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1999 | Buch

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Scaling Limits of Interacting Particle Systems

verfasst von: Claude Kipnis, Claudio Landim

Verlag: Springer Berlin Heidelberg

Buchreihe : Grundlehren der mathematischen Wissenschaften

Enthalten in: Professional Book Archive

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Über dieses Buch

The idea of writing up a book on the hydrodynamic behavior of interacting particle systems was born after a series of lectures Claude Kipnis gave at the University of Paris 7 in the spring of 1988. At this time Claude wrote some notes in French that covered Chapters 1 and 4, parts of Chapters 2, 5 and Appendix 1 of this book. His intention was to prepare a text that was as self-contained as possible. lt would include, for instance, all tools from Markov process theory ( cf. Appendix 1, Chaps. 2 and 4) necessary to enable mathematicians and mathematical physicists with some knowledge of probability, at the Ievel of Chung (1974), to understand the techniques of the theory of hydrodynamic Iimits of interacting particle systems. In the fall of 1991 Claude invited me to complete his notes with him and transform them into a book that would present to a large audience the latest developments of the theory in a simple and accessible form. To concentrate on the main ideas and to avoid unnecessary technical difficulties, we decided to consider systems evolving in finite lattice spaces and for which the equilibrium states are product measures. To illustrate the techniques we chose two well-known particle systems, the generalized exclusion processes and the zero-range processes. We also conceived the book in such a manner that most chapters can be read independently of the others. Here are some comments that might help readers find their way.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract

The problem we address in this book is to justify rigorously a method often used by physicists to establish the partial differential equations that describe the evolution of the thermodynamic characteristics of a fluid.

Claude Kipnis, Claudio Landim

1. An Introductory Example: Independent Random Walks

Abstract

The main purpose of this book is to present general methods that permit to deduce the hydrodynamic equations of interacting particle systems from the underlying stochastic dynamics, i.e., to deduce the macroscopic behavior of the system from the microscopic interaction among particles.

Claude Kipnis, Claudio Landim

2. Some Interacting Particle Systems

Abstract

We introduce in this chapter the interacting particle systems we consider throughout the book and present their main features. We shall refer constantly to Liggett (1985) for some proofs and some extensions of the results presented here.

Claude Kipnis, Claudio Landim

3. Weak Formulations of Local Equilibrium

Abstract

In Chapter 1 we presented a mathematical model to describe the evolution of a gas. In this respect we introduced the notions of local equilibrium and conservation of local equilibrium. Unfortunately, at the present state of knowledge, there is no satisfactory general proof of conservation of local equilibrium for large classes of interacting particle systems. All existing approaches to this problem rely too much on particular features of the model, such as the existence of dual processes (for symmetric simple exclusion processes by De Masi, Ianiro, Pellegrinotti and Presutti (1984), superposed with a Glauber dynamics by De Masi, Ferrari and Lebowitz (1986) or for the voter model Presutti and Spohn (1983)); explicit computations (for superpositions of independent random walks Dobrushin and Siegmund-Schultze (1982)); resolvent equation methods (for Ginzburg-Landau processes Fritz (1987a,b) and (1989a)); or on the attractiveness (Rost (1981), Andjel and Kipnis (1984) Benassi and Fouque (1987, 1988), Andjel and Vares (1987)). Furthermore, we would like to prove the hydrodynamic behavior of systems starting from initial states more general than local equilibrium measures. This leads us to weaken the concept of local equilibrium.

Claude Kipnis, Claudio Landim

4. Hydrodynamic Equation of Symmetric Simple Exclusion Processes

Abstract

In this chapter we prove the hydrodynamic behavior of nearest neighbor symmetric simple exclusion processes and show that the hydrodynamic equation is the heat equation:

$${\partial _t}\rho = \left( {1/2} \right)\Delta \rho .$$

Claude Kipnis, Claudio Landim

5. An Example of Reversible Gradient System: Symmetric Zero Range Processes

Abstract

In this chapter we investigate the hydrodynamic behavior of reversible gradient interacting particle systems. To keep notation simple and to avoid minor technical difficulties, we consider the simplest prototype of reversible gradient system: the nearest neighbor symmetric zero range process. The generator of this Markov process is given by

$$({L_N}f)(\eta ) = (1/2)\mathop \Sigma \limits_{x \in T_N^d} \sum\limits_{|z| = 1} {g(\eta (x))} [f({\eta ^{x,x + z}}) - f(\eta )]$$

(0.1)

for cylinder functions f: ℕT _N ^d →ℝ.

Claude Kipnis, Claudio Landim

6. The Relative Entropy Method

Abstract

In Chapter 1 we introduced in the context of interacting particle systems the physical concepts of local equilibrium and conservation of local equilibrium and we proved the persistence of local equilibrium in a model where particles evolve independently. Consider a particle system η _t evolving on the torus T _N ^d and possessing a family {υ _α ^N , α ≥ 0} of product invariant measures indexed by the density. Fix a profile ρ ₀: T^d → ℝ₊ and assume that the process η _t has a hydrodynamic behavior described by the solution ρ(t, u) of some partial differential equation with initial condition ρ ₀. Denote by μ ^N a sequence of initial states associated to the profile ρ ₀ and by μ _t ^N the state at the macroscopic time t of the process that started from μ ^N. The conservation of local equilibrium states that μ _t ^N should be close to the product measure υ _ρ(t,·) ^N with slowly varying parameter associated to ρ(t,·).

Claude Kipnis, Claudio Landim

7. Hydrodynamic Limit of Reversible Nongradient Systems

Abstract

We investigate in this chapter the hydrodynamic behavior of reversible nongradient systems. To fix ideas we consider one of the simplest examples, the so called symmetric generalized exclusion process. This is the Markov process introduced in section 2.4 that describes the evolution of particles on a lattice with an exclusion rule that allows at most k particles per site. Here K is a fixed positive integer greater or equal than 2. The generator of this Markov process acts on cylinder functions as

$$\left( {{L_N}f} \right)\left( \eta \right) = \left( {1/2} \right)\sum\limits_{\mathop {x,y \in {T^d}}\limits_{|x - y| = 1} } {r\left( {\eta \left( x \right),\eta \left( y \right)} \right)} \left[ {f\left( {{\eta ^{x,y}}} \right) - f\left( \eta \right)} \right],$$

(0.1)

where r(a, b) = 1{a > 0, b < k} and η ^{x, y} is the configuration obtained from η moving a particle from x to y.

Claude Kipnis, Claudio Landim

8. Hydrodynamic Limit of Asymmetric Attractive Processes

Abstract

We examine in this chapter an alternative method to prove the hydrodynamic behavior of asymmetric interacting particle systems. This approach has the advantage over the one presented in Chapter 6 that it does not require the solution of the hydrodynamic equation to be smooth. On the other hand its main inconvenience is that it assumes the process to be attractive to permit the use of coupling arguments and the initial state to be a product measure. To illustrate this approach we consider an asymmetric attractive zero range process on the discrete d—dimensional torus T _N ^d . The generator of this Markov process, denoted by L _N, is given by

$$\left( {{L_N}f} \right)\left( \eta \right)\, = \,\sum\limits_{\begin{array}{*{20}{c}}{x \in {\text{T}}_N^d} \\{y \in {\mathbb{Z}^d}}\end{array}} {p\left( y \right)g\left( {\eta \left( x \right)} \right)\left[ {f\left( {{\eta ^{x,x + y}}} \right)\, - \,f\left( \eta \right)} \right]\,,} $$

(0.1)

where p(·) is a finite range irreducible transition probability on ℤ^d. Irreducible means here that for every z in ℤ^d, there exists a positive integer m and a sequence 0 = x ₀, x ₁,…, x _m, = z such that p(x _i+1 − x _i)+p(x _i − x _i+1) > 0 for 0 ≤ i ≤ m − 1. Throughout this chapter we assume the process to be attractive. We have seen in Chapter 2 that this hypothesis corresponds to assume that the rate at which a particle leaves a site is a non decreasing function of the total number of particles at that site:

(H) g(·) is a non decreasing function.

Claude Kipnis, Claudio Landim

9. Conservation of Local Equilibrium for Attractive Systems

Abstract

In Chapter 1 we introduced the concept of local equilibrium and proved the conservation of local equilibrium for a superposition of independent random walks. Then, from Chapter 4 to Chapter 8, we proved a weaker version of local equilibrium for a large class of interacting particle systems: we showed that the empirical measure π _t ^N converges in probability to an absolutely continuous measure whose density is the solution of some partial differential equation. The purpose of this chapter is to to show that in the case of attractive processes, the conservation of local equilibrium may be deduced from a law of large numbers for local fields, i.e., from the convergence in probability of the averages

$${N^{ - d}}\sum\limits_x {H(x/N){\tau _x}} \Psi ({\eta _t})to\int_{{T^d}} {H(u)\tilde \Psi } (\rho (t,u))du$$

for every t ≥ 0, every continuous function H and every bounded cylinder function ψ. Here ρ(t, u) is the solution of the hydrodynamic equation. This statement is slightly stronger than the convergence of the empirical measures since it involves all local fields.

Claude Kipnis, Claudio Landim

10. Large Deviations from the Hydrodynamic Limit

Abstract

In Chapters 4 and 5 we proved a law of large numbers for the empirical density of reversible interacting particle systems. A natural development of the theory is to investigate the large deviations from the hydrodynamic limit.

Claude Kipnis, Claudio Landim

11. Equilibrium Fluctuations of Reversible Dynamics

Abstract

In Chapters 4 to 7 we examined the hydrodynamic behavior of several mean-zero interacting particle systems and proved a law of large numbers under diffusive resealing for the empirical measure. We now investigate the fluctuations of the empirical measure around the hydrodynamic limit starting from an equilibrium state. To fix ideas, we consider the nearest neighbor symmetric zero range process. The reader shall notice, however, that the approach presented below applies to a large class of reversible models including nongradient systems. The generator of this process is

$$\left( {{L_N}f} \right)\left( \eta \right) = \sum\limits_{x,y \in {\Bbb T}_N^d} {p\left( y \right)g\left( {\eta \left( x \right)} \right)\left[ {f\left( {{\eta ^{x,x + y}}} \right) - f\left( \eta \right)} \right]} ,$$

(0.1)

where p(y) = 1/2 if |y| = 1 and 0 otherwise and g is a rate function satisfying the assumptions of Definition 2.3.1.

Claude Kipnis, Claudio Landim

Backmatter

Titel: Scaling Limits of Interacting Particle Systems
verfasst von: Claude Kipnis
Claudio Landim
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-03752-2
Print ISBN: 978-3-642-08444-7
DOI: https://doi.org/10.1007/978-3-662-03752-2

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Introduction

1. An Introductory Example: Independent Random Walks

2. Some Interacting Particle Systems

3. Weak Formulations of Local Equilibrium

4. Hydrodynamic Equation of Symmetric Simple Exclusion Processes

5. An Example of Reversible Gradient System: Symmetric Zero Range Processes

6. The Relative Entropy Method

7. Hydrodynamic Limit of Reversible Nongradient Systems

8. Hydrodynamic Limit of Asymmetric Attractive Processes

9. Conservation of Local Equilibrium for Attractive Systems

10. Large Deviations from the Hydrodynamic Limit

11. Equilibrium Fluctuations of Reversible Dynamics

Backmatter