Series on Partial Differential Equations and Applications - V o l . 1
7
Bernard
Helffer
classical Analysis, en Laplacians, and Statistical
World Scientific
Mechanics
Semiclassical A n a l y s i s , W i t t e n Laplacians, and Statistical
Mechanics
Series on Partial Differential Equations and Applications Editorial Board Editor-in-Chief Fanghua Lin Courant Institute of Mathematical Science New York University New York, NY 10012-1110 USA Email:
[email protected] Editors Lawrence C. Evans Department of Mathematics University of California Berkeley, CA, 94720-3840 USA Email:
[email protected] Chang Shou Lin National Chung Cheng University Department of Mathematics Ming-hsiung, Chia Yi, 621 Taiwan Email:
[email protected] Paul Yang Princeton University Mathematics Department Fine Hall, Washington Road Princeton NJ 08544-1000 USA Email:
[email protected] Ma Li Department of Mathematics Tshinghua University Beijing 100084 China
Pan Xingbin Department of Mathematics National University of Singapore Singapore 119260 Email:
[email protected] Xu Xingwang Department of Mathematics National University of Singapore Singapore 119260 Email:
[email protected] Neil Trudinger School of Mathematical Sciences Australian National University Canberra, ACT 0200 Australia Email:
[email protected] Y. Giga Department of Mathematics Hokkaido University Sapporo 060-0810 Japan Email:
[email protected] Series on Partial Differential Equations and Applications - V o l . 1
Semiclassical Analysis, W i t t e n Laplacians, and Statistical
Mechanics
Bernard Helffer Universite Paris Sud, France
U J j World Scientific Vw
New Jersey London •Singapore* Singapore • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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Preface
In this book we shall analyze with techniques coming mainly from partial differential equations (PDE) and of semi-classical analysis problems coming from statistical mechanics. Our main object of analysis is a (family of) measure(s) representing the probability of presence of m particles in interaction and having the form d ^
:= Z(m, h)-1 exp
- ^
dX
(m € IN) where • Z(m, h) is a normalization constant, .
$(m)
is a C°° function defined on JR m , tending to oo at oo, with a specific structure coming from statistical mechanics (usually a perturbation of Y^?=i ^i.xi) taking account of the interaction between nearest neighbours), • ft is a strictly positive parameter playing the role of an effective planck constant, • dX is the Lebesgue measure on JRm • the integer m represents the cardinal of a set A in the lattice ZZ which will tend infinity. We have consequently two main parameters h and m. The limit h —> 0 corresponds to the so called semi-classical limit h (which can actually corresponds to the temperature) and m —>• +00 corresponds to the so called thermodynamic limit, when a large number of particles is involved. Our aim in this book is to explain, after presentation of the physical background, two techniques for analyzing these problems.
VI
Preface
The first one is the so called transfer matrix approach. In this method which takes its origin in the analysis of the Ising model, the thermodynamic limit problem is solved by reduction to the analysis of a compact operator on a one-particle Hilbert space. We shall see how in this context the semi-classical mechanics can be a useful tool in the analysis of the spectral properties of this operator. The second one is the technique of the Witten Laplacian approach which gives a new light and suggests new proofs for the analysis of Poincare estimates and log-Sobolev estimates in relation with the measure of the decay of correlations. The main difficulty will be to control constants or remainders independently of the dimension. We shall show in this context how techniques coming from the theory of partial differential equations and applied to the Witten Laplacian on one-forms can be efficient for showing the uniformity of some constants. Although the problems we are treating are strongly motivated by statistical mechanics where probability theory plays an important role, we do not assume that the reader has a strong background in this theory. We will try instead to present all the material which is needed in spectral theory, semi-classical analysis and statistical mechanics in a rather selfcontained way. Let us add that we do not pretend at exhaustivity. We have preferred to present a walk through the theory. The initial version of this manuscript was prepared when teaching in Orsay and Toulouse in spring and winter 1998. But we include also, particularly in the notes, references to more recent contributions. First of all we would like to thank J. Sjostrand who has collaborated with us on many results presented here and T. Bodineau who was the first to push us to look at these log-Sobolev estimates and to bring us his knowledge in statistical mechanics in a fruitful collaboration. For various reasons (collaborations, invitations, fruitful discussions, comments or remarks), we would also like to thank V. Bach, D. Bakry, J.D. Deuschel, T. Jecko, M. Ledoux, A. Martinez, J.-M. Roquejoffre, T. Ramond, N. Yoshida, B. Zegarlinski and all the students or colleagues who have followed or discussed with us the various versions of the material presented here. We thank also A. Bardot for his help when preparing the final version of the manuscript. Finally, we acknowleddge the support of CNRS and of the European Union TMR grant FMRX-CT 96-0001.
Contents
Preface
v
Chapter 1 Introduction 1.1 Laplace integrals 1.2 The problems in statistical mechanics 1.3 Semi-classical analysis and transfer operators 1.4 About the contents
1 1 3 6 7
Chapter 2 W i t t e n Laplacians approach 2.1 De Rham Complex 2.2 Witten Complex 2.3 Witten Laplacians 2.4 Semi-classical considerations 2.5 An alternative point of view : Dirichlet forms 2.6 A nice formula for the covariance 2.7 Notes
9 9 13 14 15 16 17 21
Chapter 3 3.1 3.2 3.3 3.4
Problems in statistical mechanics with discrete spins The Curie-Weiss model The 1-d Ising model The 2-d Ising model Notes
Chapter 4 Laplace integrals and transfer operators 4.1 Introduction vii
23 23 25 28 29 31 31
viii
Contents
4.2
Classical Laplace method 4.2.1 Standard results 4.2.2 Transition between the convex case and the double well case 4.3 The method of transfer operators 4.4 Elementary properties of operators with integral kernels . . . . 4.5 Elementary properties of the transfer operator 4.6 Operators with strictly positive kernel and application 4.7 Thermodynamic limit 4.8 Mean value 4.9 Pair correlation 4.10 2-dimensional lattices 4.11 Notes Semi-classical analysis for the transfer operators 5.1 Introduction 5.2 Explicit computations for the harmonic Kac operator 5.3 Harmonic approximation for the transfer operator 5.4 WKB constructions for the transfer operator 5.5 The case of the Schrodinger operator in dimension 1 5.6 Harmonic approximation for the transfer operator: upper bounds 5.7 First conclusions about the splitting 5.8 Some elements about the decay 5.9 Splitting revisited 5.9.1 Preliminary discussion 5.9.2 Comparison of various problems 5.9.3 Upper bound of the splitting 5.10 Notes
31 31 33 34 35 39 40 42 43 44 45 49
Chapter 5
65 67 68 70 70 70 71 73
Chapter 6 6.1 6.2 6.3 6.4 6.5
Basic facts in spectral theory and on the Schrodinger operator Introduction Selfadjoint operators, spectrum and spectral decomposition . . Discrete spectrum, essential spectrum Essentially selfadjoint operators Examples
51 51 51 55 59 63
77 77 77 83 86 88
Contents
6.6 6.7 6.8 6.9
6.5.1 The free Laplacian 6.5.2 The harmonic oscillator More on selfadjointness The max-min principle Compactness Notes
ix
88 89 91 93 94 97
Chapter 7 Log-Sobolev inequalities 7.1 Introduction 7.2 Log-Sobolev inequalities in the strictly convex case 7.3 Around Herbst's argument : necessary conditions for log-Sobolev inequalities 7.4 Extension of the Bakry-Emery argument : convexity at infinity 7.5 The case of the circle 7.6 The case of the line 7.7 General remarks 7.8 Notes
101 101 103
Chapter 8 Uniform decay of correlations 8.1 Introduction 8.2 Lower bound for the spectrum of the Witten Laplacian . . . . 8.3 Uniform estimates for a family of 1-dimensional Witten Laplacians 8.4 A proof of the decay of correlations 8.5 Generalized Brascamp-Lieb inequality 8.6 Notes
133 133 136
115 118 120 123 128 130
139 142 147 148
Chapter 9 Uniform log-Sobolev inequalities 151 9.1 Introduction and preliminaries 151 9.2 Some log-Sobolev inequality for effective single spin phase . . . 152 9.3 The role of the decay estimates for log-Sobolev inequality . . . 155 9.4 Second part of the proof of the log-Sobolev inequality 158 9.5 Conclusion 166 9.6 Notes 167 Bibliography
169
Index
177
Chapter 1
Introduction
1.1
Laplace integrals
As mentioned in the preface, our basic object will be a measure on Mm : exp — $(X) dX. Here 3> will be a real C°° function which tends sufficiently rapidly to +00 as \X\ —>• +00. Typically, we can think of $(X) = a\X\2 with a > 0, but we would like of course to analyze more general cases. In order to give a sense to what follows let us assume that *(X)>±\X\*-C,
(1.1.1)
for some C > 0. In particular, under this assumption, the quantity Z=
f
exp-${X)dX
,
(1.1.2)
is finite and we can associate to the measure exp —$(X) dX a probability measure fi: dfi = Z-1exp-$(X)dX.
(1.1.3)
Associated to this measure, we define for a function / with polynomial growth the mean value of / by (/) = Z-1 [ JM™
f(X) exp - $ ( X ) dX = f fdfi. J 1
(1.1.4)
2
Introduction
For two functions / and g, we then define the covariance of two functions with polynomial growth / and g by Cov(/, •
(1-1.5)
As simple example, let us consider :
$(X)=^2Aijxi-xj
,
(1.1.6)
where Aij is an m x m matrix which is symmetic and positive definite. The computation of the covariance of / and g with f(X) = Xk and g(X) = xi is given by {A'^u. For this given measure, we define also the variance by v a r / = Cov ( / , / ) .
(1.1.7)
Our main questions in this direction are to find natural assumptions permitting to prove the existence of the following inequalities called (P) (for Poincare) and (LS) (for logarithmic Sobolev or more shortly log-Sobolev). The first one is the existence of a constant C such that, for all / , (P)
var(/) f c - xk+1)2 + b- JT>fc - y)\xk + yf
(1.2.9)
fc=i fe=i
where a > 0, b > 0 and y > 0. This corresponds to the case
d=l.
Although this will not be the main point in this book, it is useful to have some discussion about decay of correlations and phase transition. We can for example chose to analyze as A is large the expression aA:=
var^-t-^-) '
(1.2.10)
' j'eA
When $ is even with respect t o I i 4 —X, we observe that (XJ) = 0 and we get, in the case when A is considered as periodic *A:=j^rX)CorA(M)
(1-2.11)
When we have uniform decay of the correlations like | CorA(i,j)\
< Cexp-KdA(i,j)
,
(1.2.12)
with C and K > 0 independent of A, we immediately see that lim Z d
This is typically the case when the interaction is small or when the phase is convex. Indeed, if we consider the case when is strictly convex : <j>"(x) > p > 0 ,
(1.2.14)
it is easy to see that X t-» $ A ' / " 7 ( X ) satisfies, if J > 0, the condition Hess $*•'*•* > p .
(1.2.15)
6
Introduction
We will show that this implies an uniform Poincare inequality, When applying this inequality to the function X *-> A £3,- €A Xj, one can immediately obtain : aA < - ^ .
(1.2.16)
So we have obtained a proof of (1.2.13) when <j> is strictly convex which is valid for any J > 0. On the other hand, there are cases where this limit is non zero. This will be typically the case when d > 2, the phase
