have Q 2 Q 2 q0 − = q− , αM αM where α = (M + m)/(mM) = 2/µ. For q0 , this is a circle equation with a center in Q/(αM) = ((p + q)/2)(1 − δ) and radius R2 = (q − Q/(αM))2 = |µ|2 |u|2 /4. We obtain a similar expression for heavy particles. The collision scheme shown in Figure 9.2 explains that impulses of the light and heavy particles define the spheres of equal radius (|µ| |u|)/2 with their centers to be symmetrical with respect to (p + q)/2. Here p and q—impulses of particles before collision, p0 , q0 —after; A—mean of interval p q and p0 q0 ; O—center of the sphere for impulses p0 of heavy particles with coordinate ((p + q)/2)(1 + δ); O0 —center of sphere of impulses q0 of light particles with coordinate ((p + q)/2)(1 − δ). Also the invariants and H- theorem with derivation of Maxwellian distribution for both components are considered. For these, the temperatures are found the same, and density—vary. Passing to the limit T → 0, we obtain f (q, x, t) = ρ(x, t)δ(q − mV(x, t)), F(q, x, t) = η(x, t)δ(q − MV(x, t)). Therefore, if functions ρ, η, V satisfy the system of equations: ∂ ∂ρ + (Vi ρ) = 0, ∂t ∂xi ∂η ∂ + (Vi η) = 0, ∂t ∂xi ∂Vi ∂ + (Vi Vk ) = 0, ∂t ∂xk then they are exact solutions of system (9.3.1) and the system of free movement before overlapping.

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9.3.1 Chapman-Enskog Method and Hydrodynamics in Twice Divergent Form To be able to use hydrodynamic methods, we consider a Boltzmann equation with a large parameter multiplier for collision integral: ∂f 1 ∂f = J(f , f ). + v, ∂t ∂x ε We assume f = f0 + εf1 + . . . Hence, one obtains J(f0 , f0 ) = 0 and f0 = e(ϕ,α) is locally Maxwellian distribution, ϕ is vector of five invariants, α(x, t) are parameters, through which the density, mean velocity, and energy are expressed for (explicit formulas see Section 9.5). Thus, we put ϕ0 = 1,

ϕ 1 = v1 ,

ϕ 2 = v2 ,

ϕ 3 = v3 ,

ϕ 4 = v2 .

9.3.2 Hilbert Method As the first approximation, we find ∂f0 ∂f0 + v, = L( f0 , f1 ), where L(f0 , f1 ) = J(f0 , f1 ) + J(f1 , f0 ). ∂t ∂x To solve an obtained equation with respect to f1 , an orthogonality condition for all invariants of L is required. It gives us equations for parameters α leading to Euler equations. We may write them down in the following twice divergent form [116, 118]: ∂Lαµ E αµ = 0, + divM ∂t

µ = 0, . . . , 4,

(9.3.3)

3 R P where L(α) = exp(α4 v2 + αi vi + α0 )dv, i=1

E M(α) =

Z

Ev exp(α4 v2 +

3 X

αi vi + α0 )dv.

i=1

Here α4 < 0, Lαµ —partial derivative in αµ . We see that equations are written via E functionals L(α) and M(α). This is twice divergent (Gogunov’s) form. Here and later µ = 0, 1, . . . , 4; i = 1, 2, 3. This form is very important, due to hyperbolicity of equations for the first approximation. Also one can consider it a basis for nonequilibrium Gibbs method (see details in sections 9.4, 9.5).

9.3.3 Chapman-Enskog Method ∂ ∂ ∂ = +ε , ∂t ∂t0 ∂t1 i.e., two time variables are introduced—for fast and slow times.

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We will not discuss these methods, because all corresponding material exists in many textbooks.

9.4 Quantum Kinetic Equations (Uehling-Uhlenbeck Equations) We may write quantum kinetic equations in the form similar to Boltzmann’s (1.1.1): Z ∂f ∂f + v, = BT( f )[h(v0 )h(w0 ) − h(v)h(w)]dw, ∂t ∂x where as before B = |u| σ (|u| , (u, n)/ |u|); h(v) = f (v)/(1 + θf (v)), θ = 1 for bozons, θ = −1 for fermions, θ = 0 for Boltzmann equation (1.1.1); T( f ) = (1 + θf (v))(1 + θf (w))(1 + θf (w0 ))(1 + θf (v0 )) = =

f (v)f (w)f (w0 )f (v0 ) ; h(v)h(w)h(w0 )h(v0 )

H- function is given by formulas Z H( f ) = [ f ln f − (1 + θf ) ln(1 + θ f )], with steady-state Boze and Fermi distributions defined from the condition f0 (v)/(1 + θ f0 (v)) = e(ϕ,α) , i.e., f0 (v) = e(ϕ,α) /(1 − θ e(ϕ,α) ). One can also write the corresponding hydrodynamics equations in twice divergent form. Also, more generally, when the stationary state of collision integral is an arbitrary function g of (ϕ, α), i.e., X f0 = g ϕµ (v)αµ , (9.4.1) we have the corresponding Euler equations in the form (9.3.3) for Z X L(α) = h ϕµ (v)αµ dv,

(9.4.2)

where h(λ) is antiderivative of g: h0 (λ) = g(λ). For Uehling-Ulenbek equations: Z P 1 ln 1 − θe ϕi (v)αi dv. L(α) = − θ We see the twice divergent form of notation it naturally follows from kinetic equation both in classical and quantum case.

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9.5 Peculiarity of Hydrodynamic Equations, Obtained from Kinetic Equations In the case of equilibrium functions of general form (9.4.1), we have equations of a Euler type: Z Z ∂ ∂ αµ f0 dv + vi αµ f0 dv = 0. (9.5.1) ∂t ∂xi We run summation by i = 1, 2, 3. One can rewrite these equations using two functions, instead of four in (9.3.3): ∂Lαµ ∂Rαµ αi + = 0, ∂t ∂xi

µ = 0, . . . , 4.

(9.5.2)

Here Rαµ αi —the second partial derivatives in αµ and αi , L(α) is defined by formulas (9.4.2), and Z X R(α) = q ϕµ (v)αµ dv, where q is antiderivative of function h(x) from (9.4.2): q0 = h. Such form of equations appears from ϕ1 = v1 , ϕ2 = v2 , ϕ3 = v3 , i.e., three components of velocity are invariants for any of the considered collision integrals. In the case of Boltzmann equation, R = L and equations (9.5.2) are written in terms of unique function L. Thus, variables α are expressed via ordinary macroscopic values (ρ, u, T) in explicit form. It is obtained by means of comparison of two expressions for Maxwellian distribution: X ρ (v − u)2 2 f0 (v) = exp − = exp α + α v + α v = 0 i i 4 2kT (2πkT)3/2 " # α2 α 2 = exp α0 − exp α4 v + . 4α4 2α4 Here α = (α1 , α2 , α3 ), Thus, one obtains α4 = −

1 , 2kT

αi =

α 2 = α12 + α22 + α32 . ui , kT

α0 = ln ρ −

3 u2 ln(2π kT) − . 2 2kT

These variables are typical in kinetic theory, and we can use them in gas dynamics as well. They were applied for calculation of specific problems of gas dynamics by M. A. Rydalevskaya [249]. In addition, the nonsingularity of that change was investigated. Such variables were used in paper [9] for reconstruction of formulas for entropy over minimizing distribution of the form (9.4.1). If we seek stationary state from condition of minimum of H- function (or maximum of entropy) under conservation condition for five invariants, then these variables become Lagrange multipliers.

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175

9.6 Linear Boltzmann Equation and Markovian Processes The following equation Z ∂f F(w0 )f (v0 ) − f (v)F(w) × = J(F, f ) = ∂t (u, n) × B |u| , dwdn |u|

(9.6.1)

is called a linear Boltzmann equation. It describes the situation when the component f has comparatively small density and one can neglect collisions of particles of that component in comparison with collisions with particles of component F: the particles f scatter on particles F. Here F(v) is a given function of variable v. If we consider space-nonuniform situation, then we have equation ∂f ∂f + v, = J(F, f ). ∂t ∂x Such equations are common in transport theory of neutrons in nuclear reactors and also in the problems of radiation described transport [188, 302]. In contrast to Boltzmann equation, (9.6.1) has only one linear invariant for positive function F(w)—a number of particles, namely d dt

Z

ϕ(v)f (v)dv =

Z

[ϕ(v0 ) − ϕ(v)]F(w)f (v)Bdwdvdn.

This expression is equal to zero when ϕ(v) is constant. In order to explain the behavior of the solution of equation (9.6.1), we rewrite it in the form ∂f = ∂t

Z

K(w, v)f (w)dw − a(v)f (v, t) = L( f ).

(9.6.2)

Such presentation is accepted in the theory of reactors and the problems of radiation transport. Kernel K(w, v) and coefficient a(v) can be explicitely written in terms of R functions F and B, but all we need is an equality a(v) = K(v, w)dw, which represents conservation law for numbers of the particles. One can also rewrite equation (9.6.2) in the form typical for Markovian processes or for chemical kinetics [103, 217]: ∂f = ∂t

Z

[K(w, v)f (w) − K(v, w)f (v, t)]dw.

(9.6.3)

Here K(w, v) represents the probability of transition from the state w in v (Markovian process jump).

176


If ξ(v) is a positive stationary state of equation (9.6.2) (i.e., L(ξ ) = 0), then one can prove the following form of H- theorem (see, for example [217]). R Consider function H( f ) = f ln[ f (v)/ξ(v)]dv. Due to (9.6.3), derivative (dH/dt) ≤ 0 for any positive nuclear K(w, v). Theorem is proven, applying inequality u ln u − u + 1 ≥ 0, where equality is reached for u = 1. Writing expression for dH dt and exchanging v and w in second term, one obtains f (v)ξ(w) K(w, v)f (w) ln dvdw = ξ(v)f (w) Z 1 = − K(w, v)f (w) [u ln u − u + 1]dvdw ≤ 0. u

dH = dt

Z

(9.6.4)

Here u = (f (w)ξ(v))/(ξ(w)f (v)) and Z

1 K(w, v)f (w) [1 − u]dwdv = 0, u

that follows from stationary state of ξ(v). Starting from inequality (9.6.4), it follows that solution of equation (9.6.3) tends to stationary state, which is proportional to ξ(v) with the same integral for the number of particles. From uniqueness of stationary solution follows the uniqueness for the number of particles as a linear integral for positive K(w, v). (w) f (v) Equality in (9.6.4) is reached for u = 1, i.e., fξ(v) = ξ(v) for all points (w, v), where K(w, v) > 0. The question on behavior of the solution for equation (9.6.3), hence, is reduced to existence of positive stationary solution ξ(v). Sometimes ξ(v) can be written explicitly. Thus, if Maxwellian distribution F0 is taken as F in (9.6.1), then ξ(v) is proportional to F0 . Sometimes, digitization of (9.6.3) X ∂fm X Kjm fj − Kmj fm = Ajm fj , = ∂t j

m = 1, . . . , n,

(9.6.5)

j

is said to be basic (or master) Pauli equation. It is studied in detail in the theory of Markovian processes [103]. Here matrix Ajm is obtained from matrix Kjm , representing transition probability from the state j into the state m. Let Kjm be positive for j 6= m and Kjj = 0. Then Ajm is obtained with positive off-diagonal elements, negative diagonal, and a sum of the P elements in every column equals to zero: Ajm = 0. Such form of matrix A automatm P fi = 0 ically preserves nonnegativity of fm (t) and keeps the conservation law dtd for the total number of particles fulfilled. An opposite is also correct. Any linear system with matrix complying these two proporties (off-diagonal positiveness and equality to zero of a sum of the elements of every column) allows representation via some matrix Kmj of transition

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probabilities: Amj = Kmj ,

m 6= j,

Ajj = −

X

Kjm .

m

For system (9.6.5), H- theorem can be proved with the following H- function H=

X i

fi ln

fi , ξi

where ξ —stationary solution. Such solution always exists, because determinant of matrix A is null because of conservation law (a sum of all rows equals to zero). It is important to find stationary solutions ξ of equation (9.6.5) with strongly positive components ζi to comply with H- theorem. Here we show when it is possible to obtain the stationary solutions in explicit form. 1. Symmetry of matrix Kjm ; more general, an equality to zero of the sum of the elements in every line of matrix Ajm give stationary vector ξ with equal components. 2. The case of detail equilibrium, when there exist such numbers rj that Kmj rm = Kjm rj . Then stationary vector is proportional to vector r.

Exercise 9.1. Show that all matrices 2 × 2 satisfy condition 2), but matrices 3 × 3— only for K12 K23 K31 = K13 K32 K21 . The dimension of matrices Kn×n is n(n − 1), and matrices K with condition 2)—n(n − 1)/2 + n − 1. Going back to kinetic equations, many satisfy both properties: positivity and conservation law for the number of particles, for example, all shift equations from 3.2 (Liouville equation: see part 9.7). It is evident that it does not guarantee a convergence to the stationary solutions. We could try to realize the situation making a discretization of this Liouville equation. Then we get our system (9.6.5) with sparse matrix Amj. The behavior of this discretization and of original Liouville equations are different. Nevertheless, the discretization idea originated in Markov’s processes theory is rather productive. In this way we keep the positive properties of solutions unchanged, and a number of particles is unchanged. Next section will be devoted to dissipative properties of Liouville equation and clarify such contradictions.

9.7 Time Averages and Boltzmann Extremals The maximum entropy principle under linear conservation laws gives Boltzmann extremals [51]. The stochastic ergodic theorem [238] asserts the existence of time averages or Cesa´ ro means. We prove the coincidence of time averages with Boltzmann extremals.

9.7.1 Boltzmann Extremals In [51], Boltzmann proves his H- theorem, which asserted the convergence of solutions of Boltzmann-type equations to a Maxwellian distribution. In [51] Chapter 2, the

178


maximum entropy principle was stated. Boltzmann considers the system of equations: du1 = B(u22 − u1 u3 ) dt √ du2 2 = B(u1 u3 − u22 ) dt √ du3 = B(u22 − u1 u3 ). 2 dt Boltzmann shoed that the functional √ √ E(u) = u1 lnu1 + 2u2 lnu2 + 3u3 lnu3

(9.7.1)

(9.7.2)

decreases by virtue of system (9.7.1), that is, dE u1 u3 = B(u22 − u1 u3 )ln 2 ≤ 0. dt u2

(9.7.3)

This leads one to conclude that the solutions of the system converge to their stationary. In searching for a stationary with respect to the initial condition, Boltzmann uses invariant linear functionals. In the case of (9.7.1), these are √ √ 2u2 + 3u3 = a, and √ √ B(u) = u1 + 2 2u2 + 3 3u3 = b. A(u) = u1 +

(9.7.4)

To determine the limit to which any solution of (9.7.1) converges, we must minimize functional (9.7.2) subject to (9.7.4), where the constants a and b are determined from the initial conditions. The stationaries obtained from this variational principle are precisely the Boltzmann extremals.

9.7.2 Boltzmann Extremals and the Liouville Equation Consider the system of n ordinary differential equations dx = v(x). dt

(9.7.5)

Here x = (x1 , x2 , . . . , xn ) and vi (x) are continuously differentiable functions. Consider the continuity (Liouville) equation for this system: ∂f + 5fv(x) = 0. ∂t

(9.7.6)

Suppose that system (9.7.5) has a solution on the entire time domain (i.e., a global solution) and is divergence-free (i.e., 5v(x) = 0). Then, we can write the solution of (9.7.6) in the form f (t, x) = f (0, g−t (x)), where g−t (x) is the shift of the point x in time t by virtue of system (9.7.5). We define the time averages, or Cesa´ ro means, of the

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solution (9.7.6) by 1 fT (x) = T

ZT

f (t, x)dt.

(9.7.7)

0

Neumann’s stochastic ergodic theorem asserts that the limit f C of the function fT as T tends to infinity exists in L2 (Rn ) for any initial data from this space. We define entropyR as a functional on the set of positive functions h(x) from L L2 (Rn ) by S(h) = − hlnh(x)dx. Such functionals are invariant for (9.7.6) in the divergence-free case. However, in [231], Poincare´ discussed the entropy growth for the limit function in the particular example of a collisionless gas. Kozlov and Treshchev generalized this result (see [159]); namely, they proved that the entropy of the time average is not less than that of the initial distribution for (9.7.6). In this section, we show that the solution of (9.7.6) converges “where it must:” the time averages are determined by the conditional maximum entropy principle (the Boltzmann principle). The first attempts to apply the Boltzmann principle to this situation were made in [297]. We define linear Rconservation laws for the continuity equation (9.7.6) as the linear functionals Iq (h) = q(x)h(x)dx = (q, h), which are invariant by virtue of (9.7.6), i.e. (q, f (x, t)) does not depend on time on the solutions f (t, x) of (9.7.6). Let I denote the set of such functions q from L2 (integrals). Consider the Cauchy problem for (9.7.6) with positive initial conditions f (0) from L2 (Rn ). We define the Boltzmann extremal f B = f B (f (0)) as a function maximizing the entropy subject to the given conservation laws: S(f B ) = max S(h) on the set L(I, f (0)) = {h : (q, h − f (0)) = 0 ∀q ∈ I}. Theorem 9.1. If S(h) (entropy) tends to minus infinity when norm of h tends to infinity on the set L(I, f (0)), then (i) (ii)

the Boltzmann extremal exists and is unique; the Cesáro mean coincides with the Boltzmann extremal, i.e., f Ch = f B.

(9.7.8)

This theorem follows from a more abstract result obtained further.

9.7.3 Boltzmann Extremals and the Ergodic Theorem Let U be a linear operator on a Hilbert space X with norm not exceeding 1. Then, the following theorem is valid, which is known as the stochastic ergodic theorem (of Riesz; see [238]). For each z from X, the time averages Pnm (z) =

n X 1 Uzk n−m k=m+1

strongly converge to some element PC = PC (z) as n − m tends to infinity. This element is invariant with respect to U, i.e., U(PC ) = PC . This is what is known as the stochastic

180


ergodic theorem. It determines the time averages, or Cesa´ ro means, PC = PC (z) = lim Pnm (z). n−m→∞

We define a Boltzmann extremal as an element PB for which the entropy attains a conditional maximum. More precisely, consider the set I ⊂ X consisting of linear conservation laws u ∈ I such that (Ux, u) = (x, u) for all x ∈ X. Let S(x) be a strictly concave (convex upward) functional not decreasing under the action of U, i.e., such that S(Ux) ≥ S(x) (this is an analogue of the entropy), and let Xz be the set of x ∈ X with the same constants of linear conservation laws as z, i.e., Xz = {x ∈ X : (x − z, u) = 0 ∀u ∈ I}. Consider the conditional extremal problem of finding a point at which sup S(x) is attained subject to the constraint x ∈ Xz . We denote this conditional extremum by PB (z) and call it a Boltzmann extremal. Theorem 9.2. If S(x) tends to minus infinity when norm of x tends to infinity on (i) (ii)

the extremal problem stated above has a unique solution in Xz , and the Cesáromeans coincide with the Boltzmann extremals, i.e., PC (z) = PB (z).

(9.7.9)

Proof. Here we outline the scheme of the proof: Any such functional S (tending to minus infinity when norm of x tends to infinity and strictly concave) has a unique maximum on any linear closed subspace of a Banach space. Since Xz is closed, this proves the first assertion. Since Pnm (z) ∈ Xz , it follows that PC (z) ∈ Xz . Therefore, S(PB (z)) ≥ S(PC (z)). Thanks to the uniqueness of the extremal, it only remains to show that this inequality is an equality. Following [238], consider two subspaces in X. One subspace, Y, consists of the elements x − Ux and their limits. The other subspace, Z, consists of the fixed elements: x2 ∈ Z iff U(x2 ) = x2 . It was proven in [238] that X is the direct sum of these spaces. Therefore, all points of Xz have the same time average, and, hence, PC (z) = PC (PB (z)). The convexity of S implies that, for any z, we have  S(Pnm (z)) = S 

1 n−m

n X

 Uzk  ≥

k=m+1

n X 1 S Uzk ≥ n−m k=m+1

≥

n X 1 S(z) = S(z), n−m k=m+1

therefore, S(PC )(z)) ≥ S(z). Applying this relation to PB (z), instead of z, we obtain S(PC (z)) = S(PC (PB (z))) ≥ S(PB (z)). Hence, the reverse inequality is proven and, thereby, the coincidence of the time averages with the Boltzmann extremals. Theorem 9.1 is obtained from Theorem 9.2 by considering shifts in unit time and restricting the problem to the invariant cone of positive functions. Note that the righthand side of (9.7.9) depends on the functional S, whereas the left-hand side does not. At the same time, the left-hand side depends on the operator U, whereas the right-hand side depends on it only via the conservation laws and the functional S being increasing.

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It follows that Theorem 9.1 remains valid when concave functionals, instead of the entropy, are taken. Let us show that tending to infinity of the functional on the level sets of the linear conservation laws is essential. Example 9.2. Consider the simplest Liouville equation, which corresponds to namely,

dx = 1, dt

∂f ∂f + = 0. ∂t ∂x Consider its solution f (t, x) = f (0, x − t) in L2 (R). A step of high a and width K has entropy—Kalna, but its Cesa´ ro mean vanishes. There are no invariant linear functionals (in L2 ); therefore, the Boltzmann extremal is infinity (to show this, it suffices to take positive a less than 1 and let K tend to infinity for this step). Thus, Theorems 9.1 and 9.2 are false in this case, because the entropy is not bounded above in L2 (even on steps). Note that, for numbers a smaller than 1, the entropy decreases. For a periodic problem (an equation on a circle), one conservation law (the number of particles) arises. On the corresponding set, the entropy is bounded above, and the Cesa´ ro mean does coincide with the Boltzmann extremal. Example 9.3. For the same equation and the functional “minus norm” the conditions of the theorem holds even on the line, and the Boltzmann equal to Cesaro mean. Example 9.4. (The finite-dimensional case.) The spectrum of the operator U in Theorem 9.2 must be located on the unit disk. The conservation laws correspond to the zero unit point of the spectrum. Example 9.5. (Hamiltonian systems with compact energy levels.) In this case, the entropy functional is bounded above on functions with fixed energy, and the conditions of Theorem 9.1 hold; therefore, the time average coincides with the Boltzmann extremal. Linear conservation laws are related to partitions into ergodic components [301]. It was proven in [159] that, if a solution to (9.7.6) converges, then its limit coincides with the time average. The results obtained above imply that the limit is determined by the conservation laws. Theorems 9.1 and 9.2 are also subject to generalization. They are fulfilled in finitedimensional case for Markovian processes and their nonlinear generalizations from chemical kinetics (type (9.7.1) equations). A continuous time case was studied in [297] (Boltzmann extremal convergence). Cesa´ ro means are not necessary, since a time limit at infinity exists. Considering discrete time, Markovian processes sometimes do not converge, but Cesa´ ro means should coincide with Boltzmann extremals. This detail is of great importance for computer modeling, since it always uses a discrete time scale. In the case of discrete time the convergence to the equilibrium state derived by Boltzmann from H- theorem (one of the fundamental principles) is also violated. Another application can be seen in [301]—ergodic hypotesis. Even the classical example, hard balls in the box, states that a time limit (infinite time) relates a

182


convergence of distribution function with a function depending from energy. In reality, we are studying two theoretical issues: establishing the convergence and calculating the corresponding limit. The first issue is easily solved by a Poincare´ theorem [159, 238] and corresponding integrals. Hence, we can focus our attention onto the Liouville equation (9.7.6) related to the system (9.7.5) with dx/dt = v · dv/dt = 0. Vectors x, v ∈ R3N where N is a number of the balls. Combining with a boundary reflection condition 0 < xi < l,

(xi − xj )2 > d2 ,

xi ∈ R 3 , vi ∈ R 3 ,

i = 1 · · · N,

where d is a ball diameter, l is the box length, we have to prove that in L2 such problems have solutions of the form f (x, v) = g(v2 ). In other words, they depend on energy. This reduction to the integrals in ergodic problem “statement” can also be applied to a variety of problems, if we use Theorems 9.1 and 9.2.

10 Discrete Models of Boltzmann Equation

10.1 General Discrete Models of Boltzmann Equation Let fi (t, x) distribution function of particles in the space x ∈ 0, therefore, H- theorem f is satisfied. Here we also possess nontrivial analogs of H- function, since any convex function is suitable.

186


10.3.2 Broadwell Model For H =

PR i

fi ln fi dx, we obtain

X dH =− dt

Z

i

∂ (vi fi ln fi ) dx + ∂x

Z

( f3 f4 − f2 f1 ) ln

f1 f2 dx. f3 f4

Here the second term is less than or equal to zero, but it is not possible to substitute another convex function, instead of f ln f , to preserve this inequality. For Godunov-Sultangazin model H- theorem is proven in the same manner. Since the right side of the general discrete model is obtained by summation from these three basic components, hence, H- theorem for (10.1.1), (10.1.2) follows immediately from that fact. Moreover, one may consider a Boltzmann equation as the “continuous sum” of Broadwell models.

10.4 The Class of Decreasing Functionals for Discrete Models: Uniqueness Theorem of the Boltzmann H- Function First, we have to define what kind of functionals Z HG = G ( f1 , . . . , fn ) dx decrease for discrete models of Boltzmann equation. The velocity of increase of such functional reads Z X Z ∂G ∂fi dHG X ∂G = −vi dx + Fi dx = H1 + H2 . dt ∂fi ∂x ∂fi i

i

We want this expression to be nonnegative for all functions {fi (x)} = f (x), i.e., for all periodical functions or functions equal to zero outside the initial ball H1 + H2 ≤ 0.

(10.4.1)

Consider the function f α (x) = f (αx). Making the change αx = y, we see that the first and second terms are transformed in different manner: α H1 f α = H1 (f ) |α|d 1 H2 (f ). H2 f α = |α|d Hence, H1 (f ) must be equal to zero, otherwise we can choose magnitude and sign of α so that inequality (10.4.1) will be violated. Therefore, H1 (f ) = 0 for all f (x) > 0, f ∈ C2 Rd , f is finite.

Discrete Models of Boltzmann Equation

187

The study of the condition H1 (f ) = 0 means finding the functionals conserved for free motion. This condition X Z ∂G ∂fi vi , dx = 0 ∂fi ∂x i

in one-dimensional case is a condition of the equality to zero of work in field of forces F(f )i = vi ∂G ∂fi over any closed contour (periodicity of f or equality of zero for large in modules values x denotes closure of contour). The last is satisfied, if the field of forces is potential: ∂ 2G ∂Fj ∂Fi = , i.e. vi − vj = 0, ∂fj ∂fi ∂fi ∂fj or the second partial derivatives G are equal to zero for i 6= j. Hence, ∂G ∂fi = ψi (fi ) is n P a function of one variable. Therefore, G(f ) = φj fj is a sum of functions of one j=1

variable. Investigation of collision integral for different models shows: 1. for Calerman model φ1 = φ2 = φ and any convex function is suitable. 2. for Broadwell model and Godunov-Sultangazin models theorem on uniqueness of H- function is valid. 3. a similar reasoning also applies for Boltzmann equation. The study of collision integral shows that here we also have uniqueness of Boltzmann H- function [285].

10.5 Relaxation Problem The problem on space-uniform tending of the distribution function to Maxwellian distribution is called relaxation problem. For discrete models of Boltzmann equation, we have the system of ordinary differential equations: dfi X ij = σkl fk fl − fi fj . dt klj

For H =

P

i fi ln fi

we have inequality (H- theorem): fi fj dH 1 X ij = σkl fk fl − fi fj ln ≤ 0. dt 4 fk fl k,l,i,j

From H- theorem follows that any nonnegative initial distribution tends to equilibrium cone given defined by equalities fi fj = fk fl . Taking the Calerman model as an example, we get a conservation law of the form f1 + f2 = ρ. Hence, a solution becomes ( f1 − f2 ) = ( f1 − f2 )(0)e−ρt . Therefore, it tends to the “Maxwellian distribution” cone for f1 = f2 (see Figure 10.3).

188


f2

f1 = f2

f1

Figure 10.3 Relaxation for Carleman model.

Exercise 10.1. Solve space-uniform Broadwell and Godunov-Sultangazin models and consider relaxation for them. It can be proved for Boltzmann equation that any space-uniform distribution tends to the Maxwellian. The first strong results of the kind were obtained by T. Calerman [64]. The modern state of the question can be found in [41, 95, 285], for discrete models in [264, 285] and the following section of our book.

10.6 Chemical Kinetics Equations and H- Theorem: Conditions of Chemical Equilibrium The most general form for complex chemical reactions can be written as equation [308] X dfi =− αi Kαβ f β − Kβα f α , dt

i = 1, . . . , n.

(10.6.1)

(α,β)∈J

Here, by f α , we denote a product f α = f1 α1 f2 α2 × · · · × fn αn , summation goes through finite symmetric multy-index set (α, β), α = (α1 , α2 , . . . , αn ), and β = (β1 , β2 , . . . , βn ). α’s and β’s are nonnegative integers. An addend (α, β) represents elementary chemical reaction Kβα

α1 S1 + α2 S2 + · · · + αn Sn β1 S1 + β2 S2 + · · · + βn Sn , (α, β) ∈ J, β

Kα

where Si denote symbols of reacting substances, Kβα —coefficients of chemical reaction speeds. Coefficients αi , βi are also known as stoichiometric ratios. Without loss of generality, we consider set J to be symmetrical with respect to permutations α and β. Moreover, some ratios (α, β) represent null cross-sections: Kβα = 0. On the other hand, probably Kβα > 0, which means the irreversibility of reactions. For example, inside chemical reaction 2H2 + O2 = 2H2 O, coefficients are equal to α = (2, 1, 0), β = (0, 0, 2), and pair is symmetric.


189

Example 10.1. Chemical reaction Michaelis-Menten [213] chemical reaction is defined by two reactions: S + E

K1 K−1

Q,

Q

K2 K−2

P + E,

Here S denotes source component reacting with ferment E and providing fermentcomponent complex Q. Q itself decays reversibly into product P and the same ferment E. This chemical reaction is represented by the following system of differential equations: ds = −v1 , dt

de = −v1 + v2 , dt

dq = v1 − v2 , dt

dp = v2 . dt

Quantities s, e represent concentration of component S and ferment E; v1 = K1 se − K−1 q, v2 = K2 q − K−2 ep the differences in the speed of forward and backward reactions. While composing such kind of reactions, one uses the “law of mass action,” represented by system (10.6.1). It reads as follows: The rate of a chemical reaction is directly proportional to the molecular concentrations of the reacting substances with respect to their stoichiometric ratios. This law was established by Guldberg and Waage in 1864–67 (see [175] for details). In foundations of chemical kinetics and for any statistical interactions, the first question to answer is the existence of H- function. In other words, we would like to know when system (10.6.1) possesses entropy—type functional decreasing over nonstationary solutions. And how it could beP interpreted in chemistry. The simplest case when functional H = i fi (ln fi − 1) decreases with respect to β equation (10.6.1) is symmetrical, when Kβα = Kα for all reactions (α, β) ∈ J. Usually, this family is just called symmetrical S . Another important case refers to reactions with detailed balance. It merely means that we suppose that, for our m reactions, there exists at least one solution of the system Kβα f α = Kαβ f β ,

(10.6.2)

also written as α −β1

f1 1

α −β2

· f2 2

· · · fnαn −βn =

β

Kα . Kβα

(10.6.3)

Sometimes, equility (10.6.3) is called law of mass action [170]. We denote as D applying logarithmic transformation, we obtain the relation on ln fi , i = 1, . . . , n. Family of systems (reactions) fulfilling condition of detailed balance. Solution properties can be described by H- function X fi H= fi ln − 1 . (10.6.4) ξi i

190


Here ξ is one of the positive solutions of the system (10.6.2). For the detailed description see [308] and [318]. Example 10.1 also belongs to D, but, as it will be shown later, even linear systems posessing H- function do not belong to D. Hence, we need to introduce some appropriate conditions for H- function and system (10.6.1) (compare with attempts in [122, 219]). Condition of dynamical balance (Stukelberg-Batisheva-Pirogov condition). Assume there exists at least one positive solution ξ of the following equation: X β

Kβα · ξ α =

X β

Kαβ · ξ β .

(10.6.5)

Parameter α used in (10.6.5) is chosen in such a manner that, for some β, we have β (α, β) ∈ J. Summation goes for β : Kβα 6= 0 or Kα 6= 0. These systems we denote as E , or dynamically balanced systems. Theorem 10.1. Let coefficients Kβα in system (10.6.1) provide at least one solution ξ of the equation (10.6.5). Hence, a) b)

H- function (10.6.4) does not increase along the solutions of the system (10.6.1), i.e., dH/dt ≤ 0. P k System (10.6.1) possesses n − r conservation laws of the form µi fi (t) = Ak = Const, k = 1, . . . , n −P r, where r is the dimension of the linear hull α − β, vectors µk orthogonal to all α − β : µki (αi − βi ) = 0. Fixing all constants Ak in conservation laws, we can find unique stationary solution (10.6.1) of the form X f0i = ξi e

c)

µki λk ,

k

(10.6.6)

where λk uniquely depends on Ak . k k Unique P k stationary solution (10.6.6) exists if A is defined from initial condition f (0), A = µi fi (0). Solution f (t) with introduced initial condition exists ∀t > 0, is unique and tends to stationary solution (10.6.6).

Proof. We proceed, proving all three assertions one by one. a)

We have to study properties of H- function (10.6.4). Deriving and transforming dH/dt, we obtain: X dH =− Kαβ ξ β yβ ln yβ−α = dt (α,β)∈J h i X =− Kαβ ξ β yα yβ−α ln yβ−α − yβ−α + 1 ≤ 0,

(10.6.7)

(α,β)∈J

where yi = fi /ξi . To obtain this relation, we exchanged once α and β in one addend, then used condition (10.6.5) adding zero component. Inequility (10.6.7) is valid, since u ln u − u + 1 ≥ 0. Since u ln u − u + 1 = 0 only for u = 1, then (10.6.7) becomes an equility for α−β

f0 : f0

= ξ α−β .

(10.6.8)


191

β

b)

Here Kβα 6= 0 or Kα 6= 0. Expression (10.6.8) also means that f0 complies with the same condition (10.6.5) as ξ . P One can prove the conservation property of the functional µi fi (t) for µ ⊥ α − β just by rewriting (10.6.1) in the following form: X dfi =− (αi − βi ) · Kβα · f α , dt

i = 1, . . . , n,

(10.6.9)

(α,β)∈J

due to simmetricity of the set J with respect to permutations of α and β. Applying logarithimic transformation to (10.6.8), we get that vector with components of the form ln f0i /ξi is orthogonal to all vectors α − β. Hence, relation (10.6.6) is fulfilled for all stationary solutions of the system (10.6.1) for some fixed λk . In their turn, coefficients λk are uniquely defined by parameters of the conservation laws Ak obtained from the system n−r X

∂L = Ak , ∂λk

c)

L=

n X

ξi e

k

µki λk ;

i=1

here L is a convex function. Since H- function is convex, then a solution of (10.6.1) exists for all positive initial values. Hence, the set of points f such that H(f ) ≤ H( f (0)) is a compact set. Due to convexity of H, there P exits a unique point where H gains P its minimum for the given set of parameters Ak = µki fi (0) of the conservation law µki fi (t) = Ak . According to (10.6.7), we have a strict inequility everywhere else, then after a solution tends to the minimum of H- function in other points.

Finally, we were able to construct the following classification for the equation of chemical kinetics (10.6.1) based on the entropy principle S ⊂ D ⊂ E ⊂ C.

As C here, we denote the whole class where (10.6.1) is defined for a final number of reactions. E represents systems with positive solution of the equation (10.6.5), which means that entropy increases. D is the systems (10.6.1) with detailed balance (10.6.2). And the last one S refers to the systems with symmetrical reaction constants when β Kβα = Kα . Example 10.2. Linear systems. If set J contains only unit vectors, then (10.6.1) becomes linear and was considered in Chapter 9. Linear system (10.6.1) is also called the Pauli master equation. It also is used for studying Markovian processes with finite number of states and continuous time. H- theorem, for the linear system (10.6.1), is already known (see [217] for example). But the Theorem 10.1 proven here also is usefull, since it outlines the role of the conservation laws and describes the set of stationary solutions for highly sparse matrices.

192


As a matter of fact, a certain subclass, namely, linear systems with detailed balance, has several special applications in physics. Example 10.3. Coagulation—fragmentation equations. Coagulation—fragmentation processes can be described via special equations m−1

df (m) 1 X [K(l, m − l)f (l)f (m − l) − F(l, m − l)f (m)] − = dt 2 i=1

−

∞ X

[K(l, m)f (l)f (m) − F(l, m)f (m + l)] .

i=1

Here we denote by f (m) a numerical density of particles consisting from m monomeric units, K(l, m) is a kernel (constant) of coagulation, F(l, m)—fragmentation kernel. An exact relation between the kernels arises from the condition of detailed balance F(l, m) =

K(l, m)ξ(l)ξ(m) ξ(l + m)

for some positive sequence ξ(l), l = 1, 2, . . . This relation was studied in [65, 66].

10.6.1 Conclusion The most general requirement for discrete models becomes the conservation of the properties studied for initial equations. The most important is conservativity, or just existence of the conservation laws [117, 251]. The principle of full or complete conservativity means that all conservation laws are fulfilled (see e.g. [251]). A natural question arises what does the word “all” mean? For the Boltzmann-type equations and their generalizations the linear conservation laws are widely used. This is natural, since they are used to define continuous medium equations. As a result, while translating them into discrete form, we should reject spurious invariants. This issue is discussed in Chapter 12. One can call this a principle of precise conservativity. This situation is typical for all scientists using computations for kinetic equations [11, 123, 254, 316]. From the theoretical point of view, the situation is clarified by the theorem on the uniqueness of the Boltzmann H- function [285]. From it follows in particular that for the Boltzmann equation among conservational laws there are only 5 linear standard additive invariants: Z G[g] = ϕ05 g(v)dv. Rykov (see [250]) attempted to expand the class of functionals, considering timedependent functionals of the form Z ϕ(x − vt, v, f )dvdx.


193

Here ϕ(·) is a function of seven variables. It generates eight more invariants (Rykov’s invariants). For the Broadwell model, their number is even higher and depends on arbitrary functions [10]. Another fundamental question is the correspondence of hydrodynamical equations derived from the Boltzmann equation and the results of physical experiments.

11 Method of Spherical Harmonics and Relaxation of Maxwellian Gas

In this chapter, we will solve Maxwell problem involving the derivation of moment system from a Boltzmann equation for Maxwellian molecules. This is based on the group symmetries of the Boltzmann collision integral, which has general mathematical nature. For example, if linear operator commutes with rotation in three-dimensional space, then it is multiple to unit operator. Two quadratic operators defined in 3D space and commuting with rotation group are well known—they are scalar and vector products. Do any others exist? The answer will be revealed at the end of the Section 11.2. It will be shown that there exists only one more operator of such kind, transforming three-dimensional space into five-dimensional one.

11.1 Linear Operators Commuting with Rotation Group Let C(S2 )—space of complex continuous functions on the sphere S2 and let L—linear operator transforming C(S2 ) into itself: L : C(S2 ) −→ C(S2 ). Let L commutes with rotations Tg , g ∈ SO(3): Tg L = LTg , where Tg f (S) = f g−1 S , S ∈ S2 —vector of two-dimensional sphere; g—an arbitrary rotation; and SO(3)—orthogonal group with determinant equal to unit. Then L possesses the following properties [156]: 1. spherical harmonics Ylm are eigenfunctions of operator L: L (Ylm ) = λl Ylm ;

(11.1.1)

2. eigenvalues λl do not depend on number m.

A similar situation occurs quite often in applications. As an example we refer to an angular part of Laplace operator. Such formulas are connected with the fact, that spherical harmonics Ylm (m = −l, . . . l) generate basis of irreducible representation of rotation group (dimension of representation space equals to 2l + 1). Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00011-9 c 2011 Elsevier Inc. All rights reserved.

196


Definition 11.1. Let G—group. A couple (H, T) is called a representation of group G, where H is a linear space, if T = T(g) is a collection of linear operators from H onto H depending from elements g ∈ G. Here mapping g → T(g) is homomorphism of the group G into the group of linear invertible operators L(H) of the space H. In other words, T is a homomorphism G in L(H): G → L(H). Representation is called irreducible, if there are no linear subspaces in H invariant over all operators T(g) different from zero and H. Independence of eigenvalue λl from m is formulated in Shur’s Lemma [156]. Lemma 11.1. Let G is a group, (T, H) is some irreducible representation of group G, L is a linear operator, commuting with all operators T(g). Then L is multiple to unit operator: L = λE. Exercise 11.1. Prove Shur’s Lemma. Applying Shur’s Lemma to the rotation group and using that Ylm basis of irreducible representation, we obtain formulas (11.1.1) and independence of λl from m. We will call this representation (Yl , Tl ) according to the definition, where Yl is (2l + C)-dimensional space with basis Ylm .

11.2 Bilinear Operators Commuting with Rotation Group Here we apply representation theory to quadratic operators B commuting with rotation group. Collision integral with arbitrary central interaction law between particles is such an operator. Let B be bilinear operator on the space of continuous functions on the sphere S2 : B : C(S2 ) × C(S2 ) → C(S2 ). Assume also that B commutes with rotations: B(Tg f , Tg h) = Tg B(f , h). First of all,let us see which spherical harmonics represent the decomposition of B Yl1 m1 , Yl2 m2 . The answer can be found in [108, 119, 156, 172, 285]: X l1 l2 l Bl Y . (11.2.1) B Yl1 m1 , Yl2 m2 = m1 m2 m1 + m2 l1 l2 l,m1 +m2 l

This means that summation takes place only by finite number of spherical harmonics with m = m1 + m2 , |l1 − l2 |≤ l ≤ l1 + l2 . All dependences from m are contained l l l , and information about operators is conin Klebsh-Gordon coefficients 1 2 m1 m2 m tained in coefficients Bll1 l2 independent of m. This values sometimes are called bilinear eigenvalues since they are eigenvalues numbers of some linear operator connected with B.

Method of Spherical Harmonics and Relaxation of Maxwellian Gas

197

Taking into account that mentioned formulas represent general situation, the proof will be presented in terms of representation theory for simplicity. Let (Hi , Ti ), i = 1, 2, 3—three representations of some group G, the first two of them are irreducible. Let the direct product H1 × H2 transforms to H3 by bilinear operator B, commuting with G. We continue B up to linear operator B˜ from tensor product H1 ⊗ H2 to H3 . Here we want to remind the difference between direct H1 × H2 and tensor H1 ⊗ H2 products. Let ei —basis in H1 and fj —basis in H2 . Then the basis of H1 × H2 is the ei = (ei , 0) and fj = (0, fj ), therefore dimension dim (H1 × H2 ) is equal to the sum of respective dimensions dim H1 + dim H2 . The basis of H1 ⊗ H2 are ei ⊗ fj and dimension dim (H1 ⊗ H2 ) = dim H1 dim H2 —i.e. the product of initial dimensions. Continuation is constructed by formulas B˜ ei ⊗ fj = B ei , fj and is continued by linearity. Additionally in H1 ⊗ H2 tensor product T1 ⊗ T2 of representations T1 and T2 is defined. Hence it leads to the problem how to decompose tensor product T1 ⊗ T2 of representations onto irreducible representations. The decomposition coefficients are called Klebsh-Gordon coefficients. For spherical harmonics, we obtain

Yl1 m1 ⊗ Yl2 m2 =

X |l1 −l2 |≤l≤l1 +l2

l l1 l2 Y . m1 m2 m1 + m2 l,m

(11.2.2)

Operator B˜ is linear and acts onto Yl,m by formulas B˜ Yl,m = Bll1 l2 Yl,m . Dependence of numbers Bll1 l2 from l1 and l2 is explained by restriction of the operator B˜ on Yl1 ⊗ Yl2 . We note that summing in formulas (11.2.1) and (11.2.2) is conducted by the same l found in the theorem on summing of angular momentum in quantum mechanics. An expression (11.2.1) also contains an answer for the question stated in the beginning of the chapter. Since for three-dimensional representation space we put l1 = l2 = 3, then summation in (11.2.1) and (11.2.2) becomes l = 0, 1, 2. This corresponds to the equality 9 = 1 + 3 + 5 in terms of dimensions. The scalar product represents a trace of matrix 3 × 3 of multiplications ai aj . Vector product is a skew-symmetric part. Another operator is one in five-dimensional space—traceless part of symmetrization, given by ai aj + aj ai (a11 a11 + a2 a2 + a3 a3 ) − . a matrix: 2 3

11.2.1 Computations of Eigenvalues We can compute λl in (11.1.1) using Yl0 , which is proportional to Legendre polynomial Pl . Substituting it in formulas (11.1.1) and using equality Pl (1) = 1, one obtain λl = L (Pl (cos θ )) (0, 0, 1).

198


We can calculate Bll1 ,l2 using orthogonality condition for Klebsh-Gordon coefficients: X l1 l2 l 0 l1 l2 l = δll0 δmm0 . m1 m2 m m1 m2 m0 (−l1 ,−l2 )≤(m1 ,m2 )≤(l1 ,l2 )

Hence we obtain Bll1 l2

=

4π 2l + 1

1 2

X − max(l1 ,l2 )≤m≤max(l1 ,l2 )

l1 l2 l × −m m 0

× B Yl1 ,−m , Yl2 ,m (0, 0, 1). Next, we apply these reasons for a Boltzmann equation. First, we perform a Fourier transformation of collision integral by velocity to simplify further calculations. This techniques was proposed by A. B. Bobylev [41] and is known as the simplification method of collision integral.

11.2.2 Fourier Transformation of Collision Integral Let us consider space-uniform Boltzmann equation in the form Z (u, n) ∂f1 0 0 = I[f1 , f2 ] = dwdn f1 (v )f2 (w ) − f1 (v)f2 (w) B |u|, . ∂t |u|

(11.2.3)

Multiplying this equation by e−ikv and integrating over dv we obtain new equation for ϕi (k) =

Z

fi (v)e−i(k,v) dv,

namely ∂ϕ1 = S(ϕ1 , ϕ2 ) = ∂t

Z

I [f1 , f2 ] e−i(k,v) dv.

Our task is to obtain an expression depending on ϕ1 and ϕ2 only. We transform this integral in the following manner: Z h i 0 S(ϕ1 , ϕ2 ) = f1 (v)f2 (w) e−i(k,v ) − e−i(k,v) B dwdvdn. Here we used by property of changes (v, w) → v0 , w0 from Section 9.2. v+w Taking common multiplier e−i(k, 2 ) , we obtain Z v+w S(ϕ1 , ϕ2 ) = dvdw f1 (v)f2 (w)e−i(k, 2 ) F(u, k) = S+ − S− ,

(11.2.4)


199

where Z F(u, k) =

i(k, u)  i(k, n)n − (u, n)   2 2 −e B |u|, dn e 2 

−

R and B(u, s) = ei(p, u) ψ(p, s)dp. R According to this equation obtain F(u, k) = dp ei(p, u) G(k, u, |p|), where G (k, u, |p|) =

Z

i (k, n) i (k, u) (u, n) |u| − exp − exp − ψ |p| , dn. |u| 2 2

Exercise 11.2. Prove that the function G(k, u, |p|) is symmetric with respect to arguments k and u: G(k, u, |p|) = G(u, k, |p|). Proof. If we rotate vectors k and u by the same angle, then function G does not change. It means that it depends only from |k|, |u| and (k, u). Therefore it depends symmetrically from |k| and |u|. So, trading k and u in places, one obtain expression for F

Z F(u, k) =

 dndp e

i(p, u) − i|k|

 (u, n) (u,n) (n, k) −i(p,u)−i  2 2 −e . ψ p, |k| (11.2.5)

Collecting terms with v and w in exponent, in term S+ from (11.2.4) and (11.2.5) one obtain for v k + |k|n (v, k) −i v, −p −i + i(p, v) − i|k|(n, v) 2 2 e =e and for w k − |k|n (w, k) −i w, +p −i − i(p, w) + i|k|(n, w) 2 2 e =e . Under the same assumptions in term S− one obtain: e−i

(k,v) 2

e−i

+ i(p,v)−i (k,v) 2

= e−i(v,k−p) ,

(k,w) (k,w) 2 −i(p,w) + i 2

= e−i(w,p) .

200


Integration over dv and dw gives Z (k, n) × S(ϕ1 , ϕ2 ) = dndp ψ |p|, |k| k + |k|n k − |k|n × ϕ1 − p ϕ2 + p − ϕ1 (k − p)ϕ2 (p) . 2 2 The utmost simplification is reached if B(|u|, s) does not depend on |u|—a term representing Maxwellian molecules. Then 9(p, S) = δ(p)g(S) and we obtain [41] S (ϕ1 , ϕ2 ) =

(k, n) dn dpψ |p|, × |k| k − |k|n k + |k|n ϕ2 − ϕ1 (k − p)ϕ2 (p) . × ϕ1 2 2 Z

(11.2.6) (11.2.7)

11.3 Momentum System and Maxwellian Gas Relaxation to Equilibrium. Bobylev Symmetry We obtained Boltzmann equation for Maxwellian molecules when distribution function f (v, t) is independs from space variable in Fourier representation: ∂ϕ = S(ϕ, ϕ), ∂t

(11.3.1)

where integral S is defined by formulas (11.2.6). To study the relaxation, we assume: ϕ = ϕ0 (1 + h),

(11.3.2)

−k2

where ϕ0 = e 2 —Fourier image of Maxwellian distribution, h—deviation of Maxwellian distribution. Parameter h is described by the following equation: ∂h = L(h) + S(h, h), ∂t

(11.3.3)

where L is the linearized collision operator in Fourier representation: L(h) =

Z k + |k|n k − |k|n h +h − h(k) − h(0) × 2 2 (k, n) ×g dn. |k|

Exercise 11.3. Obtain (11.3.3) from (11.3.1) taking (11.3.2).

(11.3.4)


201

Both operators: L and S commute with rotation group. This is a general property for all collision integrals. The critical property for Maxwellian molecules refere to the relation with one more Abelian group: convolution of the solution with Maxwellian distribution also makes a solution. Fourier representation for the last one is essentially simple: both operators commute with dilation group [41] also. This is the Bobylev symmetry: L (fα ) (k) = L(f )(αk),

where fα (k) = f (αk).

The same goes for S. Thanks to this property, |k|r Ylm are eigenfunctions of operator L [41]: k k r r = λrl |k| Ylm . L |k| Ylm |k| |k| Dilations possess eigenfunctions |k|r , and spherical harmonics are connected with rotations. Using formulas from section 11.2.1 we obtain the following expression for eigenvalues Z θ θ θ θ λrl = cosr Pl cos + sinr Pl cos − 1 − δr0 δl0 g(cos θ) sin θdθ. 2 2 2 2 (11.3.5) For the bilinear operator, one have: S |k|r1 Yl1 m1 , |k|r2 Yl2 m2 = |k|r1 +r2 × X k l l1 l2 Y × 3rl11lr22 l . m1 m2 m1 + m2 l,m1 +m2 |k| |l1 −l2 |≤l≤l1 +l2

Here an exact values for 3lr11lr22 l are similar to (11.3.5), see [285]. On the basis of these formulas, one can obtain momentum system for Boltzmann equation. Let X k r . h= Crlm (t)|k| Ylm |k| rlm

Such momentum system corresponds to decomposition of distribution function by Laggere polynomials multiplicated by spherical harmonics: 2 X l+1 v v l 2 f (v, t) = |v| L r−l Ylm Mrlm (t). 2 |v| 2 Hence for the momentum Crlm (t) we obtain the following system of equations: C˙ rlm (t) = λrl Crlm + ψrlm (C), X l1 l2 l ψrlm (C) = C C 3r1 ,r−r1 ,l . m1 m − m1 m r1 l1 m1 r−r1 ,l2 ,m−m1 l1 ,l2 r1 l1 l2 m1

A special feature of the obtained system is that ψrlm includes momentum of lowest index r and, therefore, it is recursively solvable. To be exact, that fact made it possible for Maxwell to solve the problem, deriving the equation of continuous medium from a Boltzmann equation. However,

202


Chapman-Enskog solution for arbitrary potentials is essentially based on given solution for Maxwellian molecules. For Maxwell-Chapman-Enskog method, it is sufficient deal with first momentum equations. The final solution of momentum system was been obtained quite recently, see [41, 285].

11.4 Exponential Series and Superposition of Travelling Waves A lot of papers devoted to the study of scattering theory (and, in particular, to the inverse problem method) revealed a lot of equations in which one can obtain an explicit analytical expressions for superposition of travelling waves [317] (n-soliton solutions). Analytical methods introducing wave superposition as series [39–41, 46, 94, 190, 204, 243, 283, 285] were also proposed. Papers [40, 46, 243] established correspondence between the series and exact solutions for the Korteweg-de Vries type equations. But this idea cannot be used freely due to existence of special relations between eigenvalues, named as resonance. In [40, 46, 243] it was proved that resonances do not arise in KdV-type equations, followed by similar results for nonlinear elliptic equations [94, 190] and isotropic Boltzmann equation [39]–[41]. Papers [283, 285] discovered this fact for anisotropic Boltzmann equation. Special efforts were made in papers [94, 190] studying convergence of this series at small (i.e. in some neighborhood of zer) for nonlinear elliptic equation. On contrast, in [39– 41, 204, 283, 285] such series were constructed for Boltzmann equation and there were proved convergence theorems in general (i.e. in the hole complex space Cn , n is a number of waves). In this section our attention is paid mainly to evolutionary equations (11.4.1). In the next subsection, 11.4.2, we introduce formal decompositions in exponential series—series representing interactions of traveling waves. Subsection 11.4.3 detects existing resonance relations. In Subsection 11.4.4, we introduced several convergence theorems at small and provided some divergence examples. Here under the name of convergence at small we understand that a function representing a superposition of travelling waves (called sometimes an interaction function) gives a solution of corresponding initial equation in some small domain. In article [155] and later publications of its author numerically was shown that for hyperbolic nonlinear equation corresponding series are fast converging on the whole plane, it seems that they do not pay attention to existing resonaces. The resonance study and convergence study are of the highest importance, since they provide a background for numerical methods solving the problem of traveling waves interaction for any nonlinear equation with coefficients independent of space and for any dimension.

11.4.1 Equations of the Form ∂u/∂t = F(u) Considering equations of the Form ∂u = F(u), ∂t

(11.4.1)


203

for u(x, t) and x ∈ R, t ∈ R; F—nonlinear operator commuting with a shift group g(a) : x → x + a, [F, g(a)] = 0. Let F(u) = L(u) + B(u) + · · · where L(u) is linear, B(u) is bilinear and the rest represent operators commuting with a shift group. Operator F of this type includes all differential equations where partial derivatives over x have constant coefficients. Our interest is focused at the second order equations, since higher order ones are studied in similar manner. Main examples are the ∂u/∂t = u(m) + ux u equation (for m = 2 it is called Burger’s equation, for m = 3 it is a KdV equation). Also we will discuss a generalised equation ∂u/∂t = u(m) + u(p) u(q) .

11.4.2 Waves Interaction Series Let vectors α = (α1 , . . . , αn ), β, γ are real or complex valued vectors. Also suppose u=

X

dk e(k,αx+βt+γ ) .

(11.4.2)

Summation goes by vectors k with nonnegative integer components; at least one component is different from zero. If we substitute (11.4.2) into (11.4.1) the resulting relation allows to define coefficients dk consequently: [(k, β) − λ((k, α))]dk =

X

3((r, α), (k − r, α))dr dk−r

(11.4.3)

0 1 relation (11.4.5) is not fulfilled. Then we introduce V as a formal interaction series (nonlinear summation, overlapping) of n travelling waves for the equation (11.4.1) Vn = V(F; n; α; z) = Vnα (z) = Vn (z1 , z2 , . . . , zn ) =

X

dk zk11 . . . zknn .

(11.4.6)

Here dk = 1 for |k| = 1. When |k| > 1, coefficients dk are defined by expressions (11.4.3), (11.4.4). Summation in (11.4.6) is defined similarly to earlier cases: over nonnegative integer vectors k : |k| = 1. Variable z connecting expressions (11.4.2) and (11.4.6) is defined by relation zi = eαi x+βi t+γi . Our goal is to verify the existence of resonance ratios (11.4.5) and convergence of series (11.4.6).

11.4.3 Investigating Resonance Relations Assume α = (α1 , . . . , αn ) a real vector with positive components. Lemma 11.2. Let λ(p)—polynom of the second or higher degree. Then condition (11.4.5) is not fulfilled for large |k|. Proof. Let assume 0 < α1 ≤ . . . ≤ αn . Hence for r ≥ 1 |k|r αnr ≥

X

kj α j

r

≥ |k|r α1r .

Comparing the sign of the difference in (11.4.5) and the sign of the coefficient at highest degree of λ(p) for |k| → ∞: if λ(p) = a0 + a1 p + · · · + ad pd (d ≥ 2, ad 6= 0), X d X then λ((k, α)) − kj λ(αj ) = ad kj αj + o |k|d , where o(x) → 0 for x → ∞. Lemma 11.3. Let λ(p)—polynom of the second or higher P degree with positive coefficients and non zero free coefficient. Then λ((k, α)) > kj λ(αj ) for |k| > 1. Proof. Taking monomial λ(p) = pr with r ≥ 1 and |k| ≥ 1 we have following inequalities λ((k, α)) =

X

kj αj

r

≥

X

kjr αjr ≥

X

kj αjr =

X

kj λ(αj ).

Let r > 1 and |k| > 1. Then if vector k has several components different from zero, then the first inequality is strict. If we have only one component different from zero, the second inequality is strict also. Calculating the sum of inequalities for different r with nonnegative coefficients, we finish the proof.


205

Lemma 11.4. Let λ(p)—polynom of the second or higher degree with complex coefficients and non zero free coefficient. If all non zero coefficients are on the same side of certain line passing through zero point, then resonances (11.4.5) do not exist. Proof. Let λ(p) = a0 + a1 p + · · · + ad pd , d ≥ 2, ad 6= 0 for ar = br + icr . Due to lemma conditions Abr + Bcr > 0 if ar 6= 0 where Ax + By = 0 if an equation of line passing through zero point. Using inequality proved in Lemma 11.3 for λ(p) = pr we make summation for r with nonnegative coefficients Abr + Bcr . Also taking into account that for r = d inequality is strict, we finally have X X X kj , αjr = (Abr + Bcr )

This inequality contradicts condition (11.4.5). Theorem 11.1. Under conditions established in Lemma’s 11.3 or 11.4 solutions (11.4.2) of the equation (11.4.1) exist ∀α ∈ Rn with positive components and ∀γ ∈ Cn . In the case of Lemma 11.2 we can guarantee the same slightly varying coefficients of operator L.

11.4.4 Convergence of the Series of n Interacting Traveling Waves The following theorem holds: Theorem 11.2. Let for equation ∂u/∂t = L(u) + B(u, u) 1. resonance relations (11.4.5) are absent; 2. there exist such constant C = C(n, α) > 0 that for some α ∈ Rn and ∀k : |k| > 1 holds an inequality sup |3[(r, α), (k − r, α)]| r < C. λ((k, α) − P kj λ(αj ))

(11.4.7)

Hence function Vnα (z) for superposition of n travelling waves is defined by a convergent series (11.4.6) in the neighborhood of zero. Proof. By induction over k we can prove that for k ≥ 1 hold an inequality (k1 + · · · + kn )! k . d ≤ C|k|−1 k1 !k2 ! · · · kn !

(11.4.8)

It is correct for k = 1 since de(j) = 1. To continue with the proof, we need the following identity for polynomial coefficients X (m1 + · · · + mn )!(r1 + · · · + rn )! m1 !m2 ! · . . . · mn r1 !r2 ! · . . . · rn

=

(k1 + · · · + kn )! k1 !k2 ! · . . . · kn

206


where summations go for vectors with nonnegative integer components r, m : r + m = k. Supposing that inequality (11.4.8) is already proved for smaller k, then for the next dk we have |k − r|!Ck−r−1 |r|!Cr = |k1 − r1 |!|k2 − r2 |! · . . . · |kn − rn |!|r1 |!|r2 |! · . . . · |rn |! k!Ck−1 . = |k1 |!|k2 |! · . . . · |kn |!

|dk | ≤

X

With respect to estimation (11.4.8) an absolute convergence of the Taylor series for function Vn follows immediately from polynomial formula and properties of geometric progression in the neighbourghood of zero. As an example, we will establish some results. Theorem 11.3. Considering equation ∂u = u(m) + u(p) u(q) . ∂t

(11.4.9)

we have 1. if m ≥ p + q, then series Vnα defining superposition of n travelling waves converges in the neighbourhood of zero for vectors α with positive components; 2. the same series diverges (even for n = 1) if max(p, q) ≥ m + 1.

Proof. 1) In this case resonances are absent: see Lemma 11.3. Then |3[(r, α), (k − r, α)]| |[(r, α)p (k − r, α)q ]| = ≤ P λ(k, α) − P kj λ(αj ) (k, α)m − kj αjm ≤

pp qq (k, α)(p+q) . P p+q (p + q) (k, α)m − k α m

(11.4.10)

j j

The inequality is obtained by ordinary differentiating and searching for maximum of the function (r, α)p (k − r, α)q over (r, α) for 0 < r < k. It equals pp qq (k, α)(p+q) . For m ≥ p + q the right side of inequality (11.4.10) is bounded (p + q)p+q by constant C(α) and thus shows the applicability of the Theorem 11.2. 2) first of all, consider equation ∂u/∂t = u(m) − u(p) u(q) which differs from (11.4.9) only in sign. But this guarantee ∀dk > 0. Moreover, taking max(p, q) > m, m ≥ 2 series V1α corresponding to this equation has dk > bk k! for some b > 0 for n = 1 and α > 0 (here α ∈ R+ and k ∈ Z+ ).

(11.4.11)


207

Now assume p ≥ q and p ≥ m + 1. The proof of (11.4.11) will by done by induction over k. Starting from k = 1 we have b < 1. Now we choose 0 < b < 1 sufficiently small to fulfill the following inequality α p+q+m (k − 1)p ≥b k2 (km−1 − 1)

(11.4.12)

for all k ≥ 2. It can be done for p ≥ m + 1. From recurrent relation for dk we obtain dk =

≥

α p+q+m X p r (k − r)q dr dk−r ≥ k(km−1 − 1) α p+q+m (k − 1)p · kdk−1 ≥ bkdr−1 ≥ bk k!. k2 (km−1 − 1)

To obtain the first inequality we reduced the expression keeping the only summand with r = 1 (since all another summands are positive). For the second inequality we used (11.4.12). P The last one is supposed to be fulfilled due to reasoning by induction. Series bk k!zk has a zero convergence radius ∀b. Hence the convergence radius of (11.4.6) also equals zero. To be able to obtain equation (11.4.9) we just note that dk ’s alternate their signs and differ from calculated only in their sign for even k. As it follows, convergence radius of this series for (11.4.9) also is zero. Theorem 11.4. Regarding equation ut = L(u) + f u, u0 , . . . , u(m)

(11.4.13)

where L is a linear operator L(u) = λ(∂/∂x)u and λ is a polynomial λ(u) =

M X

ai ui

i=0

we assume 1. resonances (11.4.5) do not exist for operator L; 2. f (0) = 0, Of (0) = 0, function f is analytical at zero; 3. M ≥ m.

Hence series Vnα = each n.

P

dk zk converges in the neighborhood of zero when ai ≥ 1 for

Proof. Using majoring technique (see [94, 190]) consider an auxiliary equation X aW (m) = f˜ W (m) , . . . , W (m) + αim zi a

208


P and corresponding series Wnα = wk zk with the same α and n. W (m) is an m-th derivative over x, zi = eαi x . Lets prove that by proper choice of a and f˜ series Wnα can be made majorant series for series Vnα ; namely, wk ≥ |dk | while Wnα is analytical at zero. P f˜ will be defined by a convergent series f˜ = |fk |rk . Now we define a > 0 to fulfill inequality X ki λ(ai ) ≥ a(k, α)m . λ((k, α)) − It can be done since resonances are absent, M ≥ m and estimations of Lemma 11.2 holds. Using series defined as above, we obtain the desired result for series Wnα and Vnα , since (a) all coefficients of polynom expressing wk in terms of wr , r < k are positive; (b) replacing arguments of function f˜ by higher derivatives, we increase the values of the coefficients since (k, α)p is a monotonic no decreasing function of variable p with αj ≥ 1; (c) choosing parameter a we reduced the values of denominator, i.e. coefficients for Wk caming from the linear term of equation.

Equation for W (m) is algebraic equation of the form F W (m) , y = 0,

y=

X

α1m z1 .

Here ∂F/∂W (m) = a > 0, then this equation can be solved according to the theorem of implicit function. Thus W (m) is analytical at zero and can be represented as W α(m) =

X

wk (k, α)m zk .

k

Being replaced by Wnα =

P

wk zk it only improves convergence.

11.4.5 Final Remarks 1. For the Burger’s equation ∂u/∂t = uxx + 2ux u we have Vnα =

z1 + · · · + zn 1 + z1 α1−1 + · · · + zn αn−1

.

In the KdV case functions Vnα a rational by z, but polynomial degrees for numerator and denominator grow along with n grows [317]. Functions Vnα constructed in Theorems 11.1– 11.3 provide us a solution of the equation (11.4.1) u(x, t) = Vn (eθ1 , eθ2 , . . . , eθn ),

θj = αj x + λ(αj )t + γj

inside the convergence domain. Theorems 11.2, 11.3 guarantee convergence ∀αj > 0 and sufficiently small x. In particular, publications [40, 46, 190, 243] proved that n-solution for the KdV-type equations just coincide with the corresponding (11.4.6)-type solutions.


209

2. Theorems 11.1–11.3 also could be considered as analogs of the Poincaré-Horn theorem for ODEs. This theorem [13, 57, 139] allow to construct particular solutions for the equations dxj = Fj (x), dt

j = 1, . . . , m.

Solutions in the form of convergent series in the neighborhood of the zero stationary point, Fj (0) = 0. Assume that linear part of operator L in (11.4.13) has n eigenvalues lying on the same side of some line passing through zero point. Under this assumption theorem says that there exist an analytical substitution translating the original system into the linear one over n dimensional invariant submanifold. Thus generating a set of particular n parameter solutions. By contrast, Theorem 11.1 acts as an existence theorem of formal series, while Theorems 11.2, and 11.3 establish convergence conditions. 3. All established results easily could be extended for multidimensional variable x.

12 Discrete Boltzmann Equation Models for Mixtures

12.1 Discrete Models with Impulses on the Lattice Recently the investigations related with study of the discrete models for Boltzmann (DM-BE) were quite active [116], [42], [43] [68], [121], [127], [209], [230], [264]. The models of mixtures which allow interchange of energy between different components are widely discussed in literature. The construction of such models has a lot of applications. We read in [43]: “In fact no model (except trivial ones) has appeared before. The simplest models are given by Monaco and Preziosi in their book [209]. They are not satisfactory (as indicated by the authors [209], p. 74), because no exchange of energy between the species occurs. It is just what we mean when we say that they are trivial.”

The construction of correct models (at least for the case of two components) is defined by exceptional difficulty to overcome the restrictions of an additional invariants (for example—energy of independent components) [43], [68], [209] which are present in discrete model, but absent in initial kinetic equation. This issue results in wrong calculated hydrodynamics if we speak about numerical modeling, since locally equilibrium distribution in this case is not Maxwellian. Below, we describe the method allowing to solve this problem effectively and construct discrete models with correct numbers of invariants (normal models). Consider a mixture of particles of r kinds with given masses m1 , . . . , mr . Ordinary Boltzmann equation (with continuous velocities) has the form (see Section 9.3): X r p ∂Fi ∂Fi + , = Qij Fi , Fj , p, x ∈ Rd , (12.1.1) ∂t mi ∂x j=1

where Fi (x, p, t)—distribution function of i-th component of mixture by coordinates x and impulses p at moment t. Assuming for simplicity Fi ≡ F, Fj ≡ f , mi ≡ M, mj ≡ m, M > m > 0 and keeping impulses as only arguments of distribution functions, collision integral Qij is written Z (u, ) Qij [F, f ] = dqd|u|σij |u|2 , × F(p0 )f (q0 ) − F(p) f (q) , |u| Rd ×Sd−1

(12.1.2) Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00012-0 c 2011 Elsevier Inc. All rights reserved.

212


p q − —relative velocity; σij —cross-section of collisions, p0 and q0 — M m impulses of the particles after collision:

where u =

p0 =

p+q 1 (1 + δ) + |u|µ, 2 2

q0 =

p+q 1 (1 − δ) − |u|µ, 2 2

M−m 2mM —reduced mass, δ = , ∈ Rd , || = 1. m+M M+m For the particles of one kind the class of Goldstein-Styurtenvant-Broadwell models [121] is used; the velocities are taken on integer lattice. In [42] was proposed the theorem on approximation of Boltzmann equation in 3D case; in [43] it was given a generalization for the case of mixture. In this book also were proposed two 2D DM-BE models for case of mixture of two kinds and energy exchange. However, the first one (with 13 discrete velocities) possesses one extra invariant. The second one (with 25 velocities)—two extra invariants. If we construct a normal DM-BE over the space of model’s linear invariants while the number of discrete impulses increases, we define the basis by the given set J of vectors α − β and thus eliminate an extra invariants. Introduce uniform grid in the space of impulses with step size h as follows: the grid nodes correspond to values of impulses pm = m · h, m—vectors with integer coordinates. Writing down conservation laws for impulse and energy: where µ =

p + q = p0 + q0 p2 − (p0 )2 (q0 )2 − q2 = , 2M 2m

(12.1.3)

we see that allowing exchange of energy between two components, the relation of masses M/m should be rational. Without energy exchange, the relation of masses can be arbitrary and the energies of isolated components are kept. Formally speaking, discrete model with impulses on the lattice is a collection of: A) masses m1 , . . . , mn (some of them can be equal); B) vectors p1 , . . . , pn and C) reactions (ij) ↔ (kl). The last one are the sets of four integer indeces, decomposed in couples. Denote this set of quadruples ((ij) ↔ (kl)) as S. It indicates non ij trivial cross-sections of collisions σkl . For a collection m1 , . . . , mn ; p1 , . . . , pn ; S; ij σkl ((ij), (kl)) ∈ S we define the corresponding system of differential equations, also called a discrete collision model: X ∂fi pi ∂fi ij + , = σkl ( fk fl − fi fj ), i = 1, . . . , n. (12.1.4) ∂t mi ∂x Summation take place for all quadruples from set S, which involve index i. In other ij words, reactions with σkl 6= 0. We emphasize the fact, that equation (12.1.4) is very convenient to operate mixture elements with coinciding masses, since it avoids twoindex notations: the real quantity of different masses r is significantly less than n. Here r = 2.

Discrete Boltzmann Equation Models for Mixtures

213

A model of collisions described above is called a discrete model of the Boltzmann equation, if and only if the reactions defined comply with conservation laws for impulse and energy: pi + pj = pk + pl , pj pk pl pi + = + , 2mi 2mj 2mk 2ml

(12.1.5)

all impulses are squared in the second expression. There are r + d + 1 invariants for Boltzmann equation (11.3.4) corresponding to conservation of r kinds of particles, d components of impulse and integral of total energy. Following [43], [68] such DM-BE with r + d + 1 invariants will be called normal. Functional I=

r X

µi fi

(12.1.6)

i=1

is unnecessary invariant here, if it is conserved by the discrete model, but it is not one of these r + d + 1 invariants, i.e., it is not presented in the form of linear combination of invariants of Boltzmann equation.

12.2 Invariants P Let us study when functional I = µi fi will be conserved according to the (12.1.4). For space-uniform version of equation (12.1.4) we have 1 X ij dI X ij = µi σkl ( fk fl − fi fj ) = σkl fk fl (µk + µl − µi − µj ). dt 2 We see that for all collisions from S condition µi + µj = µ Pk + µl is necessary and sufficient condition for I to be an integral. Condition I = ri=1 µi fi is integral of equation (12.1.4) is equivalent to orthogonality condition of vector µ to the vectors ij ij ekl = ei + ej − ek − el (ei —standard basis vector), which has σkl > 0. ij Denote J—linear hull of vectors ekl , J⊥ —an orthogonal complement (space of invariants), d(J) and d(J⊥ )—their dimensions respectively: d(J⊥ ) = n − d(J). Vecij tors ekl also will be called collision vectors. Example 12.1. Two-dimensional model with six velocities situated in the corners of proper hexagon. In that case r + d + 1 = 1 + 2 + 1 = 4. However, for the given model modules of velocities are equal to unit, hence, the number of particles coincides with energy and the correct number of invariants is lowered up to 3. Number of collisions equal to three, but due to existing relation between them d(J) = 2. Therefore, d(J⊥ ) = 6 − 2 = 4 and we have one (by the way, it is well known) extra invariant.

214


Namely, such additional invariants exist for more general models with velocities in the corners of the proper 2n—gons [264]. Exercise 12.1. Calculate the number of extra invariants for 2n—gon.

12.3 Inductive Process We shall construct the discrete model step by step inserting new particles (adding new impulse) and collisions from the set S. How the dimensions d(J) and d(J⊥ ) are changed? 1. If we add particle with new velocity, then n → n + 1, therefore d(J) → d(J) and we have new invariant d(J⊥ ) → d(J⊥ ) + 1. ij

2. If add collision with vector ekl and this vector is a. linear independent from previous ones, then we “kill” one invariant: d(J⊥ ) → d(J⊥ ) − 1; b. linear dependent from previous ones, then we have the same number of invariants: d(J⊥ ) → d(J⊥ ).

Now we show by induction that two-dimensional model on the square lattice with the side L for sufficiently large L do not have extra invatiants. This statement has been implicitly used in [42, 43, 121], but perhaps it is incorrect in one-dimensional case. We start with classical Broadwell model for the light component of mixture, when there are four discrete impulses only: p2,4 = (0, ±1) and p1,3 = (±1, 0). Admissible collision between these particles is uniquely defined in our notations: e13 24 . Since a sum of dimensions of space spanning onto collision vectors d(J) and it orthogonal complement d(J⊥ ), corresponding to quantity of linear independent invariants is equal to number of degrees of freedom of the system n, then in given case d(J⊥ ) = 4 − 1 = 3. Now if we add four heavy particles possessing the same impulses (marked by the ˜˜ wave over index) to the existing light ones, and introduce collision vector e1˜ 3˜ , then 24 we obtain six invariants (three for light and three for heavy particles). Adding col˜ ˜ 31 lision vectors between these two kinds of particles e13 ˜ , e42 ˜ , we reduce the number 24 of invariants down to four. Finally, putting the light particle in the center (0, 0), we bring to number of invariants up to five. Further increasing of quantity of discrete impulses do not lead to new invariants, i.e., we need construct linear independent vector of collision for every new particle. For light particles, it is possible to make in the following way: We add symmetrically four particles with impulses p5,6 = (1, ∓1) and p7,8 = 06 07 08 (−1, ±1) and proceed colliding them with central particle: e05 14 , e12 , e23 , e34 . Here we do not have new invariants. Then we introduce the particle with impulses p9 = (2, 0) and collision e09 56 along with axial and diagonal symmetric analogs and so on, fillng up all the surface by light particles. In order that do the same with heavy particles, ˜ 0) ˜ in center, thus increasing the number of invariants up we should put the particle (0, to six. Adding the heavy particles consequently just like we did with the light ones,


215

we will obtain one extra invariant. To eliminate it we need to introduce at least one more collision between light and heavy particles, which vector is linearly independent from previously introduced. Such collision desined with respect of (12.1.3) will fix the relation of masses M/m. Therefore, we proved a proposition that our model do not possess any extra invariants.

12.4 On Solution of Diophantine Equations of Conservation Laws and Classification of Collisions How do we solve the system of equations (12.1.3)? A rather good idea is to introduce a new vector a for the first of equations (12.1.3), such that p0 = q + a, p = q0 + a.

(12.4.1)

Substituting (12.4.1) into the second equation of (12.1.3), we obtain M 2 − 1 = 2(a, q0 − q). q0 − q2 m

(12.4.2)

Assuming q0 2 − q2 6= 0 we get M (a, q0 − q) = 1+2 2 . m q0 − q2

(12.4.3)

The relation (12.4.3) reveals the possible relations for masses of light particles with given impulses q and q0 . An earlier assumption q0 2 − q2 6= 0 corresponds to collisions without interchange of energy: from (12.4.1) it follows that p2 − p0 2 = 0, thus an amount of energy does not change. Also we should note that (a, q0 − q) = 0 means either same masses or the case without exchange of energy (12.1.3). In this case parallelogram qq0 pp0 becomes a rectangle. Therefore, collisions of the heavy and light particles one can classify in the following way: 1. 2. 3. 4.

collisions between heavy particles; collisions between light particles; collisions between different particles without interchange of energy; collisions between different particles with interchange of energy.

In one-dimensional case, (12.4.2) can be divided by q0 − q 6= 0 (because for collision is absent):

q0 − q = 0

M (q + q) − 1 = 2a. m 0

(12.4.4)

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12.5 Boltzmann Equation for the Mixture in One-Dimensional Case In one-dimensional case, sphere is replaced by two points and collision law (for heavy particle) becomes p0 = ( p + q)

mM p q M ± − . m+M m+M M m

Upper (positive) sign leads to identical transformation, and lower (negative) gives nontrivial unknown transformation. We have mM M − u, m+M m+M M mM q0 = ( p + q) + u, m+M m+M p q u= − . M m p0 = ( p + q)

(12.5.1)

If m = M, then p0 = q and q0 = p and exchange of impulses occures. Hence, collision integral J( f , f ) for the particles of the same kind is equal to zero and for onedimensional case, we have Boltzmann equation without such integral: ∂F = J( f , F) = ∂t ∂f = J(F, f ), ∂t

Z

F(p0 )f (q0 ) − F(p)f (q) σ (u)|u|dq,

where p0 and q0 are defined by formulas (12.5.1).

12.6 Models in One-Dimensional Case Considering one-dimensional case for collisions we use formula (12.4.4). Let M m = 3. Then from (12.4.4) gives a = q + q0 . A simplest nontrivial solution of that equation with a = 1, q = 0, q0 = 1 gives p = 2, p0 = 1. As result of symmetrization, we obtain the following model, represented by Figure 12.1: Model I (see [291],[293]): M m = 3, 3 light and 4 heavy particles. 0

Figure 12.1 Model I:

M = 3, 3 light and 4 heavy particles. m


217

Impulses of light particles are denoted by dark dots, heavy ones—by big circles. Light particles posses impulses—0, ±1, and heavy ones—±1, ±2. This model admits ˜

˜ 0(−2)

˜ 1(−1)

three reactions with cross-sections σ 0˜2 , σ ˜ , σ(−1)(1) ˜ different from zero. The 11 (−1)(−1) impulses of particles are enumerated, and impulses of heavy particles are denoted by tilde. Now we can check whether a given model posesse a correct number of invariants: r + d + 1 = 1 + 2 + 1 = 4. Using induction process, we start from particles ˜ −1, −1) ˜ and reactions between them σ −11˜ . Thus we have 4 − 1 = 3 invariants. (1, 1, ˜ −11

˜ Adding particles 0 and 2˜ and reaction σ 1˜1 , one obtain 3 + 2 − 1 = 4 invariants. Adding 02 ˜ and remaining reaction, one get the same number of invariants—4 and particle (−2) the corresponding system of equations. The light particles are described by functions fi , i = 0, ±1, and heavy ones by Fi , i = ±1, ±2. Then

∂f0 ˜ ˜ = σ101˜2 ( f1 F1 − f0 F2 ) + σ 0−2 ˜ ( f−1 F−1 − f0 F−2 ). −1−1 ∂t The above expression reveals that function f0 is involved in two reactions. Also we have, ∂f1 1 ∂f1 ˜ ˜ + = σ −11˜ ( f−1 F1 − f1 F−1 ) + σ012˜1 ( f0 F2 − f1 F1 ). −11 ∂t m ∂x A similar equation also holds for f−1 . The equations for heavy particles can be derived in similar manner: F1 has a veloc1 2 ity M1 = 3m and is involved in two reactions; F2 has velocity M2 = 3m and is involved in one reaction. The model presented here is a unique one-dimensional symmetric normal model constructed until now.

12.7 The Models in Two-Dimensional Cases The first example of the two-dimensional model in Figure 12.2a can be obtained by symmetrization of the one-dimensional model: Model II (see [291],[293]): of equations 5 + 8 = 13.

M m

= 3, 5 light and 8 heavy particles, with a total number

Exercise 12.2. Write collection of collisions, prove that model is normal and write the corresponding system of equations. In two-dimensional case (in contrast to the one-dimensional) one can construct model with arbitrary relation of masses using induction process, both symmetric and normal one. If we would like construct model with a small number of equations, then we should take one of the solutions of Diophantine equation (12.4.2) with energy exchange. Symmetrize it, then add possible collisions and prove that an obtained set of impulses and reactions corresponds to normal model.

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p

0 p'

q'

0

0

q (a)

(b)

(c)

Figure 12.2 (a) Model II; (b) Model III; (c) Model IV.

Model III (see [291],[293]): M m = 7, 6 light and 6 heavy particles. Model is symmetric with respect to axes, not diagonals Figure 12.2c. The basic reaction with exchange of energy is represented in Figure 12.2b, where q = (0, −1),

q0 = (−1, 1),

p = (0, 3),

p0 = (1, 1).

By symmetrization with respect to axes, one obtain the following picture, presented by Figure 12.2c. Exercise. Write all reaction, prove that model is normal and write equations. Model IV: M m = 7, 8 light and 8 heavy particles. It was obtained from the third model by symmetrization with respect to diagonals. Additionally, we would like to discuss a multiple-dimensional case, generalizing Model I and Model 2 with M m = 3. We can construct a model with 6d + 1 particles. Here 2d + 1 are the light ones with impulses q0 , . . . , q2d and 4d—heavy ones with impulses p1 , . . . , p4d , where d is a dimension of physical space. The components of impulses for light particles are: p0,i = 0,

p2k−1,i = pδi,k ,

p2k,i = −pδi,k ,

i, k = 1, . . . , d.

The components of impulses of heavy particles are given below: p4k−3,i = pδi,k ,

p4k−2,i = −pδi,k ,

p4k−1,i = 2pδi,k ,

p4k,i = −2pδi,k .

For them the following linear independent collisions are defined: d−1

2k−1,2k+1 collisions of the form e2k,2k+2 between light particles;

2d − 1

similar collisions between heavy particles;

2d + 1

4k−3,2k−1 2,2k collisions between heavy and light particles e1,2 and e4k− . ˜ ,e ˜ ˜

˜

2,1

˜

4k−1,0

˜

4k,0


219

Number of invariants for this model is equal to 6d + 1 − (5d − 2) = d + 3. Also we want to outline that in one-dimensional case d = 1 the described model is considered to be unique normal (up to present time). There exist a hypothesis that another symmetric one-dimensional normal models do not exist. Papers [69], [79], [290], [291], [293], [294] have some other examples of symmetric DM-BEs in R2 for different relation of masses equal to 2, 5, 7, etc. We have the following construction procedure for discrete models of mixtures with a small number of velocities: 1. 2. 3. 4.

Take one (basic) reaction with energy exchange between components. Making it symmetric with respect to coordinate axes, obtaining semisymmetrical model. Making it symmetric with respect to diagonals, which provides a symmetrical model. Check the quantity of invariants and choose models with correct numbers.

In two-dimensional case an introduced above model with thirteen impulses is minimal symmetrical model. Among semi-symmetrical (i.e., symmetrical with respect to axes only) models the minimal one contain nine velocities [79]. Further classification of two-dimensional models and construction more complicated ones could be found in [69], [79], [294].

12.8 Conclusions In order to construct discrete models for mixtures with correct number of invariants, we need to solve the system of Diophantine equations for conservation laws of impulse and energy (12.1.3). Just one solution with energy exchange makes possible a construction of discrete model by the following scheme: (a) symmetrization; (b) calculation of invariants; (c) adding particles and reactions in the case of necessity to eliminated extra invariants; and (d) writing down the final system of equations.

12.9 Photo-, Electro-, Magneto-, and Thermophoresis and Reactive Forces The term photophoresis was proposed by Felix Ehrenhaft [100]. In his experiments dust, silver, and copper particles in gases irradiated by light “strongly exhibited a tremendous lightnegative movement, although they ought to be most heated on the side toward the light, and would expect a movement away from the light” (see [187]). Movement toward light was called lightpositive or positive photophoresis and away from it lightnegative or negative photophoresis. “During the course of the experiment, the motion of the particle traced out a spiral path. However upon magnification of a given section of a given spiral, one saw a spiral path within the path of the larger spiral. . . . In viewing these microphotographs, one had the distinct impression that something phenomenal was happening, but no definitive explanation for the observation was presented” [187].

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Theory of photophoresis usually was considered in the framework of P.N. Lebedev irradiation pressure and heat effects. For a review, see [70]. “A unilateral aerosol particle is affected by a photophoretic force and a radiation pressure force. The former is of a radiometric nature and is a result of gas molecules interacting with the non-uniformly heated particle surface” (this is the beginning of the paper [70]). “Depending upon the size and optical properties of material of a particle both irradiated and shadowed side of a particle can become more heated. That is why both negative and positive photophoresis could take place” [313]–[315]. Our explanation is based on reactive forces. A particle evaporates molecules on irradiated side—this is the cause of positive photophoresis. This explanation was accepted in comet astronomy from the 1950s, when the American astrophysicist F. Wipple suggested reactive forces connected with sublimation of comets. In laser thermonuclear synthesis reactive forces are also well known (they call it ablation sometimes). But mathematical treatment of this kind of process goes back to Maxwell [193]. The combination of those ideas was considered in [298, 300]. Paper [312] is to some extent a direct experimental justification of this topic. Negative photophoresis is considered on the basis of counterreactive (or antireactive) forces. The light makes a surface to adhere molecules: the force of momentum of adhered molecules is less (maximum twice) than of elastically reflected ones. That is why additional pressure toward the source of light appears [298, 300]. This is a simple explanation of negative photophoresis. This movement has to be considered in the framework of more general class of movements—chemoreative ones. It was discovered that big particles moves at any physical-chemical process [196]—the motion was called chemoreative. Mathematical model of that motion was created in [20, 197, 296]. This model takes into account not only forces but torques as well: that is why together with Newton second law we considered Euler equations for solids. Exact solution of the equation showed spiral trajectories [296]. In [296] we constructed the model and got spiral paths for simplest case of spherical particles, and in [20, 21] more general case of any convex particle was studied. So we got a simple explanation for negative photophoresis and Ehrenhaft spiral paths independently (i.e., being out of knowledge of Ehrenhaft) and in more general situation. This gives explanation of mysterious Ehrenhaft helixes [100, 187]. Inversely, Ehrenhaft spirals justify our general mathematical model [20, 197, 296] experimentally. Ehrenhaft himself carried out experiments on electrophoresis and magnetophoresis as well [187] and also obtained helixes. He was an outstanding and recognized experimental physicist. He was a director of Institute of Physics in Vienna, and in 1938 he came to the United States. But his theoretical attempts to explain his helixes were criticized. For instance he tried to explain helixes in electrophoresis by introducing charges smaller than electrons. His kind of thinking is understandable as the only helixes one could extract from physic textbooks were Lorenz forced ones. But the radiuses and steps of spirals were different as we shall see from our exact solution. This explanation met strong opposition of Robert Andrews Milliken, a Noble prize winner for his


221

crucial experiment on a minimal charge. Spirals in magnetophoresis Felix Ehrenhaf tried to explain in similar manner (Lorenz forces) by magnetic monopoles. Magnetic monopoles were introduced by Pole Dirac in analogy with electrons in order to make Maxwell electromagnetism equations symmetric with respect to electric and magnetic fields. Ehrenhaft theoretical conclusions were declined (but not experiments) by Dirac, Einstein, and others [187]. Their objection was—magnetic monopoles or smaller than electron charges can be obtained experimentally (if they exist) only for much higher energies. So the experiments were recognized but not explained. We see now that Ehrenhaft examined if not so fundamental phenomena, but more close to reality, more often and much more useful for technical applications: it was an example of a spiral path of a big (1/100—1/10000000 meters) particle in any physical or chemical process. In [187] we have the following eloquent passage: Also curious is the fact that the winding shapes of some of this spirals in the microphotographs reminded me of the shapes described by Nicola Tesla with respect to Plate 48 in which Tesla wrote: “One of the streamers is wonderfully interesting on account of the curiously twisted and curved appearance. It is hard to conceive how a discharge can pass through the air in this way when there exists a strong tendency to make it take the shortest route”. The keyword here is discharges as there is no any model for twisted discharges. It seems to be also the spirals from [296] in this case of northern lights.

12.9.1 Model and a System of Equations Here we follow [296]. A model can be described as follows. A particle is considered as a ball, which is determined by coordinates of center of mass R, momentum Q, angular momentum K, and the unit vector S directed from a center of mass to the center of active zone (Fig. 12.3). The forces that act on a particle are determined by collisions and a frequency of collisions has the usual in Kinetic theory expression: d = σ · (u, n) · θ ((u, n)) · f (r, p) · dr · dp. Here σ = σ (R, Q, r, p)—is a crossection of collisions {R, Q, S, K} + {r, p} → . . . that for rigid spheres have the form 1, x > 0, σ = σ (R, Q, r, p) = δ(|r − R| − ρ), θ (x) = . 0, x ≤ 0 Q p − is a relative velocity of a molecule and a m M ball, ρ is a radius of a particle, n—is a unit vector orthogonal to the surface at the point r = R − ρn (see Fig. 12.4), f (r, p) is a distribution function of molecules over momentum p and space r. We assume that some molecules are elastically reflected from the surface of the particle, and some of them adhere (the adhesion here models chemical interaction). δ(x) is a Dirac δ—function, u =

222


K S p

Q

r

R

Q

Figure 12.3 A molecule has a momentum p and space coordinate r.

dσ

u

n

Figure 12.4 A picture of a collision.

We assume that the fraction of adhering Zparticles at the point of the ball with the internal normal n is β(n), and we put A = n · β(n)dn and orientation vector S = S2

dS 1 A . Dynamics of this vector is described by the equation = · [K, S], where J is |A| dt J a moment of inertia. A system of equations can be written in the form  dR     dt     dQ    dt dK      dt       dS dt

=

Q , M

= Fchem + Felast , (12.9.1) = Mchem , 1 = [K, S]. J


223

The first equation is the definition of a velocity; the second is a Newton law with two terms. The first term is connected with inelastic collisions and has the form ZZ 2 β(n, S){p − 2µ(u, n)n}(u, n)θ ((u, n))f (R − ρn, p)dndp. (12.9.2) Fchem = ρ R3 ×S2

The second term is determined by elastic collisions ZZ Felast = 2µρ 2 n(u, n)2 θ ((u, n))f (R − ρn, p)dndp.

(12.9.3)

R3 ×S2

The third equation describes changing of a moment dK/dt and connected only with inelastic collisions (in the case of ball): ZZ Mchem = µρ 3 β(n, S)[u, n](u, n)θ ((u, n))f (R − ρn, p)dndp. (12.9.4) R3 ×S2

mM —is the reduced mass, ρ is the radius of the ball particle, u is the M+m relative velocity. Here µ =

12.9.2 System of Equations in Maxwell Equilibrium for Small Velocities of a Particle We take destribution function as Maxwellian f (t, r, p) = n0 (2π mkT)

− 23

|p|2 − · e 2mkT .

Assume that the ratio of the particle velocity and the mean square velocity of gas molecules is small. Then we get the following expressions for integrals (12.9.2)– (12.9.4): √ 8 2π · ρ 2 n0 kT Fcomp = − · ν, 3 · (1 + ) Z Z 2ρ 2 n0 kT 1− 3− Fchem = · − β(n)ndn + √ β(n)(ν, n)ndn , 1+ 4 2 π r Z ρ 3 n0 kT 2 Mchem = − · β(n)[ν, n]ndn. 1+ π r Q m Q m Here ν = —is a ratio of velocity of a body and heat velocity, = —is a M kt M M ratio of masses of molecule and a particle.

224


Let us consider β symmetric with respect to rotations around S. That means that β is a function of only one variable β = β((n, S)).

(12.9.5)

Then the system gets the following form:  dR     dt     dQ    dt dS      dt     dK   dt

=

Q , M

= (χ0 + χ1 · (Q, S)) · S − λ · Q, 1 = [K, S], J = γ [Q, S].

Here one has by integrating with Maxwellian and taking in mind (12.9.5): 1− 2 χ0 = πρ n0 kT 1+

Z1

β(ζ )ζ dζ,

−1

√ Z1 ρ 2 n0 · 2π mkT 3 − χ1 = · · β(ζ )(3ζ 2 − 1)dζ, 2M 1+ −1

  √ Z1 ρ 2 n0 · 2π mkT  8 3 − λ= · − · β(ζ )(1 − ζ 2 )dζ  , M(1 + ) 3 2 −1

√ Z1 ρ 3 n0 · 2πmkT γ =− β(ζ )ζ dζ. M(1 + ) −1

Now we shall assume that the rotation is almost constant. Then we get the following system  Q dR   = ,    dt M   dQ = (χ0 + χ1 (Q, S))S − λQ,  dt     dS   = [ω, S]. dt

ω=

K ≈ Const, J


225

Integration of this system shows that the trajectory R(t) tends to cylindrical spiral with a constant step L and diameter D:

L=

2π (P∞ , ω) , Mω2

D=

q 2 P2∞ ω2 − (P∞ , ω)2 Mω2

.

(12.9.6)

Here we have the dependence upon momentum P in the limit as time tends to infinity. Those formulae (12.9.6) give steps and diameters of cones that Ehrenhaft could obtain and it depends upon parameters in absolutely different from Lorenz manner. For more details see [20, 21, 197, 296, 299]. The case of general form of a particle was considered by Batisheva [20, 21]. Positive and negative photophoresia are quantum effects and have to be explained through photo effect [300]. In both cases photoeffect can be internal (without emission of electrons but raising them on a higher energy level) or external: internal is perhaps more appropriate for explanation of photophoresis. If photoeffect induces photosublimation (or photodisintegration) we have positive photophoresis. And if it gives photo-adsorption one has negative photophoresis. It seems that even the same substance can exhibit negative or positive photophoresis in dependence upon frequency or intensity of light. This quantum consideration [300] differs from [312] where positive photophoresis was explained through Greenhouse effect and thermophoresis. It would be interesting to create more detailed model to explain Ehrenhafts “spirals within the parts of other spirals” [187] and to explain electrophoresis and magnetophoresis. And it has to be supported by experiments—at least to repeat Ehrenhaft’s ones. But to disentangle this is beyond the scope of this article. Let say a few words on other phenomena. 1. Electrophoresis. This is a movement of a big particle in gas or in liquid in electric fields. Sometimes they call it also photophoresis (a movement in electromagnetic field). Usually they explain it as a movement of a charged particle, but the process of ionization is difficult to explain for weak electric fields. But the former quantum mechanic view and the same mathematical model works. Again Ehrenhaft spirals give experimental justification. 2. Magnetophoresis. This is a movement in magnetic field. The model is the same. It is interesting, that those two terms seems to be introduced by Ehrenhaft also, and he wrote electrophotophoresis and magnetophotophoresis [101]. 3. Thermophoresis. This is a movement of a big particle in gas or liquid under temperature gradient. Perhaps sometimes reactive forces are important in this case also. For instant, heating may induce adhesion. In other way it is difficult to clarify negative Thermophoresis (if it exists) [19]. And to wait for helixes in experiment.

In all those cases more detailed models have to wait for experiments.

12.9.3 Applications Chemoreative motion is widely spread as we have it for any big particle, chemically or physically reacted with gas. So it must have a tremendous number of applications. Direct and indirect photophoresia have several applications.

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It was already discussed for comet astronomy and for possible explanation of forms of Northern Light. A number of applications are considered for planet astronomy in paper of G. Wurm and O. Krauss [312]. There we also read “One might think of creating an artificial surface with much stronger, optimized photophoretic forces. In analogy to solar sails based on radiation pressure solar sails based on photophoresis could be much stronger. These could, e.g., be used for propulsion of small probes on Mars or in Earth’s stratosphere.” The idea was independently discussed in [300] in the following words: “Photo-reactive engine. On the basis of direct photophoresis a scheme of photoreactive engine can be proposed. A layer of substance, sublimating under photons, covers a surface of a rocket or some part of it appearing in proper moment. Now they try to construct solar sails and to use irradiation pressure. Reactive forces can help, as they are much stronger. If to use not only photons but other components of solar wind such an engine can be called Solar-reactive.”

It is interesting for applications as well that in [312] and [300] (and here we follow [193, 298, 300]) explanations of direct photophoresis are different. G. Wurm and O. Krauss use thermophoresis and solid state Greenhouse effect. Photoeffect mechanism gives another picture from quantum mechanical point of view. Thermoeffects are connected with rotation and vibration degrees of freedom and photoeffect does not enlarge temperature. That is why it will be better to use photoeffect in those engines and perhaps even to fight against thermophoresis. So both views from [300] and [312] give complementary pictures as they work for different intensities and/or frequencies. Another application that was discussed in [300] is acceleration of particles by laser beams as it in usage for laser thermonuclear synthesis. Both indirect and direct photophoresia can be applied for dusty plasma, for Northern lights. Now it is clear that Felix Ehrenhaft spiral paths and so both positive and negative photophoresia have their explanation in the framework of chemoreative motion: former as a consequence of reactive forces and the latter of counterreactive ones. By contrast, Ehrenhaft spiral paths strongly support all mathematical theory of motion of any big particle in reacting gas (chemoreative motion), constructed in papers [20, 196, and 296]. It is interesting to explain spiral path within the path of the larger spiral—this perhaps could be done by more detailed model.

12.9.4 Conclusions 1. Mathematical theory is constructed for a movement of a big particle interacted physically or chemically with gas. Especially positive and negative photophoresia, electrophoresia, magnetophoresia, and thermophoresia got some explanation. 2. Exact solutions are constructed for system of equations of rigid body motion. Are proposed ideal helix trajectories as asymptotes for any solution as time tends to infinity. This part of job was in principle carried out by Batisheva [20, 21, 197, 296, 299]. 3. Experiments of Felix Ehrenhaft were explained and spiral paths got their mathematical justification. 4. Several new experiments are proposed. Especially to repeat Ehrenhaft ones and to compare diameters and steps of cones with exact formulae (12.9.6).

13 Quantum Hamiltonians and Kinetic Equations

In this chapter we consider the correspondence between quantum Hamiltonians and classical kinetic equations of the chemical kinetics type. Namely, it will be established on the basis of conservation laws. Both quantum Hamiltonians describing the processes of creation and annihilation of particles and kinetic equations modeling similar classical systems use conservation laws. Conservation laws linear by particles densities play important role in the theory of kinetic equations. In the case of Boltzmann equation they are fundamental macroscopic values necessary for introduction of continuous medium, when the hydrodynamics equations for mean values of density, impulse and energy are written (see Chapters 9, 10, and 12 of this book; [116], [173]). In the homogenius space case of the Boltzmann equation, these laws completely define the qualitative behavior of the system. H- theorem justify tending of the system to stationary distribution, which parameters are defined from the corresponding conservation laws. For the classical Boltzmann equation, this distribution is called the Maxwellian. The idea of this correspondence (Landau-Lifshits-Sewell-Streater [177]) allows to write down the generalizations of discrete models for Boltzmann equations based on conservation laws for quantum Hamiltonians (QH) and kinetic equations (KE). Either for cases of annihilation and particle creation or triple (and even higher) order collisions. This generalization refers to a class of equations in chemical kinetics where the H- theorem holds. In this chapter, we define the space of linear conservation laws for polynomial quantum Hamiltonians when an operator depends on a number of particles; we also consider their classical analogs. Second, we consider similar conservation laws for kinetic equations, revealing the correspondence between QH ↔ KE and proving H- theorem for them. We study conservation laws for quantum and classical cases to describe the process of Raman scattering throughout the study of spectrum of polynomial Hamiltonian.

13.1 Conservation Laws for Polynomial Hamiltonians Let us consider tensor product F = Fp (Fock space) of p Gilbert spaces with the basis en1 ⊗ en2 · · · ⊗ enp , where eni , ni = 0, 1, . . .—a usual unit vector (for the details of tensor Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00013-2 c 2011 Elsevier Inc. All rights reserved.

228


+ multiplication, see [38] or Chapter 11). Also we define conjugate operators a− k , ak acting over F by the rule √ a− k |n1 i|n2 i · · · |np i = nk |n1 i|n2 i · · · |nk − 1i · · · |np i, p (13.1.1) a+ k |n1 i|n2 i · · · |np i = nk + 1|n1 i|n2 i · · · |nk + 1i · · · |np i. + Here operators a− k , ak do satisfy the standard bozon rules of commutation: h i + a− j , ak = δjk .

(13.1.2)

These operators usually are called the operators of annihilation and creation, while − operator a+ ˆ k is named as an operator of number of particles, since its eigenk ak ≡ n values are integer numbers nk -s, and eigenvectors represent the already mentioned standard basis. Many problems of quantum optics are described by Hamiltonians which are polynomial by operators of creation and annihilation. For example, the classical model of Raman scattering for the one-dimensional radiation on excitations (phonons) of crystal lattice reads as H=

2 X

+ + − − + − − ωk a+ k ak + B a0 a1 a2 + a1 a2 a0 ,

k=0

where ωk and B are real constantes. The generalizations of this model are studied in Section 13.3. Even for the simplest nonlinear model of this kind, we need to know the conservation laws to be able to study the spectrum of Hamiltonian and its asymptotic properties. Let us consider polynomial Hermitian operators of the general form X ˆ =H ˆ0+ H bαβ a+α a−β + h.c. (13.1.3) (α,β)∈J

ˆ 0 = H0 nˆ 1 , . . . , nˆ p is a diagonal operator over the given basis: Here H ˆ 0 |n1 i · · · |np i = H0 n1 , . . . , np |n1 i · · · |np i, H α

bαβ —real valued coefficients; aα = aα1 1 · · · ap p ; (α, β) ∈ J ⊂ Z+ —nonnegative integer multi-indices, i.e., the vectors from some subset J of integer 2p-dimensional lattice 2p Z+ ; letters h.c. denote Hermitian conjunction. By J1 we denote the set of vectors α − β ∈ Z p , and L—its linear hull. What kind of conditions must be set upon the set J of values of multi-indices to guarantee that operator (13.1.3) possess conservation laws linear in number of particles? The answer has been obtained in [292]: Iˆ =

p X k=1

µk nˆ k

2p

(13.1.4)

Quantum Hamiltonians and Kinetic Equations

229

is an operator that commutes with Hamiltonian (13.1.3), if vector µ = µ1 , . . . , µp ) is orthogonal to linear hull L of vectors α − β, i.e., belongs to the orthogonal complement L⊥ in Rp . Indeed, from equations (13.1.1), (13.1.2) follows: h i X X ˆ = Iˆ, H µk (αk − βk ) bαβ a+α a−β − h.c. .

(13.1.5)

(α,β)∈J k

ˆ = 0. Thus, Hamiltonian (13.1.3) has Then for arbitrary vector µ ∈ L⊥ , we obtain [Iˆ, H] p − d (d ≡ d (J1 ) = dim L) linear independent conservation laws of the form (13.1.4). Which interpretation has this conservation laws in the case of classical Hamilton function? We associate a function of 2p variables with operator (13.1.3): X H = H0 |z|2 + bαβ z¯α zβ + c.c., (13.1.6) (α,β)∈J

where c.c. denotes complex conjunction, and canonical variables z, z¯ satisfy Hamilton equation: z˙k = −i

∂H , ∂ z¯k

∂H z˙¯k = i . ∂zk

Then expression I=

p X

µk |zk |2

(13.1.7)

k=1

is the first integral of the system (13.1.6) taken for vectors µ, which for all nontrivial indices α, β from (13.1.6) satisfy condition (µ, α − β) = 0. Conservation laws (13.1.7) are well-known in classical mechanics (see [58], [212] for example). They were used to study the stability of the solutions and the study of integrability.

13.2 Conservation Laws for Kinetic Equations Kinetic equations describe changes of distribution function of the particles in chemical reactions, collisions, and so on. A simplest example of this equations are the balance equations for density of particles in binary collisions, when transition of the particles of one kind into another one is proportional to the density of colliding particles. The so-called four-component Maxwell-Braodwell model serves as a perfect example: dn1 = n3 n4 − n2 n1 ≡ f1 , dt

dn2 = f1 , dt

dn3 dn4 = = −f1 . dt dt

The discrete models of Boltzmann equation (DMBE) are the generalizations of this model (see [116], [264], and Chapters 10 and 12 of this book): dni X ij = Bkm (nk nm − ni nj ), dt jkm

i = 1, . . . , p.

(13.2.1)

230

Kinetic Boltzmann, Vlasov and Related Equations p

ij

Here n ∈ R+ —vector of p-dimensional linear space with positive components, Bkm = ji Bijkm = Bkm > 0—positive constants (cross-sections of collisions) for reaction of the form (i, j) → (k, m). Summation in (13.2.1) goes over all reactions in which participates i-th matter. H- theorem is satisfied for this system as well as for Boltzmann equation: functional p P ∂H ≤ 0. ni ln ni decreases due to system (13.2.1), i.e. H= ∂t i=1 P It is easy to check that µi ni is conserved for system (13.2.1) if for all nontrivial cross-sections of collisions holds the equality µk + µm = µi + µj . With every reaction ij with nontrivial cross-Section Bkm we associate two vectors eij and ekm ∈ Z p . They have ones in the places indicated by indices, and all other elements are zero. Hence an existence condition of linear conservation laws µk + µm = µi + µj is written as (µ, eij − ekm ) = 0. The mentioned expressions has the same form as conservation laws (13.1.4) and (13.1.7) representing quantum and classical systems. Therefore, one can establish correspondence between quantum Hamiltonians and kinetic equations based on existence of the similar conservation laws. Noting that quantum Hamiltonian modeling point wise collision of two particles (of the k and m kinds), providing as the result particles of i and j kinds (and vice versa too) is written as X ij + − − ˆ =H ˆ0+ H bkm a+ k am aj ai + h.c. ijkm

One can obtain a Hamiltonian in the form (13.1.3) just by denoting α = eij , β = ekm . Moreover, as it follows from (13.1.5) operator (13.1.4) commutes with H when ij (µ, eij − ekm ) = 0 for all indices (k, m), (i, j) where bkm > 0. From here follows the natural comparison of kinetic equations of the form (13.2.1) and quantum Hamiltonians (13.1.3), making possible the generalization of DMBE for an arbitrary power reactions. For each Hamiltonian of the form (13.1.3), we establish the correspondence with the system of equations with the same conservation laws: X dnk = (αk − βk )Bβα (nβ − nα ). dt

(13.2.2)

(α,β)∈J

It is easy to see that (13.2.2) has the same conservation laws as for (13.1.4): I=

p X

µk nk = Const,

µ ∈ L⊥ .

(13.2.3)

k=1

However, system (13.2.2) becomes an exact kinetic analog of quantum evolution equations for operators, depending on the number of particles (or corresponding P classical dynamical systems), because (13.2.2) has decreasing functional H = (nk ln nk − nk ), which complies with the H- theorem: k


X dH X = ln nk Bβα (αk − βk )(nβ − nα ) = dt k (α,β)∈J α X n α α β = Bβ (n − n )ln β ≤ 0. n

231

(13.2.4)

(α,β)∈J

Now we are ready to extend the correspondense (13.1.3)↔(13.2.2) onto continuous case. 1. Scatttering of particles with energy E(p) over potential Z 1 eipx V(p)dp U(x) = 2π has a corresponding Hamiltonian Z ZZ ˆ = E(p)ˆn(p)dp + H V(p − p0 )a+ (p)a− (p0 )dpdp0 and transport equation: Z df (p) = σ (p, p0 ) f (p0 ) − f (p) dp0 . dt 2. Four-particle Hamiltonian ˆ =H ˆ 0+ H Z + dp1 dp2 dp3 dp4 w(1, 2 → 3, 4)a+ (p4 )a+ (p3 )a− (p2 )a− (p1 ) + h.c. has a corresponding equation of Boltzmann type: Z df (p1 ) = dp2 dp3 dp4 σ (p1 , . . . , p4 )( f (p3 ) f (p4 ) − f (p2 ) f (p1 )). dt 3. Hamiltonians of quantum electrodynamics are introduced in a convenient form in the book [104]. Using the introduced technique we can introduce the kinetic equations related to them. But the problem for calculating kinetic coefficients Bαβ introduced by constantes of interaction bαβ is still actual: for example, finding velocities of chemical reactions in chemical kinetics is a fundamental problem. The relation with quantum electrodynamics is quite useful, since we can find an approximate solution from perturbation theory. By contrast, kinetic radiation transport equations are widely used nowadays, but calculation of crosssections by formulas of perturbation theory is still under study. 4. Chemical kinetics. Equations like (13.2.2) are equations describing chemical reactions for which velocities of direct and inverse reactions coincide. However in reality, direct and inverse reactions oftenly have different velocities. Therefore it is interesting to consider them from point of view of H- theorem and conservation laws.

Quantum kinetic equations need further generalizations. In [252], [263] are introduced discrete models of quantum Boltzmann equation (Uehling-Uhlenbeck equation). In [177] the analogy between kinetic equations and quantum Hamiltonians

232


were used for description of properties of dialectrics. The general discrete models of Uehling-Uhlenbeck equation are introduced and studied in [199], [201], [287]. In particular cases mentioned in [242], such models appeared in modeling of chemical kinetics. Here we give necessary generalizations of the mentioned above results, being able to derive an analog of the law (13.2.4). ∂H

Let H(n) is some function and hi = e ∂ni . Let Bβα (n)—the collection of positive functions. Consider the system of differential equations: X dni = (αi − βi )Bβα (hβ − hα ). (13.2.5) dt (α,β)

P This system also possesses conservation laws of the form µi ni , for (µ, β − α) = 0. Functional H also decreases: X dH =− Bβα (5H, α − β)(e(5H,α) − e(5H,β) ) ≤ 0. dt (α,β)

Therefore, one can construct the functional H over arbitrary collection of stationary points gaining its minimum in those points and obtain many examples of equations from chemical kinetics with given attraction points. For example, assume N = 3 in (13.2.5). Taking the first matter—boson, second, and third—fermions. Second matter denotes fermion in excited state. It emits photons (bosons), hence we obtain the third matter—nonexcited fermion. Under this assumptions we obtain the system: dn1 n2 n1 n3 =B ≡ g; − dt 1 − n2 1 + n1 1 − n3 dn3 dn2 =− = −g; dt dt 010 B ≡ B101 = b(1 + n1 )(1 − n2 )(1 − n3 ),

(13.2.6) b > 0.

Dynamical system (13.2.6) is considered in semicylinder P : n1 ≥ 0, 0 ≤ n2 , n3 ≤ 1. These restrictions are permanent, H- function is convex in P. An introduced system have two conservation laws: n1 + n2 = C1 , n2 + n3 = C2 . The second one expresses conservation for the number of fermions. Hamiltonian from Section 13.1 corresponding to system (13.2.6), has the form ˆ =H ˆ 0 + B a+ f + f − + f − f + a− , H 1 2 1 2 ± where a± are related with boson, and f1,2 —with fermions. In the same manner, one can construct a relaxation model (13.2.5) for each quantum Hamiltonian (13.1.3) for bosons or fermions with H- function of the form (see 9.4): X H= Hθj ( fj ), Hθ ( f ) = f ln f − θ (1 + θf ) ln(1 + θ f ), j

where θ = +1 for bosons, θ = −1 for fermions and θ = 0 for Boltzmann statistics.


233

13.3 The Asymptotics of Spectrum for Hamiltonians of Raman Scattering In this section, we consider the problem on asymptotics of spectrum for Hamiltonian of the form (13.1.3), for the case when conservation laws (13.1.4) convert the problem into finite-dimensional.

13.3.1 Stokes Scattering In quantum optics to describe a Raman scattering of one-mode radiation of frequency ω0 on excitations (phonons) of various kinds arising in medium, the following model (see for example [228]) is commonly used: ! p p X Y − + H= ωk a+ a− (13.3.1) k ak + B a0 k + h.c. , k=0

k=1

where ωk , B are real valued constantes. According to (13.1.5), this system has p conservation laws, written in the form Iˆ0 = nˆ 0 + nˆ p , Iˆk = nˆ k − nˆ p ,

k = 1, . . . p − 1.

(13.3.2)

We will seek eigenvector of Hamiltonian decomposing it over standard tensor multiplication basis of one-particle Fock spaces: X λj0 ... jp |j0 i · · · | jp i. (13.3.3) |ψi{3} = j0 ... jp ∈3

Since operators (13.3.2) commute with Hamiltonian, the space of eigenvalues of these operators is invariant to Hamiltonian acting (13.3.1). Hence applying Hamiltonian to every term of (13.3.3), values j0 + jp and jk − jp do not change. For simplicity we choose these constants in (13.3.3) to obtain j0 + jp = N, j1 = j2 = . . . = jp = j. Conservation laws provide establish bounds to the range of indices jk , therefore 3 in (13.3.3) is a finite set of admissible values of completing numbers for the given constants in conservation laws (13.3.2). This choice uniquely defines the set 3 by the value of the constant N, and equation (13.3.3) becomes |ψiN =

N X

λj |N − ji0 | ji1 · · · | jip .

(13.3.4)

j=0

Then the eigenvalues problem for Hamiltonian (13.3.1) over this subspace is writ(N) |ψi or ˆ ten in the form of equation to determine value λj : H|ψi N =E N p p N − j(j + 1)p/2 λj+1 + N − j + 1(j)p/2 λj−1 = " !# p X E(N) − ω0 N j = + −ω0 + ωk λj . (13.3.5) B B k=1

234


System (13.3.5) written with respect to j = 0. N also can be presented in matrix form Dλ = xλ,

x ≡ x(N) =

E(N) − ω0 N , B

(13.3.6)

where D—symmetric three-diagonal matrix of dimension (N + 1) × (N + 1) matrix with elements (no summing by j): Djk =

√ √ qj−1 δj,k+1 + qj δj,k−1 + cj δjk,

qk = kp (N − k + 1), ! p X k ck = ω0 − ωi . B

(13.3.7)

i=1

Our first problem is study the behavior of eigenvalues of matrix Djk while N → +∞. This study [228] shows the domain of applicability of such models in quantum (N) optics. Let xj eigenvalues of matrix Djk for any fixed N. To be able to detect major ˜ with elements Djk N −(p+1)/2 . Introducing the asymptotics in N, we introduce matrix D corresponding eigenvalues s(N) = x(N) N −(p+1)/2 ,

(13.3.8)

˜ and consider the traces µn of the nth degree of matrix D: N

1 X µn = N +1 j=0

(N)

xj

N p+1

!n =

1 ˜ jk n . tr D N +1

(13.3.9)

Tending N → +∞ and setting Q(z) = zp (1 − z), from (13.3.7) we obtain (detailed calculations are provided in [220]), Z1 2n 1 µ2n = Q(z)n dz + O = n N 0 2n (pn)!n! 1 = +O , N n [(p + 1)n + 1]! 1 µ2n+1 = O . N

(13.3.10)

Hence, when N → +∞ momentums of properly normalized eigenvalues (13.3.8) are defined and are finite. Then according to the theorem of spectrum radius s Ri = sup s(N) = lim µ1/n n =2 n→∞

pp . (p + 1)p+1

(13.3.11)


235 (N)

According to Xelli theorem [103] a distribution function for zeros sk N

ρN (s) =

1 X (N) δ(s − sk ) N +1

(13.3.12)

k=0

has a limit point ρ(s), and as it follows from (13.3.10) momentums of the function ρN (s) have a limit at N → +∞ σn(N)

Z+∞ Z+∞ n (N) = ρN (s)s ds, µn = lim σn = ρ(s)sn ds. N→∞

−∞

(13.3.13)

−∞

Relation (13.3.11) shows that Kalerman’s theorem [103] about one-to-one reconstruction of distribution function from its momentums is suitable in our case, too. Therefore, there exists a weak limit ρ(s) = lim ρN (s). Applying Kalerman’s theoN→∞

rem to the odd part of ρ(s), one obtain that odd part of ρ(s) vanishes due to (13.3.10). Therefore, the measure ρ(s) is symmetric with respect to zero. Returning now to a matrix presentation D, we obtain from (13.3.8) the existence of such eigenvalues that x(N) ≈ ±BRN (p+1)/2 . Then from (13.3.6) follows that for p > 1 spectrum of Hamiltonian (13.3.1) is not separated from minus infinity for any sign of constant B. This result is important, because it restricts the domain of validity for polynomial models in quantum optics, describing the interaction of radiation with matter.

13.3.2 Raman Scattering Considering Anti-Stokes Component As discussed earlier, p-dimensional system (13.3.1) have p − 1 additional conservation laws (13.3.2). However, even regarding a lower number of conservation laws, the system allows a similar aproach to study spectrum asymptotics. The key factor here is fixing the signs of constantes for conservation laws. Doing so, we get the finitedimensional system of equations on eigenfunctions. For example, for Hamiltonian generalizing, the model of so-called Stokes and anti-Stokes scattering [228]:

H=

p+1 X

− + ωk a+ k ak + BS a0

k=0

p Y

+ a− k + BAS ap+1

k=1

p−1 Y

a− k + h.c.

k=0

This Hamiltonian describes interaction of p + 2 quasiparticles, and possesses p conservation laws: nˆ k + nˆ p+1 − nˆ p = Jˆ k ,

nˆ 0 + nˆ p + nˆ p+1 = Jˆ 0 .

(13.3.14)

By analogy with “Stokes”-only approach we introduce normalized eigenvalues (N) = xk N −(p+1)/2 and consider asymptotics of momentum

(N)

sk

k = 1, . . . , p − 1;

236

Kinetic Boltzmann, Vlasov and Related Equations r(N)

1 X (N) n µn = sk r(N) k=1

for N → +∞, where r(N)—the number of integer points on polygon defined by fixing conservation laws. We denote these constants eigenvalues of operators in (13.1.4), as Nk and N, respectively. We’ll study asymptotics of momentum assuming N → +∞,

Ni → +∞,

Ni → Ai , N

i = 1, . . . , p − 1.

Using the same techniques as before, we obtain 1 , N X n 2 2n n

µ2n+1 = O µ2n =

n

k=0

ZZ C2n =

k

2n−2k C2n + O B2k S BAS

xk yn−k (1 − x − y)n

x+y ≤ 1 y − x ≤ Ai

1 N

(13.3.15)

p−1 Y

(Ai + x − y)dxdy.

i=1

Since the number of conservation laws (13.3.14) is lower by 2 than a dimension of the system, we obtained in (13.3.15) a double integral instead of single-dimensional in (13.3.10). And now once again Xelli and Calerman’s theorems give the existence of limit measure and it finiteness for normalized eigenvalues.

13.4 The Systems of Special Polynomials in the Problems of Quantum Optics In this section, we study the system of eigenfunctions for the spectrum problem considered in Section 13.3 for Hamiltonian (13.3.1). A similar calculations can be done in more general case. According to Section 13.3, eigenvector of Hamiltonian (13.3.1) has the form (13.3.4). To determine its components λk , k = 1, . . . , r(N) we use system (13.3.5). It can be simplified by introducing Pk (x) defined by formulas (N)

Pk (x) ≡ Pk (x) = uk λk (x),

√ uk+1 = uk qk+1

(13.4.1)

where qk = kp (N − k + 1) according to (13.3.7). Then (13.4.1) and (13.3.5) introduce the following system for Pk (x) (upper index is omitted to simplify the expressions): Pk+1 (x) = (x − ck )Pk (x) − qk Pk−1 (x),

P0 = 0,

P1 = 1.

(13.4.2)


237

Since (13.4.2) represents decomposition formulas for (k + 1)-th minor by row and denoting Mk (x) major minor of k-order for matrix |D − xI| (13.3.6), we obtain (N) Pk+1 (x) = (−1)k Mk (x). Therefore, the problem of finding the eigenfunctions of spectrum problem is reduced to recurrent relations (13.4.2), also known from the theory of orthogonal polynomials [269]. A major difference between (13.4.2) and classical polynomials involves numbers qk , which can be zero and negative values. Calculations are completed at step N + (N) 1, because qN+1 = 0. Then eigenvalues are the roots of polynomial PN+1 (x), equal to the matrix determinant in (13.3.6). For example, for p = 1 (the case of quadratic Hamiltonian) the roots of polynomials are integer numbers: (2N−1) P2N (x) = x

N Y

(x − 4k ), 2

2

(2N−2) P2N−1 (x) =

k=1

N Y

(x2 − (2k − 1)2 ).

k=1

13.5 Representation of General Commutation Relations Section 13.1 studied conservation laws for quantum Hamiltonians in the case of classical boson commutation relations (13.1.2). It was shown that a quantity of conservation laws, which are linear by operator depending on the number of particles, is defined by dimension of linear hull of vectors α − β, composed from degrees of polynomials. How do these results depend on the form of commutation relations between conjugate operators of creation and annihilation? How does the spectrum asymptotics depend on them? First, we consider which representations of generalizations of standard commutation relation can appear. The questions of representation theory of noncanonical commutation relations has been investigated in many papers (see, for example, [202], [280], [281], [282], [284], [288]). Let us consider commutation relations of the form a− a+ = f (a+ a− ).

(13.5.1)

We construct representations of relations (13.5.1) with a− and a+ —mutually conjugate in some Hilbert space and (13.5.1) are satisfied on the basis of eigenvectors of operators a+ a− and a− a+ . The corresponding eigenvalues are nonnegative. So, let some real function f is defined on nonnegative semiaxes. Depending on its behavior, we differ between five different types of representations for relations (13.5.1). Denote as {en } some orthogonal basis of linear space (finite-dimensional or infinitedimensional) consisting from eigenvectors of operator a+ a− . Actions of operators a± over the basis, we define in the following manner: a− en =

p

λn en−1

a+ en =

p

λn+1 en+1 ,

where collection of indices n depends from representation.

(13.5.2)

238


Substituting (13.5.2) into (13.5.1), we obtain recurrent relation defining eigenvalues λn : λn+1 = f (λn ).

(13.5.3)

Depending on behavior of sequence (13.5.3), we obtain the following representations of relations (13.5.1). 1. Representation of boson type. It is given by formulas (13.5.2), (13.5.3) for n ≥ 0, if λ0 = 0 and function f guarantees that λn > 0 for all n > 0. 2. Representation of antiboson type. Also it is given by formulas (13.5.2), (13.5.2), but for n ≤ 0. Moreover, λ0 = 0 and function f guarantees that λn > 0 for all n < 0. 3. Two-sided representation is given by formulas (13.5.2), (13.5.3) for all integer n, if function f gives λn > 0 for all integer n. 4. Representation of fermion type. This representation is constructed in Rm+1 by the same formulas (13.5.2), (13.5.3) taking 0 ≤ n ≤ m. Here m is a positive integer, such that condition λ0 = 0; λn > 0, 1 ≤ n ≤ m; λm+1 = 0 holds. 5. Periodical representation appears if there exists such m, that λ0 = λm+1 > 0. Then we construct representation in finite-dimensional space Rm+1 , having complemented formulas (13.5.2), (13.5.3) by conditions p p (13.5.4) a+ em = λm+1 e0 , a− e0 = λ0 em .

The methods of investigation of the spectrum for polynomial Hamiltonians given in Section 13.3 remains valid for nonstandard commutation relations, when the particles from different Hilbert spaces commute between each other (so-called generalized Paulions). Examples of such study are given in [202], [288], [289], where the separability criterion of the spectrum from minus infinity also was obtained.

13.6 Tower of Mathematical Physics We obtain the following construction scheme for the structure of mathematical physics: Lagrangian

Quantum Hamiltonians

Vlasov-type Kinetic equation

Boltzmann-type Kinetic equation

Continuous medium

Starting from Lorentz lagrangian (Vlasov-Maxwell; see Chapter 3), we are able to construct nearly all physical phenomena. Nearly all refer to the existence of two another “physics”: l

l

micro-world, described by Vlasov-Yang-Mills lagrangian and its generalizations; macro-world, described by Vlasov-Einstein lagrangian.


239

To obtain “nearly all physics” we have to get sparse gases, liquids, and hard bodies, as well as of plasma state of matter. Sparse gases are well described by Boltzmann equation, but are applicable only for short-range forces. Speaking about radiation and neitron transport (see Chapters 9, 10, and 11) we obtain linear Boltzmann equations and the problem of cross-sections for them is still actual. This aspect of the problem we discussed in this chapter and may be an explanation can be obtained in the field of quantum Hamiltonians. Correspondence principle quantum Hamiltonians—Kinetic equations can perhaps be applied to some questions of chemical kinetics. It was explained in detail in this chapter, too. Navier-Stokes equations for liquids also can be derived from Boltzmann equation, but there are a lot of situations where these basic equations are not applicable (see numerous paradoxes in the 6-th volume of Landau-Lifshits textbook [171]). For example, the dependence of viscosity from spatial gradients of velocity was derived in [200] for big spatial gradients, and it is not Navier-Stokes one. The modern hydrodinamics theory definitely show that basic equations are not applicable (see comments in the 6-th volume of the Landau and Lifshitz monograph). Some viscosity phenomena, especially for high velocity gradients were solved; see [200], for example. By contrast, discrete Boltzmann equations, especially lattice gases automata (LGA), are widely used today to describe complex properties of liquids. Moreover, LGAs can be efficiently used for other mediums, like hard bodies or magnetic hydrodinamics. In general, any object where collisions are described in terms of probality, can be represented as LGAs. Evidently such applications should be revised first, to avoid the appearence of extra invariants, see Sections 5.7, 6.6 and Chapter 9 in general. There are some other examples of applications [9], [10], [44], [95], [250], [285] of the conservation laws in kinetic equations. In [95] we study relations of kinetic equations with hydrodinamic approximations. Papers [9], [285] propose an approach to restore formula for entropy from equilibrium states. In particular, paper [9] states that this restoration is uniquily defined if the equation has more than one conservation law. Results [285], [44] devoted to classification of conservation laws for colisionless kinetic equations. For the free particle motion in [285] and for an arbitrary external forces in [44]. Publications [10], [250] considered time-dependent conservation laws and decreasing functionals. It was shown [250] that Boltzmann equation has eight additional conservation laws (except five classical); for the Broadwell model [10] their number is continual. Paper [199] continue the study of diagonalization of quantum Hamiltonian (they call it Orlov—Vedenyapin diagonalization problem) for the model of frequency transformation.

13.7 Conclusions 1. This chapter established the relation between quantum Hamiltonians and kinetic equations when conservation laws liner in terms of particle numbers are translated into conservation laws linear in terms of distribution functions. This analogy is valid for chemical kinetics,

240


discrete Boltzmann system of equations, triple (or higher order) collisions and coagulation— fragmentation equations, satisfying H- theorem. 2. Why do the systems of equations in mechanics are Hamiltonian (Lagrangian) ones, i.e., comply with some least action principles? The answer on this question is rather simple, but nontrivial: because elementary particles are being created and annihilated. Speaking about creation and annihilation of particles we can writedown a quantum Hamiltonian, followed by related classical one. Moving from quantum to classical representation we have the unique correspondence, while a reciprocal relation is not unique due to uncertanity of quantization. 3. “In the giant manufacture of Nature the principle of entropy plays a role of a director, which prescribes the form and providing of all bargains. The Energy conservational law plays a role of only accountant, which calculates debit and credit” (Robert Emden). Which elementary actions lead to the increase of entropy in nature? Studying the relationship between quantum Hamiltonians and Kinetic equations, one can find that the most fundamental act is the creation and annihilation of particles: the director-entropy in such a manner gambles pair of dice (according to A. Einstein).

14 Modeling of the Limit Problem for the Magnetically Noninsulated Diode

14.1 Introduction This chapter is aimed at studying the stationary self-consistent problem of magnetic insulation under space-charge limitation via the asymptotics of the Vlasov-Maxwell system. This approach has been introduced by Langmuir and Compton [173] and recently developed by Degond and Raviart [87], N. Ben Abdallah, P. Degond, and F. M’ehats [35] to analyze the space charge limited operation of a vacuum diode. In a dimensionless form of the Vlasov-Poisson system, the ratio of the typical particle velocity at the cathode to that reached at the anode appears as a small parameter [87]. The associated perturbation analysis provides a mathematical framework to the results of Langmuir and Compton [173], stating that the current flowing through the diode cannot exceed a certain value called the Child-Langmuir current. We study the extension of this approach, based on the Child-Langmuir asymptotics to magnetized flows [35]. In particular, the value of the space charge limited current is determined when the magnetic field is small (noninsulated diode). Since the arising model could not be solved analytically, it is very important to discover its properties in noninsulated and nearly-insulated cases first. For better understanding of the discussed mathematical problem and especially the correspondence of the numerical modeling results with a rising physical effects in vacuum diode first we need to introduce the description of how it really works. The other related important thing is a brief discussion of the physical processes giving rize to the diode current fluctuations. The better understanding of these properties will be needed for examining current instabilities in the nearly-magnetic insulated diode. These issues will be discussed at the end of Section 14.6. The excellent description of this process found in [154] is discussed now.

14.2 Description of Vacuum Diode The vacuum diode consists of a hot cathode surrounded by a metal anode inside an evacuated enclosure. At sufficiently high temperatures electrons are emitted from the Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00014-4 c 2011 Elsevier Inc. All rights reserved.

242


cathode and are attracted to the positive anode. Electrons moving from the cathode to the anode constitute a current; they do so when the anode is positive with respect to the cathode. When the anode is negative with respect to the cathode, electrons are repelled by the anode and the reverse current is almost zero (due to the tail of the Maxwellian distribution of the electrons it is greater than zero). The space between the anode and the cathode is evacuated, so that electrons may move between the electrodes unimpeded by collisions with gas molecules. If Vf = 0 (no filament voltage) then no emission takes place, the diode may be regarded as a parallel-plate capacitor whose potential difference is Vp . In this case, the potential distribution in the cathode-plate space is represented by a straight line which joins the points corresponding to cathode potential Vk = 0 and the plate potential Vp . When the filament voltage rises, the electrons leaving the cathode gang up in the interelectrode space as a cloud called a space charge. This charge alters the potential distribution. Since the electrons making up the space charge are negative, the potential in the cathode-plate space goes up, though all points remain at positive potential. The vector of the electric field is directed from the plate to the cathode, so all the electrons escaping from the cathode make for the plate. In this case, the plate current equals the emission current. One could say the all electrons are being sucked away from the cathode by the anode. This region is known as the emission-limited region. As the filament voltage is increased, emission increases, and so does the space charge. Electrons having low initial velocities are driven back to the cathode by the negative space charge due to the electrons. The density of the electron cloud near the cathode increases to the point where it forms a negative potential region whose minimum, Vmin , is usually within a few hundredth or tenths of a millimeter of the cathode surface. Thus, there is a high retarding electric field near the cathode (0 < x < xmin ); the vector is directed away from the cathode to the plate. To overcome this field, an initial velocity v0 of the r electrons leaving the e cathode should exceed a certain value determined by Vmin , v0 > 2 Vmin . m If the electron is below this value, the electron will not be able to overcome the potential barrier. It will slow down to a stop, and the field will push it back to the cathode. Accordingly, the retarding field region (from 0 to xmin ) contains not only electrons traveling away from the cathode, but also those falling back toward the cathode. At a constant filament voltage, a dynamic equilibrium sets in, so that the number of electrons reaching the plate and the number falling back to the cathode is equal to the number of electrons emitted by the cathode. Therefore, plate current is smaller than emission current, or the cathode produces more electrons than the anode can. We know that the minimum signal observable in an electronic circuit is set by the level of electrical noise in the system. This is caused by a random fluctuation of voltage and current or electromagnetic fields. Shot noise in a diode is the random fluctuation in a diode current, I, due to the discrete nature of electronic charge. Noise causes the signal to fluctuate around a given value. The average of these fluctuations are zero, due to their randomness. But the root mean square of the fluctuation is measurable. Perhaps a discussion of the various types of electrical noise giving rise to degradation of the observed signal, would be in order. Electrical noise may persist even

Modeling of the Limit Problem for the Magnetically Noninsulated Diode

243

after the input signal has been removed from the electronic circuit. This implies the existence of a basic limit below which signals are no longer distinguishable. The signal-to-noise ratio quantities the observability of an output signal. Hence a measure whether satisfactory amplification can be obtained is given by this ratio. Therefore in order to ensure the maximum observability of an amplified weak signal we must ensure that the noise power introduced by the circuit devices and components should be as small as possible. External sources of noise can produce electrical interference in circuits. This may be done by electromagnetic radiation. Examples of this would be narrow-frequency band sources such as radio transmitters, local oscillators and powersupply cables and also broad-band sources such as lightning and fluorescent lamps. Another means by which electrical noise may be induced in an electronic circuit from an external source is electromagnetic induction. Since magnetic fields arise from alternating currents, thus by electromagnetic induction corresponding noise signals may be induced into other circuits or different parts of the same electronic system. In order to reduce such effects we take care in the positioning of critical circuit components to take advantage of the short range of such magnetic fields. Such effects can be greatly reduced by electrostatic screening (i.e., placing the entire circuit, or at least the sensitive portions of it, inside a closed metal box and connecting the box to earth potential). It is important that the total electrostatic screening for a system is earthed at one point only this ensures that no large-area circuit earth-loops can exist in which signal may again be induced by electromagnetic induction. The main types of internal sources of noise present in electronic devices are thermal noise and shot noise. Thermal noise is due to the random motion of the current carriers in a metal or semiconductor which increases with temperature. Thermal noise arises from the random motion of electrons in materials due to their thermal energy of 3kT/2 and therefore occurs even in the absence of an applied electric field. Shot noise is due to the random flow of electrons in an electric current and is due to the particle nature of electric charge. The current flows in a vacuum diode is due to emission of electrons from the cathode which then travel to the anode. Each electron carries a discrete amount of charge and produces a small current pulse. The average anode current, Ia , is the summation of all the current pulses. The emission of electrons is a random process depending on the surface condition of the cathode, shape of electrodes, and potential between the electrodes. This gives rise to random fluctuations in the number of electrons emitted and so the diode current contains a time-varying component. Since each electron arriving at the anode is like a “shot,” the fluctuating current gives rise to a mean-square shot noise current i2s .

14.3 Description of the Mathematical Model We consider a plane diode consisting of two perfectly conducting electrodes, a cathode (X = 0) and an anode (X = L) supposed to be infinite planes, parallel to (Y; Z). The electrons, with charge −e and mass m, are emitted at the cathode and submitted to an applied electromagnetic field Eext = Eext X; Bext = Bext Z such that Eext ≤ 0

244


and Bext ≥ 0. Such an electromagnetic field does not act on the PZ component of the particle momentum. Hence, we shall consider a situation where this component vanishes, leading to a confinement of electrons to the plane Z = 0. The relationship between momentum and velocity is then given by the relativistic relations    

V(P) =

  

V = (VX , VY ),

s P , γm

γ=

1+

|P|2 m 2 c2

P = ( PX , PY ),

(14.3.1) 2

|P|

= P2X

+ P2Y ,

which also can be written V(P) = 5P E (P),

(14.3.2)

where E is the relativistic kinetic energy E (P) = mc2 (γ − 1),

(14.3.3)

and c is the speed of light. We shall, moreover, assume that the electron distribution function F does not depend on Y and that the flow is stationary and collisionless. The injection profile G( PX , PY ) at the cathode is assumed to be given whereas no electron is injected at the anode. The system is then described by the so-called 1.5 dimensional Vlasov-Maxwell model ∂F d8 dA ∂F dA ∂F +e − VY + eVX =0 (14.3.4) VX ∂X dX dX ∂PX dX ∂PY e d2 8 = N(X), X ∈ (0, L), 2 ε0 dX 2 d A = −µ0 JY (X), X ∈ (0, L), dX 2

(14.3.5) (14.3.6)

subject to the following boundary conditions: F(0, PX , PY ) = G( PX , PY ), PX > 0, F(L, PX , PY ) = 0, PX < 0, 8(0) = 0, 8(L) = 8L = −LEext , A(0) = 0, A(L) = AL = LBext .

(14.3.7) (14.3.8) (14.3.9) (14.3.10)

In this system, the macroscopic quantities, namely, the particle density N, X, and Y are the components of the current density JX , JY . In the above equations, ε0 and µ0 are respectively the vacuum permittivity and permeability. The boundary conditions are justified by the fact that the electric field E = −d8/dX and the magnetic field B = −dA/dX are exactly equal to the external fields when selfconsistent effects are ignored (N = JY = 0).


245

The 1.5 dimensional model (14.3.4)–(14.3.10) ignores the self-consistent magnetic field due to JX , which would introduce two-dimensional effects, and is only an approximation of the complete stationary Vlasov-Maxwell system. In this chapter we especially interested in the case, when the applied magnetic field is not strong enough to insulate the diode, JX does not vanish and our model can be viewed as an approximation of the Maxwell equations. In order to get a better insight in the behavior of the diode, we write the model in dimensionless variables in the spirit of [87, 88]. We first introduce the following units respectively for position, velocity, momentum, electrostatic potential, vector potential, particle density, current, and distribution function: X¯ = L, V¯ = c, P¯ = mc, E = mc2 , ¯ ε0 8 N¯ mc2 ¯ mc ¯ = ¯ F¯ = , A= , N¯ = , J¯ = −ecN, , 8 2 ¯ ¯ e e xX P2 and the corresponding dimensionless variables X P , p = = ( px , py ), X¯ P¯ q p V E v = (vx , vy ) = = p , = = 1 + p2 − 1, V¯ E¯ 1 + p2 A N J F 8 ϕ= , a= , n= , j= , f = . ¯ N¯ J¯ F¯ 8 A¯

x=

The next step is to express that particle emission at the cathode occurs in the ChildLangmuir regime: in such a situation, the thermal velocity VG is much smaller than the typical drift velocity supposed to be of the order of the speed of light c. Letting ε = VG /c, we shall assume that f (0, px , py ) = gε ( px , py ) =

1 px py g , px > 0 , ε ε ε3

where g is a given profile. The scaling factor ε3 ensures that the incoming current remains finite independently of ε, whereas the dependence on pε expresses the fact that electrons are emitted at the cathode with a very small velocity. We refer to [87, 88] for a detailed discussion of the scaling. The dimensionless system reads ε ∂f ε dϕ daε ∂f ε daε ∂f ε vx + − vy + vx = 0, (14.3.11) ∂x dx dx ∂px dx ∂py (x, px , py ) ∈ (0, 1) × R2 , d2 ϕ ε = nε (x), x ∈ (0, 1) dx2 d 2 aε = jεy (x), x ∈ (0, 1). dx2

(14.3.12) (14.3.13)

246


Here nε (x) is a particle density, jεy (x) is a current density in Y direction. The initial and boundary conditions are also transformed f ε (0, px , py ) = gε ( px , py ) = f ε (1, px , py ) = 0, px < 0, ϕ ε (0) = 0, ϕ ε (1) = ϕL , aε (0) = 0, aε (1) = ϕL .

1 px py g , , ε ε ε3

px > 0,

(14.3.14) (14.3.15) (14.3.16) (14.3.17)

Omitting the complete derivation of the limit system, when ε → 0, we need to introduce some notions and notations used ahead. Definition 14.1. We define θ (x) = (1 + ϕ(x))2 − 1 − a2 (x) as an effective potential. It is readily seen that electrons do not enter the diode unless the effective potential θ is nonnegative in the vicinity of the cathode. Therefore, we always have θ 0 (0) ≥ 0. The limiting case θ 0 (0) = 0 is the space charge limited or the Child-Langmuir regime. In view (14.3.16), (14.3.17) (it still hold in the limit ε → 0), this condition is equivalent to the standard Child-Langmuir condition dϕ dx (0) = 0. Let θL be the value of θ at the anode θL = (1 + ϕL )2 − 1 − a2L . If θL < 0, electrons cannot reach the anode x = 1, they are reflected by the magnetic forces back to the cathode and the diode is said to be magnetically insulated. This enables us to define the Hull cut-off magnetic field, which is the relativistic version of the critical field introduced in [144] in the nonrelativistic case: aH L =

q

ϕL2 + 2ϕL .

H The diode is magnetically insulated if aL > aH L , and is not insulated if aL < aL In dimensional variables, the Hull cut-off magnetic field is given by

s 2mc2 1 82L + 8L . BH = Lc e Thus our primary goal is a stugy of noninsulated, or nearly insulated diodes, which means Bext < BH . The complete derivation of the model is given in [35], while we need only its formal expressions d2 ϕ (x) = jx q dx2 d2 a (x) = jx q dx2

1 + ϕ(x) (1 + ϕ(x))2 − 1 − a2 (x) a(x) (1 + ϕ(x))2 − 1 − a2 (x)

,

(14.3.18)

,

(14.3.19)


247

with a corresponding Cauchy and boundary conditions ϕ(0) = 0, ϕ(1) = ϕL dϕ (0) = 0 dx a(0) = 0, a(1) = aL

(14.3.20) (14.3.21) (14.3.22)

Let us recall that the unknowns are the electrostatic potential ϕ, the magnetic potential a and the current jx (which does not depend on x). It is to be noticed that the whole construction of this model depends heavily on the assumption that the effective potential is positive. Actually, θ could vanish at some points in the diode, leading to closed trajectories and trapped particles. Apart of heuristic discussions there also could be made some analytical remarks about the parametric dependences jx , β. In particular, they are (1 + ϕ(x))a0 (x) − ϕ 0 (x)a(x) = β p 2 2 2jx θ(x) − (ϕ 0 (x)) + (a0 (x)) = β 2

(14.3.23) (14.3.24)

The analysis of this equations were made in [35] but the proposed approach do not provide any information to be immediately used in numerical computations. Nevertheless, this relations could be treated as an auxiliary method for verification of any jx , β made. Vector ( jx , β) hereinafter is usually refered as a parameter vector, depending on the boundary condition of the problem (14.3.18), (14.3.19). Since the analysis of the couple arbitrary chosen boundary conditions ϕL , aL is not very useful, we refer √ √ √ to θ(x) and θL especially as a distance measure. The quantities (ϕL , aL ) or (ϕL , θL ) are algebraic equivalent on R+ to define the boundary √ conditions, thus we evaluate a sets of equally-distant point and refer to them as (z, θL ), z = ϕL . Keeping in mind the above remarks we devote Section 14.4 to the analysis of the solution trajectories, their relation with the lower and upper estimations obtained by A.V. Sinitsyn and better solution approximations. In Section 14.6, we introduce the results of numerical experiments, describing the properties of the parameter vector for different “distances” θL . The numerical experiments shown that the character of the parameter curves highly depends on the quantity θL .

14.4 Solution Trajectory, Upper and Lower Solutions Finally, the limit model of magnetically noninsulation diode is described by the system of two second order ordinary differential equations (14.3.18), (14.3.19) with conditions (14.3.20)–(14.3.22). Let us introduce the definition of cone in a Banach space X. Definition 14.2. Let X be a Banach space. A nonempty convex closed set P ⊂ X is called a cone, if it satisfies the conditions:

248

(i) (ii)


x ∈ P, λ ≥ 0 implies λx ∈ P; x ∈ P, −x ∈ P implies x = O, where O denotes zero element of X.

Here ≤ is the order in X induced by P, i.e., x ≤ y if and only if y − x is an element of P. We will also assume that the cone P is normal in X, i.e., order intervals are norm bounded. ¯ u = v = 0} we introduce the norm |U|X = |u|C1 + In X ≡ {(u, v) : u, v ∈ C1 (), |v|C1 , and the norm |U|X = |u|∞ + |v|∞ in C, where U = (u, v). Here a cone P is given by P = {(u, v) ∈ X : u ≥ 0, v ≥ 0 for all x ∈ }. So, if u 6= 0, v 6= 0 belong to P, then −u, −v does not belong. We will work with classical spaces on the intervals I¯ = [a, b], Iˆ =]a, b], I = (a, b): C(I¯) with norm k u k∞ = max{|u(x)| : x ∈ I¯}; C1 (I¯) =k u k∞ + k u0 k∞ ; Cloc (I), which contains all functions that are locally absolutely continuous in I. We introduce a space Cloc (I) because the limit problem is singular for ϕ = 0. The order ≤ in cone P is understood in the weak sense, i.e., y is increasing if a ≤ b implies y(a) ≤ y(b) and y is decreasing if a ≤ b implies y(a) ≥ y(b). T Theorem 14.1 ((comparison principle in cone)). Let y ∈ C(I¯) Cloc (I). The function f is defined on I × R. Let f (x, y) is increasing in y function, then v00 − f (x, v) ≥ w00 − f (x, w) in mean on I,

(14.4.1)

v(a) ≤ w(a), v(b) ≤ w(b) implies v ≤ w on I¯. For the convenience of defining an ordering relation in cone P, we make a transformation for the problem (14.3.18)–(14.3.22). Let F(ϕ, a) and G(ϕ, a) be defined by (14.3.18)–(14.3.22). Then through the transformation ϕ = −u the limit problem is reduced to the form −

d2 u 1−u 4 ˜ jx , u, a), =jx p = F( 2 dx (1 − u)2 − 1 − a2

u(0) = 0,

a d2 a 4 ˜ jx , u, a), =jx p = G( 2 2 2 dx (1 − u) − 1 − a

a(0) = 0,

u(1) = ϕL , (14.4.2) a(1) = aL .

We note that all solutions of the initial problem, as well the problem (14.4.2), are symmetric with respect to the transformation of sign for the magnetic potential a : (ϕ, a) = (ϕ, −a) or the same (u, a) = (u, −a). Thus we must search only positive solutions ϕ > 0, a > 0 in cone P or only negative ones: ϕ < 0, a < 0. Thanks to the symmetry of problem it is equivalently and does not yields the extension of the types of sign-defined solutions of the problem (14.3.18)–(14.3.22) (respect. (14.4.2)). Once more, we note that introduction of negative electrostatic potential in problem (14.4.2) is connected with more convenient relation between order in cone and positiveness of Green function for operator −u00 that we use below.


249

Definition 14.3. A pair [(ϕ0 , a0 ), (ϕ 0 , a0 )] is called (a)

sub-super solution of the problem (14.3.18)–(14.3.22) is relative to P, if the following conditions are satisfied  \ \  (ϕ0 , a0 ) ∈Cloc (I) C(I¯) × Cloc (I) C(I¯), (14.4.3) \ \  (ϕ 0 , a0 ) ∈C (I) C(I¯) × C (I) C(I¯) loc loc 1 + ϕ0 4 ϕ000 − jx p = F(ϕ0 , a) ≤ 0 (1 + ϕ0 )2 − 1 − a2

in I,

1 + ϕ0 4 (ϕ 0 )00 − jx p = F(ϕ 0 , a) ≥ 0 in I (1 + ϕ 0 )2 − 1 − a2 a0 4 = G(ϕ, a0 ) ≤ 0 in I, a000 − jx q (1 + ϕ)2 − 1 − a20 (a0 )00 − jx p ϕ0 ≤ ϕ 0 ,

a0

4

(1 + ϕ)2 − 1 − (a0 )2

a0 ≤ a0

= G(ϕ, a0 ) ≥ 0

in I

∀a ∈ [a0 , a0 ];

∀ϕ ∈ [ϕ0 , ϕ 0 ];

in I

and on the boundary

(b)

ϕ0 (0) ≤ 0 ≤ ϕ 0 (0),

ϕ0 (1) ≤ ϕL ≤ ϕ 0 (1),

a0 (0) ≤ 0 ≤ a0 (0),

a0 (1) ≤ aL ≤ a0 (1);

sub-sub solution of the problem (14.3.18)–(14.3.22) is relative to P, if a condition (3.4) is satisfied and ϕ000 − F( jx , ϕ0 , a0 ) ≤ 0

in I,

a000 − G( jx , ϕ0 , a0 ) ≤ 0

in I

(14.4.4)

and on the boundary ϕ0 (0) ≤ 0,

ϕ0 (1) ≤ ϕL ,

a0 (0) ≤ 0,

a0 (1) ≤ aL .

(14.4.5)

Remark 14.1. In Definition 14.3 the expressions with square roots we take by modulus of effective potential θ (·). By analogy with (14.4.4), (14.4.5), we may introduce the definition of super-super solution in cone. Definition 14.4. The functions 8(x, xai , jx ), 81 (x, xϕj , jx ) we shall call a semitrivial solutions of the problem (14.3.18)–(14.3.22), if 8(x, xai , jx ) is a solution of the scalar boundary value problem ϕ 00 = F( jx , ϕ, xai ) = jx p ϕ(0) = 0,

ϕ(1) = ϕL ,

1+ϕ (1 + ϕ)2 − 1 − (xai )2

,

(14.4.6)

250


and 81 (x, xϕj , jx ) is a solution of the scalar boundary value problem a00 = G(jx , xϕj , a) = jx q

a (1 + xϕj )2 − 1 − a2

, (14.4.7)

a(0) = 0, a(1) = aL . Here xai , i = 1, 2, 3 and xϕj , j = 1, 2 are respectively, the indicators of semitrivial solutions 8(x, xai , jx ), 81 (x, xϕj , jx ) defined by the following way: xa1 = 0, if a(x) = 0; xa2 = a0 , if a = a0 be upper solution of the problem (14.4.7); xa3 = a0 , if a = a0 be lower solution of the problem (14.4.7); xϕ1 = ϕ 0 , if ϕ = ϕ 0 be upper solution of the problem (14.4.6); xϕ2 = ϕ0 , if ϕ = ϕ0 be lower solution of the problem (14.4.6). From Definition 14.4, we obtain the following types of scalar boundary value problems for semitrivial (in sense of Definition 14.4) solutions are 1+ϕ

, ϕ(0) = 0, ϕ(1) = ϕL . (1 + ϕ)2 − 1 1+ϕ ϕ 00 = F(ϕ, a0 ) = jx p , ϕ(0) = 0, ϕ(1) = ϕL . (1 + ϕ)2 − 1 − (a0 )2 1+ϕ , ϕ(0) = 0, ϕ(1) = ϕL . ϕ 00 = F(ϕ, a0 ) = jx p (1 + ϕ)2 − 1 − (a0 )2 a a00 = G(ϕ 0 , a) = jx p , a(0) = 0, a(1) = aL . 0 (1 + ϕ )2 − 1 − a2 a a00 = G(ϕ0 , a) = jx p , a(0) = 0, a(1) = aL . (1 + ϕ0 )2 − 1 − a2 ϕ 00 = F(ϕ, 0) = jx p

(A1 ) (A2 ) (A3 ) (A4 ) (A5 )

We shall find the solutions of problems (A1 )–(A3 ) for ϕ0 < ϕ 0 , where ϕ0 (xa1 ), are respectively, lower and upper solutions of problem (A1 ). The solution (ϕ, a) of the initial problem should belong to the interval \ \ ϕ ∈ 8(ϕ, 0) 8(ϕ, a0 ) 8(ϕ, a0 ), \ a ∈ 81 (ϕ 0 , a) 81 (ϕ0 , a).

ϕ 0 (xa2 )

Moreover, the ordering of lower and upper solutions of problems (A1 )–(A3 ) is satisfied ϕ0 (xa1 ) < ϕ0 (xa2 ) < ϕ0 (xa3 ) < ϕ 0 (xa2 ) < ϕ 0 (xa1 ). We shall seek the solution of problems (A4 )–(A5 ) for a0 < a0 . In this case the following ordering of lower and upper solutions of problems (A4 )–(A5 ) a0 (xϕ1 ) < a0 (xϕ2 ) < a0 (xϕ2 ) < a0 (xϕ1 ). is satisfied.


251

We go over to the direct study of the problem (14.4.6) which includes the cases (A1 )–(A3 ). Let us consider the boundary value problem (14.4.6) with F(x, ϕ) : (0, 1] × (0, ∞) → (0, ∞).

(B1 )

In condition (B1 ) for F(x, ϕ) we dropped index ai , considering a general case of nonlinear dependence F of x. We shall assume that F is a Caratheodory function, i.e., F(·, s) measurable for all s ∈ R, F(x, ·) is continuous a.e. for x ∈]0, 1],

(B2 ) (B3 )

and the following conditions hold Z1

s(1 − s)Fds < ∞.

(B4 )

0

∂F/∂ϕ > 0, i.e., F is increasing in ϕ.

(B5 )

There are γ (x) ∈ L1 (]0, 1]) and α ∈ R, 0 < α < 1 such that |F(x, s)| ≤ γ (x)(1 + |s|−α ), ∀(x, s) ∈]0, 1] × R.

(B6 )

We are intersted in a positiveT classical solution of equation (14.4.6), i.e., ϕ > 0 in P for x ∈]0, 1] and ϕ ∈ C([0, 1]) C2 (]0, 1]). The problem (14.4.6) is singular, therefore, condition (B1 ) is not fulfilled on the interval ϕ ∈ (0, ∞) and in this connection, the well-known theorems on existence of lower and upper solution in cone P does not work. It follows from Theorem 14.1, since F in (14.4.6) is increasing in ϕ, then ϕ < w for x ∈]0, 1], where ϕ and w satisfy the differential inequality (14.4.1). Theorem 14.2. T Assume conditions (B2 )–(B6 ). Then there exists a positive solution ϕ ∈ C([0, 1]) C2 (]0, 1]) of the boundary value problem (14.4.6). Proof. Let ϕ > 0 is a solution of problem (14.4.6). According to the Theorem 14.1 ϕ < w for x ∈]0, 1]. Take > 0 and consider equation 1 + ϕ + 4 = F ( jx , ϕ + , xai ), ϕ00 = jx p (1 + ϕ + )2 − 1 − (xai )2 ϕ (0) = 0,

(14.4.8)

ϕ (1) = ϕL .

Let w and ϕ are upper and lower solutions of equation (14.4.8) (below, in Proposition 14.1 is shown that such solutions really exist). Hence the theorem on monotone iterations (see [130]) gives an existence of classical solution ϕ of equation (14.4.8), which satisfies w > ϕ > ϕ for x ∈ (0, 1] and is bounded in C. Thus F ( jx , ϕ + , xai ) is bounded and there exists uniform limit lim→0 ϕ = ϕ. It follows from the last, if 0 < η < 21 , then lim→0 F ( jx , ϕ + , xai ) = F( jx , ϕ, xai ) uniformly on [η, 1 − η] and ϕ > 0 for x ∈ [η, 1 − η].

252


Since ϕ is uniformly converged on [0, 1], then it implies existence lim→0 ϕ0 (η). Therefore there exists lim→0 ϕ00 (x) on the compact subspaces (0,1) and {ϕ0 } is uniformly converged on (0,1) to a differentiable function ϕ 0 on [η, 1 − η]. From the last it follows that ϕ isTtwice differentiable on [η, 1 − η], ϕ 00 = F( jx , ϕ, xai ), x ∈ [η, 1 − η] and u ∈ C([0, 1]) C2 ((0, 1]) is a positive solution of the problem (14.4.6). Remark 14.2. A delicate moment in the proof of Theorem 14.2 is connected with the finding of a lower ϕ and an upper w solutions for perturbed problem (14.4.8). As a lesser solution, we can take the solution of equation (A1 ) (semitrivial solution ϕ), then an upper solution will be, for example, maximal solution of equation (A1 ). Application of monotone iteration techniques to the equation (14.4.6) gives an existence of maximal solution ϕ(x, ¯ jx ) such that ϕ(x, xj ) ≤ ϕ(x, ¯ xj ) < w(x) for x ∈]0, 1]. Proposition 14.1. Let 0 < c ≤ jx ≤ jmax x . Then equation (A1 ) ϕ 00 = F( jx , ϕ, 0) = jx √ ϕ(0) = 0,

1+ϕ , ϕ(2 + ϕ)

ϕ(1) = ϕL

has a lower positive solution u0 = δ 2 x4/3 ,

(14.4.9)

if p 2 2 4δ 3 ≥ 9jmax x (1 + δ )/ 2 + δ

(14.4.10)

and an upper positive solution u0 = α + βx (α, β > 0)

(14.4.11)

ϕL ≥ δ 2 ,

(14.4.12)

with

where δ is defined from (14.4.10). √ Remark 14.3. Square root is taking as |ϕ(2 + ϕ)| in the case of negative solutions. Here u0 = −x is an upper solution, and u0 = −2 + is a lower solution (0 < < 1). Hence equation (A1 ) has the negative solution only for 0 < ϕL < −2 because F(x, −2) = −∞.


253

It follows from (14.4.10), (14.4.12) that a value of current is limited by the value of electrostatic potential on the anode ϕL jx ≤ jmax ≤ F (ϕL ). x

(14.4.13)

Analysis of lower and upper solutions (14.4.9), (14.4.11) exhibits that for δ 2 = ϕL > 2 and α = β ≤ 1 interval in x between lower and upper solutions is decreased, and for the large values of the potential ϕL diode makes on regime ϕL x4/3 . Proposition 14.2. Let 0 < c ≤ jx ≤ jmax x . Then equation (A4 ) a00 = G( jx , ϕ 0 , a) = jx p

a (1 + ϕ 0 )2 − 1 − a2

,

a(0) = 0,

a(1) = aL

with a lower solution a0 = 0 and an upper solution a0 = u0 > 0, conditions (3.14), (3.16) has an unique solution a(x, jx , c), which is positive, moreover q 0 ≤ aL ≤ ϕ 0 (2 + ϕ). Proof. The positive solution of problem (A4 ) is concave and be found as a solution of initial problem with a(0) = 0, a0 (0) = c, where c is a shooting parameter. The solution a = a(x, jx , c) is unique and strongly decreasing in c because the right part of differential equation q is decreasing in a. The least nonnegative solution is f (x, jx , 0) = 0 and for 0 ≤ aL ≤ ϕL0 (2 + ϕL0 ) there exists only one solution and no positive solutions for other values aL . Remark 14.4. The problem (A5 ) is considered by analogy with problem (A4 ), change √ of an upper solution a0 = u0 to a lower a0 = u0 one and 0 ≤ aL ≤ ϕ0L (2 + ϕ0L ). Following to the definition 14.3 and Propositions 14.1, 14.2, solutions of the problems (14.4.6), (14.4.7) we can write in the form lower-lower (ϕ0 , a0 )): ϕ0 = u0 = δ 2 x4/3 ,

a0 = 0,

ϕL ≥ δ 2 ;

ϕ, a

ϕ0

ϕ0

a0 a0 0

Figure 14.1 Location of lower (ϕ0 , a0 ) and upper (ϕ 0 , a0 ) solutions

1

x

254


upper-lower (ϕ 0 , a0 ): ϕ 0 = u0 = α + βx,

a0 = 0,

δ 2 ≤ ϕL ≤ C ,

C = max{α, β};

lower-upper (ϕ0 , a0 ): ϕ0 = u0 = δ 2 x4/3 ,

a0 = u0 ,

ϕL ≥ δ 2 ,

aL ≤

p

(u0 (2 + u0 );

upper-upper (ϕ 0 , a0 ): ϕ 0 = u0 = α + βx,

a0 = u0 ,

ϕL ≤ C ,

aL ≤ a0 ≤ u0 .

14.5 Existence of Solutions for System (14.3.18)–(14.3.22) In the previous section we demonstrated the existence of semitrivial solutions of system (14.3.18)–(14.3.22). Here we show the existence of solutions for the complete system (14.3.18)–(14.3.22) using the following McKenna-Walter theorem. Theorem 14.3 (see McKenna, Walter [194]). Assume conditions (B1 )–(B6 ). We assume that there exists the ordered pair (u, u¯ ) of lower and upper solutions, i.e., u, u¯ ∈ Cloc ((0, 1])2

\

u(0) ≤ 0 ≤ u¯ (0),

u(1) ≤ uL ≤ u¯ (1);

C([0, 1])2 ,

u ≤ u¯ on (0, 1] 4

uL = (ϕL , aL ),

∀x ∈ (0, 1] : ∀z ∈ R2 , u(x) ≤ z ≤ u¯ (x), zk = uk (x); −u00k (x) ≥ hk (x, z))

(14.5.1)

∀x ∈ (0, 1] : ∀z ∈ R2 , u(x) ≤ z ≤ u¯ (x), zk = u¯ k (x) : − u¯ 00k (x) ≤ hk (x, z)

(14.5.2)

and

for all k ∈ {1, 2}. Then there exists a solution u ∈ C2 ((0, 1])2 problem −u00 = h(·, u(·)) (0, 1] u(0) = 0, u(1) = uL .

T

C([0, 1])2 of the


255

For keeping of ordering of lower and upper solutions in Theorem 14.3 (in cone P) we write differential inequalities (14.5.1), (14.5.2) in the following form ∀z ∈ [v(x), w(x)], ±

w001 (x) T ±

∀z ∈ [v(x), w(x)],

z1 = w1 (x): F1 (w1 (x), z2 ) z1 = v1 (x):

± v001 (x) S ± F1 (v1 (x), z2 ) ∀z ∈ [v(x), w(x)]; ±

w002 (x) T ±

∀z ∈ [v(x), w(x)]; ±

v002 (x) S ±

z2 = w2 (x): F2 (z1 , w2 ) z2 = v2 (x): F2 (z1 , v2 ).

Remark 14.5. The change of signs with (+) to (−) in differential inequalities is connected with adjustment of signs and ordering (≤) of lower (upper) solutions of system (14.3.18)–(14.3.22) in Definition 14.3 and lower (upper) solutions in Theorem 14.3. From the last relations we obtain   w00 (x) = F1 (w1 (x), 0) ≤ F1 (w1 , z2 ) , 00  v1 (x) ≥ sup F1 (v1 (x), z2 ) z2

  w002 (x) ≤ F2 (z1 , w2 )

 v2 (x) ≥ sup F2 (z1 , v2 ) 00

.

z1

From inequality v002 (x) ≥ supz1 F2 (z1 , v2 ), we get estimations to the value of magnetic field on the anode aL aL ≤

jx jmax F (ϕL ) ≤ x ≤ 2 2 2

(14.5.3)

taking account of (14.4.13) and θL > 0. Under realization of estimation (14.5.3) the diode works in noninsulated regime, moreover, the value aL is limited by value of electrostatic potential on the anode ϕL with a critical value ϕL = 2. In increasing of magnetic potential aL the diode transfers in isolated regime that leads to more complicated problem with free boundary. Thus, we have the following main result of this paper. Theorem 14.4. Assume conditions (B2 ), (B3 ), (B6 ), and inequalities (14.4.10), (14.4.13) and aL ≤

jx jmax F (ϕL ) ≤ x ≤ 2 2 2

256


fulfilled. Then the problem (14.3.18)–(14.3.22) possesses a positive solution in cone P such that ( ϕ000 ≥ jx F(ϕ0 , z2 ), z2 ∈ [0, ϕ 0 ] , (ϕ 0 )00 ≤ jx F(ϕ 0 , z2 ), z2 ∈ [0, ϕ 0 ] ( a000 ≥G( jx , z1 , a0 ), z1 ∈ [ϕ0 , ϕ 0 ] , (a0 )00 ≤G( jx , z1 , a0 ), z1 ∈ [ϕ0 , ϕ 0 ] where ϕ0 = δ 2 x4/3 is a lower solution of problem (A1 ), ϕ 0 = α + βx (α, β > 0) is an upper solution of problem (A1 ) with condition ϕL ≥ δ 2 ; a0 = 0 is a lower solution of p problem (A4 ) with condition 0 ≤ aL ≤ ϕ 0 (2 + ϕ 0 ). Theorem 14.4 may be used to the construction of the minimal and maximal solution of (14.3.18)–(14.3.22) on the basis of monotone-iteration method in Heikkila [130].

14.6 Analysis of the Known Upper and Lower Solutions Up to this moment the analytical solution of the ODE system defined by (14.3.18), (14.3.19) with respect to the conditions (14.3.20)–(14.3.22) is unknown. The only nown result partially describing the form of the solution trajectory was given 14.1, see also [261]. According to it, both solution trajectories are bounded by the upper and lower solutions yUP (x) = kx + b, k, b > 0 yLOW (x) = c x 2

4 3

(14.6.1) (14.6.2)

Using the boundary conditions (14.3.20), (14.3.22) one can obtain a quite good solution trajectory estimations 4

c2 ϕL x 3 ≤ ϕ(x) ≤ ϕL x,

0 ≤ a(x) ≤ aL x

defined on x ∈ [0, 1]. Here and everywhere we assume boundary conditions ϕl , aL correctly defined, i.e. θL > 0. Looking forward and leaving the discussion of numerical solution methods for next the sections, here we provide some numerical solution trajectory examples both for ϕ(x) and a(x) evaluated for different boundary conditions. The straightforward analysis of the trajectories Figures 14.2–14.4 shows that the lower solutions obtained in Section 14.4 could be made significantly better and the upper solutions are exactly ϕL x and aL x. The lower solution obviously could be written as yLOW (x) = y(1)xγ ,

γ > 1.

(14.6.3)


257

a (x) 1

0.8

0.8

0.6

0.6

a (x)

Phi (x)

ϕ (x) 1

0.4

0.4

0.2

0.2

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

x

0.6

0.8

1

x

Figure 14.2 Numerical solution for ϕL = aL = 1; numerical integration error εϕ = 2.4611389315421e − 17, εa = 6.13116328540553e − 17; estimated jx = 0.534075023488271, da (0) = 0.879738089874635. Function ϕ(x): upper solution y = x and lower solution dx 4 7 4 y = x 3 . Function a(x): upper solution y = x and lower solution y = x 3 . 10

ϕ (x)

a (x)

8

3 2.5

6

a (x)

Phi (x)

2 4

1.5 1

2 0.5 0

0 0

0.2

0.6

0.4 x

0.8

1

0

0.2

0.4

0.6

0.8

1

x

Figure 14.3 Numerical solution for ϕL = 8, aL = 3; numerical integration error εϕ = 3.27429056090622e − 16, εa = 1.19262238973405e − 17; estimated jx = 8.93859989164142, 4 da (0) = 1.72776197665836. Function ϕ(x): upper solution y = 8x and lower solution y = 5x 3 . dx 5 4 Function a(x): upper solution y = 3x and lower solution y = x 3 . 2

258


ϕ (x)

a (x) 0.8

0.3 0.25

0.6 a (x)

Phi (x)

0.2 0.15

0.4

0.1 0.2 0.05 0

0 0

0.2

0.6

0.4

0.8

0

1

0.2

0.4

0.6

0.8

1

x

x

Figure 14.4 Numerical solution for ϕL = 0.3, aL = 0.8; numerical integration error εϕ = 1.71574993795831e − 17, εa = 5.97937498125756e − 17; estimated jx = da (0) = 0.759092882499624. Function ϕ(x): upper solution y = 0.3x 0.0761231763035411, dx 4 and lower solution y = 0.18x 3 . Function a(x): upper solution y = 0.8x and lower solution 4 y = 0.8x 3 .

18 16 14

Jx, beta

12 10 8 6 4 2 20

40 z

60

80

Figure 14.5 jx , β parameter curves for θL = 1.

Here y(1) is ether ϕL or aL . The value of the parameter γ depends only on ϕL , aL and could be found numericaly. Since √ it is much easy to view the modeling results, first we draw them in a pictures for (z, θL ) = (z, 1): a) b)

Figure 14.5 assume z = ϕL = 2, . . . , 30, 35, 40, 45, 50, 60, 70, 80; Figure 14.6 assume z = ϕL = 0.45, 0.5.1, 1.25.1.5, 1.75, 2, 2.5, 3.


259

2.5

Jx, beta

2

1.5

1

0.5

0.5

1

1.5

2

2.5

3

z

Figure 14.6 jx , β parameter curves for θL = 1.

The upper curve correspond to parameter jx at Figure 14.5 and to β at Figure 14.6; the lower—to parameter β at Figure 14.5 and to jx at Figure 14.6. This two figures were provided separately for two purposes: a) b)

β trajectory seems to keep the same character over the whole interval. This character is likely to be (ax)b , b < 1; The jx trajectory character is different for z ≤ 2.5 and z > 2.5. The right branch is the (2) convex function, but at the left the sign of the jx (ϕL , aL ) is changed. The corresponding xb approximation curve could be similar to a c , for example. x +d

14.7 Conclusions One of the main conclusions to be made is the fact that a statement of limit problem does not comply with a Child-Langmuir regime that a current density jx is saturated in the case of noninsulated and nearly-insulated diode. Moreover, with a respect to the rough numerical approximations made, the current density jx will infinitely grow if the voltage applied to the diode also grows. On the other hand, the experimental data show that jx grow faster for noninsulated diode than for a nearly-insulated one. This could be seen as a preliminary numerical proof that a Child-Langmuir regime could be achieved only in magnetically insulated diode. The second conclusion refers to the nearly-insulated diode. According to the earlier made comments, the discovered properties described by a limit problem in mathematical statement fully comply with their physical expectations described in introduction section. Jointly with a first conclusion, it characterize the obtained limit model as a reliable one, that comply with the physical processes, underlying the thermo-vacuum diode with a plane cathode and anode.

15 Generalized Liouville Equation and Approximate Orthogonal Decomposition Methods

15.1 Introduction At the beginning we have to outline some aspects, concerning classic Liouville theorem (Liouville 1838 [178]) and equation for the ODE system x˙ = X(x, t),

x(t0 ) = x0 ∈ Rn

(15.1.1) (1,1)

where x ∈ Rn ; J = {t : t0 ≤ t < +∞}; Xi (x, t) ∈ Cxt (G); G = × J; ⊆ Rn —a bounded domain. If the named conditions are fulfilled, then at arbitrary moment t0 a unique solution x(t) = x(x0 , t0 , t) of the Cauchy problem (15.1.1) passes through each point x0 ∈ Rn [162, 233]. It is well known (see Nemitskij, Stepanov 1949 [215], Kaplan 1953 [153]), that the ODE system (15.1.1) has a corresponding Liouville equation: ∂ f (x, t) = Lf (x, t), ∂t

f (x, t0 ) = f0 (x).

(15.1.2)

Here L· = −

n X ∂ [Xi (x, t)·] = −div[X(x, t)·] ∂xi

(15.1.3)

i=1

is a Liouville operator. Assume f (x, t) ∈ L2 (R), t ∈ J and suppose L acting like L : C0∞ (Rn ) → L2 (Rn ).

(15.1.4)

Here function f0 (x) is defined below: f0 (x) ≥ 0,

f0 (x) ∈ C0∞ (Rn ),

Z

f0 (x)dx = 1.

Rn


(15.1.5)

262

l

l

l

l


χ(x, t) = divX(x, t)—divergence h i of vector field for ODE (15.1.1); ∂x(x0 ,t0 ,t) 0 D(x(x , t0 , t), t) = det —Jacobian of inverse transition x0 → x(x(x0 , t0 , t); ∂x0 h 0 i 0) S(x, t) = det ∂x (x,t,t —Jacobian of transition x(x0 , t0 , t) → x0 ; ∂x χ(x(x0 , t0 , t), t)—divergence of vector field for ODE (15.1.1) calculated along the solution trayectory x(t) = x(x0 , t0 , t).

As usual, we treat the Gibbs ensemble of represented points (Gibbs 1902 [109]) for the system of equations (15.1.1) as a set of the identical systems (15.1.1) having different initial states. Let t0 ⊂ be a compact Lebesgue mes t0 measure set filled by a Gibbs ensemble of represented points for the ODE system (15.1.1) at the moment t = t0 . Each of the represented points x0 ∈ t0 , moving along with the ODE (15.1.1) trayectories, is shifted to a new state x(x0 , t0 , t) = T(t, t0 )x0 ∈ t ⊂ starting from moment t0 to t. Here T(t, t0 ) is a shift operator along the ODE system (15.1.1) trayectories (Krasnoselskij 1966 [162]); t = {x(x0 , t0 , t) = T(t, t0 )x0 : x0 ∈ t0 }—an image of the set t0 according to ODE system (15.1.1). It means that t = T(t, t0 )t0 . Let mes t is a Lebesgue measure of the set t ⊂ n . Function f0 (x) which comply with conditions (15.1.5) can be considered as a distribution density function for the Gibbs ensemble of represented points from t0 , system (15.1.1). The current value of distribution density function f (x, t) ∈ L2 (Rn ), t ∈ J, is defined by the initial conditions of the Cauchy problem (15.1.2), (15.1.3). It describes the state of the Gibbs ensemble of represented points for the ODE system (15.1.1) in the image t of the set t0 . To indroduce a uniqueness and existence results we need to introduce the following assumption: ASSUMPTION (A) holds for the ODE system (15.1.1), if the solution x(t) = x(x0 , t0 , t) is nonlocally continuable (Krasnoselskij 1966 [162]) on J for all represented points x0 ⊂ t0 and is kept in ∀t ≥ t0 . Definition 15.1. We call a function f (x, t) ∈ L2 (Rn ) a classical solution of the Cauchy problem (15.1.2) with operator (15.1.3) acting according to (15.1.4), if the substitution of this function f (x, t) into the Liouville equation (15.1.2) turns it into identity. Then theorem holds [246]. Theorem 15.1. First. Let assumption (A) holds for the ODE system (15.1.1). Second. The corresponding ensemble of Gibbs represented points has an initial distribution density function f0 (x) satisfying conditions (15.1.5) in the compact set t0 ⊂ . Third. Let t = {x(x0 , t0 , t) = T(t, t0 )x0 : x0 ∈ t0 } be an image of the set t0 defined according to system (15.1.1) and D(x(x0 , t0 , t), t) 6= 0. Hence the shift operator T(t, t0 ) along the solution trayectories (15.1.1) defines a set homeomorphism t0 ⊂ onto the set t ⊂ ; There exists the unique solution of the Cauchy problem (15.1.2)–(15.1.4) ∀t ∈ J and it complies with conditions Z n ∞ f (x, t) ≥ 0, f (x, t) ∈ C0 (R ), f (x, t)dx = 1, (15.1.6) Rn

Generalized Liouville Equation and Approximate Orthogonal Decomposition Methods " −

f (x(x0 , t0 , t), t) = f0 (x0 )e

Rt

# χ(x(x0 ,t0 ,t),t)dt

t0

"

= f0 (x0 )/D(x(x0 , t0 , t), t),

Rt − χ(x( p(x,t,t0 ),t0 ,τ ),τ )dτ

f (x, t) = f0 ( p(x, t, t0 ))e

263

t0

(15.1.7)

#

= f0 ( p(x, t, t0 ))S(x, t). (15.1.8)

If we denote L—Liouville operator (15.1.3); p(x, t, t0 ) = T −1 (t, t0 )x = x0 , then there hold the relations introduced below: d ln D(x(x0 , t0 , t), t) = χ (x(x0 , t0 , t), t), D(x(x0 , t0 , t), t) t=t = 1, 0 dt ∂S(x, t) = LS(x, t), S(x, t) t=t = 1, 0 ∂t " # t R 0 Z χ(x(x ,t0 ,t),t)dt mes t = e t0 dx0 ,

(15.1.9) (15.1.10)

(15.1.11)

t0

mes t =

Zt Z

χ (x, τ ) dxdτ + mes t0 .

(15.1.12)

t0 t

The above theorem 15.1 has some other interpretations. For example, Nemitskij, Stepanov 1949 [215]; Zubov 1982 [323] treated function ρ(x, t) satisfying Liouville equation (15.1.2), (15.1.3) as a kernel or the density of the integral invariant. Resolving n equations x = x(x0 , t0 , t) with respect to n initial conditions x0 , we have x0 = T −1 (t, t0 )x ≡ p(x, t, t0 ).

(15.1.13)

Here functions p(x, t, t0 ) are the first n independent integrals of the ODE system (15.1.1). The transformation mentioned above could be made since a transition involved by shift operator T(t, t0 ) is homeomorphic and conditions on the implicit function are hold. Theorem 15.2 (Zubov 1982, [323]). Assume 1. 2.

Let the solution x = x(x0 , t0 , t) of the system (15.1.1) exists for t ∈ (−∞, +∞), t0 ∈ (−∞, +∞), x0 ∈ Rn ; Let vector function (15.1.13) exists for t ∈ (−∞, +∞), t0 ∈ (−∞, +∞), x ∈ Rn ,

then each nonnegative function ρ0 (x) 6= 0 on x ∈ Rn is continuously differentiable by all arguments and possesses a unique nonnegative solution ρ(x, t) of the equation ∂ρ(x, t) + 5 · [X(x, t)ρ(x, t)] = 0, ∂t such that ρ(x, t) = ρ0 (x) at t = t0 . Function ρ(x, t) is a kernel of integral invariant for the system (15.1.1) also.

264


Liouville theorem and equation play a great role (Hinchin 1943 [133]; Krylov 1950 [164]; Bogolubov [49]; Prigozhin 1964 [234]) in statistical mechanics. Providing a statistical justification of the principles and by discovering the structure of the multiple body and/or multiple process systems tending to the equilibrium state. Invariant measure Liouville theorem (Arnold, Kozlov, Ne˘ıshtadt 1985 [14]) is the basis for qualitative studying methods of the n–body problem (Hilmi 1951 [132]). Liouville equation was taken as an initial point for the ergodicity theory (Cornfeld, Fomin, Sinai 1980 [78]); for the kinetic theory of irreversible processes; in derivation of Vlasov-Maxwell (VM) integro-differential equations (Vlasov 1950 [305]). Exactly speaking, Vlasov equation [305] could be derived from Liouville equation for the charged particles distributuion function, neglecting particle correlations and supposing many-particle distribution function as a direct product of proper single-particle distribution functions. An application of Liouville theorem for studying VM equation also could be found in (Maslov, Fedoryuk 1985 [191]; Lewis, Barnes, Melendez 1987 [176]; Horst 1990 [143]) and some recent papers. An infinite dimensional formal Hamiltonian approach for the infinite dimensional VM system was developed by (Morrison 1980 [211]; Marsden, Weinstein 1982 [186]). The mentioned papers introduce Poisson bracket evaluation technique for the VM system and prove that it is an infinite dimensional Hamiltonian system, i.e. could be written as Liouville equation. It seems that Bogolubov was the first mathematician who introduced the “classical” representation of the Liouville equation ∂ f (q, p, t) = [H(q, p, t), f (q, p, t)], ∂t

f (q, p, t0 ) = f0 (q, p)

(15.1.14)

to describe the probabilistic properties of the canonical Hamilton equation system q˙ i =

∂ H(q, p, t), ∂pi

p˙ i = −

∂ H(q, p, t), ∂qi

qi (t0 ) = q0i ,

pi (t0 ) = p0i (15.1.15)

with an arbitrary initial states distributed in R2n phase space. Here q, p ∈ Rn —are the generalized coordinate and generalized conjugate impulse vectors correspondingly; t ∈ R = (−∞, +∞); H(q, p, t) : G → R, G ⊂ R2n+1 — Hamilton function from C2 with respect to coordinates q, p; n X ∂H ∂f ∂H ∂f [H, f ] = − ∂qi ∂pi ∂pi ∂qi i=1

is the Poisson bracket; f0 (q, p) and f (q, p, t) are the Gibbs ensemble of represented points [109] for the system (15.1.15) in R2n . This functions satisfy the probability conditions Z Z f0 (q, p) ≥ 0, f0 (q, p)dqdp = 1, f (q, p, t)dqdp = 1. R2n

R2n

Generalized Liouville Equation and Approximate Orthogonal Decomposition Methods

265

The “classical” equation (15.1.14) is a particular case of the so-called generalized Liouville equation (15.2.1), which deals with compressible or dissipative dynamics. This equation is discussed futher. One of the most reperesentative modern digests concerning generalized Liouville equations was made by Ezra [102]. An interested reader also can find there a wide list of basic and recent bibliographic references. At present Liouville theorem and generalized Liouville equation are widely used for proving existence theorems (Povzner [233]), optimal control synthesis for beam trajectories (Ovsiannikov [97, 222]); stability researches (Fronteau [105]; Rudykh [245]; Zhukov [322]); dynamic properties analysis (Misra [207]; Steeb [268]; Fronteau [106]; Rudykh [246]); qualitative investigation of dynamic systems (Cornfeld, Fomin, Sinai [78]); discovering the stochastic behavior of dynamic systems (Sinai [260]) and molecular dynamics with applications to chemistry (Tuckerman, Martyna [275]. Look for some other developments of Tuckerman’s group in Section 15.3). Due to the importance of the problem, our primary goal becomes the construction of iterative analytical integration methods. In Section 15.2, we provide a complete problem statement and introduce some remarks about analytical iterative solutions. In Section 15.3 we introduce some of the latest results to be discussed and compared later in the text. Section 15.4 contains new results concerning the other classical approach for evaluating asymptotical orthonormal decompositions—eigenvalue/eigenvector operator decomposition. Section 15.5 is a reminder for the results published in [99] necessory for the numerical modeling presented in the next section.

15.2 Problem Statement In this chapter, we consider a method of approximate integration for Cauchy problem of the generalized Liouville equation [50, 225] ∂ f (q, p, t) = Lf (q, p, t), ∂t

f (q, p, t)|t=0 = f0 (q, p)

(15.2.1)

corresponding to the autonomous system of quasicanonical Hamilton equations q˙ i =

∂ H(q, p), ∂pi

p˙ i = −

∂ H(q, p) + Q∗i (q, p), ∂qi

(15.2.2)

qi (t)|t=0 = q0i , pi (t)|t=0 = p0i . An additive inclusion of the nonpotential term Q∗ (q, p) allows to construct proper measure for existence theorem more easily. Definitely speaking, L· = [H(q, p), ·] −

n X ∂ {Q∗ (q, p)·} ∂pi i

(15.2.3)

i=1

∂ {Q∗ (q, p)·} term corresponds ∂pi i to the divergent criterium of the solution stability–instability issue. is a Liouville operator, f (q, p, t) ∈ L2 (R2n ). Here the

266


Assume the transition rule as given below: L· = D(L) = C0∞ (R2n ) → R(L) = L2 (R2n ).

(15.2.4)

q, p ∈ Rn are a vector of generalized coordinates and generalized conjugate impulse (2,2) (1,1) vector; H(q, p) ∈ Cqp (R2n )—Hamiltonian of the system; Q∗i (q, p) ∈ Cqp (R2n ) P 4 nonpotential generalized forces; χ (q, p) = ni=1 ∂p∂ i Q∗i (q, p) divergence of vector field for the system (15.2.2); [·, ·] is Poisson bracket; f0 (q, p), f (q, p, t) are the initial and current values of probability density function for the Gibbs ensemble of represented points in the system of equations (15.2.2) in R2n ; Z

f0 (q, p)dqdp = 1,

4

t ∈ R+ = {t : 0 ≤ t ≤ ∞}.

R2n

The equation (15.2.1) and corresponding operator (15.2.3), (15.2.4) are widely studied. Research papers [225, 226] seem to be the most earlier ones. In the semigroup theory the Cauchy problem (15.2.1) with operator (15.2.3) acting as C0∞ () → L2 (), ⊂ R2n was studied in papers [244]. In particular, there was proved a existence and uniquines theorem for the problem (15.2.1). But this theorem can not provide an effective algorithm to the analytical problem solution. Such development is of a great practical importance for the complex multidimensional generalized Liouville equation (15.2.1) and it’s solution f (q, p, t). To obtain a solution one can integrate (15.2.1) numerically [62], but it is hardly acceptable, since f (q, p, t) complitely describes the system properties dependending on q, p, t. Thus it is more valuable to obtain such a relationship in analitical form. Moreover, a numerical integration takes a lot of computational time even for small dimensions n and advanced integration methods as quasi-Monte Carlo (q-MC) on the low discrepancy lattices. On the other hand, since the probability density function f (q, p, t) completely describes a system (15.2.2), we need it to describe the time dependences of the mean and dispersion for generalized coordinate and impulse vectors q, p also. Using this prior statistical knowledge we can try to reduce the dimension of the problem, keeping the statistical properties unchanged, applying specific group renormalization method. Or to replace the initial problem with an equivalent one (also of smaller dimention). The term “equal” should be defined very carefully for each problem type. Nevertheless, this idea could be applied more effectively if we want to study the local means, for example. This method, i.e., modeling the mean properties of the dynamic systems, combined with advanced Monte-Carlo (MC) and q-MC numerical integration techniques proved to be very usefull for the group of Berkley scientists lead by Chorin (see articles [71]–[73] for example). Called “Stochastic optimal prediction” in general, it is compatible with Hamiltonian formalism and becomes very usefull for a preliminary research.


267

Therefore we are interested in evaluating f (q, p, t) as analytical expression: converging series which coefficients could be simply evaluated and the convergence properties could be analytically studied. If the orthonormal function system 2n {hk (q, p)}∞ k=0 ∈ L2 (R ) is apriory known or already constructed, the solution of the initial problem (15.2.1) is translated into the solution of the infinite system of differential equations to determine the dk coefficient values in expansion f (q, p, t) = P∞ d (t)h k (q, p). On the first glance this approach seems to be pure analytical, but k=0 k it does not void the applicability of numerical methods, especially numerical integration methods. One of the main joints between them concerns the convergence study. Since the majority of applications can be studied only in terms of truncated series P d f (q, p, t) ≈ N k=0 k (t)hk (q, p), and as it will be shown later—the formal convergence criteria are hard to check, just taking care of formal condition (15.1.5) becomes a real headache. But it can be a unique available computational criteria. On the other hand, we can find the solution f (q, p, t) of the problem (15.2.1) on the basis of iterative operator method in the small time space over the apriori constructed 2n system of orthonormal functions {9k (q, p)}∞ k=0 ∈ L2 (R ): f (q, p, τ ) =

∞ X

ak (τ )9k (q, p).

k=0

A proposed approach makes it possible to bypass the solution of the infinite system of differential equations and establish some convergence propositions. The method introduced below could be treated as a combination of the mentioned approaches for the equation (15.2.1).

15.3 The Overview of Preceeding Results Here we state some earlier results for the problem (15.2.1). Only some of them were published before internationally. Since the approaches used there are rather simple, we do not focus on their details.

15.3.1 An Approach of M.E. Tuckerman Group on the Classical Mechanics of Non-Hamiltonian Systems Here we will outline some basic results obtained by Professor M.E. Tuckerman and his workgroup (see [276, 277] for example) on the generalized Liouville equation and non-Hamiltonian dynamical systems. It is known that Hamiltonian flow preserve the measure of phase space treated as a Euclidean manifold. It differes for non-Hamiltonian flow, since measure is not preserved in general case. Thus we could state another question: “Is there an invariant measure keeping phase volume unchanged for non-Hamiltonian flows?” A partial answer could be found in the discussed articles. It is shown that such a key concepts of Hamiltonian systems such as “invariant measure” and “continuity” can be generalized

268


to the non-Hamiltonian case by a proper treatment of the geometry of the phase space and that an invariant measure on the phase space manifold can be derived. Thus, we introduce a general Riemannian manifold to derive the generalized Liouville equation for non-Hamiltonian systems of the type ∂ √ √ ( f g) + ∇ · ( f g˙x) = 0, ∂t

(15.3.1)

x˙ i = ξ i (x, t)

(15.3.2)

with

a non-Hamiltonian dynamical system for the evolution of the n coordinates x = x1 , . . . , xn with initial values x01 , . . . , x0n . The n coordinates describe a point P of an n-dimensional Riemannian manifold G with metric G. The phase space must be treated as a general Riemannian manifold with arbitrary curvature, and the volume n-form, which determines the volume element in an arbitrary coordinate system, should be √ expressed as w˜ = gdx1 ∧ . . . ∧ dxn . The general statement of Liouville’s theorem for non-Hamiltonian system becomes f (xt , t)e−w(xt ) dxt1 . . . dxtn = f (x0 , 0)e−w(x0 ) dx01 . . . dx0n

(15.3.3)

√ with g = e−w(x) . Right now we are able to establish the correspondence between (15.3.1)–(15.3.3) and the problem stated is this paper. Liouville equation (15.1.3) for the ODE sys√ tem (15.1.2) coincides with (15.3.2) for g = 1, and the generalized Liouville equation (15.2.1) for Non-Hamiltonian system (15.2.2) is a partial case. Hence the Liouville theorem for non-Hamiltonian dynamics allows us to find more than just an invariant measure construction. We are able to derive the initial distribution function (15.3.3) f0 used for Liouville operator Lf0 . All that we need is a Liouville operator Lf0 . While discussing the possibilities one should have note invariant measure construction for the generalized Liouville equation written for a real non-Hamiltonian system usually appears to be very difficult. Hence we take an initial distribution function first without paying attention on the existance of such invariant measure. Second, we evaluate the Poisson bracket or Liouville operator to study the dynamics of the initial Hamiltonian or Non-Hamiltonian system. Moreover, Tukermann’s approach is not concerned with the solution of the Liouville equation. One just use the generalized law on the conservation of the phaze volume. It allows to construct the invariant measure without the proper solution of the Liouville equation. Differentely speaking, we are dealing with statistical ensembles for the non-Hamiltonian systems, using only the conservation laws. Such an approach used for derivation of (15.3.2) is based on the differential geometry and is well known for ages. The new term is its application for the nonequilibrium dissipative thermodynamics.


269

15.3.2 Small–Time Parameter Method Considering the formal asymptotic solution of Cauchy problem (15.2.1) by the method of the small time space, proposed in [223] and defined by mapping τ = 1 − e−st , s > 0 4

we transform the initial infinite time interval R+ in a small, finite one J = {τ : 0 ≤ τ < 1}. This technique is quite general and was applied to Liouville equation by G. Rudykh and A. Sinitsyn at the earlier 1980s. This results never were translated into english earlier, but still are valuable. The small–time transformation is also well known to the applied mathematitians, becouse it suits for numerical integration over the semi-infinite intervals. The obvious benefit for a such kind of transform lies in power series expansion over time scale. Using finite interval [0, 1] one has |τ k | ≤ 1, k = 0, 1, . . ., thus all attention while studying convergence of (15.3.5) is payed to the bounds of series coefficients. In our particular case a transformed Cauchy problem becomes (1 − τ )

1 ∂ f (q, p, τ ) = Lf (q, p, τ ), ∂τ s

f (q, p, τ )|τ =0 = f0 (q, p).

(15.3.4)

A solution of the Cauchy problem (15.3.4) in the small time space was studied in the form of asymptotic expansion f (q, p, τ ) =

∞ X

fk (q, p) · τ k .

(15.3.5)

k=0

Substituting (15.3.5) into (15.3.4) and equating the coefficients at the same τ orders, one obtain fk (q, p) =

1 k−1 fk−1 (q, p) + [H(q, p), fk−1 (q, p)]− k sk n 1 X ∂ − {Q∗ (q, p) · fk−1 (q, p)}. sk ∂pi i i=1

Hence "k−1 # Y 1 1 fk (q, p) = k L 1 + r L f0 (q, p), s s

k = 2, 3, . . .

r=1

with k−1 Y r=1

1 1 1 1 + r L = ak−1 + ak−2 L + · · · + k−1 a0 Lk−1 s s s

(15.3.6)

270


where a0 = 1/(k − 1)!, ak−1 = 1, and s1 s2 s (−1)i a0 · 3 ai = i! ... ... si si =

k−1 X

zil ,

1 s1 s2 ... ... si−1

0 2 s1 ... ... si−2

0 0 3 ... ... si−3

... ... ... ... ... ...

0 0 0 , . . . . . . s1

i = 1, 2, . . . , k − 1,

zl = −l.

l=1

So, the expression (15.3.6) can be written in brief form fk (q, p) = b1k Lf0 (q, p) + b2k L2 f0 (q, p) + · · · + bkk Lk f0 (q, p).

(15.3.7)

P ak−j > 0, kj=1 bjk sj = 1 for k = 1, 2, . . .. j ks The recurrent relation (15.3.7) express fk (q, p) in terms of initial probability density function f0 (q, p). Consequent evaluation of fk (q, p) terms by equation (15.3.7) with further backward substitution into (15.3.5) gives a formal asymptotic solution of the Cauchy problem (15.3.4). Using the definition of a small time parameter method, one can derive the solution f (q, p, t) of the initial Cauchy problem (15.2.1). To study the convergence properties of the developed asymptotic solution, we need one assumption to be made. Let there exist such constant c > 0, that ∀k = 1, 2, . . . there holds the following inequality Here bjk =

kLk f0 (q, p)kL2 (R2n ) ≤ ck · k f0 (q, p)kL2 (R2n ) .

(15.3.8)

Honestly speaking, this condition is rather rigid and will be disscused further. Nevertheless, such a formal assumptions are typical both in pure and applied math. A general example is a Kantorovitch’s lemma on the local convergence of Newton method. Everyone knows that there exist a set of conditions to guaranty the convergence, but it is hard to check them. Another remark concerns the “width” of this local domain. It could happen that an initial point selected upon this conditions could be treated as a perfect approximation in hardware computatons. Here and that follows we denote Lk f0 (q, p) = L(Lk−1 f0 (q, p))—an embedded Liouville operator. Remark 15.1. Since the generalized Liouville operator is unbounded, it’s embedded degree Lk has a restricted definition domain and in general, more restricted then a domain of the initial operator L. Proposition 15.1. Let for the generalized Liouville operator L defined by formula (15.2.3) acts according to (15.2.4); The inequality (15.3.8) and condition cτ/s < 1 are


271

P k hold. Then the series ∞ k=0 fk (q, p)τ is weakly convergent to the element f (q, p, τ ) ∈ 2n L2 (R )—a solution of the Cauchy problem (15.3.4). Besides, there holds an estimation

∞

X

k fk (q, p)τ

k=0

≤ L2 (R2n )

s · k f0 (q, p)kL2 (R2n ) . (1 − τ )(s − cτ )

(15.3.9)

Proposition 15.2. Let the generalized Liouville operator (15.2.3) act according to (15.2.4); The inequality (15.3.8) and condition c/s < 1 are satisfied. Then the series P ∞ k 2n k=0 fk (q, p)τ strongly converges to the element f (q, p, τ ) ∈ L2 (R )—the solution of Cauchy problem (15.3.4) and the relation ( fk , fn )L2 (R2n ) ≤

c2 · k f0 kL2 (R2n ) = B (s − c)2

holds. Remark 15.2. If the inequality c/s < 1 holds, then the inequality cτ < 1 holds either. P∞ k Differently speaking, if series k=0 fk (q, p)τ converges strongly to the element 2n f (q, p, τ ) ∈ L2 (R ), then it is weakly convergent also. 2n Let the orthonormalized function system {9k (q, p)}∞ k=0 ∈ L2 (R ) be constructed ∞ from the linear independent elements of the sequence { fk (q, p)}k=0 ∈ L2 (R2n ) by the Gram-Schmidt orthogonalization process [152]. Below we provide strong converP gence conditions of a series ∞ a k=0 k (τ )9k (q, p), where

ak (τ ) =

∞ X

! fi τ , 9k

.

i

L2 (R2n )

i=k

Proposition 15.3. Let the generalized Liouville equation (15.2.3) act according to (15.2.4); P∞ The inequality (15.3.8) and condition cτ/s < 1 are fulfilled. Then the series k=0 ak (τ )9k (q, p) based upon the an orthonormal system of functions 2n 2n {9k (q, p)}∞ k=0 ∈ L2 (R ) is strong convergent to the element f (q, p, τ ) ∈ L2 (R )—the solution of the Cauchy problem (15.3.4), and

∞

X

ak (τ )9k (q, p)

k=0

L2 (R2n )

∞

X

≤ fk (q, p)τ k

≤

∞ X k=0

|ak (τ )|2 ≤

≤

L2 (R2n )

k=0

s · k f0 (q, p)kL2 (R2n ) , (1 − τ )(s − cτ ) s2 (1 − τ )2 (s − cτ )2

· k f0 (q, p)k2L

2 (R

2n )

.

272


P ∞ 2n 2 Since ∞ k=0 |ak (τ )| < +∞ and {9k (q, p)}k=0 ∈ L2 (R ), then according to RieszFisher theorem [2, 147] there exits a unique function f (q, p, τ ) ∈ L2 (R2n ) such that ak (τ ) = (f , 9k )L2 (R2n ) ,

∞ X

|ak (τ )|2 = k f k2L

2 (R

2n )

.

k=0

P It is well known that ak (τ ) and ∞ k=0 ak (τ )9k (q, p) are exactly the coeficients and Fourier series of the function f (q, p, τ ) ∈ L2 (R2n ). On the other hand, the Gram matrix 4 8 = 9ij = (9i , 9j )L2 (R2n ) ,

i, j = 0, 1, 2, . . .

2n of {9k (q, p)}∞ and positive definite. Then k=0 ∈ L2 (R ) is bounded P P∞the sequence ∞ 2 |a (τ )| < +∞, and according to [270] the series k k=0 k=0 ak (τ )9k (q, p) strongly 2n converges to the function f (q, p, τ ) ∈ L2 (R ). 2n The constructed above orthonormal system of functions {9k (q, p)}∞ k=0 ∈ L2 (R ) 2n is not complete in L2 (R ). However it is known [152], that every incomplete orthonormal system of functions could be extended to get the complete one by associating a number of proper functions. In practice to realize such an extension is fairly complicated. Since Gram matrix 8 = {9ij } is bounded and positive definite, the count¯ able set N = {9k (q, p)}∞ k=0 is an orthonormal basis of the subset [9k ] [270], i.e., every element from [9¯k ] is expanded in a unique strong convergent series. Here [9¯k ] is a closure of the linear hull [9k ]. This approach, to be said—recurrent relation (15.3.4), (15.3.5) also could be used for symbolic processing. But as it will be shown later, there exist an other decomposition with a simpler expression. Moreover, the numerical modeling for the small-time transformation could be done for the known parameter s only. This limitation occures due to the convergence conditions stated in the above propositions 15.1, 15.2, and 15.3 where the transformation parameter s depends on boundness constant c.

15.3.3 Hermite Polynomial Decomposition Since the choice of the basis system of functions for the solution decomposition could be quite arbitrary, we revised the applicability of the Herimite time–space polynomial decomposition to the problem (15.2.1). Hence the basic problem statement could be written as ∞ X k=0

fk (q, p)Hk0 (t) =

∞ X

Lfk (q, p)Hk (t)

k=0

with respect to the definition and the properties of the Poisson bracket.

(15.3.10)


273

Denote Hk (x)—Hermite polynomial defined by recursive expression Hk+1 (x) = 2xHk (x) − 2kHk−1 (x),

H0 = 1,

H1 = 2x;

Z+∞ √ 2 e−x Hk2 (x)dx = π 2k k!. −∞

Proposition 15.4. A solution of the problem (15.2.1) in series (15.3.10) based on orthogonal Hermite polynomial is f (q, p, t) =

∞ X

fk (q, p)Hk (t)

(15.3.11)

k=0

fk (q, p) =

1 1 Lfk−1 (q, p) = k Lk f0 (q, p). 2k 2 k!

(15.3.12)

Here Lk f = L(Lk−1 f ) is an embedded Liouville operator. Since Hermite polynomials are orthogonal on R we can deal with their orthonormal analogs hk (t) =

Hk (t) , k 1√ 2 2 π 4 k!

Z

2

e−t h2k (t)dt = 1.

R

The corresponding problem solution (15.3.11), (15.3.12) becomes f (q, p, t) =

∞ X

fˆk (q, p)hk (t).

(15.3.13)

k=0

fˆk (q, p) decomposition coefficients linearly depends form fk (q, p): 1 fˆk (q, p) = √ Lk f0 (q, p). 2k k!

(15.3.14)

To discuss the convergence properties of the (15.3.13) decomposition, we need to remind some basic definitions, see [45, 84, 152] and [269] for example. Definition 15.2. Let f (x) ∈ L2 (V). Then the norm in L2 (V) is defined as

k fk =

p

 1 2 Z (f , f ) =  ω(x)| f (x)|2 dx V

with respect to weight function ω(x) > 0 such that

2 V |ω(x)| dx < ∞.

R

274


Definition 15.3. Let functions f (x), h(x) ∈ L2 and ω(x) > 0—real function are defined in domain V. Then  1 2 Z 2 d( f , h) = k f − hk =  ω(x)| f (x) − h(x)| dx V

is called the distance between functions f and h. Definition 15.4. Assume that relation Z d2 (sn , s) = ω(x)|sn (x) − s(x)|2 dx → 0,

n → ∞.

V

is fulfilled for the functions s0 (x), s1 (x), . . . , ∈ L2 (V). Then the sequence {sn }∞ n=0 converge to the function s(x) ∈ L2 (V) in mean. First of all we have to note that Hermite polynomials do not belong to L2 (R). Nevertheless, assuming Vˆ = R2n × R, we will be able to obtain some useful converˆ gence estimations in L2 (V). Theorem 15.3. Let the boundness condition (15.3.8) kLk f0 (q, p)kL2 (R2n ) ≤ ck k f0 (q, p)kL2 (R2n ) , c > 0 holds ∀k = 1, 2, . . .. Then

2 ˆ k=0 k fk (q, p)kL2 (R2n+1 )

P∞

< ∞ and the series (15.3.13) con2

verge in mean with respect to weighting function e−t to the solution f (q, p, t) ∈ L2 (R2n+1 ) of the problem (15.2.1) and for partial sums sn (q, p, t) of the decomposition (15.3.13) holds an estimation c2 1 ksn (q, p, t) − f (q, p, t)kL2 (R2n+1 ) ≤ √ k f0 (q, p)kL2 (R2n ) e 2 . 2n

P Proof. Denote again sn (q, p, t) = nk=0 fˆk (q, p)hk (t). Hence, according to the definition of the convergence in mean we have to prove that lim ksn (q, p, t) − f (q, p, t)k2L

2n+1 ) 2 (R

n→∞

= 0.

Since f (q, p, t) can be written as series (15.3.13), the limit relation becomes

lim k

n→∞

∞ X

 fˆk (q, p)hk (t)k2L

2 (R

k=n+1

2n+1 )

≤ lim  n→∞

∞ X k=n+1

2 k fˆk (q, p)hk (t)kL2 (R2n+1 ) 


k fˆk (q, p)hk (t)k2L

2

Z

275

2

2

e−t | fˆk (q, p)hk (t)| dqdpdt ≤

= (R2n+1 ) R2n+1

Z ≤

2

| fˆk (q, p)| dqdp ·

R2n c2k

≤

Z

2

e−t h2k (t)dt = k fˆk (q, p)k2L

2 (R

2n )

·1 ≤

R

2k k!

k f0 (q, p)k2L

2n 2 (R )

Hence  lim d2 (sn (q, p, t), f (q, p, t)) ≤ lim 

∞ X

ck

2

k f0 (q, p)kL2 (R2n )  = √ k k! 2 k=n+1  2 ∞ k X c  = k f0 (q, p)k2L (R2n ) lim  √ 2 n→∞ k k=n+1 2 k!

n→∞

n→∞

According to the Cauchy–Buniakovsky inequality 2

   ∞ ∞ 2k X X 1 c ×  ≤  ≤ √ k k! k! 2k 2 k=n+1 k=n+1 k=n+1 "∞ # "∞ # 2 X 1 X c2k 1 ec 1 c2 × = · 2 · e = ≤ n+1 k! 2n 2 2n+1 2k 

∞ X

ck



k=0

k=0

and  lim 

n→∞

∞ X

ck

2

 ≤ ec2 lim 1 = 0. √ n→∞ 2n k k=n+1 2 k!

15.3.4 Advanced Convergence Results Using ideas introduced above, we can also assert a generalized result concerning properties for the mean convergence. First of all, assume that ∀k = 1, 2, . . . kLk f0 (q, p)kL2 (R2n ) ≤ ϕ(k) · k f0 (q, p)kL2 (R2n ) ,

ϕ(k) ≥ 0.

(15.3.15)

276


Function ϕ(k) also could depend upon a number of additional parameters. Hence a new limit relation becomes ∞ X

lim d2 (sn (q, p, t), f (q, p, t)) = lim k

n→∞

n→∞

2 (R

2n+1 )

≤

k=n+1

 ≤ lim  n→∞

fˆk (q, p)hk (t)k2L

2

∞ X

ϕ(k) k f0 (q, p)kL2 (R2n )  = √ k k=n+1 2 k!

2 ϕ(k)  ≤ = k f0 (q, p)k2L (R2n ) lim  √ 2 n→∞ k k! 2 k=n+1 

∞ X

∞ X ϕ 2 (k)

≤ k f0 (q, p)k2L

2

(R2n ) k=0

= k f0 (q, p)k2L

2

k!

∞ X ϕ 2 (k) (R2n ) k=0

k!

× lim

n→∞

lim

∞ X 1 = 2k

k=n+1

1

n→∞ 2n

The last relation reveals the fact that our sequence converge in mean with respect to 2 weighting function e−t to the solution f (q, p, t) ∈ L2 (R2n+1 ) of the problem (15.2.1) if series ϕ 2 (k)/k! is convergent. Then k f0 (q, p)k2L

∞ X ϕ 2 (k)

2n 2 (R )

k=0

1 1 ≤ Const · k f0 (q, p)k2L (R2n ) lim n = 0 n 2 n→∞ 2 k! n→∞ 2 lim

and there holds the following theorem: Theorem 15.4. Let the boundness condition (15.3.15) kLk f0 (q, p)kL2 (R2n ) ≤ ϕ(k) · k f0 (q, p)kL2 (R2n ) , ϕ(k) ≥ 0 P∞ P 2 ϕ 2 (k) ˆ holds ∀k = 1, 2, . . . and ∞ k=0 k! = Const ∞. Then k=0 k fk (q, p)kL2 (R2n+1 ) < P ˆ ∞ and a series ∞ k=0 fk (q, p)hk (t) converge in mean with respect to weighting function 2 −t e to the solution f (q, p, t) ∈ L2 (R2n+1 ) of the problem (15.2.1). Remark 15.3. Comparing the results provided by theorems 15.3 and 15.4 one can see, that from the formal point of view a boundness assumption (15.3.15) is more flexible then the initial condition (15.3.8). Nevertheless, the appropriate choice of ϕ(k) function lead us to different results. For example, using the standard series when k is interpreted as series coefficient


277

index only, the application of D’Alambert and Raabe convergence criteria give us the following results: ϕ(k + 1)
n k!

1k

√ n = lim d M = d. n→∞

According to the definition of the power series convergence radius, assume 0 < x2 < 1/d. Then we get the following result: ck ↔ M

dk k! ⇐⇒ ck ↔ Mk!, dk

k = 1, 2, . . .

with corresponding factorial (extended) boundness condition for mean convergence: kLk f0 (q, p)kL2 (R2n ) ≤ Mk! · k f0 (q, p)kL2 (R2n ) ,

0 < M ∞.

(15.3.16)

Since boundness parameters c and M are finite numbers, and factorial grows asymptotically faster then a number’s degree, then we can combine theorems 15.3, 15.4 in the following manner: Corollary 15.1. Let the boundness condition kLk f0 (q, p)kL2 (R2n ) ≤ max(M1 ck , M2 k!) · k f0 (q, p)kL2 (R2n ) , 0 < c; M1 , M2 ∞ holds ∀k = 1, 2, . . . Then

P∞ 2 ˆ ˆ k=0 k fk (q, p)kL2 (R2n+1 ) < ∞ and a series k=0 fk (q, p)hk (t) 2 −t to the solution f (q, p, t) ∈ respect to weighting function e P∞

converge in mean with L2 (R2n+1 ) of the problem (15.2.1).

It is evident that boundness condition appeared in corollary 15.1 is much more flexible and extends the applicability of the proposed solution technique.

15.3.5 Partial Conclusions The advantages of the used polynomial decompositions are the relative simplicity of coefficient recurrence relations and their convergence properties both in the infinite initial and finite transformed solution time domains. Assuming Liouville operator L to be bounded (15.3.8), we can choose a proper small time space transformation parameter s:τ = 1 − e−st to guarantee strong or/and weak asymptotic series convergence to the solution of the problem (15.3.4) in R2n . By contrast we have a fast growing computational complexity of evaluating embedded Liouville opearator even for two and three dimensional generalized vectors q, p. Hence all this techinques mainly could be used for qualitative solution analysis. Nevertheless, once we have some analytic truncated series approximation, we are free to choose any kind of trayectories and impulse vectors dependencies for numerical modeling and/or visualization.


279

15.4 Eigen Expansion of Generalized Liouville Operator The other classical approach to solve Cauchy problem (15.3.4) assumes that expansion of the solution f (q, p, τ ) over the system of eigenfunctions {gk (q, p)}∞ k=0 to be found. Here we needs the generalized Liouville operator (15.2.3) defined on the test functions C0∞ (R2n ). Thus, translating them into complex Hilbert space L2 (R2n ) we are able to fullfill our task. Without loss of generality let us assume {gk (q, p)}∞ k=0 to be an orthonormal sequence of eigenfunctions of operator (15.2.3), complete in R(L) ⊂ L2 (R2n ); {λk }∞ k=0 is corresponding sequence of eigenvalues; Z

gk (q, p)¯gn (q, p)dqdp = δkn

R2n

Lgk (q, p) = λk gk (q, p),

gk (q, p) ∈ D(L) = C0∞ (R2n ).

The solution of the Cauchy problem (15.3.4) we will seek in the form of asymptotic expansion f (q, p, τ ) =

∞ X

vk (τ )gk (q, p),

vk (τ ) =

k=0

Z

f (q, p, τ )gk (q, p)dqdp.

(15.4.1)

R2n

Let τ = 0. Then (15.4.1) becomes f (q, p, 0) = f0 (q, p) =

∞ X

vk (0)gk (q, p).

k=0

To find the functions vk (τ ) we pose the Cauchy problem (1 − τ )

1 ∂ {vk (τ )gk (q, p)} = L{vk (τ )gk (q, p)}, ∂τ s 4

vk (τ )|τ =0 = vk (0) = v0k whose solution has the form vk (τ ) = v0k · (1 − τ )−λk /s .

(15.4.2)

Formula (15.4.2) could be derived directly from (15.4.1). In fact, differentiating (15.4.1), we get d 1 vk (τ ) = · (Lf (q, p, τ ), gk (q, p))L2 (R2n ) . dτ s(1 − τ )

280


Since there holds an identity (Lf (q, p, τ ), gk (q, p))L2 (R2n ) = vk (τ ) · (Lgk (q, p), gk (q, p))L2 (R2n ) , then the relation λk d vk (τ ) = · vk (τ ) dτ s(1 − τ ) is fulfilled. Taking vk (τ )|τ =0 = v0k , it immediately follows (15.4.2). Function f (q, p, τ ) defined by a series (15.4.1) with a coefficients defined by expression (15.4.2) is the solution of the problem (15.3.4). PCauchy ∞ A series (f , g )L2 (R2n ) · gk will be convergent independently of the terms k k=0 ¯ where L = order. Moreover, the sum is equal to f (q, p, τ ) for f (q, p, τ ) ∈ [L], {gk (q, p)}∞ . k=0 Therefore a series (15.4.1) can be differentiated termwise ∞

X ∂ f (q, p, τ ) = v˙ k (τ )gk (q, p), ∂τ

v˙ k (τ ) =

k=0

d vk (τ ). dτ

Multiplying both sides of previous equality by (1 − τ ) and taking into account that 1 (1 − τ )gk (q, p) · v˙ k (τ ) = vk (τ )Lgk (q, p) s we obtain ∞

∂ 1X f (q, p, τ ) = vk (τ )Lgk (q, p) = ∂τ s k=0 (∞ ) 1 X 1 L vk (τ )gk (q, p) = Lf (q, p, τ ). s s (1 − τ ) ·

k=0

Here f (q, p, τ )|τ =0 = f0 (q, p). Thus we derived the formula f (q, p, τ ) =

∞ X

v0k · (1 − τ )−λk /s · gk (q, p)

(15.4.3)

k=0

representing the solution expansion of the generalized Liouville equation with respect to complete orthonormal system of eigenfunctions {gk (q, p)}∞ k=0 of operator L in R(L). Making the backward time scale substitution we transform (15.4.3) into f (q, p, t) =

∞ X k=0

v0k · eλk t · gk (q, p).


281

Now we have had to pay attention for evaluating eigenfunctions gk (q, p) and eigenvalues λk of the operator L. To this end we take an element gi (q, p) and consider the expansion with respect to orthonormal function system gi (q, p) = r0i ψ0 (q, p) + r1i ψ1 (q, p) + · · · + rij ψj (q, p) + · · ·

(15.4.4)

where rji = (gi , ψj )L2 (R2n ) . The equality (15.4.4) holds for gi (q, p) ∈ [ψ¯k ]. Since a series on the right of (15.4.4) is convergent independently of the terms order, there holds an equility Lgi (q, p) = r0i Lψ0 (q, p) + r1i Lψ1 (q, p) + · · · + rji Lψj (q, p) + · · · or λi gi (q, p) = r0i Lψ0 (q, p) + r1i Lψ1 (q, p) + · · · + rji Lψj (q, p) + · · · .

(15.4.5)

Denote xik = (gi , ψk )L2 (R2n ) ,

aik = (Lψi , ψk )L2 (R2n ) .

(15.4.6)

Scalar product of (15.4.5) and function ψk (q, p) could be written λi xik = r0i a0k + r1i a1k + · · · + rji ajk + · · · or λi xik = xi0 a0k + xi1 a1k + · · · + xij ajk + · · · in notations (15.4.6). The last relation could be expressed in the standard matrix form λi Xi = AXi .

(15.4.7) 4

Thus expansion coefficients rji = (gi , ψj )L2 (R2n ) = xij of the function gi (q, p) in a series (15.4.4) are set by a linear algebraic problem on eigenvalues (15.4.7) for infinite matrix A. Taking some given number of elements in truncated series (15.4.1) the problem (15.4.7) could be solved for certain finite matrix A. Here λi and gi (q, p) are complex in general, matrix A—non simmetrical one. But under certain suppositions an operator iL, i2 = −1 will be simmetrical and even selfajoint. In this particular case λi are real and gi (q, p) are imaginal. Assuming this, one can derive an efficient algorithms for 4

solving (15.4.7) for a coefficients ajk = (Lψj , ψk )L2 (R2n ) = akj . Using natural notations for the Gram-Schmidt orthonormalization process we get ψi (q, p) = α0 f0 (q, p) + α1 f1 (q, p) + · · · + αi fi (q, p), fi (q, p) = γ0 ψ0 (q, p) + γ1 ψ1 (q, p) + · · · + γi ψi (q, p).

282


Therefore the following chain of equalities holds Lψi (q, p) = α0 Lf0 (q, p) + α1 f1 (q, p) + · · · + αi Lfi (q, p) = = ξ1 f1 (q, p) + ξ2 f2 (q, p) + · · · + ξi fi (q, p) + ξi+1 fi+1 (q, p) = = β0 ψ0 (q, p) + β1 ψ1 (q, p) + · · · + βi ψi (q, p) + βi+1 ψi+1 (q, p), and if j > i + 1, then (Lψi , ψj )L2 (R2n ) = (ψi , Lψj )L2 (R2n ) = 0. Hence matrix A is a band matrix, or exactly speaking, it has a three diagonal form

a1

b1

A=

b1 a2 b2

0 b2 a3 b3

0

b3 a4 .. .

.. .. ..

. . .

.

.. .

.. .

(15.4.8)

The eigenvalue problem for the banded matrices a well studied and a proper algorithms could be found in hundreds of books and research articles. The standard numerically stable approach like QR-shift method could be directly applied to the matrix A of the given structure. Fixing the number of series (15.4.3) terms we solve problem (15.4.7) for matrix A (15.4.8) of some finite dimension. For larger dimensions—the case of the special interest in practice—the computational error grows and elements ψi (q, p) loose their orthogonal property in numerics. Making a short resume, the eigen decomposition technique hardly could be recommended for general practical applications especially for complicated analitical initial distribution functions f0 (q, p) since they generate linear systems of complex valued nonsymmetric matrices turning the truncated solution to be numerical one. The numerical stability characteristics are uncertain. Also we have a tripled (at least) computational requirements.

15.5 Hermitian Function Expansion Since the standard Hermite polynomial provides only a convergence in mean (see theorem 15.1) and Hk (t), hk (t) do not belong to L2 (R), we are interested in obtaining some other expansions over a certain set of functions {uk }∞ k=0 ∈ L2 (R). Such functions based on Hermite orthogonal polynomials are usually called Hermitian [269], or associated Hermite functions, see Chapter 22, “Orthogonal Polynomials” [1]. Formally they are constructed as a parametric family of orthonormal functions with additional


283

useful properties un (t) = Z

r √

2 a − at2 H (at)e , n πn!2n

0 0, a > 0 to be some constants and conditions (15.3.8), (15.5.9) are fulfilled. Denote  √ !2s  √ !2s k k X X c 2 c 2  < ∞. ξ(c, a) = max  sup λ(2k) , sup λ(2k+1) s s a a k→∞ k→∞ s=0

s=0

Using an introduced notation ξ(c, a), the estimation (15.5.8) becomes ∞ X

k fk (q, p)k2L

2

(R2n )

≤ k f0 (q, p)k2L

2

k=0

(R2n )

ξ 2 (c, a)

∞ X (2k + 1) + (cν)2 k=0

= k f0 (q, p)k2L

2

(2k + 1)!

=

2 2 ξ (c, a) cosh(1) + (cν) sinh(1) c 2. Then P ∞ 2 k=0 k fk (q, p)kL2 (R2n ) < ∞ and according to Riesz–Fisher theorem the series are convergent ∞ X

fk (q, p)uk (t) → f (q, p, t),

f (q, p, t) ∈ L2 (R2n+1 )

k=0

to the unique function. Comparing results stated in propositions 15.1, 15.2, 15.3, and theorems 15.3, 15.5 there arise some limitations for it’s analytical/numerical applications as expansion series: 1. The formal boundness assumption (15.3.8) is the sufficient and necessory condition to ensure the convergence of the expansion series for small–time space method or Hermite–based decompositions; 2. To ensure the applicability of Riesz–Fisher theorem, one should exactly know the value of the bounding constant c. This value is needed for evaluating the correct small–time parameter s or Hermite associated function parameter a; 3. Supposing Liuoville operator boundness constant c to be finite unknown number, we can guarantie only the convergence in mean for standard Hermite polynomial decomposition.

15.6 Another Application Example for Hermite Polynomial Decomposition A novel approach proposing decomposition of the distribution function fj (t, x, v), index letter j = i, e for ions and electrons respectively was proposed and successfully

286


applied by Japanese scientists A. Suzuki and T. Shigeyama [266]. They took VlasovMaxwell system of equations in the R × R3 × R3 phase space as a model of collisionless plasmas. Assuming plasma to be stationary and homogeneous in one direction we have physical variables independent from time t and selected direction, for example z. Hence we can introduce Hermitian decomposition depending on relation vz /vj , j = i, e. And once again the orthogonal property of Hermite polynomials produced a recurrent relation for decomposition coefficients. Studying several different types of equilibrium conditions they were able to derive analytical solutions as well as for nonMaxwellian equilibrium making a deep insight into the properties of self-consistent plasma configurations. Since this paper was an initial step it does not provide any kind of general convergence conditions or theorems. But in our case the most important is the applicability of the Hermite-type decompositions for a wide range of fundamental equations describing the building of modern physics.

Glossary of Terms and Symbols 1D 3D BEq BVP DM-BE KdV KE LGA MC ODE QH RSS VM VP

one-(single) dimensional three-dimensional Branching equation Boundary-value problem Discrete models of Boltzmann equation Korteweg-de Vries Kinetic equation Lattice gases automata Monte-Carlo Ordinary differential equation Quantum Hamiltonian Rudykh-Sidorov-Sinitsyn Vlasov-Maxwell Vlasov-Poisson

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