Scaling Limits of Interacting Particle Systems

by Kipnis, Claude; Landim, Claudio

• ISBN13: 9783540649137
• ISBN10: 3540649131
• Format: Hardcover
• Copyright: 1999-03-01
• Publisher: Springer Verlag

Summary

This book presents in a progressive way the techniques used in the proof of the hydrodynamic behavior of interacting particle systems. It starts with introductory material on independent particles and goes all the way to nongradient systems, covering the entropy and the relative entropy methods, asymmetric processes from which hyperbolic equations emerge, the equilibrium fluctuations and the large deviations theory for short-range stochastic dynamics. It reviews, in appendices, some tools of Markov process theory and derives estimates on the spectral gap of reversible, conservative generators. The book is self-contained and can be read by graduate students in mathematics or mathematical physics with standard probability background. It can be used as a support for a graduate on stochastic processes.

Table of Contents

Introduction	1	(6)
An Introductory Example: Independent Random Walks	7	(14)
Equilibrium States	7	(4)
Local Equilibrium	11	(2)
Hydrodynamic Equation	13	(4)
Equivalence of Ensembles	17	(1)
Comments and References	18	(3)
Some Interacting Particle Systems	21	(20)
Some Remarks on the Topology of NZd and M1(NZd)	21	(4)
Simple Exclusion Processes	25	(3)
Zero Range Processes	28	(7)
Generalized Exclusion Processes	35	(1)
Attractive Systems	36	(2)
Zero Range Processes in Infinite Volume	38	(3)
Weak Formulations of Local Equilibrium	41	(6)
Hydrodynamic Equation of Symmetric Simple Exclusion Processes	47	(20)
Topology and Compactness	49	(6)
The Hydrodynamic Equation	55	(7)
Comments and References	62	(5)
An Example of Reversible Gradient System: Symmetric Zero Range Processes	67	(48)
The Law of Large Numbers	69	(10)
Entropy Production	79	(2)
Proof of the Replacement Lemma	81	(3)
The One Block Estimate	84	(8)
The Two Blocks Estimate	92	(6)
A L2 Estimate	98	(5)
An Energy Estimate	103	(6)
Comments and References	109	(6)
The Relative Entropy Method	115	(26)
Weak Conservation of Local Equilibrium	116	(14)
Comments and References	130	(11)
Hydrodynamic Limit of Reversible Nongradient Systems	141	(50)
Replacing Currents by Gradients	147	(4)
An Integration by Parts Formula	151	(3)
Nongradient Large Deviation Estimates	154	(7)
Central Limit Theorem Variances	161	(10)
The Diffusion Coefficient	171	(9)
Compactness	180	(4)
Comments and References	184	(7)
Hydrodynamic Limit of Asymmetric Attractive Processes	191	(40)
Young Measures	200	(8)
An Entropy Inequality at Microscopic Level	208	(10)
Law of Large Numbers for the Empirical Measure	218	(3)
Comments and References	221	(10)
Conservation of Local Equilibrium for Attractive Systems	231	(26)
Replacement Lemma for Attractive Processes	233	(2)
One Block Estimate Without Time Average	235	(11)
Conservation of Local Equilibrium	246	(5)
Comments and References	251	(6)
Large Deviations from the Hydrodynamic Limit	257	(30)
The Rate Function	260	(3)
Weakly Asymmetric Simple Exclusion Processes	263	(3)
A Superexponential Estimate	266	(2)
Large Deviations Upper Bound	268	(5)
Large Deviations Lower Bound	273	(6)
Comments and References	279	(8)
Equilibrium Fluctuations of Reversible Dynamics	287	(24)
The Boltzmann-Gibbs Principle	292	(5)
The Martingale Problem	297	(2)
Tightness	299	(8)
Generalized Ornstein-Uhlenbeck Processes	307	(2)
Comments and References	309	(2)
Appendices
Markov Chains on a Countable Space	311	(38)
Discrete Time Markov Chains	311	(3)
Continuous Time Markov Chains	314	(7)
Kolmogorov's Equations, Generators	321	(5)
Invariant Measures, Reversibility and Adjoint Processes	326	(4)
Some Martingales in the Context of Markov Processes	330	(2)
Estimates on the Variance of Additive Functionals of Markov Processes	332	(2)
The Feynman-Kac Formula	334	(4)
Relative Entropy	338	(2)
Entropy and Markov Processes	340	(3)
Dirichlet Form	343	(3)
A Maximal Inequality for Reversible Markov Processes	346	(3)
The Equivalence of Ensembles, Large Deviation Tools and Weak Solutions of Quasi-Linear Differential Equations	349	(24)
Local Central Limit Theorem and Equivalence of Ensembles	349	(8)
On the Local Central Limit Theorem	357	(5)
Remarks on Large Deviations	362	(3)
Weak Solutions of Nonlinear Parabolic Equations	365	(4)
Entropy Solutions of Quasi-Linear Hyperbolic Equations	369	(4)
Nongradient Tools: Spectral Gap and Closed Forms	373	(52)
On the Spectrum of Reversible Markov Processes	375	(3)
Spectral Gap for Generalized Exclusion Processes	378	(16)
Spectral Gap in Dimension d ≥ 2	394	(4)
Closed and Exact Forms	398	(21)
Comments and References	419	(6)
References	425	(16)
Subject Index	441