The Boltzmann Equation Seminar 1970 to 1971

Courant Institute of Mathematical Sciences

The Boltzmann Equation F. Alberto Grunbaum

0

New York University

THE BOLTZMANN EQUATION

Seminar 1970 -1971

Edited by F. Alberto Granbaum

HONG KONG POLYTECHNIC LIBRARY

4

Courant Institute of Mathematical Sciences New York University

ii

The Courant Institute publishes a number of sets of lecture notes. A list of titles

currently available will be sent upon request.

Courant Institute of Mathematical Sciences 251 Mercer Street, New York, New York 10012

Copyright Courant Institute of Mathematical Sciences 1972

111

Table of Contents

PREFACE .

.

INTRODUCTION

.

v

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

. .

.

.

.

.

.

.

. .

.

.

.

.

. .

.

.

.

. vii

.

.

.

CASE, K. M., The Rockefeller University The Soluble Boundary Value Problem of Transport Theory

1

CHORIN, ALEXANDRE J., Courant Institute of Mathematical Sciences Numerical Solution of the Boltzmann Equation .

.

.

.

.

.

.

25

.

55

DORFMAN, J. R., University of Maryland, College Park, and COHEN, E. G. D., The Rockefeller University Velocity Correlation Functions in 2- and 3-Dimensions, I. Low Density . . . . . . . . . . . . . . . . . . . . .

GRf`JNBAUM, F. ALBERTO, Courant Institute of Mathematical Sciences

On the Existence of a "Wave Operator" for the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . LAX, PETER D., Courant Institute of Mathematical Sciences Exponential Modes of the Linear Boltzmann Equation .

.

NICOLAENKO, B., Courant Institute of Mathematical Sciences, and University Heights . . Dispersion Laws for Plane Wave Propagation . .

.

.

.

.

.

103

.

111

.

125

NICOLAENKO, B., Courant Institute of Mathematical Sciences, and University Heights Operator-Valued Analytic Continuation and the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . 173

THURBER, J. K. Purdue University Spectral Concentration and High Frequency Sound Propagation . . . . . . . . . . . . . . . . . .

. .

.

.

.

211

V

PREFACE

These notes are based upon a series of lectures given at a seminar on the Boltzmann equation, held at the Courant Institute during 1970-1971.

This seminar was organized by F. A. Griinbaum

and P. D. Lax.

An effort was made to keep the lectures and the notes accessible to a wide audience, particularly to people not too familiar with the physical setup.

Thus the emphasis in most of

the lectures is on the mathematical analysis of the equation. All of the lectures are presentations of current research,

and in many cases they indicate directions that should be explored.

Not all of the talks in the seminar have been included here.

In particular, H. Grad, H. P. McKean, Jr. and J. Percus

gave some lectures based on work of their own that had been already published.

For completeness we list here the references.

H. Grad, Asymptotic theory of the Boltzmann equation, Phys.

Fluids, 6 (1963). Asymptotic theory of the Boltzmann equation, II, in Rarified Gas Dynamics, vol. 1, Academic Press (1963).

Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, Symposia in Applied Math., vol. 17, AMS (1965). High frequency sound according to the Boltzmann equation, Jour. SIAM, 14 (1966).

vi

Singular and nonuniform limits of solutions of the Boltzmann equation, in Transport Theory, SIAM-AMS Proc., vol. 1,

(1969).

H. P. McKean, Jr., A simple model of the derivation of fluid mechanics from the Boltzmann equation, Bull. Am. Math. Soc., vol. 75, No. 1,

Jan. 1969.

J. Liebowitz, J. Percus, J. Sykes, Time evolution of the total distribution function of a one-dimensional system of hard rods, The Physical Review, vol. 171, No. 1, July 1968.

vii

INTRODUCTION

Consider a dilute gas composed of a very large number of

molecules moving in space according to the laws of classical mechanics, and colliding in pairs from time to time.

Assume

that we can disregard all external effects, such as gravity, so that the motion is completely specified by giving the intermolecular forces.

One is interested in the number of molecules which at time t have position r and velocity v, within drdv.

This is given by

n(t, r, v) = Nf(t, r,v)drdv where f is called the density function and N is the total number It is clear that this quantity is going to change

of molecules.

in time due to the motion of the molecules and to the effect of the collisions.

Boltzmann derived an equation for the rate of change of f with time.

It has the form of a non-linear integro-differential

equation: (l)

of

al+ vof Tr =

JJ

=

ff(v2)f(vi)- f(vl)f(v2)llvl-v21I(Iv1-v2I,e)sin Oded$dv2

Here f stands for f(t,r,vl).

The integral part, containing the

non-linearity reflects the effect of the collisions between

ff

molecules; the term vl 7Fr reflects the motion of the molecules between collisions.

viii

For a smooth introduction to the subject the reader can consult a number of sources; we just mention a few:

BOLTZMANN, L., Lectures on gas theory, Berkeley, University of California Press (1964).

CERCIGNANI, C., Mathematical methods in kinetic theory, New York, Plenum Press (1969),

GRAD, H., Principles in the kinetic theory of gases, Handbuck der Physik, Vol. XII, Springer (1958).

UHLENBECK, G. and FORD, G., Lectures in statistical mechanics, American Mathematical Society (1963).

1

The Soluble Boundary Value Problems of Transport Theory

K. M. Case* The Rockefeller University New York, New York 10021

I.

Introduction

Here we wish to discuss closed form solutions of the linearized transport equation

( Ftt

+V+v(v)) .(r,v,t)=cfd3v'f(v,v't) . v r v,-

(1)

..,

+9.(r,v,t) Here the distribution function

,

reV

is to be found from Eq. (1) in some

volume V bounded by a surface S subject to appropriate boundary

conditions, e. g.

I'(r,v,t) = T The scattering kernel presumed given.

reS f

v inwards ,

source function q

(2)

,

and ts

are all

2

In the last five years or so there has been a spate of papers(1) introducing apparently new and powerful mathematical methods to deal with such problems. Unfortunately, I contend that these methods

are neither as powerful nor as useful as the simplest and oldest techniques we have. These are the Fourier Transform and Wiener-Hopf

decomposition -- nearly 50 years old. These older methods solve all

problems that the new ones do, give the answers in neater forms, and show how to obtain solutions for problems that would be extremely unlikely to be suggested by the new methods.

Our program is then the following: In Section II we briefly

discuss the application of the theory of several complex variables. In Section III the application of the theory of generalized analytic func-

tions is considered. The following sections are devoted to a survey of the soluble problems. In particular we show when a problem will be soluble and when it will not be. In the course of this we hope to

verify the contention stated above. U.

Functions of Several Complex Variables

To see why this seems to be of little use let us consider the simplest (and only) non-trivial problem solved in the past. This

consists of the solution of Eq. (1) where V is a half-space. The technique is that of Wiener-Hopf. Two properties are required:

3

1)

The possibility of decomposing a function into the product of two

functions which are analytic in complementary half-planes. 2)

The fact that one equation can determine two functions if they are

analytic in complementary half-planes and satisfy the equation in some common strip. For more than one complex variable (say two) we have an analog

of the first property. This is the Bochner theorem. Stripped of mathematical refinements this is fairly obvious. Suppose F(kl, k2)

is well behaved in the vicinity of the real kl and k2 axes. By applying Cauchy's theorem in both variables we can express F(zl, z2) in the vicinity of the real axes in terms of four double integrals where the paths are either above or below the real axes. The individual integrals then can be used to continue the four components into func-

tions analytic in the upper-upper, lower-upper, etc. regions. To decompose a function into a product we merely need apply the above

procedure to its logarithm. However, we do not have the analog of the second property. One equation will not determine four functions. It may be noted that

this could be avoided if it were possible for example to decompose a function of two variables in the form

4

++(k

A(kl'k2) -

l' k 2)

(3)

+2 (kl, k2)

where the superscripts indicate analyticity in the appropriate upper or lower half-planes. Unfortunately the Bochner construction clearly shows that this is in general impossible.

M. Generalized Analytic Functions A number of authors

(2)

have suggested applying the theory of

generalized analytic functions to such problems. This approach

comes about fairly naturally on trying to generalize the singular eigenfunction method to other than one-dimensional one-velocity problems.

These methods are not wrong. They are merely cumbersome and yield results in unwieldy form. (3) Further it is very unlikely that some of the complicated problems whose solutions are sketched below

would ever have been solved by such methods. As indicated elsewhere (3)

the trouble with these approaches is that they lead directly to a very unpleasant dispersion function, namely one which is non-analytic

almost everywhere in a finite area of the complex plane. That it is possible to avoid such a function is fairly obvious intuitively.

The

dispersion function is analytic in much of the plane. Now an analytic function usually admits continuation. (While the possibility exists,

5

of course, of a natural boundary this would be extraordinarily sur-

prising to occur in a physical problem.) Indeed it is just the analytically continued function which we find plays the key role in the solution.

One final remark. Since there is much obfuscation in the literature about the "Theory of Generalized Analytic Functions"

it is perhaps well to note that the key "theorem" here is the statement that a

i

=

where

= x + iy

.

(4)

This in turn we remember is, in disguised notation, the statement that

1n In

is the Green's function for the two dimensional Laplacian.

IV. A General Formulation For time dependent problems we introduce, the Fourier transform

*(r, v, w) = fO

e-1wtdtT(r,

v, t)

(5)

Then Eq. (1) becomes {v(v)+iw + v V }

= c fd3v'f(v, v') % (r,v')+q'(r,v)

reV

(6)

6

and n

(r-is , ) _

(r , v) s Ns _

(Here we have lumped into g'

r e S , v inwards "s

(7)

the given volume sources and the

initial value contribution - T(t=0). ) There are several ways to reduce the problem to an integral equation in fewer variables. One such is to introduce the infinite

(This eventually leads to an equation for

medium Green's function.

the outgoing surface distribution.) For our present purposes, however, it is pedagogically preferable to use the collisionless Green's function. (This is equivalent to the usual procedure of obtaining an

integral equation for the density. With

G(r'v)

dike-ik- r

1

(ZR)3 r v(v)+ii+ik v

(8)

and using Green's theorem we obtain

'I(r, ) =Vf

ti

nd3r'G(r-r', v)(cp(r'v) + a(rr', v)]

+v fn. d 2 r G(r SN

$s

(9 )

ti

s

s

,v)

7

where we have abbreviated PQ, v) = f d3v'f(v,

(10)

Now let us require Eq. (9) to hold for all r A

(This merely defines

.

for rAV.) Taking three dimensional Fourier transforms and denoting trans-

forms by a superscript tilde we obtain cf' (k v) + 1(kv y) y) _

v(v)+iiil-ik v

ti d2 v , v(v)+iiv-ik v f nn

e

*k'r is A ,l, (r s

v)

Q

where fVd3r

Pv=

11

eik ~p(N,N)

We note that for one-velocity, isotropic scattering PV is independent of v

.

The velocity average of Eq. (11) then gives a

relation between P and PV problems PV gives the complete solution.

.

For whole space and half-space

(12)

8

More generally let us now assume that we have an expansion of the form f(v, y') = g(w') afn0n(V)On*(K')

Then p(k, r) = Mn fnP" (k On (v)

(14)

with Pn(k) =

fd3v

Then multiplying Eq. (11) by go

(k, v)

(15)

and integrating over v gives an

algebraic system of relations between pn and pn where v Pn V = fd3k' Ov(k-k')Pn(k')

(16)

n { nm - 11nm(k) } pm (1) + Bn(

(17)

-1

Indeed we obtain Pn(k) =

9

Here Bn(k) is given in terms of known contributions from the initial conditions, boundary data, and sources. The dispersion matrix is given by

A

nm

nm

- cf

mfd 3tiv g(vu '

0n (,,)0m(Y) (18)

Eq. (17) is a (possibly infinite) set of coupled integral equations

for the

pV(k)

.

If we can solve these we have p

and then from

Eq. (11) we have T . The question therefore arises under what circumstances can we determine the pV from Eq. (17)? One situation in which the problem is in principle soluble is

..

clearly when V is all space. Then Ov (k) = 6(k) and the equation reduces to EA pm= Bn m nm Here the problem reduces to the algebraic task of inverting Anm At least for a finite number of terms in the expansion of Eq. (13)

this is always possible. The next (and only) non-trivial case we can handle are half-

space problems.

(19)

10

In the next section we will see that if the scattering kernel is

separable (i. e. there is only one on ) the half-space problem is always soluble. For the more general case a matrix factorization

of A nm would be needed. There are a few very special situations where this is possible. For example if each element Anm of the n'th

row has the same cuts in the complex k tin plane and if the ratio of boundary value

nm /A-nm

A+

across each cut is independent of m

then the procedure for the separable case is applicable. This is clearly a very special situation and hardly of physical interest.

Another possibility is that the matrix which transforms A to

Jordan canonical form is a rational function of k n

.

Indeed,

there is at least one case of this kind of mild physical. interest -we sketch it below.

The conclusion is that Eq. (17) is usually not amenable to analytic solution. There is, however, one class of kernels, in addi-

tion to the separable case which is not terribly special and yet for which Eq. (17) can be solved. These are kernels of the form of Eq. (13)

where the 0 n or appropriate linear combinations 'm satisfy a three term recursion relation of the form vk n(v) =

bn(k)$n_l(n)

(20)

11

V.

Separable Kernels

Let us consider in detail the case where only one 0n= 0(v) ,.. appears in Eq. (13). Our fundamental Eq. (17) becomes p(k) = Pv(k)[l - A(k)] + B(k)

,

(21)

where P(k) = Id3v' g(v')0

(22)

and the dispersion function is

A(k) = 1 - f1

d3_v, g(V) 1 $ (v) 2

(23)

This is an appropriate place to single out the essential difference between methods using the "theory of generalized analytic functions"

and the present one. As I am sure most of you know all methods involve the investigation of analytic properties of the dispersion function. Here we have defined A for real k., only to begin with.

Methods of generalized analytic functions use a A defined by Eq. (23)

for all complex k

.

We instead note that in almost all applications

the function defined by Eq. (23) for real 15 permits a continuation

12

into the complex plane -- with a much simpler resulting function (usually one with only a few branch point singularities).

The solution of the infinite space problem is, of course,

trivial. Namely PV = P

(24)

P (k) = A(k)

(25)

and therefore from Eq. (21)

For half-space problems let the normal to the boundary plane be in the x-direction. Writing k = k and usually omitting the x implicit dependence on kt a (k , kZ) we have Y

aV(.

1

'(-t)

(26)

2iri k+io

Hence

Pv(k) = p +(k)

where + refers to a decomposition p = p+ + p

(27)

into functions

analytic above and below the real k-axis. Our fundamental equation

13

may now be written A(k)P+(k) = - P_(k) + B(k)

(28)

The problem is of a standard form readily solved by function-theoretic

arguments. The solution is 1

P

V (k) = P+(k) = 21riX (k)

f- B(k')X (k')dk' -cc

k' -k-io

(29)

where

1nX (k) =

1

21ri

-

-oo

.tnA k' dkt k' -k

(30)

Two points may be noted: 1)

Here we have simultaneously constructed the solutions of the

problems(1) considered by Cercignani, Kaper, and Klinc and Kuscer.

2) The solution is not obtained in the usual (and convenient) normal mode form. In particular cases this is, however, readily found by

modifying the contour integrals using the properties of our A . VI.

Non-Separable Kernels

As mentioned earlier there is a class of non-separable kernels

14

which do lead to closed solutions.

Namely those kernels such that

the 0n satisfy a three term recursion relation of the form given in Eq. (20).

The most important example is that of time dependent,

one-speed arbitrary anisotropic scattering for a half-space. Since the algebra for this is somewhat tedious and will be presented in detail elsewhere (3) we treat instead a model for disturbance propagation in a gas.

The model is so: We are to solve (l + v - Ni (x, v) = c f dv' f (v, v') 4J (x, VI) (31)

+ I(x,v)

x>, 0

subject to

4i (o, v) = "P4 (v) ,

v>0

(32)

(It may be noted that time dependence could be readily included.

We do not do so merely for reasons of simplicity in presentation).

Taking Fourier transforms we obtain

CP (k,v) 4, (k, v) =

1

ikv

+ B(k, v)

(33)

where B is known and (34)

Pv= fdv'f(v,v')Lv(k,v')

For the scattering kernel f we use a slight generalization of one due to Kac. (4) 2

f(v,v') = e-VI Ft illi(v)H.(v')

(35)

where our previous 0's are now the Hermite polynomials Hi Following our general procedure we define p (k) by

2

pi(k) =

fdv'e-v

(36)

so that

Pv(k, v) = jf ipv (k) Hi (v) Continuing as before we find that the dispersion matrix is 2

dv e-v HIHm Aim= tm - cfmf 1 - ikv

(37)

Our problem would be solved if we could determine the pv using Eq. (17). However, this is straightforward only for the infinite medium

case.

16

For the half-space we are fortunate in that the recursion relations for the Hermite polynomials v Hn(v) = n Hn-1(v) +

2

Hn+1(v)

p(o) enable us to express all of the pR v in terms of v

(38)

.

The zeroth

component of Eq. (17) P(o)(k) = m (tom-Aom) pm + B(o)(k)

(39)

is then an equation identical in form to the separable case and may be

treated similarly. Two points should be noted: 1) For the one speed anisotropic scattering problem the analogous property is the recursion relation for the Legendre polynomials

k rw P (k n ti. N) = Ik I wh

k-

(n+l) Pn (-' r) +nPn-l(k r) (40)

2n+1

k

(41)

The appropriate recursion relations for the pm are just extensions to v the case of x, y,.z dependence of those obtained by Mika(5) for the

17

purely one dimensional problem.

2) Actually in the solution for the pm v in terms of pv(o) some unknown constants occur. These can, however, be determined after

the formal solution has been obtained by a consistency check.

(6)

Let us sketch the procedure: Suppose we multiply Eq. (31) by

eilcx

and integrate over V . We obtain (1-ikv) v

= cPv +

v [eikx q] S

(42)

2

Multiplying by e-v H n(v) and using both the recursion relations

and orthogonality properties of the H's we get n-1 n - ik(npv PV

with fn = 2n n! ; Tr fn

+

1

2

n ) = c '`fnpv + qn+ S n

n+l

pv

(43)

and 2

S n= [eikx f dv e-v Hn(v) v 4 (x, v)IS

(44)

can then be written as The solution for pn v n-1

Pv = gnPv + E bn1(g1 +S 1) 1=0

(45)

18

where (1-c fn) gn - ik (ngn-1 +

2

gn+1) = 0

n>1 (46)

g 0= 1

g_1= 0 and

(1 cfa)bn4 ik(nba-1, Q+

2

n> I + 2

bn+1, J= 0

(47)

2

b1+1,1=

ik2

' b1+2,1= - (ik)2(1-cfF H)

Inserting this into Eq. (39) then yields P (o)

(k)

1 bn.Q(g1+ S1) + $(k) - [i- A(k)]Pv(k) = cEn fnGn I=Eo

(48)

Here A(k) = 1 - c E n InGn(k) gn(k) with

(49)

(7)

fdve-v

Gn(k)

1

2

1Hn v)

1-iv

(50)

We see that, as promised, the problem has been reduced to that of the previous section.

19

V.U.

Multi-Group Problems(8)

As indicated earlier there is a class of problems with nonseparable kernels which may be solved analytically with our general formulation and which does not make use of recursion relations. Thus suppose we consider a set of one dimensional functions 4,(x, u) which satisfy the equations 7

U8 (a-I. + 2 x)

1

E ci f 1 du o (x, u)

IPi

j

(51)

+ qi(x, u) s

4i.(x, v) _ 4 (x, v) , x e S

,

,

xev

v inwards

(The indices here may be understood to refer to a discrete set of

speeds v

i.

)

This is a straightforward generalization of our original

equations. Fourier transforming we obtain j

v-iku 3 cijPv

+

B. i

(52)

(Again known quantities are lumped in the B. .) The functions Vv are then to be found from the analog of Eq. (17) Pi =

where now

ij

- Ai.) pv + B.

(53)

20

A ()k ij

fl

S

du

ij - cij -1 v-iku i

(54)

For arbitrary c it is once again clear that only the infinite space case will be soluble. However, for the particular case defined by c

ij

=0

i<j

(55)

the half-space problem is readily solved. (This is a case of some

physical interest -- scattering only decreases particle energy. Indeed, here Eqs. (53) imply 1"(k) =

1111(k)] pr(1)(k) + B1(k)

(56)

This is just the problem of our separable kernel. Further, once p 1 )(k) is found the remainder follow successively. For each p1(k) we have an equation like Eq. (56) with inhomogeneous terms

depending on the pI found earlier.

We remark further that with the restriction of Eq. (55) the case involving three dimensional anisotropic scattering in a half space may also be solved in closed form.

21

VIII.

Conclusion

It is hoped that it has been shown that most, if not all, of the soluble problems of linear transport theory can be treated by the relatively simple (and old fashioned) methods presented here. Among the soluble half-space problems permitting closed form solutions are those with a) Separable kernels,

or

b) Pure down scattering,

or

c) Kernels whose component functions satisfy three term re-

cursion relations of the form indicated.

It should be emphasized that the three term nature of the re-

cursion relation is essential. Even a slight modification messes things up fully. As an example consider the three dimensional

scattering kernel of Kac.

(9)

f(v, v') = e

This has v

i2 4

io figiv) gi (v')

(57)

with 2(v2 -

go= 1 , 91,2,3=

Z

(vx,vy,vz), g4=

3 )

2

d3

(58)

22

While formally this seems but a small modification of the kernel treated in Section VI it is not. Four term recursion relations exist

but not three term ones. As a consequence we are unable . to reduce Eq. (17) to an equation involving only one I and its corP

responding pv

.

The methods presented here then fail to give a

closed form solution. Indeed, I am convinced that no other methods

will either.

23

Footnotes

We are describing work done with Dr. R. D. Hazeltine. (1)

See, for example C. Cercignani, Ann. Phys. 40, 454 (1966). H.

J. Kaper, J. Math. Phys. 10, 286 (1969).

R. C. Erdmann and A. Sotoodehnia, J. Math. Anal. and Appl. 31, 603 (1970).

A. Leonard, submitted to J. Math. Phys. J. Sanchez and R. C. Erdmann, Submitted to J. Math. Phys. (2)

Cercignani, Ibid; Kaper, Ibid; Erdmann and Sotoodehnia, Ibid.

(3)

See, for example

K. M. Case and R. D. Hazeltine, J. Math. Phys. 11, 1126 (1970) (I)

K. M. Case and R. D. Hazeltine, submitted to J. Math. Phys. (II) (4)

M. Kac, Proceedings of the Third Berkeley Symposium on Mathematics, Statistics and Probability, Vol. 3, pp. 171-193.

(5)

J. Mika, Nucl. Sci. Eng. 11, 415 (1961).

(6)

For the cognoscenti this is equivalent to the fact that the "solution" of the corresponding singular integral equation

is itself a Fredholm equation -- but with degenerate kernel.

24

(7)

For many purposes it is helpful to note that the G n also

satisfy recursion relations. These are G-ik(nG1+ n n- 2 (8)

Gn+1)

S

Problems of the type we discuss here have been considered by

C. E. Siewert and P. S. Shieh, J. Nucl. Energy 21, 383 (1967). (9)

See G. E. Uhlenbeck and G. W. Ford, Lectures in Statistical Mechanics (American Mathematical Society, Providence, R. 1963), p. 93.

See also P. L. Bhatnagor, E. P. Gross and M. Krook, Phys. Rev. 94, 511 (1954).

I.

no

25

NUMERICAL SOLUTION OF THE BOLTZMANN EQUATION

Alexandre J. Chorin Courant Institute of Mathematical Sciences

Abstract

A numerical method for solving the full non-linear Boltzmann equation is presented, and applied to the problem of shock structure in a gas of elastic spheres.

The success of the method hinges on

the systematic use of Gaussian quadrature and Hermite interpolation.

26

Introduction. The Boltzmann equation describes the evolution of the one-particle distribution function f = f(x,u,t), where x, with components (x1, x2, x3), is the position vector, u with components (u1, u2, u3), is the velocity vector, and t is the time.

In the

case of a gas of elastic spheres it has the form 2

(1)

'E +

m( -

where m is the mass of

)f = 2

fIV.- el(f +f+-ff')du'dw

a particle, a its radius,

4x denotes the

gradient operator with respect to the x variables, Vu denotes the gradient operator with respect to the u variables, F is the external force, e is a unit vector pointing in the direction of the solid angle element dw, V = u' - u, a bar under a symbol denotes a vector quantity, and

f = f(x,u,t)

f' = f(x,u',t)

f+ E f(x,ut) f+ = f(x,u+ ,t) where

u+ = u + u+

u+, u+

= u -

are the velocities before collision of those spheres which

after collision have the velocities u and u'.

Analogous expressions

can be written for other kinds of interparticle force.

For an

elementary discussion of this equation, see [141; for a thorough

27

discussion see e.g. [2] and [5].

The right hand side of

equation (1) will be called the collision integral. It is the purpose of this paper to present a numerical algorithm for solving Equation (1) and to apply it to the study of the structure of a shock in one space dimension.

Generalizations

of this method to problems involving very strong shocks, more space dimensions, and other molecular models, will be also disIt will be seen that the solution of the shock problem

cussed.

provides a key to the solution of the other problems; the main difficulty has been overcome in the program discussed in this paper.

Furthermore, the numerical solution provides insight into

some approximate procedures, in particular Grad's thirteen moment approximation [6] and Mott-Smith's bimodal approximation [12]. Unlike the work presented here, most previous numerical treatments of the Boltzmann equation relied on a Monte-Carlo technique; some of these treatments are ingenious and interesting, but none can be considered accurate.

See [1], [8], [9],

and [13].

Reference [9] is particularly helpful. For

any function 4(x,u), let (x) denote the integral

4(x) = f4(x,u)f(x,u)du

.

Some of the quantities of interest in the solution of the Boltzmann equation are the following moments of f:

mean velocity u, the pressure p =

the density p(x) = 1, the

3pw2, where w = u - u, the

temperature T = p/pR, where R is the universal gas constant, the

28

pressure tensor pig =Pwiwj., and the heat flux vector S = 2pw2w

Other quantities of interest are the Boltzmann H function

H = f f log f du and in the shock wave problem,

,

various geometric parameters which

characterize the shock.

In the case of a gas of elastic spheres, the mean free path is

2

= 1/( / H P

o2)

We shall now specialize equation (1) to a form appropriate to the shock problem.

Let T

be a reference temperature, and u0 a 0

reference thermal velocity, u0 = T

= i'u0, and

2RT0.

P0 a reference density.

Let T

be a collision time,

Introduce the non-dimensional

variables

x* = x/2 , u* = u/u0, t* = t/T, f* = u3f/PO

;

substitute them into equation (1) and drop the stars.

pick units in which a reference mean free path 1/( f 7 1.

Furthermore, p0a 2) is

Assume F = 0, (no external forces), and allow f to depend only

on one space variable x1 = x, and two velocity variables u1 = u and ur, where u is in the direction of x and ur is in a direction orthogonal to u.

These assumptions imply that the flow is invariant

29

Under these assumptions

under rotation around the x-axis.

f = f(x,u,ur,t) satisfies

(2)

f7# sin 0

+ u 8x

!f` dX -! o

da' l+`° dur(f+f+-ff') I V-ei /2v n

where cos X, sin

e = (cos $, sin

sin X),

u' = (u',urcos X, ursin X), u = (u, ur, 0), V =

u+ = u + u+

(u+,u2,u3), (u+

= u -

,

u2

,

u3 ),

u+ _ vu+2+u+2

r

ur'

f'

2 /(

3

u2')2+(u3

_

= f(x,u',ur),

f = f(x,u,ur),

f+ = f(x,u+,ur+,), ,

,

f+ = f(x,u

ur+

This is the form of the equation we shall use below, although the method of solution applies to the general equation (1) as well.

30

Principle of the method. There are several major difficulties in the solution of (1) o'-

(2).

The function f depends on a

relatively large number of independent variables-three plus time in the case of one-dimensional flow, six plus time

in the general

case-so that if (2) is replaced by a system of algebraic equations, their number will be large.

The presence of the fourfold non-

linear integral ensures that the algebraic equations will be not only numerous, but also very cumbersome. clearly needed.

Efficient computation is

Another difficulty stems from the nature of the

collision term, more specifically, from the integration over the angular variables.

Suppose f is represented by a discrete set of The

values assumed on a discrete set Z of points in phase space.

integration over u', ur becomes a sum over the values assumed by f on Z.

The integration with respect to 6,X becomes a sum

discrete set 0 of values of 9,X. Z and 0,

Thus

over a

For any reasonable choice of

the arguments of f+, f+ will include points not in Z.

interpolation, both accurate and stable, between values of

f on Z, will be required.

(Monte-Carlo methods avoid the inter-

polation problem at a heavy price in accuracy.) These difficulties can be resolved as follows:

once one

resigns oneself to the need for interpolation, there is no need to identify Z with the nodes of a regular mesh.

One can then

evaluate f at points (xk, ui, ur,j) where the ui, ur,j are at one's disposal.

In particular, one can choose ui, ur,j to be the

roots of a polynomial PN(u), where PN is the N-th degree member

31

a sequence of polynomials P0, P1,...,Pn,..., orthogonal with respect to a weight W(u).

With this choice of u1,, u

one r,j

can interpolate between the values of f at these points using the orthogonal polynomials Pn(x). see details below.

Such interpolation is stable;

Furthermore, with this choice of integration

points, integrals can be evaluated by an appropriate variant of Gaussian quadrature.

In the case of the Boltzmann equation,

it is natural to use the Hermite polynomials Hn(u) given by n

eu2

Hn(u) = (-1)ncn

d

e_u2

,

cn = (2nn!)-

1/2

dun

which are orthonormal with respect to the weight W(u) =

7T

-1/2e-u2

i.e.

Tr

d

n,m

-1/2 f Hn(u)Hm(u)e-u 2 du = do m ;

the Kronecker delta.

The set {H (u)e-u /2} is complete in n

L2See [10]. The preceding remarks lead to the following step-by-step procedure for solving (2):

Let At be the time step.

Assume

that at time t = nAt f is given by a series L1 (3)

L2

ai(x,t)

f(x,u,ur,nLt) = 7-l(ux)-2 i=0 j=0

3

Hi((u-vn)/uf)Hj(ur/un)exp(_((u-vn)2 + u2)/(um)2)

32

where vn is the center of the expansion and un is its scale.

The

subscript x in un and vn indicates that both parameters are allowed to vary with x, and it is assumed that aij = 0 for i > L1, j

> L2.

Appropriate vx, ux, L1, L2 will be determined below.

It is adequate to evaluate aij(x) at the points x = kAx, k integer, Ax a spatial increment, and we can write

an

ijk

= a.

(kAx,nAt)

The density at time not is

Pn(x) = a00(x,nAt) = a00(x)

the temperature is

Tn(x) = (un)2((3/2)a00 + 2_1/2a20 + 21/2a02)/3RPn x

(4)

and the mean velocity is

un = vn + 2-1/2unan x 10 /Pn x

(5)

Our aim is to obtain f(x,u,ur,(n+l)At) as a series of the form (3),

but possibly

with a new scale uX+1 and a new center vX+l

To

n+1 achieve this aim we evaluate the values fn+l of f(x,u,ur,(n+l)At) ijk where at the points xk = kAx, ui = vn+l + un+l i ur,j =

i,j

are roots of HN(u) = 0.

The value of N remains to be chosen.

33

The algorithm for evaluating fink will be described below. Given f(x,u,ur,(n+l)pt), the coefficients aij are defined by

(6)

an l(x) _ 7r-1(un+l)-2 If

f(x,u,ur,(n+l)At)H.((u-vn+l)/un+1).

'H(u/un+l )dudur j r

Tr

-1(un+l)-2 If f Hi ((u-vn+1)/un+l)H (ur/uX+1x

x

j

exp((u-vn+l)/uX+1)2 exp(ur/ux+1)2

'exp(-(u-vn+l)/un+1)2 exp

-

(ur/un+l)2 dudur

An obvious change of variables reduces the last integral to the form 2 -u2

r dudur

If g(u,ur)e-u e

which can be evaluated by Gauss-Hermite quadrature (see [151), i.e. using a formula

(7)

If g(u,ur

)e-u

2 -u2 e rdudu

r

N

N g( ., )w1 .wj

=

,

i=0 j=0

where 6i, j are the roots of HN(u) = 0 and wi, wj are appropriate weights.

Because of the choice of quadrature points, g is already

34

Formula (7) is exact

known at the appropriate points

if f(x,u,ur) has a Hermite expansion of the form (3) with Ll = N, L2 = N (see [11]).

Thus as L1, L2 are increased, N

should be increased.

There remains only the task of deciding how to evaluate fink.

This is done in the present paper by an explicit formula

of the form

(8)

fljk = A fnjk + At Q(f,f)

where A is a linear operator such that fn+l - Afn approximates

at - u a , and Q(f,f) is an approximation to the collision integral. We use

Af

i,j,k

_

+ u At (1-ut)f Ax Ax i,j,k

where s(u) = 1 if u < 0 and s(u) = -1 if u > 0.

f

i,j,k+s(u)

This A has an

obvious inituitive appeal, but since it is only of first order accuracy a more accurate approximation may be in order in future work.

(9)

For stability we must of course

x maxluI

have

< 1.

i

In the evaluation of the collision integral, the crucial fact is that fn is given as a continuous function of u and ur, and thus further interpolation problems will arise. of ener;y and mgr"ni+m,

the rpnreaentation

no

Using the conservation (3) of f, and an

35

obvious charge of variables, the collision term at (x,u.,ur,j) can be reduced to the form ,2 -u'2

C(X,Ui,u

r,j

f++

)

1

d f++ldX f++' du' J

G(e,X,u',ur)e-u

du'

e

where C is a constant which depends on x, ui, u

r

This integral

r,j.

can be approximated by the mixed Gauss and Gauss-Hermite quadrature formula Ni (10)

N2

N3

N4

= = = G(ei'Xj'uk'ur 2)WiWjwkw2 i=O j =O k=0 2=0 '

where the ei, Xj are the roots of the Legendre polynomials of degree N1, N2; uk, u.,,, are the roots of HN (u) = 0 and HN (u) = 0, and 3

4

Wi, Wj, wk,wQ are the quadrature weights, see [15]. Our method can thus be summarized as follows: ning of each step,

At the begin-

the solution f is given as a Hermite series

of the form (3); f at the next level is evaluated at appropriate points using a difference scheme for the linear terms and weighted Gaussian quadrature for the collision term. then synthesized into a Hermite series.

The new values are

A variant of this method,

using Monte-Carlo quadrature rather than Gaussian quadrature for the collision term, was presented in [3].

The fundamental difference between our approach and Hermite expansion methods such as Grad's thirteen moment approximation [6] lies in the fact that the number of Hermite polynomials used is not fixed in advance but depends on the course of the computation.

36

In particular, the adequacy of the representation at each step can be checked by using one more term and verifying that its effect is small.

In order to apply the algorithm just described, one needs an initial function f0, as

well as boundary conditions on fn.

Care

must be exercised when the boundary conditions are imposed: f(u,ur) at a boundary may be imposed only for values of

u, ur

such that the vector (u,ur) points from the boundary into the gas. of the particles coming from the

The distribution of the velocities

fluid and hitting the boundary depends on the imposed arbitrarily (see[7]).

If this obvious condition is not

respected, numerical instability Application to a shock problem.

flow and cannot be

will result.

Consider a gas of elastic spheres

flowing in -- < x < +°°, with

(11)

f(--,u,ur) = P1TF-1U12exp(-((u-vl)2+ur)/U1)

(12)

f(+-,u,u) r = Pv-1U22exp(-((u-v 2 )2+ur)/U2 2

clearly u(--) = v1, u(+=) = v2.

The Mach number M is defined by

(13)

M = V vl/U1

.

37

If M > 0 a shock will develop.

be a steady shock if

There may

the following conservation laws are satisfied

(14)

P1v1 = P2v2

(15)

P1(v1+ 2U1) = P2(v2+ 2 U 2)

(16)

l 1 pv(vl+ 2Ui)

= p 2 v 2 (v22 + 2U 1

where it is assumed that the ratio of specific heats is

Y = 5/3

.

From equation (14), (15), (16) we may deduce

(17)

(U2/U1)2 = (M2+3)(5M2-1)/16M2

(18)

(v2/v1)

=(M2+3)/4M2

An important parameter in the shock problem is the shook thickness conventionally (and awkwardly) defined by v2. ,v (19)

x=

1

du mxjdx

38

We pick p1 = 1, V1 = 1. yield U13 p2,v2, U2.

Given M, equation (13), (17), (18)

We call the left hand end of the shock

the upstream side; we thus chose the units so that the mean free path upstream is one.

In those units X is the ratio of the shock

strength to the upstream mean free path; several authors have studied X 1 as a function of the Mach number M.

For practical reasons we replace the region - < x
0

We divide [-a,+a] into K - 1 segments, with a spatial increment

Ax = 2a/K

.

Our aim it to obtain the steady shock profile as the limit, when the time tends to infinity, of an unsteady flow starting from

an initial function f0 = f(x,u,u0). r

This initial function should

be chosen so that the steady limit is achieved as fast as possible. We first tried initial function f0 resulting from an approximate

39

solution of the Boltzmann equation, in particular we tried the solution of the Mott-Smith u2 theory [12]. a very poor choice.

This turned out to be

It is clear that the convergence to the steady

limit is inherently slow-if we use K points across the shock, and if the stability condition (9) is respected, it takes at least K steps for the fastest particles to cross the shock.

If f0 is the

Mott-Smith solution, the initial values assumed by at are very small, and the relaxation to equilibrium takes an extremely long time, (showing, by the way, that the Mott-Smith solution is not a very good approximation to the real f). appear:

In addition, some odd effects

at low M the Mott-Smith theory overestimates the shock

width, yet with Mott-Smith initial data the shock at first appears to widen; this effect can also be observed in the work of Haviland [9].

After considerable experimentation, it was found that an appropriate f0 is the one which corresponds to a shock of zero width

(20)

for x < 0

.

for x > 0

.

f(x , u , ur, 0) [f(+oo,u,ur)

The initial f given by (20) is particularly appropriate when one tries to determine the shock width X as defined by (19). local property of the shock center, and with the data (20)

approaches equilibrium values long before at

X is a X

becomes close to

zero.

It is worth noting that from the numerical point of view the

40

determination of the shock width X is a comparatively difficult undertaking, since it requires high accuracy in the region of fastest variation of f.

In a variety of other problems, e.g.

problems involving the interaction of a shock with a boundary, the choice of initial data is less critical and the computation is easier to carry out.

We now apply the method outlined earlier to the study of the shock wave.

There is a considerable number of numerical

parameters to be chosen:

the centers vn and scales un of the

expansion (3), as well as the number (Ll+l)(L2+1) on nonzero terms;

the size 2a of the region of integration, the spatial

increment Ax, the time step At, the number of quadrature points N N 2 N N 4 in each evaluation of the collision integral and the 1

3

number N2 of points at which fn+l is evaluated given fn We choose vn and un as follows:

vn+l = un(x) (21)

n+l =

2RTn(x)

i.e. we expand at each step around the mean velocity at the preceding step and using a scale determined by the temperature at the preceding step.

un, Tn, are given by (4) and (5).

This choice

is not the only reasonable one, and will be further discussed below. The width 2a of the region of integration was chosen by trial and error, generally around 25 mean free paths. Ax is chosen small

41

enough so that any further decrease in Ax will not affect the outccme of the calculation.

We proceed as follows:

We evaluate

dx which enters the definition (19) of X using both the formula

du v uk+l-uk-1

(22)

dx

2Ax

and

du N 7k+l-uk

(23)

dx

Ax

which are of different orders in Ax; when they are in substantial agreement Ax can be considered sufficiently small.

It was

found that Ax of order 1 (i.e. one mean free path) is generally adequate; under these circumstances, X evaluated with the use of (23) is a more reliable estimate of the true X, since X is a local property of the shock center and an estimate using (23) depends on the values of f in a smaller neighborhood. The stability condition (9) gives a good estimate of the appropriate value of At.

We usually choose At to be 0.8 times

the maximum value allowed by (9).

Higher values of At may give

rise to instability in the presence of temperature overshoots while lower values lengthen the computation without increasing its accuracy.

At this point we have to introduce an additional

numerical parameter.

It is readily seen that the stability of the

scheme fn+1 = Afn would imply the stability of the complete scheme (8) if only the integrand on the right hand side had compact support.

This last condition is not satisfied, but f does decrease rapidly with increasing lul,

lurl, so that one might assume that condition

42

(9) is sufficient for stability.

Numerical experimentation shows

this to be the case whenever Ll< 3 and L2 < 3.

However, when

L1 or L2 is larger, the range of u, ur over which f is not negligible increases, and it is necessary to truncate the support of f. Jul

This can be done by setting fn = 0 whenever

> vm + EXun,iur1 > XCun' where un is the scale of the expansion,

vn its center,

is the largest root of HN(u) = 0 and A is a constant

larger than 1.

When A > 1 such truncation leads to no decrease in

accuracy, since the expansion in Hermite polynomials is not uniformly valid in u, ur.

We generally chose A - 1.1.

We generally took

L1 = L2, equal to an integer L.

Clearly we

must have L < N; on the other hand if N were much larger than L, information would be generated and immediately discarded; so we generally chose N = L +

even) and N = L + 2 (L odd).

The

difference between the odd and even cases is due to programming consideration and is of no particular significance.

This leaves open the choice of L, the number of Hermite polynomials in each of the variables u, ur.

It would be natural to

choose L so large that aij = 0 for either i or j larger than L. It turns out however that aij decays much more slowly with i and j than expected, but that the presence of the higher terms in the

expansion affects but little the computed values of X and of the density, mean velocity and temperature.

For example, at Mach

number M = 1.6, a04 near the center of the shock tends to the steady value a04

-. 4, yet within computational error there is no difference

between the value of X computed with L = 4 and the value computed

43

with L = 3, i.e. neglecting a04.

It does appear therefore that

the lower moments of f are almost independent of the higher moments, a result both surprising and natural.

It also appears

that the assumption underlying Grad's thirteen moment approximation [5], namely that the coefficients of the Hermite polynomials of degree greater than 3 aae small, is not correct in itself but could lead to correct answers. We made runs with both L = 4.

L = 3 and

It must be added that although the values of X do not seem

to depend on L provided L > 3, when'M < 2, the initial rate of change in X does depend on L.

This is probably of no

physical significance, since the initial data are wholly unrealistic. The relationship between our method and Grad's will be the object of further investigation elsewhere.

It should be noted that

when

L = 3 our f is represented by 8 coefficients aij, taking into account the fact that by symmetry aij = 0 for odd j; when L = 4 our f is represented by 15 functions.

This compares with 5 functions

for the one-dimensional case of Grad's expansion. N13 N2, N3, N4 are also chosen by trial and error. have

We must

N3 > L/2, N4 > L/2, so that the highest moments of f used enter

the collision integral.

It was generally found that with L = 3 or

L = 4, the choice N1 = N2 = N3 = N4 = 3, i.e. 81 integration points for every evaluation of the collision integral, was quite adequate. The fact that such low values are adequate is testimony both to the power of Gaussian quadrature and to the aptness of the representation (3).

The existence of conservation laws

affords a natural check on

.44

accuracy, since no exact conservation is built into our scheme. With the initial data (20), and with a large enough, the mass, momentum and energy in the shock region are constant.

The ma.vni-

tude of the numerically induced variations in, say, the mass provides a reasonable indication of the accuracy of the computation. In tables I and II we display the relaxation from the initial data (20). of

In table I the mean velocity is tabulated as a function

x for low values of t/dt and at Mach number 2; this should give

a qualitative picture of the behavior of the numerical, process.

In

table II the instantaneous value X 1 of the reciprocal of the shock width, the maximum of

atl

,

the location of that maximum, and the

computed total mass Q in the shock region, are tabulated as functions of t/tt for M = 2.

It is seen that 1atl does not decay

to zero fast, if at all, and that X 1 oscillates.

In each run we

therefore estimated the range of values assumed by X-1, defined as the range between the last maximum and last minimum of X-1. not clear whether the oscillations ever die out.

It is

They are amplified

if the width of the region of integration 2a is chosen too small, but they can no longer be decreased by a further increase in 2a. The location of the maximum of 1atj recedes in time, showing that

upstream convergence is slower than downstream. observations were made by Haviland [ 9]. evaluated by K (24)

Q = >p(i0x)dx i=0

Similar

Q, the total mass, is

45

Table I

u as a function of x and t. M = 2, At = .413, Ax = 1.5

t/At = 1

t/ot = 4

t/at = 8

-12.75

1.000

1.000

1.000

-11.25

1.000

1.000

1.000

-9.75

1.000

1.000

1.000

-8.25

1.000

1.000

1.000

-6.75

1.000

1.000

..999

-5.25

1.000

.999

.999

-3.75

1.000

.999

.998

-2.25

1.000

.995

.961

-0.75

.979

.873

.726

0.75

.461

.499

.535

2.25

.437

.458

.478

3.75

.437

.444

.460

5.25

.437

.438

.451

6.75

.437

.437

444

8.25

.437

.437

.440

.437

.437

.438

11.25

.437

.437

.437

12.75

.437

.437

.437

x

46

Table II

Relaxation to a steady shock

M = 2, At = .413, Ax = 1.5 location of t/At

X 1

maxIIII

maxjatj

1

.61

.056

+ .75

44.31

2

.56

.069

- .75

44.29

3

.50

.088

- .75

44.29

4

.44

.097

- .75

44.31

5

.38

.099

- .75

44.33

6

.32

.096

- .75

44.35

7

.27

.087

- .75

44.37

8

.27

.072'

- .75

44.40

9

.28

.057

44.45

10

.28

.043

- .75 -2.25

11

.28

.049

-2.25

44.51

12

.26

.055

-2.25

44.54

40

.20

.056

-6.75

44.30

41

.19

o6o

-6.75

44.27

42

.19

.061

-6.75

44.25

43

.21

.060

-6.75

44.22

44

.23

.057

45

.24

.052

-6.75 -6.75

44.17

46

.25

.045

.24

.o45

-6.75 -8.25

44.15

47 48

.23

.051

-8.25

44.11

49

.21

.056

-8.25

44.10

Q

44.48

44.19

44.13

47

Table III

Structure of a shock M =2, t = 9.5192 H/p

x

u

p

-12.75

1.000

1.000

T

.300

-1.398

-11.25

.999

1.000

.300

-1.398

- 9.75

.999

1.000

.300

-1.398

- 8.25

.999

1.000

.300

-1.398

- 6.75

.998

1.001

.301

-1.403

- 5.25

.977

1.020

.316

-1.467

- 3.75

.831

1.165

.431

-1.787

- 2.25

.670

1.371

.622

-2.270

-

.75

.660

1.392

.680

-2.479

+

.75

.616

1.587

.660

-2-299

2.25

.546

1.937

.628

-2.011

3.75

.501

2.213

.609

-1.802

5.25

.485

2.342

.602

-1.712

6.75

.479

2.381

.601

-1.692

8.25

.477

2.379

.605

-1.708

9.75

.472

2.367

.612

-1.743

11.25

.467

2.349

.619

-1.788

12.75

.437

2.285

.623

-1.883

48

Table IV

Coefficients a..

t = 10.76

Mach number = 1.6, x = L5,

i= 0 j = 0

1.447

i= 1 -.0002

i= 3

.48

-.04

-.38 .01

= 2

.24

-.19

-.005

.05

j = 4

.11

.08

.0o4

-.02

j

i= 4

1= 2

a.. = 0 for odd j.

-.006

49

Table V

Reciprocal shock width X-1 as a function of Mach number M.

M = 1.4

L = 3 .12 to .13

M = 1.6

.22 to .24

L = 4

Gilbarg and Paolucci .136

.22 to .24

Mott-Smith

Ziering et al.

.116

.181

.164

.238

.205

.284

.235

.324

.222

M = 1.8

.18 to .21

M = 2.0

.19 to .25

.23 to .29

.38'.

50

it is seen that Q varies little; whatever variations there are can be ascribed to the inaccuracy of the formula (24). The

In table III we display the structure of a typical shock. mean velocity u, density p by p

,

,

temperature T, and Boltzmann H divided

are given as functions of x, for M = 2 and t = 9.5192.

familiar features of the shock appear:

The

u and p vary in a monotone

fashion;T exhibits an overshoot, see [16]; H/p, which is determined up to an additive constant, displays a dip.

H is evaluated from f

using, as usual, Gauss-Hermite quadrature. In table IV we present the coefficients aij for x = 1.5, M = 1.6, t = 10.76.

The purpose of the table is to show that a40

at that point is not small.

Some of the more interesting results are grouped on table V, where the ranges of oscillation of X-1 for Mach numbers 1.4, 1.6,

1.8, and 2.0 are given, with both L = 3 and L = 4, and compared with the values of X 1 computed by Gilbarg and Paolucci using the NavierStokes equations, and by Mott-Smith and Ziering et al using their respective theories.

As expected, at M = 1.4 the computed X-1 is

very close to the Navier-Stokes result.

At M = 1.6, where the

result is seen to be independent of L > 3, the shock is thinner

than the Navier-Stokes shock, with X-1 close to the value given by Ziering et al.

Although our

is clearly inspired by Grad's

work, and although some of Grad's ideas are resoundingly vindicated, the numerical results do not agree with Grad's, whose shocks are always thicker than the Navier-Stokes shocks.

It seems that five

moments are just one or two short of giving an accurate description of the shock.

51

Between M = 1.6 and M = 1.8 there seems to be a change of regime; suggestively this occurs in the region where Grad's approximation breaks down.

Above M = 1.8 the results seem to agree

with the Mott-Smith predictions.

Comparison of these results with available Monte-Carlo results is difficult, since the Monte-Carlo calculations in the literature cover time spans too short to be of any significance.

The results

contradict the conclusions of Bird [1], whose shocks are always thicker than the Navier-Stokes shocks, and they are in some qualitative agreement with the conclusion of Haviland [9], but one may wonder whether this is more than coincidence. Generalizations and comments. It is quite clear that the procedure of the preceding section will break down, for a fixed number of terms in the Hermite expansion, whenever the Mach number is large enough; certainly by the time all the velocities u = vx + i =

0,

of the same sign.

With ux,

vx given by (21) and N = 5 this breakdown occurs just above Mach number 2.

One could keep increasing the number of polynomials as

M increases; it is more reasonable to systematize the Mott-Smith and Ziering et al procedures by representing f as a sum of two series of the form (3), with scales and centers determined respectively be the conditions upstream and downstream from the shock. Other changes in the scaling (21) may be justified:

for example,

52

it is probably beneficial to introduce two distinct scalings for the variables u and ur.

Another modification our basic method was explained in [3]: the evaluation of the collision integral may be performed by MonteCarlo quadrature, with the possible help of the variance reduction technique introduced in [3].

This should be particularly effective

close to equilibrium when the integrand of the collision term is small, provided this term is not separated into gain and loss terms, as was done by Nordsieck [13].

The methods of this paper are readily generalized to problems in more dimensions and with other types of interparticle force.

53

Bibliography

[1]

G.A. Bird, Shock Wave Structure in a Rigid Sphere Gas, Rarefied Gas Dynamics, Suppl 3, Vol. I (1965).

[2]

S. Chapman and T.G. Cowling, The Mathematical Theory of NonUniform Gases, Cambridge University Press (1958).

[3]

A.J. Chorin, Hermite Expansions in Monte-Carlo Computation, to appear in J. Comput. Physics.

[4]

D. Gilbarg and D. Paolucci, The Structure of Shock Waves in the Continuum Theory o

[5]

Fluids, J. Rat. Mech. Anal., 2, 617.

H. Grad, On the Kinetic Theory of Rarefied Gases, Comm. Pure Appl. Math., 2, 311 (1949).

[6]

H. Grad, The Profile of a Steady Plane Shock Wave, Comm. Pure Appl. Math., 5, 257, (1952).

[7]

H. Grad, Principles of the Kinetic Theory of Gases, Handbuch der Physik, Vol. XII, Springer-Verlag (1958).

[8]

J.K. Haviland, Determination of Shock-Wave Thickness by the Monte-Carlo Method, Proc. 3rd Symp. Rarefied Gas Dynamics, Academic Press (1963).

[9]

J.K. Haviland, The Solution of Two Molecular Flow Problems by the Monte-Carlo Method, Methods in Computational Physics, Vol. 4, p. 109 (1965).

54

[10] S. Kaczmarz

and H. Steinhaus, Theorie der Orthogonalreitan,

Warsaw (1935).

[11] C. Lanczos, Applied Analysis, Prentice Hall (1956).

[12] H.M. Mott-Smith, The Solution of the Boltzmann Equation for a Shock Wave, Physical Review, 82, 885 (1951).

[13] A. Nordsieck and B.L. Hicks, Monte-Carlo Evaluation of Boltzmann Collision Integral,

the

Proc. 5th Symp. Rarefied Gas

Dynamics, Academic Press (1967). [14] A. Sommerfeld, Thermodynamics and Statistical Mechanics, Academic Press (1964).

[15] A.M. Stroud and D.Secrest, Gaussian Quadrature Formulas, Prentice Hall (1966). [16] S.M. Yen, Temperature

Overshoots in Shock Waves, Phys. Fluids,

9, 1417 (1966).

[17] S. Ziering, F. Ek and P. Koch, Two-Fluid Models For the Structure of Neutral Shock Waves, Phys. Fluids, 4, 975 (1961).

55

VELOCITY CORRELATION FUNCTIONS IN 2- AND 3- DIMENSIONS I.

Low Density

by

J. R. Dorfman University of Maryland, College Park, Md. 20742 and

E. G. D. Cohen

The Rockefeller University, New York, N. Y. 10021

ABSTRACT

The long time behavior of velocity correlation functions p(d)(t) characteristic for self diffusion, viscosity and heat conductivity is calculated for a gas of hard disks and hard spheres on the basis of the kinetic theory of dense gases.

In d dimensions one finds that p(d)(t) after an initial exponential decay for a few mean free times t0 exhibits for times up to at least ti 40 t0 a decay ti a(d)(p)

(t0/t)d/2

where a(d) is of the order of pd-l, p = nad with n the number of density and a the hard disk or hard sphere diameter. a (d) (p)

The

are determined by the same dynamical events that are

responsible for the divergences in the virial expansion of the transport coefficients.

In this paper the a (d) (p) are

calculated to lowest order in p. In this order, they are identical to the low density limit of the a(d)(p) that have been obtained by other authors on the basis of hydrodynamical considerations.

56

1.

Introduction.

Recently Alder and Wainwrightl),2) computed the velocity autocorrelation function, p(2)(t), for a system of 500 hard disk particles using computer simulated molecular dynamics. The particles were studied for about 30 mean free times t0 for a range of densities from 0.2 to 0.5 of the density at close packing.

Although for a few mean free times Alder and Wainwright

found the exponential decay that would be predicted on the basis of the Boltzmann -- or Enskog equation, they noted that for times

t

in the range

pD2)(t) showed a

10t0 < t < 30t0,

non-exponential slowly decaying behavior.

Similar results over

a comparable range of densities and times were obtained later for the velocity correlation functions

p(2)(t) and

p(2)(t),

characteristic for viscosity and heat conductivity. 3)

Although they reported only one result in three dimensions for the velocity auto-correlation function p(3)(t), there seems to be little doubt that both the two-dimensional and the threedimensional results can be represented for 10t0 < t < 30t0 by:

(d) pD

(t)

=



ti

t

(d) (p) (-7F-) 0 D

d/2

Here v(t) is the velocity at time t of a chosen particle in the fluid, whose initial valocity is of space,

v(0),

d is the dimension

p = nad, where n is the number density and

diameter of the hard disks or hard spheres.

a

is the

The brackets denote

57

a molecular dynamic time average over all particles in the system.

4)

For d = 2,

(1.1) describes p(2)(t) over the entire reported

range of densities and time within the "experimental error" which is estimated to be on the order of 10%. For d = 2 and d = 3, theory of p(d)(t).

(1.1) also agrees with a hydrodynamical

A hydrodynamical description of their

results for p(d)(t) was presented by Alder and Wainwright,

based on a numerical solution of the Navier-Stokes equations, which was in good agreement with the molecular dynamics calculations.l),2}

Also, the molecular dynamics calculation for d = 2

exhibited a vortex-type of velocity correlation between a chosen molecule and the surrounding molecules, which is very similar to the hydrodynamical flow field surrounding a moving volume element in a fluid which is initially at rest. Furthermore, using hydrodynamical arguments based on an analytic solution of the linearized Navier-Stokes equations, Alder and Wainwrightl),2),4)

and Ernst, Hauge and van Leeuwen5) were

able to derive theoretical expressions for the asymptotic time behavior of p(d)(t) as well as of p(d)(t) and p(d)(t) which are in agreement with equation (1.1) and lead to expressions for a(d)(p), a(d)(p), and a(d)(p)

that are numerically consistent

with the available computer calculations. for p(d)(t) have been obtained by Kawasaki

The same results 6)

and by Ernst,7)

using the hydrodynamical mode-mode coupling theory.

58

The purpose of this paper is to elaborate on a discussion of Alder and Wainwright results using the methods of the kinetic theory of gases, and an analysis originated by Pomeau.8)

A preliminary version of this work has been reported elsewhere.

9)

We shall illustrate our calculations of the long time behavior of p(d)(t)

in detail, while we only sketch the very

similar calculations for ppd)(t) and p(d)(t).

Our starting point is the definition of p(d)(t) given by equation (1.1), where the average is now interpreted as that over a canonical ensemble in the thermodynamical limit.

Such

an average is assumed to be identical with the average used in the computer calculations, if the number of particles used in these calculations is sufficiently large.

In this paper we

will only consider the low density limit of a(d)(p); in a subsequent paper, the extension of the present calculations to higher densities will be given. 10)

We shall formulate the

theory for a general short ranged intermolecular potential.

The formulae will be applied, however, to hard disks and hard spheres only.

In Section 2, we outline the cluster expansion on which our discussion of p(d)(t) is based.

In Section 3, we discuss

a rearrangement of this cluster expansion which is necessary if one wants to find the long time behavior of p(d)(t).

In

Section 4, the hydrodynamical modes of the linearized LorentzBoltzmann equation and the Boltzmann equation, which are needed to find the long time behavior of p(d)(t), are summarized.

59

In Section 5, the t- d/2 time dependence and the coefficient a(d)(p) are obtained for hard disks and hard spheres in lowest

order of the density.

In Section 6, the corresponding expressions for the long time behavior of p(d)(t) and p(d)(t) are given.

The calculation

leading to these expressions is outlined in the Appendix. some apsects of the results obtained in this

In Section 7

paper are discussed.

Cluster Expansion for p(d)(t)

2.

.

We consider N particles in a volume V at temperature T = (BkB)-1 where kB is Boltzmann's constant.

of PD(d) (t)

p (d) D

11)

The definition

is

(t) =

2



=

lim

N,V+



.1

N/V=n

dv1 !

°lx

Dd)

(2.1)

(v1,t)

where

4)(d)(vl,t) =

lim N , V soo

N/V=n

ff

x'N-1

(2.2)

6o

Here xN = xI'x2...xN stands for the phases xi = ri, xi

of

the N particles 1,...,N; m is the mass of a particle.

The

N-particle streaming operator S_t(x_N), when acting on a

function f(xN) of the phases of the N-particles, transforms this function into:

S_t (xN) f (xN) = f (xN (-t) ) where xN(-t) = xl(-t) the particles time t.

... xN(-t)

1,...,N

,

are the initial phases of

which lead to the phases xN after a

They can be obtained from xN by solving the equations

of motion of the N-particle system with the Hamilton function: N

H(xN)

Pi2

N

i=1 2m + i<j=l 4(r i .)

(2.3)

where the interparticle potential 4(rij) is short ranged and

between the two

depends only on the distance rij = Iri rjI

For hard disks and hard spheres of

particles i and j. diameter a, one has

r a

The operator S_t(xN) can be formally written S_t(xN) = exp (-t H(x N))

,

(2.5)

where

Note script letters e.g. A B C D E F G H.. as distinguished from regular typescript e.g. A B C D E F G H...

61

H (xN)

H (xN) ?

l

N

-

HO (x N)

1 i<j

ei

(2.6)

.

Here H0(X

=

m1

2

.

(2.7e)

Vr.

and

aij =

(rij) a

Sri

-+

api

a

(2. 7b)

arJ apj

and the curly brackets {

,

H(xN)}

denote the Poisson

brackets with the Hamilton function H(XN).

For hard disks

and hard spheres, the operator e.. and consequently H(xN) are 13 However, in this case it is still possible to obtain singular. a suitable representation of the streaming operator S_t(xN). Since we do not need this representation in this paper, we 12)

shall not give it here but refer for it to the literature N is the probability density in the canonical ensemble p(x )

P

where Z =

(x )

=

Z-1 exp (-s H (xN) )

dx exp (-6 H(x )).

Finally

v1x is the x-

J

component of the velocity of the chosen particle, 1. We note that due to the time translational invariance of the equilibrium average p(d)(t) = P(d)(-t)

(cf. eq.

(1.1)).

In order to avoid the necessity of solving the N-body problem in the computation of p(d)(t), one expands p(d)(t) in a systematic way in terms of the solution of the 2-, 3-, 4-, body problem.

This can be achieved by generalizing the cluster

expansions used in equilibrium statistical mechanics13) to

62

obtain virial expansions of the thermodynamic quantities and reduced distribution functions to the case of non-equilibrium statistical mechanics.

14)

A cluster expansion p(d)(t) can be obtained in a variety of ways.

Our starting point will be the following cluster

expansion of the N-particle streaming operator S_t(xN) in terms of 1-, 2-, 3-, ... particle streaming operators. 14)

S_t(xN-1)

S_t(xN) = U(xl.t) S-t)

N

+

U(xl'xi't) S-t(xN-2)

iL=2

U(xl,xi,xj,t) S-t(xN-3) + ...

+

(2.8)

,

2> t0 is considerably

simplified, in particular in connection with the resummations to be carried out later, if we first compute the Laplace transform of p(d)(t), which we denote by p(d)(e), which is defined by

pDd) (e)

dt exp (-et)

pDd) (t) = 1 dv1 vlx CD U(d) (vl,E) (2.14)

0

where

D(Dd) (vl,e) =

lim R (m)d+1 V ( dxN-1 G(xN,e) p(xN) vlx J

N,V-*w

N/V=n

(2.15)

and the operator

G(xN,e)

= [e + H(xN)]-1 =

I

0

dt exp (-et) exp (-t H(xN)) (2.16)

64

is the Laplace transform of the streaming operator S_t(xN). The operator G(xN,e) should always be interpreted as given by the right-hand side of the equation (2.16).

If we take the Laplace transform of equation (2.9), we can obtain a cluster expansion of 0Dd)(vl,e) similar to that for (D Dd)(vl,t).

In so doing we will encounter the Laplace transform

of the cluster operator U(xl,.... xs,t), which we denote by

We show elsewhere

U(xl,.... xse).

11),16)

that from the

application of Liouville's theorem, spatial homogeneity, and repeated use of identities like

[e + H(xl,x2,x2)]-1 = [e + H(xl,x2) +

+[e + H(xl,x2) + (813+023)

H(x3)l-1

H(x3)l-1

H(xl,x2,x3)l-1

[e +

the following equality obtains: 1 (s-1)!

r

(

s

d2...

dS U(xl,x2,...,xs) g(r1,...,rs) T7 yvi)

i

1

J

1 11

s

ds A(xl,x2,...,xs) g(r1,...,rs) T7 0(vi)

d2 ...

= s

i

J

,

(2.17)

where the left and right hand side of equation (2.17) are operators which act only on functions of

vi but not rl

and where

A(xl,x2,e) = 812 G(x1,x2,e)

A(xi,x2,x3,e) = 012 G(xi,x2, A(xl,x2,...xs,

)

)

(813+823) G(xl,x2,x3,e)

= 812 G(xl,x2,e) (813+023) G(xl,x2,x3,e)... (01s+02s+... 8s-l,s) G(xl,x2,...,xS,£). (2.18)

65

We remark that for hard disk and hard sphere particles, it is possible to give a representation of these

operators, in

terms of binary collision operators, which avoids the use of eij.16),17) As a result of equation (2.17) we obtain the following cluster expansion for (D Dd)(vl,E):

'Dd)

(v1,E) = s£m

1 + n f d2 A(xl,x2,e) g(r],r2)

0(v2)

+ n2 Jd2 Jd3 A(xl,x2,x3,

ni Jd2... Jdi A(x1,x2,...,x,,E)g(rl,...rQ i

0(v1) vlx

40(vi) +

(2.19)

i

If we further expand the g(r1,...,rs) in powers of n, using (2.12), the following formal density expansion for 0Dd)(vl,E) results: (DD(d)(vl,e)

=

E

nQ AR+1(vl,E)

I 1 +

40(v1) vlx, (2.20)

Q=1

where A

D,-

1

=J d2

A

A

±

,

...

( di

Q

[A(xl... ,xQ,e) g0(r1,...IrQ)

J

+ A(xl,...,x,-1'E) g1(r1,.... r-1IrQ) + ...

12 + A(xl,x2,E)

gk-2(rl,r21r3...rk)

Yvi) (2.21)

.

66

The cluster expansions (2.9) or (2.20) cannot be used to determine the behavior of 0Dd)(v1,t) for times t > t0; or equivalently, to determine the small e behavior of 0Dd)(v1,t). For in addition to the factor 1/c on the right side of equation (2.19), a dynamic analysis of AD(vl,e) with k > 2 reveals that each term diverges as c - 0, and that the most divergent contribution to each AD comes from sequences of k-1 uncorrelated binary collisions 19) among k particles, divergence of AD.

leading to an

An improved expression, which eliminates the above mentioned divergences, can be obtained by regarding 0Dd)(v1,e) to be determined by the equation

20)

-1

30

Ie +

nk Ak+1(vl,e)

I

4) Dd)(vl,e) = am 00(v1 )

k=l

vlx

(2.22)

We may define a new set of

rather than by equation (2.20).

operators BD(vl,e) by means of the density expansion of the inverse operator appearing on the left-hand side of (2.22) i.e. by:

21) 1

Cl +

9

nQ Ak+1(vile

= 1 -

nk

kl

k+l(vit

(2.23)

which yields: k

BD k+1 (Vile) =

1

(-1)]+i

z

}

J{a

J=1

AD a2+1 ... ADa+1 (2.24) a1+1 AD

i

ai=k i=1

This leads to the following equation for 0Dd) (Vile):

67

1 r

n

Dd)(v1,E) = em l6 -

l

E

BD+1(vl'E)

j=l

L

0(v1) vlx (2.25)

f

The operators BD(vl,E) may easily be obtained successively from equation (2.24) as

B2(vl,E) = A2(vl,E) =

J

d2 812 G(xl,x2,E) g0(r1,r2) 40(v2) (2.26a)

83(vl,E) = A3(v1,E) - [A2(vl,E)]2

d2 f d3 612 G(xl'x2'E){(813+e 23)G(xl,x2,x31E)

= !

g0(rl,r2,r3) - g0(r1,r2) 813 G(x1,x3,E)

g0(r1,r3) + 91(rl'r21r3)1 0(v2) 00(v3)

(2. 26b)

and so on. Although in three dimensions there are phase space arguments, based on the dynamics of 2 and 3 particles, which indicate that

EBD

and

EB3

exist in the limit E -* 0, these same

dynamical phase space arguments suggest that for Q > 4 the EBD diverges as

19),22) 1

e

->

0 when acting on a general function of

In fact these phase space arguments suggest that

for E -r 0, EB4

'

log e, while

EBD

ti

E-(Q-4) for pt >4, for

particles interacting with a short range repulsive potential.

Similarly for d = 2, although

EBD exists, phase space -(k-3)

arguments give that for c } 0, for k > 3.

EBD ti log e while EBD '

E:

For a gas of hard disks, the phase space arguments

for the divergence

EBD

have been substantiated by Sengers23)

68

and others, 24) and similar results have been obtained for a

variety of Lorentz models. tion of the operators BD

25),26)

Thus, although the introduc-

removes the most divergent

contributions for e 3 0 in each order of n in the density expansion (2.20) of cD Dd)(vile), there still remain divergent

contributions in the BQ operators if e } 0.

A further

rearrangement of the expansion (2.25) is therefore necessary. Since no rigorous proof of these divergences in the eBD has been given, other than for eBD for hard disks, we will carry out a rearrangement of the expansion (2.25) in the following section based on the assumptions:

(a) the qualitative behavior

of eBD for small e is the one quoted above, and (b) this behavior is due to the dynamical events discussed in the next section.

3.

The Binary Collision Expansion of BR(vl,e); Resummation. It is generally assumed, although not proven, that for a

short ranged repulsive potential, such as that given by equation (2.4), sequences of i-binary collisions are responsible for the most divergent contributions to the operators

BQ(vl,e) in the limit e } 0.

In order to isolate

these most divergent contributions in each BQ

and then to sum

them up into a well behaved operator, we introduce in this section an expansion of 8 (vl,e) in terms of sequences of binary collision:

the binary collision expansion of the BQD 4. (vl,e).

69

The basic binary collision expansion which we use reads for a general potential 12),16),20)

ea G(xl,...,xs,C) = Cs(a,E)11+

E

Cs(S,E)

$7a +

Z

Cs(a,E) C5(Y,E) + ...]

(3.1)

S#a,YP6s

Here

a,S,y, ...

denote pairs of particles chosen from the

particles (1,2,...,s) and the operator

CS(a,e) is defined by

(3.2a)

Cs(a,e) = ea Gs (a, E)

where Gs(a,E) = [c + H0(xs) - ea]1

(3.2b)

In deriving the expansion (3.1), repeated use has been made of identities similar to that preceding equation (2.17). We further define a binary collision operator Ts(a,E) in terms of Cs(e,e) by

Cs (a, 6) = Ts (a, 6) G0 (xs)

(3.3)

with -1

G0(xs) = [e + H0(xl,x2,...,xs)]

(3.4)

The following developments are all based on the binary collision expansion (3.1), using the binary collision operator Ts(a,E) given by (3.3). Ts(a,e)

The operator defined in (3.3) is denoted by

in order to conform with the literature, where a repre-

sentation of this operator is discussed for hard disks and hard spheres.

121

70

An analysis of

eB3

,

eB4

,

...

on the basis of the binary

collision expansion leads to expressions for these operators as

22),27)

(3.5a)

CBD(vl,e) = e f d2 C2(xl,x2,e) g0(Y1,r2) Yv2) eB3(vl,e) = C

d3 C2(xl,x2,e) AD (xl,x2lx3;e)

d2 1

C2((xl,x2,e) r0(v2) 0(v3) + L.D.T. CB4(vl,e) = e

I

(3.5b)

d2 J d3 J d4 C2(xl,x2,e) A3(xl,x2Ix3;e)

J

A3(xl,x21x4;e) C2(xl,x2.6) 1 0(vi) + L.D.T., i=2 (3.5c)

and so on where

AD(xl x21xj;e) = C3(xiIxj,C) + C3(x2,xj,e)

(1 + P2j),(3.6)

with Pij , the permutation operator, which exchanges particle indices i and j.

In obtaining equation (3.5b) etc, one uses

apart from Liouville's theorem, that when acting on a function of velocities of the particles 1,2,3,...

the operator

Cs(xl,x2,e) can be replaced by the operator C2(xl,x2,e), and that a sequence like C2(xl,x2,e) C3(xl,x3,e) C2(x2,x3,e), which ends in a C-operator that does not contain particle 1 in the interacting pair, is not a most divergent contribution to CBD.

The terms which we have explicitly written out in

equation (3.5b), contain the contributions from those sequences of i-binary collisions among k particles that we assume to be the most divergent in each BD in the limit e - 0.19) Due to

71

the graphical representation of these terms given by Kawasaki and Oppenheim,22) these terms are generally referred to as the "ring events". by L. D. T.

The remaining parts of e8D are indicated

("less divergent terms") and contain (a) sequences

of more than k binary collisions among k particles, which are all less divegent or convergent,

(b) terms that contain

equilibrium cluster functions (g0(r1,...,rk)-l, g1(r1,...,rk), etc.) and (c) terms where more than two particles are within a distance of the

0(a) at the same time.

These latter terms

may involve genuine n-tuple collisions for n > 2, or a number of binary collisions which take place within a few collision times tc.

For a further discussion of these points we refer

to the literature.

17),19)

Although in three dimensions

eBD is finite as c - 0, it

will be convenient for the determination of the long time behavior

p(d)(t) to include this term, as given in

equation (3.5b), in the resummation to be performed below. We are now in a position to consider the summation of the most divergent terms, or ring events, in the BD-series appearing in equation (2.25). We write

e

D nk e BD +l = e - n e B2 - n e RD(v1,e) + L.D.T.

(3.7)

with 1-3

ERD (v1,e) = C

r

d2 C2(x1,x2,E)I1 - n J

f

d3 AD(x1,x2lxg%e)O0(v3).J

C2(x1,x2,e) 40(v2).

(3.8)

To obtain (3.8) we have added and subtracted a finite term,

72

e n f d2 C2(xl,x2,e)

C2(xl,x2,e) c0(v2), to the geometric

series used to obtain RD(vl,e).

The subtracted term together

with all the less divergent (and finite) terms of order n2 RD(vl,e), are

and higher, which have not been included in

collected in the term which we denote by L.D.T. in equation (3.7).

The expression for eRD(vl,e) may be further simplified by using the relation between C and T operator given by equation (3.3), and by using the fact that eRD acts only on functions of the velocity of particle 1.

eRD(vire)

= J d2 T2(xl,x2,e)

Thus we may write

27)

Ie + H0(xi x2) 1

-n f d3 AD(xl,x21x3;e"0(v3)1 (3.9)

where D A33

(xl,x2I x3'- 6) = T3(xiIx316) + T3(x2,x3,6)

(1 + P23).(3.10)

The determination of the long time behavior of p(d)(t) to D

be carried out later will be greatly facilitated if we go over to a Fourier representation of the fact that

eRD(vi,e).

To do this we use

eRD(vl,e) does not depend on r1, and write

eRD(vl,e) = j dv2

dr1 j dr2 S(rl) T2(xi,x2,e)

E: + H0(xl,x2)

J

4. } -n jdr3 jdv3

-r-1 D

T2(x1,x2,s)c0(v2). (3.11)

Then by inserting S-functions, using their Fourier representation and that 20)

73

drl... J

l

(2Tr)

k r.)

diR exp (-i

j=1

1

d (Q-1)

j=1 2

8 (kl+k2-kl-k2) T7 6

one finds

28)

eRD(vl,e) = J

dkd (27r)

Ie

(3.12)

+ ik

v4.

T2(xl,x2,e)

f e-2

1-1

(3.13)

,

where

kvl,e)

d3 0,

w(k=0) = 0

and ((k=0) = 0.

Using the fact that XQ(vl) has only one zero eigenvalue, one has, writing

w (k) = w0 + w1 k + w 2 k 2 and

+ ...

,

X (w) (k,v) = X0w) (v) + k X1w) (v)+

(4.3a)

... ,

(4. 3b)

77

that the one hydrodynamic mode of the operator

n AD(v) 0

is a diffusive mode which is to 0(k2) given by: (4.4a) (4.4b)

(4.5) where DO is the value for the self diffusion coefficient obtained by the basis of the Boltzmann equation.

Similarly, using the

fact that X0(v) has d+2 zero eigenvalues and writing

Q W = 00 + k 01 + k 2Q 2

+ ...

,

(k,v) = 0(V) + k

(v) +

...

(4.6)

and

Oi

(4.7)

,

to find that to 0(k2) the eigenvalues are given by: (Vi)

1(H)

Q

00 = Q1

= 0

,

i = (1,...,d-1),

a + -) _ + ic0

(4.8b)

1 (Vi)

(4.8c)

v0 = fl0/n.m

Q2 0(H)

= DT

(4.8a)

=

(4.8d)

0

Q2±)

= 1 rs0 = 2 ( 2(dd-1) vO + (y0-l)DTO]

.

(4.8e)

Here H denotes a heat-mode, Vi, i = 1,...,d-1, the d-1 shear(or viscous) modes and ±, the two sound modes. c0

d+d2

(em)-1]1/2

Furthermore

is the ideal gas sound velocity in

d-dimensions, yO = cpO/cv0 = (d+2)/d where cp0 and cv0 are the ideal gas specific heats at constant pressure and volume, respectively;

r10 and A0

,

are the values for the coefficients

78

of viscosity and thermal conductivity respectively obtained We note that the bulk viscosity

from the Boltzmann equation.

The subscript 0 denotes

vanishes in the low density limit.

that the low density limit has been taken.

The corresponding

eigenfunctions 0(k) are:

(V.) 00 1 (k,v) = (Rm)1/2 k(1).V 00(H)(k,v)

,

2

= (2)1/2

- 2)

m2v 2

00(H) (k,v)

= (5)1/2

0±) 0(k,v)

=

60})(k,v)

= (10)1/2 [(imv2/3 +

Here k,

(4.9a)

m2v

-

2)

,

d = 2,

(4.9b)

d = 3,

(4.9c)

2 [Smv2/2 + smc0(k.v)],

kil),

(d-1)

(4. 9d)

d = 2,

d = 3

(4. 9e)

form a Cartesian set of

mutually orthogonal unit vectors,

We use the hydrodynamic modes to express the operator

}

D-.

->

[s+ik..v12- na0(v1) - na0(v2)]

-1

for small k, when acting

on a function of the form [c +

D f(vi,v2) SH

as

na0(v2)]-1

na0(v1) -

f(v1,v2) 0(v1) 0(v2)

*0(v1) g0(V2) + SD f(v1,v2) c0(V1)

0(v2),(4.10a)

where

SH f (vl,v2) 0 (v1) 0 (v2)

w,8

(E: +w (k)+8 (k)) -1X (w) (k,v 1)

k,v2)

(dv1 (dv2 X(w) (k,v1) f(v1,v2) 0(v1) c0(v2)

.

(4.10b)

79

Here the prime on the summation symbol indicates that only the hydrodynamic modes X(w) and 0(Q) in the sum.

The other operator,

SD

are to be included ,

contains the

contribution from non-hydrodynamic eigenfunctions i.e. from perturbed eigenfunctions obtained from non-zero eigenvalues of XD and A0.

The Behavior of pD(d) (t) in Time:

5 .

The t

" 2 Dependence.

In this section we shall compute the behavior of pD(d) (t)

in time for hard disks and hard spheres by iterating the operator on the right-hand side of equation (3.18) about [e - n X0(v1)]-1.

In this way we shall obtain an initial

exponential decay, which in the low density limit can be derived from the Boltzmann equation, as well as a long time behavior ti

t-d/2

Using the equations (3.18) and (3.13) we have then:

P

(d) D

(e) = pDd) (e) + pD(d) ,l (e) + ... ,O

,

(5.1)

with PDd0(e) = gm f dv1 vlx [e - na0(vl)]-1 vlx

0(v1(5.2a)

and PD,l(e) = (3mn Jdv1vlx[e-nX0(vl)1-1R0D (vl,e)[e-nX0(v1)] (5.3)

Here we have assumed that one can drop the less divergent terms in the equation (3.18) for the computation of the long time behavior of p(d)(t) for hard disks and hard spheres

80

at low density.

An indication for the correctness of this

assumption can be found in a subsequent paper.10)

Since we

shall only calculate the first two iterates, the time interval over which our results are valid may be restricted. This point will be further discussed in

Section 7.

leads to an expression The Laplace inversion of p(d)(e) D,O for p(d)(t) for all t of the form D,O pDdO(t) = Rm f dv1 vlx exp

(na0(v1)t) vix 0(v1)

(5.4)

Although the expression may be evaluated in terms of the eigenvalues and eigenfunctions of the operator na0(v) as a sum of exponentials, it is a sufficiently good approximation

to replace (5.4) by pDdO(t)

exp [- t(RmD00)-

(5.5)

where D00 is the self diffusion coefficient obtained from the Lorentz-Boltzmann equation in first Enskog approximation.

32)

Since SmD00 is proportional to the mean free time t0, pDd)(t) decays over a few mean free times.

33)

We now evaluate p(di(t) equation (5.3), and divide the k-integral, appearing in this expression, into two pieces for which 0

< k < k

and k0 < k < - respectively, where k01

is on the order of a mean free path.

The contribution to

pDd)(e) coming from k > k0 will be neglected, for we assume

that this part of the k-integration incorporates the effects of collisions which take place on a space scale small compared to a mean free path, in which we are not interested here.

81

The region for which k < k0 is the region where the perturbation theory outlined in the previous section is valid. Using the equations (4.10a,b) we may express this part of p(d)(s) as: D,l

dk d

pDd1() ti nsm

(2n)

f k to, pDdi(t) can be expressed as:

(d) pD,l (t)

ti

Sm

(

n

0

dk (27r)

d exp [- (52 (k) +W (k) ) t]

dv1 vix 00k,v1)

0(v1))2

(5.10)

.

Considering the tensorial character of the hydrodynamical modes (D0M)(k,v1), we see that only the second and shear modes of

00(0)

give a contribution to equation (5.10). Of these modes,

the shear modes give the dominant contribution to p(di(t),

since the presence of the + ikc0

in the sound mode eigenvalues

can be shown to result in a faster time decay than that given by the shear modes.

16)

We therefore obtain

d-1 pDdi (t) ti Rn z

dk

f i=1 k< k 0

d exp (-tk2 (D0+v0) ]

(2TT)

(vi )

dv1 vlx 00

[

} ->

2

(5.11)

J

Using the fact that for d = 2,

k1 = k (ky-kx), we obtain,

with (4.9), for d = 2

P(2) D,l (t)

ti

87rn(11

0+v 0)t

[1 - exp [-(D0+v0)k2t1]

(5.12)

84

or, for t >> t0

pD21(t) ti For hard disks of

[8Trn(D0+v0)t0]-1

diameter

(t0/t)

(5.13)

t0 is in the low density

a,

limit given by 1/2

t0 = (WIT)

(5.14)

= 2na

while D0 and v0 are in first Enskog approximation

D0 =

v0 =

(2na(am1r)1/2)-1

(5.15a)

(2na(Smir)1/2)-1

(5.15b)

so that, for such particles p

Similarly, for d = 3,

nat. t 0

(2)(t) ,,

D,1

(5.16)

t

4

p(3)(t) becomes D,l

2

(V.)

d}

exp [-k2t(D0+v0)][ dvlvlx

P(3)(t) ti $m

i t

k< k 0 J

(V) [

2

d.k(k(1) + kit) 8Tr1n

!r

1(-k,v)

0

T

(

;

2

j dv1 vlx 00 1 (-k,v)$0(v1)l k0

2

dk k2 exp[-k2t(D0+v0)l

)

J

110

(5.17)

where kiX) and k(2) are the x-components of the two mutually orthogonal unit vectors which, together with k form a Cartesian set.

The k-integral may be shown to be equal to 87r/3 so that

with (4.9):

85 k0

pD3i(t)

1 ti

3Tr n

f dk k2 exp [-k2t(D0+v0)l 0

,,-I- [Tr (D0+v0) t0F3/2 for t >> t0.

(t0/t) 3/2

(5.18)

For hard spheres of diameter a, t0 is for

low densities given by

29)

t0 = (Sm/Tr) 1/2 / 4na2

(5.19)

and using the values of D0 and v0 in first Enskog approximation D0 =

32

29)

(a MR)-1/2

8na

v0 =

5

2

(Sm7r)-1/2

16na

we obtain

35)

P(3)(t) - 12 (14)3/2 (na3)2 (tO)3/2 = 1.17 (na3)2 (t0/t) 3/2 D,l

(5.20) Equations (5.16) and (5.20) exhibit the t-d/2 behavior found by Alder and Wainwright and are consistent with the computer results extrapolated to low density.

6.

The Behavior of p(d)(t) and p(d)(t) in Time. Using similar procedures as those for p(d)(t), the behavior

of other velocity correlation functions with time can be determined.

In this section we discuss those velocity

correlation functions that give the kinetic contributions to the coefficients of shear viscosity and thermal conductivity.

86

36)

In particular we shall consider functions of the form N P (d) (t)

=


J(vi(-t))> / -lmd V r dxN-1

S

J

t (xN) p(XN)

i=l

J(v) i

N/V=n (6.3b)

In view of the great similarity of the equations (6.1)

and (6.3) to (2.1) and (2.2), it will be clear that the time behavior of p(d)(t) can be determined in a manner similar to that used for p(d)(t).

We will briefly outline this

procedure in the Appendix and only give the main results here. Corresponding to equation (5.2) we may expand p(d)(e) as

Pi(d) (6) = P (d) (e) + PJd) (a) + The Laplace inversion of p. J,O(e) leads to an exponentially

(6.4)

87

decaying function similar to that given by equation (5.4)

for pDd) (t). <J2(vI)>-1

pJdO(t) _

J dvl J(-V)',)

exp [na0(v1)t] J(vl) 0(v1)

(6.5) In the first Enskog approximation

for all t.

pnd) (t) = exp

(6. 6a)

[-t(Rmv00)-1]

and

exp [-t(4mDT00)-ll

p( d)(t) = X'O

where

v00 and DT

(6.6b)

are the first Enskog approximations 00

to V0 and DT

,

that have been defined in the equations

0

(4.8c,d), respectively.

We remark that p(d)(t) decays over a period of a few mean J'O

free times.

A treatment of p(d)(t) similar to that given for p(d)(t) shows that for long times, the dominant behavior is contained in the expression:

P

(d) (t) J'i

<J2 (V 2n (

Q,Q

J

dk (27

exp (-t (U (k)+U' (k)) ]

dvl J(vl) 00") (k,vl)

p(St

)

(6.7)

where the prime on the summation sign means that only the hydrodynamic eigenfunctions 0

and

0(Q1)

are to be included,

and the subscript zero on the eigenfunctions refers to them in the approximation given by equation (4.9). Of all the

88

hydrodynamic modes in the summation in equation (6.7) the comes from those combinations dominant contribution to p(d)(t) J,l

which are such that the sum S2 (k) + S2' (k) is ti k2.

of S2 and S2'

These combinations are easily seen to arise from (a) two shear modes, (d)

(b) two heat modes, (c) a heat and a shear mode, and

two sound modes such that one has eigenvalue

ikc0 +

2

r50

and the other

k2

k2.

-ikc0 +

rs0 2

Inserting the expressions for S2(k) given by equation (4.8) and for

8(,Q)

given by (4.9) we find that only combinations

of two shear modes, and of two "opposite" sound medes contribute to ppd)(t), while combinations of one shear and one heat mode, and of two "opposite" sound modes contribute to pXdi(t), because of the tensorial character functions Jn(v1)

of the

respectively.

and J(v1)

For d = 2, the long time behavior of p(2)(t) is given by

ri'l

2

pndi(t)

dk

ti 1

f

k< k0

(2tr)

2

k k

2

x4y [2 exp (-2v0+k2)+ exp(-Ps0k2t)],

k

(6.8)

where the first term in the brackets incorporates the contributions of the shear modes, while the second term contains the sound mode contribution.

Carrying out the k-integrals, we obtain for t >> t0 pn2)(t)

(32Trnt0)1 [v01 + (vo+,l0/2nkB)-

'

(t0/t)

(6.9a)

or

pp2) (t)

ti

6 (na2) (t0/t)

where we have used that = (Sm)-2

(6.9b)

and the values for

89

X

0

and n0 for hard disks of diameter a in first Enskog

approximation.

Similarly

pX2) (t) ti 2n

dk

2kj2

(2ir) 2

k2

exp[-k2t(v0+a0/2nkB)I 2k +

2

(6.10)

k2 exp-rsOk2t]

where the first term in the brackets incorporates the contributions from combinations of shear and heat modes, while the second term contains the sound mode contribution. Thus P(2l)(t) ti

(47rnt0)-1 (v0 + a0/2nkB)-1 (t0/t)

(6.lla)

or

p(2) (t) " 3 (na2) (t0/t)

(6.llb)

where the values for n0 and X0 in first approximation have been used.

For d = 3 we find that

35)

4v3t0)-3/2](t0/t) 415?OtOnk

p(3) (t)

ti

3/2

+

(6.12a)

or using the values for a0 and no for hard spheres of diameter a in first Enskog approximation: 29)

P

64 [7(10) -3/2 + (3/35) 3/2]

(3) (t) n,l

15 I

1.05 (na3)2

(t0/t)3/2

while for p(3)(t) we find that

35)

(na3) 2

(t 0/t) 3/2 (6.12b)

90

tO)-3/2 +

P (3) (t)

ti

(12n7t 3/2)-1[(vt+ o0 5nk B 1

4a t

+ 1 (15nk0 + 3

v0t0)-3/21 (t0/t)3/2 (6.13a)

B

or p(3i(t) ti 1.32

(na3)2 (t0/t)

3/2

(6.13b)

if the first Enskog approximation to n0 and a0 for hard spheres is used.

The equations (6.9), (6.11),

(6.12) and (6.13) are

consistent with the computer results of Alder and Wainwright, extrapolated to low densities.

7.

Discussion.

A number of remarks can be made in connection with the results presented here. (1)

The expressions given by the equations (5.13), (5.18),

(6.9a), (6.lla), (6.12a) and (6.13a) are identical with those derived by Alder and

Wainwright,l),2),4)

Ernst, Hauge, and

van Leeuwen,5) and Kawasaki for p(d)(t), pnd)(t) and p(d)(t)

on the basis of hydrodynamical considerations, if one replaces the transport coefficients in the expressions given by the above mentioned authors by their low density values. (2)

The results of sections 5 and 6 for the long time

behavior of p(d)(t) seem also to apply to a general class of systems with short range interparticle forces. This obtains,

in spite of the k- and e- dependence of the Fourier representation of the T-,

D. and A- operators in this case.

For, this

91

k- and s- dependence seems to incorporate effects on the scale of the range of the interparticle forces and of the duration of a collision, which both should lead to corrections of 0(n) compared to the effects on the scale of the mean free path and mean free time, considered here. This expectation seems to be borne out by machine calculations of P(3)(t) by Verlet and Levesque for systems of particles interacting with a 12 - 6 Lennard-Jones potential. (3)

37)

As remarked before, the results obtained here are

consistent with the machine calculations of Alder and Wainwright1)'2),4)

extrapolated to low density.

In this connection it is interesting to note that in two dimensions a

(1/t) time dependence is obtained for p(2)(t),

without carrying out the rearrangement discussed in Section 3. This is due to the log e behavior of eBD and eB3 -- which is defined in the Appendix -- for small c.

Using the results of

Sengers23) for the coefficients of the log e terms in e$D and EB3 , we have computed the coefficient of (t0/t),

which would be obtained from eBD and e83.

A comparison of

the results for the coefficients of the (t0/t) term in p(2)(t) before and after resummation is presented in Table I.

It is

clear that the unresummed coefficient is inconsistent with the machine calculations of Alder and Wainwright, having the opposite sign in two cases, and being between 5 and 20 times smaller than the coefficient obtained in the resummed theory.

This agreement with the Alder and Wainwright machine computations can be taken as an a posteriori justification for the

92

rearrangement carried out in Section 3 and as consistent with the existence of the divergences in which necessitate such an arrangement.

sBQ

as s - 0,

In fact, the same

dynamical events responsible for the most divergent contributions to the

sSQ and

c8

are, after resummation, responsible

for the (t0/t)d/2 tails in the velocity correlation functions p

(d)

(t),

(4)

Ppd) (t)

and

pfd)

(t).

We have considered here only the first two terms in

the iteration method to determine p(d)(t).

This may set an

upper limit for the time interval over which the results obtained here are valid.

The higher iterates involve more

complicated dynamical events than considered here.

A rough

estimate of the terms we have neglected suggests that they may make themselves felt for times longer than about 40 t0. This would imply that the (t0/t)d/2 - terms should be dominant in p(d)(t) for the times relevant in the Alder and Wainwright machine computations. (5)

Physically the long time tails of the correlation

functions are caused in our calculation by the slowly decaying hydrodynamic modes.

Kinetically this is due amongst others,

to the possibility of re-collisions, i.e., collisions between two particles that have collided before. They lead to a much slower decay of the initial state of a particle than if they are excluded, since they can still "remind" the particle of its initial state after many collisions have taken place. (6)

Since the transport coefficients are related to time

integrals of the time correlation functions p(d)(t), the

93

results (5.13), (6.9a) and (6.lla), if valid for all t >> t0 would imply that the time correlation function expressions for the Navier-Stokes transport coefficients do not exist in two dimensions.

Similarly, the results (5.18), (6.12a),

and (6.13a) would imply that the time correlation function expressions for the Burnett-transport coefficients do not exist in three dimensions, since integrals of the form 5),38) J

dt t p (d) (t)

0

occur.

However, we stress that in view of the fact that the

results obtained here may only hold over a restricted time interval, the existence or non-existence of these transport coefficients in two dimensions is an open question. (7)

In view of the long tail of the time correlation

functions p(d)(t), a sharp separation of kinetic and hydrodynamic time scales is not possible. Therefore the precise range of validity of even the Navier-Stokes equations is not clear, since in their derivation it is tacitly assumed that the transport coefficients attain their full value on a kinetic time scale which is much shorter than the hydrodynamical time scales to which the equations apply.

In

particular, it is not clear to what extent these equations can be used for phenomena which are not infinitely slowly varying in space and time. In a subsequent paper we shall discuss how the present considerations can be generalized to higher densities. We will obtain there, in a similar fashion as in this paper,

an initial exponential decay followed by a decay (t0/t)d/2

94

with coefficients that reduce to those obtained here in the low density limit and which can also be compared with the results of the hydrodynamical theories.

These results will

also allow a comparison with the computer data of Alder and Wainwright over the whole range of densities for which they are available.

Acknowledgement

The authors gratefully acknowledge stimulating discussions with Drs. Alder and Wainwright, Ernst, Hauge and van Leeuwen, Wood and Erpenbeck, Hemmer and Dufty.

One of us (J.R.D.) would like to thank

the Rockefeller University for its hospitality during the year 1969-1970, as well as the National Science Foundation for its support under Grant NSF-GP-29385.

95

Appendix Here we outline the method which leads to equations (6.9a) (6.10), (6.12a) and (6.13a) for ppd)(t) and p(d)(t). X,l

Since

,l

the method closely parallels that used to obtain p(di(t) we will only indicate the essential modifications. By taking the Laplace transform of equations (6.3a) and (6.3b) one can write

pJd) (e) = J dv1 J (v1) 4) (d) (vl,e)

(A.1)

with N dxN-1 G(xN,c) p(xN) i=l ZJ(v i)

r 4)j

(vl,e) =

lim V and <J2(v )>-1 1

N , V--

I

11

N/v=n

(A.2)

Using the method outlined in Section 2, one obtains for (D (d)(vile) the equation (see eq.

(P

( id) (v1,') =

I1 + I nk

(2.20)) J(v1)<J2(v1)>-1

Ak+1(vl,e)]$0(v1)

E where

(A.3)

Ak(vl,e) are given by the equations (2.21), provided k

that in the operators

AR(vl,e), the product JT 40(vi) i=2

k

k

is replaced by E 00(vi) i=1

Defining the operator

Pli(o0(vl))-1

i=1

Bk(v,e)

by an identity similar

to (2.23) and using Ak instead of AD

one can give

an

(d) } (vile) in a form similar to equation (2.25) expression for 4) J

for (D

Dd)

(vl , e)

(D Jd)(vl,e) = <J2(v1)>-1 [e-

I

nke 8

-1 k+1(v1,e)]

4.

0(vl) J(vl).

k=1 (A.4)

96

Here the BQ bear the same relation to the AZ as the BD do to the AQ, of equation (2.4).

the operator

For hard disks and hard spheres,

eB2(vl,e) is the linearized Boltzmann collision

operator X0(v1) given by equation (3.17).

Using the binary collision expansion, one can sum the most divergent terms in the B-expansion with the result nQ e BR+1(vl,E) = n e R(vl,e) + L.D.T.,

(A.5a)

P,=2

with dk d r dv2

f

[e+ikv12-nak(v1)-na-kv2)]

0(v2). (A.5b)

For reasons identical to those given in Section 3, we replace in (A.5b) the operator [e +

nAj(vl)

- nX_j(v2)]

by the operator [e+ ik v12 - nA0(vl) - na0(v2)]-1 to obtain

eRO(v1 E)

Equations (6.4) and (6.5) for p (d) (c) and

(E) respectively

are obtained by writing <J2(v1)>-1

Jd)(vi,e) =

[E-nA0(v1)-n

J(v1)00(v1)

(A.6) by iterating about the operator [e-na0(v1)] (A.1).

(d)

is obtained from equation (A.6) as

p,.

pJ'l(e) = n<J (v1)>

and then using

2 4-

+

1

f

4

dv1 J(v1)

[c-nA0(v1)]

[E-nX0(v1)]

1

1

-, cR0(vl.e)

4.

(A.7)

97

Proceeding as in Section 5 and using identifies similar to those employed in the transition from equations (5.8) to (5.9),

the following expression for p(d)(e) is obtained 16),39) J'l

(cf. eq. (5.9)): }

r

<J2(v1)>-1

pidl(e) ti 2

(2r)

-1

d [e + Q(k) + W'(k)]

c,c'

[

J dv1 J(v1) X0(v1)

00Q)(k,v1)

[e-na0(v1)]

0(v1)]2.

(A.8)

Neglecting in the square brackets terms of O(e) or higher,

Laplace inversion of (A.8) leads for t > t0 to the equation (6.7), from which all further results of Section 6 can be derived.

98

References

1. B. J. Alder and T. E. Wainwright, Phys. Rev. Letters 18, 988 (1967); J. Phys. Soc. Japan 26 Suppl. 267 (19697 2. B. J. Alder and T. E. Wainwright, Phys. Rev. Al, 18, (1970). 3. T. E. Wainwright, B. J. Alder and D. M. Gass, Phys. Rev. A4, 233 (1971). 4. B. J. Alder, D. M. Gass and T. E. Wainwright, J. Chem. Phys. 53, 3813 (1970). 5. M. H. Ernst, E. H. Hauge, and J. M. J. van Leeuwen, Phys. Rev. Letters 25, 1254 (1970); Phys. Rev. A4, 2055 (1971); Phys. Letters 34A, 419 (1971). 6. K. Kawasaki, Progr. Theoret. Phys. (Kyoto) 45, 1691 (1971). 7. M. H. Ernst, preprint (1970).

8. Y. Pomeau, Phys. Rev. A3, 1174 (1971); see also Phys. Letters, 27A, 601 (1968). 9. J. R. Dorfman and E. G. D. Cohen, Phys. Rev. Letters 25, 1257 (1970). 10. J. R. Dorfman and E. G. D. Cohen (in preparation). 11. Cf. M. H. Ernst, L. K. Haines, and J. R. Dorfman, Rev. Mod. Phys. 41, 296 (1969), esp. for a bibliography.

12. M. H. Ernst, J. R. Dorfman, W. R. Hoegy, and J. M. J. van Leeuwen, Physica, 45, 127 (1969), 13. Cf. J. De Boer, Rept. Prog. Phys. 12, 305 (1949); and G. E. Uhlenbeck and G. W. Ford, Studies in Statistical Mechanics, I. G. E. Uhlenbeck and J. De Boer eds. North Holland Publ. Co. Amsterdam (1961), p. 123. 14. Cf. E. G. D. Cohen, Physica 28, 1025, 1045, 1060 (1962); J. Math. Phys. 4, 143 (1963), and M. S. Green and R. A. Piccirelli, Phys. Rev. 132, 1388 (1963). See in particular, E. G. D. Cohen in Fundamental Problems of Statistical Mechanics II, E. G. D. Cohen ed., North Holland Pub-l. Co., Amsterdam (1968), p. 228. 15. T. Hill, Statistical Mechanics, McGraw-Hill, New (1956), p. 184. 16. W. W. Wood, J. Erpenbeck, J. R. Dorfman, and E. G. D. Cohen, (in preparation).

York,

99

17.

J. V. Sengers and W. Hoegy, Phys. Rev. 2A, 2461 (1970).

18.

Cf. refs. 11, 14, and N. N. Bogolubov, Studies in Statistical Mechanics, I, G. E. Uhlenbeck and J. De Boer, eds., North Holland Publ. Co., Amsterdam (1961), p. 11.

19.

See J. R. Dorfman and E. G. D. Cohen, J. Math. Phys. 8,

20.

R. Zwanzig, Phys. Rev. 129, 486 (1963).

21.

This inversion procedure is equivalent to the equilibrium-like inversion procedure used by Cohen (cf. refs. 11, 14) which is based on the method used to derive density expansion from fugacity expansions in equilibrium statistical mechanics.

22.

K. Kawasaki and I. Oppenheim, Phys. Rev. 139A, 1763 (1965); in Statistical Mechanics, T. Bak, ed., W. A. Benjamin Inc. New York, (1967), p. 313, Cf. ref. 11 for a bibliography.

23.

J. V. Sengers, Phys. Fluids 9, 1685 (1966).

24.

L. K, Raines, J. R. Dorfman, and M. H. Ernst, Phys. Rev. 144, 207 (1966),

25.

Cf. J. M. J. van Leeuwen and A. Weijland, Physica 36, 457 (1967), 38, 35 (1968); E. H. Hauge and E. G. D. Cohen, J, Math. Phys. 10, 397 (1969); P. Resibois and M. G. Verlarde, Physica 51, 541 (1971).

26.

W. W. Wood and F. Lado, J. Comput. Phys. 7, 528 (1971).

27.

Cf. J. R. Dorfman, Lectures in Theoretical Physics (Boulder))9C, W. E. Brittin, ed., Gordon & Breach, New York (1967), p. 443.

28.

This form is closely related to that given by Kawasaki and Oppenheim, ref. 22; see also ref. 27.

29,

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd ed., Cambridge Univ. Press, London 1970

282 (1967).

.

30,

Cf. J. D. Foch and G. W. Ford, Studies in Statistical Mechanics, V, J. De Boer and G. E. Uhlenbeck eds., North Holland Publ, Co., Amsterdam (1970), p. 103.

31.

Although we will apply the results of this section to a gas of hard spheres or disks only, all the results of this section, in particular the equations (4.4), (4.5), C4.8a-e), and (4.9a-e) are valid for a general short range repulsive potential. One needs only identify

100

XD and a0 with the Lorentz-Boltzmann and linearized Boltzmann ecollision operators appropriate for such a potential, and D0, n0 and a0 as the corresponding coefficients of self-diffusion, shear-viscosity and thermal conductivity, respectively. 32.

That is,} we treat vlx as an approximate eigenfunction of n a0(vl) with appropriate eigenvalue (R m D00) See also ref. 29.

33.

One can show that the initial slope of p(d)(t) for hard disks and hard spheres is given by: = SmnX

(dpDd)(t)/dt)

t=0

1

16)

dv1 vlx A0(vl) vlx 0(v1

where X is the equilibrium radial distribution function evaluated at an interparticle distance equal to their 29) We are indebted to Drs. E. H. Hauge diameter. and W. W. Wood for this remark. 34.

Cf. J. V. Sengers ref. 23, and Lectures in Theoretical Physics, (Boulder) 9C, W. E. Brittin, ed., Gordon & Breach, New York (1967), p. 335.

35.

Equation (5.18) for p(3)(t) has been obtained by Dufty on the basis of a procedure which does not involve the iteration method used in this section, but other approximations. He determines the approximate eigenvalues of the operator [E- na0 - nER01-1.

obtains by his method equation (6.12a) for

He also p(3)(t)

and equation (6.13a) for pX3)(t). See J. Dufty (to be published). 36.

Cf. W. Steele in, Transport Phenomena in Fluids, H. J. M. Hanley ed., Marcel Dekker, New York (1969), p. 209.

37.

L. Verlet and D. Levesque, private communication.

38.

This point will be further discussed in M. H. Ernst and J. R. Dorfman "On Non-Analytic Dispersion Relation in Fluids" (in preparation).

39.

For a proof see M. H. Ernst and J. R. Dorfman, "On Non-Analytic Dispersion Relation for Hard sphere Gases," Physica, to appear.

101

Table I

Comparison of the coefficients of (t0/t) as obtained (a) From the divergence of the three-body collision term eB3 or EB3

23),24)

for hard disks, and

(b) From the method outlined in this paper, after

a resummation of the BQ- and B

series has been

carried out.

Before Resummation

(2)

(t)

- 0 . 06

(2)(t)

- 0 . 22

P

D

p

n

p

(2) a

(t)

n a2 4

0

t

t

n a2 t0 6

+ 0 . 18 n 3 a2 t0

After Resuinmation

na 4

0

t

n a2 t0 6

t

n a t0 3

t

103

ON THE EXISTENCE OF A "WAVE OPERATOR" FOR THE BOLTZMANN EQUATION

F. Alberto Griinbaum Courant Institute of Mathematical Sciences

Abstract The Boltzmann equation is considered on the appropriate Hilbert space.

The nonlinear problem is looked at as a

perturbation of its linearized version.

Thus, one deals with

a pair of contractive semigroups, and a "wave operator" for this pair is studied.

We find a subspace of finite codimension

where the corresponding limit exists.

104

The Boltzmann equation for a monoatomic gas is

+v1-grad f =Bf =

(1)

eded4dv2. ° 1J (f (_2)f(vl)-

Here f(t,r,v) is the velocity distribution function at time t at

the point r, and the star on vl and v2 denotes the effect of a binary collision.

I(jvl-v2j,e) is the differential scattering

cross-section corresponding to the turning of the relative velocity v1-v2 in an interaction.

We are concerned with the spatially homogeneous case and moreover we assume that we are dealing with a cut-off interaction, so that

f I(v,e) sin eded4 < oo

(2)

.

Under these restrictions the initial value problem for the Boltzmann equation has been much studied. There is one molelular interaction, proposed by Maxwell, which simplifies the mathematics in (1) a bit.

One proposes a

central potential inversely proportional to r4 and one finds that vI(v,O) is a function of a alone, with a pole at 0 = 0. is removed by the cut-off assumption (2). be written as

This pole

Thus the equation can

105

t = f*f - f

(3)

with (4)

(f*f)(v1) = ff f(v2)f(vi)lvl-v2 II(Ivl-v21,e)sin eded4dv2

In (3) we are taking the total cross section to be unity. (2ir)-3/2 Define g(v) = exp - v2/2. Then (3) can be considered

as an initial value problem on the submanifold of L2(g-1) given by those functions which are positive and satisfy the five scalar conditions

f f(v)dv = 1 ,

f f(v)vdv

(5)

=0

f f(v) v2dv = 1 .

It turns out that (3) is well posed in a sufficiently small

ball centered aroung g.

This result is well known and not

particularly hard to prove.

One also proves that g is an attrac-

tive center for the flow given by (3), so that any initial datum in the vicinity of g approaches it as time increases. Our aim is to compare the actual flow (3), subject to the conditions (5), with its linearized version around g.

f = g+h, notices that g*g = g and then drops from

h = f*f- f = (g*h+h*g -g) +h*h the nonlinear term in h, obtaining

(6)

h=Ah=g*h+h*g-h.

One writes

106

The treatment of (6) is greatly simplified by the fact that A is a negative selfadjoint operator having a purely discrete spectrum. See (1) and [2].

Let Qt and Tt denote the semigroups relating data at time 0 to its evolution at time t for equations (3) and (6) respectively. We are interested in proving the existence of a nonlinear change of coordinates, around g, that would convert the nonlinear problem (3) into the linear one given by (6).

Explicitly, we

want to find a very smooth mapping V from a neighborhood of g into itself, that leaves g invariant, coincides with the identity up to first order, and satisfies

Qt =

(7)

*_1Ttjfr

for all positive times.

One shows easily that such a 7P is readily available if lim

(8)

T_tQt

t-> +oo exists and is invertible close to g.

This is a common procedure

in scattering theory where one deals with two unitary groups.

The author found that the trick works for the Boltzmann equation too.

Here one is dealing with contractive semigroups and one of

them is nonlinear.

The main differences with the unitary case

are that a) even the finite dimensional case is interesting,

b) limits can exist only in one sense and not as t -±co. Limits like that in (8) are called wave operators in scattering theory.

A mapping like the i in (7) was first

considered by Poincare.

See [3] and its references.

Actually we want Tt(g+h) = g+ht where ht is the evolution of h according to (6).

107

In [3] a complete study of a simpler model of (3), introduced by Kac [4], is done and the existence of the limit (8) is established.

In the 3-dimensional case the situation is much more involved and the result is changed a bit.

Although a fi satisfying (7) can

still be found the limit in (8) does not exist for a general initial data.

Remarkably enough one can exhibit a large subspace

where the limit does exist.

It turns out to be a subspace of

codimension 3, in the appropriate Hilbert space, given in a rather simple way. THEOREM:

This is the content of the following

If f e 0(g-1), is close enough to g, and satisfies not

only (5) but also the three extra scalar requirements

f f(v) vv2dv = 0 ,

(9)

i.e. the "heat flow vector" vanishes, then lim

T-tQtf

exists

.

t + OD Condition (9) is both sufficient and necessary for the existence The necessity could already have been established

of the limit.

by Maxwell himself.

Indeed, using the notation in [1] or [2], the

eigenvalues of A are w

'r2 = 2w

f sin 0

rr

8 F(e)delcos2r+2 e P2(cos LL

2)

+sin2r+2 e P2(sin 2)- (l+so25or)1

108

fTr

Set A2k = 2rrJ

(sin e)2k+1F(e)de and conclude that

0

T22=-8A2+1

Now condition (9) would be unnece.41;;.ry only if one could prove the inequality 2?11 < T22.

This is equivalent to 2A2 < 3A4.

Maxwell [5] had a.1:_oady computed A2 and A4 with such an accuracy

that he could have ruled out the inequality above. The sufficiency of (9) is, of course, harder to establish.

We cannot, unfortunately, give any physical explanation for the result above.

It is not even clear that there should be any.

Instead, the proof is based on a careful study of the spectral properties of the operator A in (6).

The proof depends heavily

on the use of the Talmi transformation [6] and the numerical computation of a large number of eigenvalues of A done by Alterman, Frankowski and Pekeris [7].

The transformation referred to above was introduced by Talmi in a study of the harmonic oscillator shell model of nuclear physics.

The connection hinges on the fact that the

eigenfunctions of A are those of the harmonic oscillator. Kumar [8] introduced the Talmi transformation in kinetic theory.

The numerical computation of the eigenvalues of A - only the first 559: - turns out to be very useful to supplement analytical facts in proving some crucial "eigenvalue inequalities". See [31 and [9].

log

Bibliography

[1]

C. S. Wang Chang and G. E. Uhlenbeck, The kinetic theory of gases, in Studies of Statistical Mechanics, Vol. V, North Holland (1970).

[2]

G. E. Uhlenberk and G. W. Ford, Lectures in Statistical Mechanics, Ar::rican Math. Society, Providence, R. I. (196)).

[3]

F. A. Gri'inbaum, Trans. American Math. Soc., 165 (March 1972).

(4]

M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. III (1955)-

(5)

J. C. Maxwell, Scientific Papers, Dover Publisher, New York n.d.

[6]

I. Talmi, Helv. Phys. Acta 25, 185 (1952).

[7]

Z. Alterman, K. Frankowski, C. L. Pekeris, Astrophys. J. Suppl., Series 7, 291 (1962).

[8]

K. Kumar, Annals of Physics, 37, 113 (1966).

[9]

F. A. GrUribaum, Proc. Nat. Acad. of Sciences, 67, No. 2, 959 (1970).

111

Exponential modes of the linear Boltzmann equation Peter D. Lax Courant Institute of Mathematical Sciences

There is an interesting formal relation between the purely decaying modes of the linear Boltzmann equation of transport theory and those of the wave equation.

This relation

comes about by connecting each one of these rather dissimilar equations with a pair of very similar looking integral equations.

The situation we consider first is that of an homogeneous convex medium 0 where a mono-energetic beam of neutrons undergoes isotropic scattering and fission. Let f = f(t,x,v) be the distribution function of the neutron density, where x is position in space, v velocity. Since the particles are assumed Mono-energetic, the velocities all have the same absolute value, say 1.

The transport equation asserts that

(1)

1lvl=l

f dv = 0

.

Here X is the removal cross section and c the combined scattering and fission cross section; A and c are constants. The equation asserts that the rate of change of f along a particle path is the sum of two terms:

one, proportional to

f, is due to the removal of scattered particles from the

beam, the other, proportional to

dv, is due to particles Jf

scattered into the beam or created by fission; both processes

112

are assumed to be isotropic.

The boundary conditions are that no particles enter 0 from the outside; for convex 0 this means that

f(t,x,v) = 0

(2)

at all boundary points for those velocities v which point into 0.

Let us look at exponential solutions f = e Ptg(x,v); we get 0

(3)

,

where w = w(x) stands for

(4)

g (x,v) dv .

W (X) j

Abbreviating u-X by a we can write

(

3

as

)

ag+ ds g(x-sv,v) = -cw

Multiplying by eas and integrating with respect to s from 0 to d, where d is the distance of x to the boundary in the direction v, we get, using the boundary conditions, d

g(x,v) = c(

easw(x-sv)ds 0

Substituting this into the definition

(

4

)

of w we obtain

113

d

w(x) = c( easw(x-sv)ds dv 1Ivf=1J0

(5)

We introduce instead of s and v a new variable of integration y:

y = x-sv Since dy = s2ds dv and s = Ix-yI,

1 c w(x)

.

5

(

becomes

)

= f eaIX 2Iw(y)dy Ix-yl2

f

We can summarize this result as follows: The Boltzmann equation

(

1 ),

(

2

)

has an exponential

mode a utg if and only if 1/c is an eigenvalue of HG(0),

o = u-a, where Ha(0) is the integral operator over 0 with kernel

eoIx-YI2 I x-y We turn now to investigating the eigenvalues of K. K Q

The kernel of Ko is only mildly singular.

K6 is a compact

operator, it has a purely discrete spectrum accumulating at 0 and each eigenvalue different from 0 has finite multiplicity.

Also, the kernel of K

a

is real and symmetric so that these

eigenvalues are real.

Denote by Kn(a) the n-th positive eigenvalue of Ka with the understanding that

KI(Q) > K2(0) > ...> 0

.

We study now the behavior of Kn(a) as a function of a.

114

One can prove that Ka is positive definite for a 0 this is no

However we will not make use of these facts.

The information that we need is contained in Kn(a) is a continuous function of a for each n.

LEMMA 1. PROOF:

It follows from the minimax characterization of

eigenvalues that

1Kn (0) -Kn (T) I < IIKa-KT)I It can be shown easily that K

depends continuously on a

a

in the norm topology, from which Lemma 1 follows. For all n

LEMMA 2.

lim Kn (a) _ +00.

6-PROOF:

We choose a ball inside 0; denote its diameter by D.

We select n distinct points x1,...,xn on the surface of the ball, so chosen that no two are antipodal.

Denote by d the

maximum distance of any point xi to any point xj or xj, j

i, where xj denotes the antipode of xj. d < D

By construction

.

Let B. be the ball of radius r and center xi, Bi its antipode..

For r small enough these balls belong to 0 and

are disjoint.

Define n functions uj, j = 1,...,n, as follows

uj (x) = a in BjU Bj

115

a so chosen that the L2 norm of u. is 1. disjoint supports, they are orthogonal.

Since the u. have

We claim that for

any u belonging to the span of the {uj}, for a

large enough

and for r small enough,

(u,K U)

(6)

a

e(D-e)cs u

-

>

The assertion of Lemma 2 follows from this via the minimax principle.

We have

u=

;

E aj uj

since the uj are orthonormal,

J1u1I2 = Eat Now

(7)

(u,K6u) = = (ui,KQuj)aiaj i,j

(ui,Kaui)a2

+ 7

(ui,K6uj)aiaj

,

and by definition, (ui,KOui) = (r e J!

a x-yI 2 ui(x)ui(y)dx dy

Ix-yi

Since the support of ui consists of two antipodal balls whose minimum distance is D-2r, and since ui was chosen as

even, we see that for a large enough (8)

(ui,Kaui) >

const.

e-2r) a (D

116

On the other hand, the maximum distance of any point x in the support of ui to any point y in the support of uj is, for i

3,6

Therefore,

j, less than d+2r.

(ui,Kau ) < const. e6

(9)

Substituting

(

8

)

and

r < !(D-d) inequality

(

6

into

)

)

( 7

), we see that for

holds.

For all n

LEMMA 3.

lim

PROOF:

9

(

(d+2r)

Kn(a) = 0

According to an inequality of Eiulmgren the L2 norm

of an integral operator with a symmetric kernel K(x,y) is bounded by

max JIK(x,y)Idx y

Combining now the three lemmas we see that each Kn(a) is a continuous function whose values range from 0 to +- as a goes from -- to +-.

Therefore, we conclude that for each

n, the equation

has at least one solution Qn

This proves

117

Theorem:

There are infinitely many exponential modes.

The exponential mode e increasing if p

< 0.

lit

is decaying if p> 0, and

Recall that we have p = a+X.

The system is called subcritical if all modes are decaying and supercritical if there is some increasing mode. A glance at the diagram shows that for given a all the p are positive for c small enough.

On the other hand for c large

enough there will always be some negative U.

This shows that the system is subcritical for c small enough and supercritical for c large enough. fission cross-section this is to be expected.

Since c is the

118

The exponentially decaying modes of the wave equation We consider the equation Utt = Au-qU

to be satisfied by a function U(x,t) in the exterior of a

region 0.

(10)

We consider the simple type of potential

q (x)

fc in 0 =

0 outside 0

We look for solutions U of the form

U(x,t) = e-at w(x)

,

6 > 0.

The function w(x) satisfies the relation

a2w-OwIgw = 0

and we further impose that w should be outgoing at -.

if

one abbreviates

La = 62 - A Then we have L

a w+qw= 0

.

A useful formulation of the outgoing property of w is expressed by saying that w should be a superposition of

translates of the fundamental outgoing solution ea of Lc. More explicitly one has

119

e0 (x) _

eaIXI

IxI

and w should be a superposition of

ea ,y (x) = ea (x-y) with y inside 0.

For such w, v =

e-at

w is a superposition

of functions of the form e

Clearly this reIx-yI

presents waves which propagate to infinity as t goes to infinity. A use of Green's formula gives de

la D (e a,P

o,Y - e do

de

cry

6,P du )

for any domain D, and pry both in D.

dS = 0 The main relation is

contained in LEMMA 4.

Suppose that w is an outgoing solution of Law = 0

outside 0; then for any domain D such that 9D lies outside 0

!a D

PROOF:

(e a,p

dw

do -`a

dea,P )dS

do

=0

Outgoing means that, outside of 0, w is a superposition

of functions ea,Y, y in 0.

Forming a superposition of the

integrals asserted to be zero just before the lemma the proof is done.

Now one derives an integral equation for the outgoing solutions of

Law + qw = 0

.

We apply Green's formula to w and ea,p, over any domain containing 0.

According to the Lemma the boundary

120

integral is zero; therefore so is the volume integral.

Since

Laea'p = 6(x-p) and, according to (11 ), Law = -qw, that volume integral is

w(p) +fD e6,pgw dx = 0

.

Recalling the definitions of e6 p and of q we can rewrite this as

(12)

- c1 w(p) = (Gl

eo I x-p I Ix-pi

w(x) dx

Conversely, let w be a function defined in 0 which satisfies equation (12 ) for p in 0; let us define w outside of 0 by (12).

Clearly, the function w thus extended satisfies

L w + qw = 0 6

and, being a superposition of function eo,x, x in 0, is outgoing. What we have shown so far can be summarized as The wave equation with the potential (10

)

has an

exponential mode e atw if and only if -1/c is an eigenvalue of K a(0), an integral operator over 0 with kernel

eaIx-PI x-pI

The analysis of the kernel K a

is very similar to the one

for the Boltzmann equation, only that this time one clearly wants to study the negative eigenvalues.

The operator at hand is again compact, with a nice

121

discrete spectrum.

Denote by Kn(o,O) the n-th negative

eigenvalue of Ka when Ka has n negative eigenvalues. wise set Kn(a) = 0.

Other-

Kn(a) is again a continuous function

of a, and one has LEMMA 4. PROOF:

Kn(a) = 0 for a 0 for a 0 such

that, for every closed operator B satisfying D(B)7 D(A) and

IIB[4](J

Thus, considering the column-vector (,(iPrmn4>)' the equation can be rewritten as a matrix system:

(43b)

with

I (< 'rmn' >) = (Tr,rn n)

(

rmn>)

159

Tr,in,n

(43c)

rmn

=

/

N

'rmn

rmn \ 'rmn'

iw0-k p+v

and

F(w0,k) = detl2 - TrmnI -

(43d)

was the desired dispersion law

(

0

a transcendental function of

w0, k, as opposed to polynomials). None of these works considered the convergence of infinite determinant-type dispersion laws, corresponding to the replacement of the collision operator by the exact infinite dyadic expansion. The very existence of exact dispersion laws for the rigorous Boltzmann equation has been an open problem, up to now.(l) present a new approach to this problem.

We

First, using a perturbation

scheme in a neighborhood of w0 = 0, k = 0, we establish the existence of an. exact local dispersion law.

Next, using the

theory of meromorphic compact operator valued functions, we show that such a local dispersion law is in fact globally true outside the continuous spectrum.

To fix ideas, consider the free wave operator (eq. 17) and the equivalent equation (37):

(37a)

{I -

2

(),,k0)} = a-ik( )+v

We look for a dispersion function F(k0,A) __

know that

{I-;(x,0)}-1

0.

For k0 = 0, we

160

has a discrete set of real eigenvalues {ai}

accumulating at i=0

= -v(0).

k0 = 0

,

Start a perturbation scheme in the neighborhood of

A = a0 (i.e. any one of the{a0}°

).

Consider the

i=0

projection operator P associated with the eigenspace of X necessary,

.

If

define P through a Dunford-Schwartz integral (16):

X0,0)}-1dz

P =

(44)

Recall that z = 1 is an isolated eigenvalue of the compact operator $(A ,0), a c

x©).

(caution: k0 and A are kept constant, k0 =

0,

P is a projection on MP = P' 1, along M1_P = (I-P)' l,and:

`141

(45)

= MP C Ml-P

From the classical Riesz-Schauder

operator I - (XO,0).

theory(12), P reduces the to M1_P is

The restriction

invertible; the restriction of{I-c(X,0)}

to MP is a nilpotent

operator (i.e. its spectral radius is zero). p. 178-181).

(See Kato, Ref. 16,

Thus, we claim that:

Proposition 1. The operator' {I-;(X ,0)-aP} has a bounded inverse, for all complex a

0.

Proof. It is sufficient to show that the restrictions of {I-"(X0,0)-aP} to MP and M1_P are separately invertible (Ref. 23, p. 270).

On MP:

{I-P0,(a0,0)} -aI

161

the latter is invertible, if a

of I - P(ai,0) is zero.

0, since the spectral radius

On MP-1

(I-P) {I-ID(A,o)-aP} =_

°,o)}

which is invertible.

Q.E.D.

Now,'{I-'D(X,k0)-aP} is a bounded operator-valued analytic

function of the two complex variables a and k0; since it is invertible at (X=a0, k0=0), it must have a bounded inverse in

some neighborhood

0i of the latter points:

{X,k0

a-a° I ,,k0)-aP} + aP

setting (47)

I-;(a,k0) -aP = B(A,k0)

162

this is equivalent, in

Qi, to:

{I+aPB-1(x,k0)}B(X,k0)

(46b)

B(a,k0) is invertible in

ci{aPB 1(a,k0)

.

is a degenerate

operator, with a finite range contained in MP.

Recall Lemma A,

and get:

(48)

{I-;(X,k0)}-1 = B-1(X,k0) {I+aPB-1(X,k0)}

-1

Thus, the wave-operator is invertible in Qi if and only if 1(X,k0)}-1

{I+aPB

exists.

From lemma A, a necessary and

sufficient condition is:

Ai(a,k0) = detli+aPB

1(X,k0)l

X 0

of. definition C

Thus Ai(A,k0) s 0 gives all X-poles of the resolvent {I-4 (a,k0)}

for a fixed k0, within

Q i.

This is the desired dispersion

function,In Qi, it is an analytic function of both X and k0. far, its validity is local (in ai).

So

To show that it is globally

correct, consider the identity (48):

(48)

{I-

iX,k0)}-1

_ B 1(A,k0)

{I+aPB1(A,k0)}-1

Investigate each component independently, for all (A,k0) outside the continuous spectrum:

163

a)

{ (X,k0)} is holomorphic-compact; thus

(49a)

{I4

(a,k0)}-1

_

I-R(X,k0)

where R(X,k0) is essentially-meromorphic (Definition E, theorem F). b)

The same remark applies to {(a,k0)+aP}; thus, if

(47)

B(X,k0) = I - 0 (a,k0) - aP

then:

(49b)

B-1(A,k0)

=

I - V(a,k0)

where V(A,k0) is essentially-meromorphic (w.r.t. X, for fixed k0). c)

I + aPB 1(X,k0) = I + aP - aPV(X,k0)

.

{aPV (a,k0)-aP} is an essentially meromorphic function with range

in a finite domain contained in MP:

thus it is a degenerate

operator valued meromorphic function, and theorem D applies:

{I+aP-aPV(X,k0)}-1

(49c)

= I - W(A,k0)

where W(a,k0) is also degenerate meromorphic (w.r.t.X). Thus, we are lead to attempt the proof of the following identity, equivalent to relation 48: 1

(50)

-R(x,k0) _

{I-V(X,k0)}{I-W(A,k0)}

164

which is an identity between meromorphic compact operator valued functions of 1. domain c

To do so, we show that it is true on an open

in the X-plane, corresponding, roughly, to'{a}

large positive real part, and a small imaginary part.

with a

Then we

proceed to analytically continue this identity between meromorphic (operator-valued) functions, outside c

This is done

by generalizing the following theorem: Proposition 2.

(Cf. Saks, p. 152, Ref. 10, 2nd paper): If two mero-

morphic functions in a given region [here I-R(X) and (I-V(X))(I-W(a)) outside the continuous spectrum] assume identical values at the points of a set having a point of accumulation in that region, then these functions are identical in the region. Conclusion: {I-

(X,k0)}-1

_ B

1(X,k0){I+aPB-1(A,k0)}-

for all a,k0 not belonging to the continuous spectrum.

Poles

of {I+aPB 1(X,k0)}-1 are poles of the resolvent of the Boltzmann wave operator.

Zeros of

pi(a,k0) = detll+aPB-1(X,ko)I = 0

are poles of the same resolvent. Remembering that:

(49b)

B 1(X,k0) = I - V(A,k0)

165

the latter being essentially meromorphic, we see that pi(a,k0) is, a priori, a meromorphic function of a for fixed k0 (Theorem D).

Reversing the whole scheme of proof by fixing a = iw0 and

letting k0 = k free, we see that it is also meromorphic w.r.t. k for fixed iw0 = X. So:

Proposition 3. The Boltzmann wave operator possesses an infinite number of exact dispersion functions

pi(a,k) = 0

which are transcendental meromorphic functions of

, and k, outside

the continuous spectrum. Notice that all the roots of a single function Ai(a,k0) s

0

are not likely to give the complete set of poles of the resolvent, because of extra poles appearing in B-1(X,k0), in the identity (448).

The relationship between the {pi} is an open question.

It is

likely that each pi corresponds to a a-eigenvalue curve originating from a = a9, k = 0, unless such curves cross-over. i

Proof of the existence of For k0 fixed (not limited to small k0), and for x with a

large positive real-value, smallimaginary part:

I1;(a,k0)Il

< lal

So IIB

X,k0)II
E

note that PQ(Z) is a Legendre polynomial:

Pi(Z) =

(32)

(keeping in mind the expression of QQ(Z), eq. 19).

One now integrates

, isolating the expression: by parts the kernel of Hn'm 1,J Q

k by

1 log Z+1 {2

ZZ-1 + P Z-Zp

}

b {2 log(Z±1) + ,Lw)(Z-Z

_

p=1

0

((w*-iv)/E1) ) !0 { 2 (w*_1v +k)

1 2

1

b ((w*-iv)/k w (

))

gn,m p

)

(-)a

((w*-iv)/ 1) ) (w-1y -k) *1

_

1v - Z )1

kEl

p

where the prime symbol means derivative w.r.t. l.

Integration by

190

parts is justified by the absolute continuity of the functions involved, especially the

m) "

(

n

More precisely, as a function

of of two real variables E and C1, 117i] is absolutely continuous in 1, has-a bounded, continuous partial-derivative w.r.t. 1; is absolutely continuous in F, and has a bounded, continuous partial derivative w.r.t. F; as a consequence, Bi'j,k satisfies a Lipshitz condition and l > O,except,"t erhaps,

(All of this holds for

of order one in -1.

_1 = 0).

in a small neighborhood of

The kernel of H.'. in the complex k-plane.

has now a simple pile as a singularity To apply Plemelj

(13)

formulae to such

generalized Cauchy-integrals(13), further manipulations are necessary; recall the fundamental analytic mappings (eqs. 4a-b,5a-b):

iv Q l)

-.4

k = Yt+ (1)

(4a)

1 +

*

k

(4b)

1

lv (1) 1

+ X1

which map the positive real E1 axis onto the boundaries 7n+ and Ik of continuous spectrum (see Fig. 1); the corresponding inverse mappings

being

(k) and ,N (k) .

(34a)

(j]

Hn'm

Then:

(

+

d 1 dal

dal

c1

dm

i k by p=0

_

2 (h1+tkZp) 1

Bi.j.2(

,

2

i)

R

Ck b p da

p=0

[i]d1

l

2 (7h --kZp) 2

191

Switching from the positive real

and

C1-axis to path integrals in

the complex k-plane: r (34b)

`

=

tl-k

2 _

p=Q +1

I

11

j

2

{

k b

)

__

(tl+kZp) 2

+

p

+

p=0

kb p (t1-kZp) 2

tl-k LP=0 +

_

_

(t

_

_

_

F (tl)) [V,i]dt1

where ttt ,t1 are complex variables restricted to the boundary arcs 7Y

and-)'YU- respectively (in the complex k-plane) and such that:

(35a)

t+

_

+()

(35b)

ti

=

,9t (Sl)

Recall that Zp

=+(t+) E1

=+(ti)

±1, where Zp is a zero of PZ(z)(cf. Eq. 19).

Thus, we have essentially reduced the boundary arcs "n ,m

etc.

111+ and )t.

to Cauchy integrals on

in the complex k-plane.

Since

is Lipschitz continuous w.r.t. 1, it has the same property

Bi,j,9, [ipi]

w.r.t. t1, through the mapping:

1.

tl

This enables one to use Plemelj-boundary value formulas

f (t,t k)

(36) 7Y

tl-k

dtl = (PV) ( l

f (t,t ,k*) 1

tl-k*

(13)

of the type:

did Sri f(t,k*,k*)

192

when the eigenvalue k 3 k* on either

r`+

or-A7.

The preceding

computations are the direct application of the general method of Melnik(14) and Mikhailov(15) as discussed in Gakhov(13) pp. 66-68 and relations 9.1-9.8).

(see Gakhov

This method relates to generalized

Cauchy-path integrals of the form:

(37)

F (k) =

1

P(l,k)

(

Q( l,k)

1

1

L

where P and Q are holomorphic in k for all values of satisfy a Holder condition w.r.t. El.

lE.L and

In the complex k-plane, the

line of singularities is:

0

(El,-_L)

(in our case, L is the positive real axis in the 1 plane).

The

detailed justification of Plemelj formulas will be found in the work quoted above (there is a subtlety to the extent that P(C1,k) depends

upon both k andin our case, since F$i) depends upon k, the j

same follows for Bi,j,R(i)). Finally, when k -} k* on the edges of the continuous spectrum, (for w -r w* from below), the above reduction of

to generalized

Cauchy-integrals, shows that the integral operators T, themselves split into Cauchy principal value type integral operators plus "residue" integral operators:

193

(38)

=n,m ^j Ti'j [Pil

R=n+i n,m =2i*; G Gi Q=0

i,jQ[^3]i

PV)Hn'm

±

Sri Resn'm

1,j k i

I

With:

ResijR[;il = Bi,j,2(

(39)

,

'(k*)) [ail

(i.e. one setsl Since B., 3. j

had two different analytic expressions, depending on ,R

whether

1f the. same follows for Resn'm

according to:

(k*) y

or

(k*)

Recall that

,

(cf. fig. 3)

(k) is the inverse mapping of the analytic map

and gives a one-to-one correspondence of the arcs %Ytt onto the positive,

real velocity axis.

(40a)

Now; if

Res1lil

=

r

< :

e(k*)

hr()

(

(k

+ ( jr(,) (°° hr()PR()1i()cdC

l94

If E


(k*): [n/2]

(41b)

r=0

hr(

(.

(k

)

+

(k)

+

(k*)

[n/2]

+ r=0 = 7r(

)+w

The full matrix system of operators becomes:

198

{m

n,m ^j

(42)

Ti,j

n,m

`R 0 Gi,j.k

'n,m Hi,

2,i

Where:

Hi,j,k[ it =

(43a)

+ /(+°'{ E

r

{

0

(r)

hr(

L (r)

and

QQ (Z) = QR { w+ky (d

(43b)

}

(on the positive real velocities axis)

The integral operator F. is no longer singular.

The above analytic continuation is formal:

a more delicate

task remains, finding the proper function, analytic setting of the continued system of operators.

One immediately notices that the

"residue operators" map analytic functions of complex-c, because of the complex path integrals:

j(k*) !

S(k)

Thus, one must first define the proper Banach space of locally analytic

(im)n=N n n=0

;

then show that all operators involved

are bounded, compact in this new Banach space; finally verify that

199

they depend analytically upon the "eigenvalue parameter" k.

This

will enable us to claim that the continued operator:

m (I - ni,J,2(k)

is essentially meromorphic (w.r.t. k) and that its only singularities are isolated, finite-order poles in k.

(From the Gohberg-Harazov-

Vidav theory of compact operator valued holomorphic functions). (see first paper, section IV).

U The Composite Banach Space The continued operators still involve integral equations on the positive, real c-axis:

hence, one must keep some square integrability

L2 norm on the real c-axis.

Complex velocities are involved only in

a small neighborhood of , 1(k*), including

1 (k): call (Dc) such an

open domain, with restrictions that (D) must be simply connected, and its boundary be a simple closed curve. (Dc).

Let (D ) be the closure of

One might consider the following Banach space:

200

m=+n;n=N l (44)

m -n;n=0

GmII

L2 +

(sup D

i.e. a Banach space of functions square integrable on the positive real c-axis, analytic in (Dc), continuous and bounded on the boundary of (D ).

Provided one takes a sup-norm on (Dc), this is a

Banach space indeed.

The seemingly complicated conditions on the

boundary of (Dc) are necessary if one wishes to prove compactness of the operators involved w.r.t. the complex-c sup-norm.

Unfortunately, things are more involved because of the analytic nature of the kernels of the residue operators (Eqs. 4la-b):

they

are entire in the complex c-variable, (Eq. 29) but take different functional shapes, depending on whether

is continued in the complex

plane, starting with Real E f(k*);

201

(The domain (D-) in the

Riemann Surface -)

A

(The domain

in the

Riemann Surface t)

fig. 4

Domains D-and D+

202

Thus, the residue-operators map an analytic function of into two different analytic functions.

(in

One must define two distinct

domains (D+) and (D ), situated in two different Riemann surfaces(+) and(-); the Riemann sheet(

real interval 0

as in common with the original s-plane the



The kinetic models are then defined as

23 2

as

(47)

+ v. aax + vOP = vOPNHPNP

The case N = 5 is the model given by Cercignani(27) and later investigated by Thurber(2b) in connection with the asymptotic behavior of solutions with smooth initial data and also the high frequency sound problem.

In neutron transport theory

(49,50,9)

closely related family of models have been investigated.

a

In

neutron transport theory the conservation laws do not in general hold because of the neutron interaction with background material (i.e. the moderator) which scatters neutrons as well as absorbs and produces them.

Nonetheless the basic mathematical features

are essentially the same.

Workers in kinetic theory have

generally had more experience with models for which N > 2 but for which v0 is constant, with some exceptions(27'26,51).

In

neutron transport much of the work has been limited to the case N = 1, however) v0 = total cross section times velocitylis taken

to be velocity dependent and the form of the velocity dependence is often quite complex. and York

(52)

Thus we find the comment by Sirovich

(first paragraph) strange and hard to understand!

233

The Analytic Structure of Model Operators

III.

The two main problems which naturally arise are the initial value problem(28) and the boundary value problem(1).

In order

to sjmplify the analysis and to concentrate on essential features, both problems will be idealized.

For the initial value problem

we will assume plane wave initial data. r

That isjat time t=O;

(see eq. 47) will have a simple exponential space dependence,

depending upon a single wave number.

The boundary value problem

will be idealized by considering a delta function source located at the plane x = 0.

As has been pointed out by K. Case (53)

the

class of exact solutions available in transport theory are quite limited in scope.

More realistic closed form solutions of

boundary value problems except under artificial assumptions like constant density or temperature are not available and are unlikely to emerge without more sophisticated and as yet unavailable tools. For the plane wave initial value problem, a will be the Laplace transform variable corresponding to the time variable t, and k is the wave number variable which is the Fourier transform variable corresponding to the space variable x.

We will here

take k and x to be one-dimensional scalar variables.

The trans-

formed version of (47) is

(48)

(a-iv 1k+v0)r = v0PNHPNr + r0

where r0 is the value of the Fourier transform of r at t=0 and v1 is the first or x component of the vector v.

In eq.

(48)

234

above r is used also to denote the Fourier-Laplace transform of F. In eq.

(48) k is a parameter and the Fourier inversion is a

triviality.

The basic problem is the compute the inverse Laplace We choose the Bromwich path of inversion to the

transform of P.

right of all of the singularities of r such as poles, branch points and essential singularities.

For all of the models that

we consider it can be shown that there is a real number a < 0 such that for Re 6 > a, there are no singularities in the complex o-plane.

Thus in such a region we can solve for r as follows:

First from (48) V0PNHPNr (49)

r0

a-iv 1k+v0

a-iv 1k+v0

Second defining sn7

by the relations n

(50)

(see Eqs.

HXn

=sn7x7 J=1 .

(40) and (43)).

Thirdly defining the moments an by N

PNr = = anXn

(51)

n=1 (see (44)).

From (49) we have

(

52

)

v O P N HPN r r P NHPNr = P N HP N c-ivlk+v0) + PNHPN(o-iv1k+v0) (

.

235

Next defining the Si an the aji by the following relations N

PO

PNHPN(6-ivlk+vO)

(53)

= = SjXj

and N

V X.

PNHPN(

(54)

-iv k+ 1

v0

)

= = a.X 7i i i=l

one obtains N

N

N

anSnjXj =

(55)

ajiXi(injaj

j,n=1

311 ,n=1

+ j=1 = SjXj

Equating coefficients of Xj we obtain equations for the

moments an, namely N

(56)

N

= ansnj = n=1

for 1 < j



.

These equations can be solved for the an and the results substituted into eq.

(49) for r.

This completes the solution

for r in the transform representation.

Thus the basic difficulty

lies in inverting the transform.

The determinant

A = det Dnj

NN

Dnj = Snj -anj i-1 Sin

236

occurs in the denominator of the solutions an obtained from equations

(

-56) above.

The anj are complicated integrals

which result in transcendental expressions in o and k which are also multivalued.

As a preliminary step to inverting the Laplace

transform the Riemann surface for the multivalued function p must be investigated.

Typically there are infinitely many values of a

for which p = 0 for fixed k.

However whether or not a zero gives

rise to a residue contribution depends upon which Riemann sheet of p the zero a is located.

This is complicated because of the

parametric dependence of each root a upon k.

As k varies the

roots a can shift from one Riemann sheet to another.

In addition

roots can bifurcate or coalesce as branch points are encountered. This in fact occurs for the initial value problem. poisoning problem

(9,36)

In the neutron

where a different part of the Riemann

surface of the Boltzmann operator is involved this phenomena of bifurcation also occurs.

Further the Riemann surface of p will

generally contain essential singularities for finite values of a and k.

The existence of essential singularities is an important

feature of boundary value problems as we shall shortly see.

In

addition to residues coming from zeros of p,one also has contributions to the total solution arising from the branch points of the an, which usually are among the branch points of A.

In some problems

there may be additional source related branch points and poles coming from P0.

In the computation of P) as opposed to just

computing the moments anyone has additional poles coming from the zeros of a-iv 1k+v0'

237

For our idealized boundary value problem having simple harmonic time dependence of the form elwt and a delta function source at x = 0, equation (48) is modified as follows

(59)

(iw-iv 1k+v0)r = vOPNHPNr + r0

where now r0 is the Fourier transform of the source at x = 0

which we take to be a multiple of 5(x)' and

whose coefficient is

a function of the components of the velocity vector v, times e 1Wt The source could be generalized to include a linear .

combination of 5(x), 5'(x), 6"(x) etc., however we will be concerned here only with a simple 5(x) source term.

We are thus

investigating a certain class of fundamental solutions of the model equations.

The solution now will depend upon w in a para-

metrical manner and the time dependence is trivial.

The basic

problem is to invert the Fourier transform and obtain the spatial or x-representation of the solution r and or any of its moments.

We will describe the Riemann surfaces of the fundamental solutions for some model equations.

Some typical source terms

and the corresponding asymptotic forms of the fundamental solutions will be given later on.

For the sake of simplicity rather than

necessity the collision frequency v0 will be taken to be linear in the magnitude of the dimensionless velocity v.

Namely

(60)

vO = 1 + by

(b>0)

238

The model with VO = l+bv and satisfying the conservation laws of mass, momentum and energy has been discussed in ref. (see also ref.

(51)).

(26)

For the case of mass conservation only

in the context of neutron transport theory see refs.

(9,50).

We begin with the initial value problem which corresponds to figure #1.

The dispersion law A = 0 has for each value of k

infinitely many roots a. a

For k = 0 we have a triple root at

= 0, these are the hydrodynamic roots, two sound propagation

As k increases these roots move

and the heat diffusion mode. monotonically to the left.

The diffusion root moves on the

negative real axis and at some definite value of kjsay k

it

reaches the logarithmic branch point at a = -1 and then bifurcates into two roots which move onto adjacent Riemann sheets off from the principal Riemann sheet.

For values of k of the order

of magnitude of k) the trajectory of the roots remain close to the branch cut which is taken from -1 to -- along the negative real axis.

At Cr = -1 the trajectories of the bifurcated roots

are tangent to the real axis.

The sound modes have a non-zero imaginary part for k

0

and the real and imaginary parts both become infinite as k -} =.

The two sound modes are complex conjugates of one another.

In

addition to the hydrodynamic modes there are infinitely many additional roots.

At k = 0 for integer values of N >>l the

location of these roots are given by the following asymptotic formula

(61)

aN

-1 -

b + i 21TN b

239

I complex a-plane

\additional roots of

7 `Dispersion Law on adjoint $iemann sheet

\a = -1-bv+ikv

/

a = -1-bv-ikv

Figure

I

Riemann Surface for the initial value problem for model equation with collision frequency having a velocity dependence of the form v0 = l+bv.

24 0

case x >

0

-OD

8 = tan-1w Singularity, Branch-Point and Limit Point of Roots Analytically Continued 1 Dispersion Law.

Figure 2 Riemann surface for the boundary value problem for model equation with collision frequency having a velocity dependence of the form v0 = 1+bv.

Figure 3

-0D

Riemann surface for the poisoning problem which occurs in Neutron Transport Theory.

complex k-plane

242

These roots are all located on the Riemann sheets adjacent to the principal Riemann sheet. the left in the a-plane.

As k increases these roots move to

However they also eventually cross the

branch cut and move onto the principal Riemann sheet. For a fixed value of k only finitely many roots will have crossed onto the principal Riemann sheet.

For k sufficiently large the following

asymptotic formula for the location of the roots is valid.

CF

N

1 -b -log k-

flog k) 2+4Tr2N2

-k i/log k + /(log k) 2+4Tr2N2 (62) ±ik

-log k+ /(log k)2+4Tr2N2

+ib

log k +

(log k) 2+4r2N2

where N is an arbitrary integer >0.

The special case N = 0 reduces to

(63)

S0

1 -k /2 log k + ib /2 log k

This corresponds to the bifurcated diffusion mode.

Note that

when b = 0 there is only one root and in fact the branch point does not occur.

This is consistent with earlier calculations

on Maxwell molecule models (8,54) which assumed a constant collision frequency v0.

sound modes.

For N = 1 we get the analytic continuation of the For N > 2 we obtain the additional roots.

That

243

infinitely many roots exist and that their number in a given circle of radius R about the origin in the complex a plane is consistent with the formula of relation (62) can be verified by winding number analysis.

In inverting the Laplace transform

by a Bromwich integral we have different cases to consider. k

For

0 but sufficiently small we have the 3 residue contributions

from the hydrodynamic modes plus a branch integral contributions arising from the branch cut from -1 to -..

For k

0 sufficiently

small the real part of the hydrodynamic roots will be greater than -1.

it is immediately evident then that for large t the

residues coming from the hydrodynamic roots will give the asymptotically dominant contribution to the moments an, and to the full distribution function r.

This is the region of validity

of the Chapmann-Enskog expansion.

An important point must be

made however, although the Chapmann-Enskog expansion in the sense of McLennan (55) can be shown to be convergent, it will

not in general converge to a solution of the model equation but rather to an asymptotic solution.

The total solution consists

of the residue plus branch integral contributions, only under exceptional conditions will the branch integral contributions equal zero.

the

full

This same observation

(10)

has been shown to hold for

linearized hard-sphere Boltzmann equation, where one

has infinitely many dispersion laws each of which gives rise to a class of residue contributions to the total solution plus

branch integral contributions.

Although we regard McLennan's

analysis of the Chapmann-Enskog expansion to be incomplete,

244

nonetheless following his analysis we agree on the convergence of the Chapmann-Enskog expansion for the linearized hard-sphere Boltzmann equation.

However we find that in general that the

convergence is not to a solution of the Boltzmann equation but

rather to an asymptotic solution corresponding to the residue contributions from the hydrodynamic dispersion law which is

only one of infinitely many dispersion laws for the Boltzmann equation (see ref. point).

(10), for a more complete discussion of this

Thus Hilbert's paradox is seen to arise out of the

confusion of convergence of the Chapmann-Enskog or Hilbert

expansion, with convergence to a solution of the Boltzmann equation

.

(On this point see H. Grad ref.

(56)).

As k increases and the diffusion mode bifurcates the residue contribution from the diffusion mode disappears.

However when

the bifurcated roots remain close to the branch cut we have the phenomena of a Friedrichs(13) type spectral concentration.

By

means of a Laplace type asymptotic expansion of the branch integral contributions we obtain for some intermediate time scale an

asymptotically dominant contribution arising from the presence of the nearby bifurcated root. of ing.

This is analogous to the phenomena

resonance states which occur in the quantum theory of (57)

scatter-

Such resonances behave like a bound state for a

limited interval of time, eventually the energy is dissipated from such a state and it dies out.

In our situation for

sufficiently large time the dominant contribution from the branch integral must come from the end point at c = -1.

For the boundary value problem one also has the phenomena of

245

spectral concentration but for different reasons however. geometry for this problem is given in figure 2.

The

We restrict our

attention to the case x > 0 and w > 0, analogous diagrams can be drawn for the three other possibilities. roots at k = 0.

For w = 0 we have two

A sound propagation root which as w increases

leaves the origin along a curve tangent to the real positive k-axis.

The diffusion root leaves the origin along a line at

an angle of 45° with the real k-axis.

At the point k = -ib the

dispersion law has a logarithmic branch point which happens also to be an essential singularity.

limit point

(26,58)

persion law.

The point k = -ib is also a

of roots of the analytically continued dis-

The dashed line emanating from k = -ib with a slope

of -1/w is the edge of the continuous or essential spectrum, a concept introduced originally by Lehner and Wing

(59)

into transport

theory/ and in particular into energy dependent neutron transport theory,by N. Corngold(60)j and in the context of kinetic theory by H. Grad(6).

For

w sufficiently large the hydrodynamic roots

will cross the above mentioned dashed line, whose position as indicated also depends on w.

In previous discussions both in the

context of neutron transport and kinetic theory there has been some confusion in properly understanding this phenomena both purely mathematically as well as in relation to experimental interpretation.

It was believed that somehow the discrete modes

disappeared into the continuous spectrum and that therefore discrete exponential-like behavior could never be observed experimentally for w beyond some critical value w*.

Part of the

confusion was a result of the failure to distinguish between a

246

distribution function r and a moment say an)of r. r

is a function

of v as well as x and t; however the moment an or for that matter p, u, P or T are functions of x and t only.

Thus the dashed line

mentioned above which arises from the nature of the v-dependence of r) simply has no meaning for moments of r where the velocity v is integrated out.

As

w increases one simply analytically

continues across the dashed line.

In inverting the Fourier trans-

form,one simply pushes the path of integration down below the real axis and picks up residue contributions from whatever poles one crosses and as usual one folds the contour around the branch point at k = -ib.

In order to investigate the asymptotic

behavior of our fundamental solutions)we deform the folded path about -ib until it coincides with a line of constant

phase(9'25).

As is indicated in figure 21 the line of constant phase leaves the

essential singularity at k = -ib at an angle of 8 = tan-1w from the imaginary axis.

At the saddle point the tangent to the path

of constant phase makes an angle which is asymptotically equal, to

8/3jfor large w,with the imaginary axis.

As we move further

down1the path of constant phase becomes parallel to the imaginary axis.

There are infinitely many additional roots(9'26) of the

dispersion law which are poles of the transformed moments and distribution function, which converge to k = -ib as a limit point along the dashed line emanating from -ib which is at a 45° angle from the tangent to the path of constant phase.

The location of

the poles or roots of p is dependent only upon the value of the parameter w.

The location of the saddle point on the path of

constant phase is however dependent both upon the value of x as well as w.

247

We will now assume that the frequency parameter w is sufficiently large, so that the hydrodynamic roots have crossed the dashed line

representing the edge of the continuous spectrum, which is the situation relevant to the high-frequency sound experiments of Greenspan(3) and Meyer and Sessler(4).

There is no

longer the

possibility of obtaining exact plane wave solutions for such values of w, but corresponding residue contributions always exist, contrary to an assertion of H. Grad(6).

Such residue contributions

may or may not give rise to a Friedrichs(13) type spectral concentration.

Under the conditions

1

(64)

4< x