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1985 | Buch

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Interacting Particle Systems

verfasst von: Thomas M. Liggett

Verlag: Springer New York

Buchreihe : Grundlehren der mathematischen Wissenschaften

Enthalten in: Professional Book Archive

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Über dieses Buch

At what point in the development of a new field should a book be written about it? This question is seldom easy to answer. In the case of interacting particle systems, important progress continues to be made at a substantial pace. A number of problems which are nearly as old as the subject itself remain open, and new problem areas continue to arise and develop. Thus one might argue that the time is not yet ripe for a book on this subject. On the other hand, this field is now about fifteen years old. Many important of several basic models is problems have been solved and the analysis almost complete. The papers written on this subject number in the hundreds. It has become increasingly difficult for newcomers to master the proliferating literature, and for workers in allied areas to make effective use of it. Thus I have concluded that this is an appropriate time to pause and take stock of the progress made to date. It is my hope that this book will not only provide a useful account of much of this progress, but that it will also help stimulate the future vigorous development of this field.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract

The field of interacting particle systems began as a branch of probability theory in the late 1960's. Much of the original impetus came from the work of F. Spitzer in the United States and of R. L. Dobrushin in the Soviet Union. (For examples of their early work, see Spitzer (1969a, 1970) and Dobrushin (1971a, b).) During the decade and a half since then, this area has grown and developed rapidly, establishing unexpected connections with a number of other fields.

Thomas M. Liggett

Chapter I. The Construction, and Other General Results

Abstract

The interacting particle systems which are the subject of this book are continuous-time Markov processes on certain spaces of configurations of particles. These processes are normally specified by giving the infinitesimal rates at which transitions occur. In general there are infinitely many particles in the system, and infinitely many of them make transitions in any interval of time. The transition rates for individual particles can depend on the entire configuration. Consequently, it is not immediately clear whether the specification of the local dynamics determines the evolution of the system as a whole in a unique way. Before proceeding to analyze the infinite system, it is therefore necessary to examine this question of existence and uniqueness of the process.

Thomas M. Liggett

Chapter II. Some Basic Tools

Abstract

This chapter has two objectives. The first is to introduce some tools which will be used frequently in succeeding chapters. These tools will be illustrated here with applications to countable state Markov chains. The hope is that after seeing how they are applied in this familiar setting, the reader will be better prepared to appreciate their usefulness in the context of particle systems. The second objective of this chapter is to prove some nonstandard results about Markov chains which we will need in later chapters. The material in the first five sections relates primarily to the first objective, while the latter sections relate to the second objective.

Thomas M. Liggett

Chapter III. Spin Systems

Abstract

A spin system is an interacting particle system in which each coordinate has two possible values, and only one coordinate changes in each transition. Throughout this chapter, the state space of the system will be taken to be X = {0, l}^s where S is a finite or countable set. There are numerous possible interpretations of the two possible values 0 and 1. The next three chapters deal with three classes of spin systems in which the transition mechanisms have a particular form, and each of these classes corresponds to a different interpretation for 0 and 1. In the stochastic Ising model, they represent the two possible spins of an iron atom (for example). In the case of the voter model, they denote two possible positions of a “voter” on some political issue. In the case of the contact process, 0 and 1 represent healthy and infected individuals respectively.

Thomas M. Liggett

Chapter IV. Stochastic Ising Models

Abstract

Stochastic Ising models can be thought of loosely as reversible spin systems with strictly positive rates. (For a more precise version of this statement, see Theorem 2.13.) The measures with respect to which they are reversible are the Gibbs states of classical statistical mechanics. Thus stochastic Ising models can be viewed as models for nonequilibrium statistical mechanics. These were among the first spin systems to be studied, because of their close connections with physics. While no prior knowledge of statistical mechanics will be assumed in this chapter, the reader may find it useful to refer occasionally to more comprehensive treatments of that subject. Recommended for this purpose are Ruelle (1969, 1978), Griffiths (1972), Preston (1974b), Sinai (1982), and Simon (1985).

Thomas M. Liggett

Chapter V. The Voter Model

Abstract

The voter model is the spin system with rates c(x,η) given by

$$ c(x,\eta ) = \left\{ {\begin{array}{*{20}c} {\sum\limits_y {p(x,y)\eta (y)} } & {if{\text{ }}\eta (x) = 0,} \\ {\sum\limits_y {p(x,y)[1 - \eta (y)]} } & {f{\text{ }}\eta (x) = 1,} \\ \end{array} } \right. $$

(0.1)

where p(x,y) ≥ 0 for x, y ∈ S and

$$ \begin{array}{*{20}c} {\sum\limits_y {p(x,y) = 1} } & {for{\text{ }}x{\text{ }} \in {\text{ }}S.} \\ \end{array} $$

Thomas M. Liggett

Chapter VI. The Contact Process

Abstract

The contact process is the spin system in which S = Z^d and

$$ c(x,\eta ) = \left\{ {\begin{array}{*{20}c} {\lambda \sum\limits_{|y - x| = 1} {\eta (y)} } & {if{\text{ }}\eta (x) = 0,} \\ 1 & {f{\text{ }}\eta (x) = 1,} \\ \end{array} } \right. $$

(0.1)

where λ is a nonnegative parameter. One interpretation of this process is as a model for the spread of an infection. An individual at x ∈ S is infected if η(x) = 1 and healthy if η (x) = 0. Healthy individuals become infected at a rate which is proportional to the number of infected neighbors. Infected individuals recover at a constant rate, which is normalized to be 1.

Thomas M. Liggett

Chapter VII. Nearest-Particle Systems

Abstract

Nearest-particle systems are one-dimensional spin systems in which the flip rates depend on η in a certain way which we will now describe. Configurations η ∈ X = {0,1}^Z1 will be given an occupancy interpretation: η(x) = 1 means that there is a particle at x, and η( x) = 0 means that x is vacant. For x ∈ Z¹ and η ∈ X, let l_x (η) and r_x (η) be the distances from x to the nearest particle to the left and right respectively:

$$ \begin{array}{*{20}c} {lx(\eta ) = x - \max \{ y < x:\eta (y) = 1\} ,{\text{ }}and} \\ {rx(\eta ) = \min (y > x:\eta (y) = 1\} - x.} \\ \end{array} $$

Thomas M. Liggett

Chapter VIII. The Exclusion Process

Abstract

The exclusion process differs from the spin systems which are the subject of the previous five chapters in that two (rather than one) coordinates of η₁ change at a time. In order to describe this process, let p(x, y) be the transition probabilities for a discrete time Markov chain on the countable set 5:

$$ p(x,y) \geq 0,{\text{ }}and{\text{ }}\sum\limits_y {p(x,y) = 1.} $$

Thomas M. Liggett

Chapter IX. Linear Systems with Values in [0, ∞) s

Abstract

All the processes considered in previous chapters have the property that each coordinate η(x) can take on only two values. When the set of possible values per site is allowed to be noncompact, new problems and different phenomena occur. The literature contains many types of models in which the set of possible values per site is either the nonnegative integers or the nonnegative real numbers. The oldest and simplest of these is a system of particles which move independently on 5. This process has been modified by adding a speed change interaction and/or by allowing branching. In these cases, η (x) is interpreted as the number of particles at x. In other models, one can view η (x) as being a nonnegative real-valued characteristic of the particle at x, which is updated in some way which involves interactions among the various sites.

Thomas M. Liggett

Backmatter

Titel: Interacting Particle Systems
verfasst von: Thomas M. Liggett
Verlag: Springer New York
Electronic ISBN: 978-1-4613-8542-4
Print ISBN: 978-1-4613-8544-8
DOI: https://doi.org/10.1007/978-1-4613-8542-4

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Introduction

Chapter I. The Construction, and Other General Results

Chapter II. Some Basic Tools

Chapter III. Spin Systems

Chapter IV. Stochastic Ising Models

Chapter V. The Voter Model

Chapter VI. The Contact Process

Chapter VII. Nearest-Particle Systems

Chapter VIII. The Exclusion Process

Chapter IX. Linear Systems with Values in [0, ∞) s

Backmatter