Infinite Particle Systems Research Articles

Ergodic theorems for the asymmetric simple exclusion process

Consider the infinite particle system on the integers with the simple exclusion interaction and one-particle motion determined by p ( x , x + 1 ) = p p(x,x + 1) = p and p ( x , x − 1 ) = q p(x,x - 1) = q for x ∈ Z x \in Z , where p + q = 1 p + q = 1 and p > q p > q . If μ \mu is the initial distribution of the system, let μ t {\mu _t} be the distribution at time t. The main results determine the limiting behavior of μ t {\mu _t} as t → ∞ t \to \infty for simple choices of μ \mu . For example, it is shown that if μ \mu is the pointmass on the configuration in which all sites to the left of the origin are occupied, while those to the right are vacant, then the system converges as t → ∞ t \to \infty to the product measure on { 0 , 1 } Z {\{ 0,1\} ^Z} with density 1 / 2 1/2 . For the proof, an auxiliary process is introduced which is of interest in its own right. It is a process on the positive integers in which particles move according to the simple exclusion process, but with the additional feature that there can be creation and destruction of particles at one. Ergodic theorems are proved for this process also.

Convergence to Total Occupancy in an Infinite Particle System with Interactions

Let $p(x, y)$ be the transition function for an irreducible, positive recurrent, reversible Markov chain on the countable set $S$. Let $\eta_t$ be the infinite particle system on $S$ with the simple exclusion interaction and one-particle motion determined by $p$. The principal result is that there are no nontrivial invariant measures for $\eta_t$ which concentrate on infinite configurations of particles on $S$. Furthermore, it is proved that if the system begins with an arbitrary infinite configuration, then it converges in probability to the configuration in which all sites are occupied.

Functionals of an infinite particle system with independent motions, creation, and annihilation

Particles enter a state space at random times. Each particle travels in the space independent of the other particles until its death. Functionals of the particle system are studied with strong laws and central limit theorems being obtained.

Functionals of an infinite particle system with independent motions, creation, and annihilation

Particles enter a state space at random times. Each particle travels in the space independent of the other particles until its death. Functionals of the particle system are studied with strong laws and central limit theorems being obtained.

Recurrent Random Walk of an Infinite Particle System

Let $p(x,y)$ be the transition function for a symmetric irreducible recurrent Markov chain on a countable set $S$. Let ${\eta _t}$ be the infinite particle system on $S$ moving according to simple exclusion interaction with the one particle motion determined by $p$. Assume that $p$ is such that any two particles moving independently on $S$ will sooner or later meet. Then it is shown that every invariant measure for ${\eta _t}$ is a convex combination of Bernoulli product measures ${\mu _\alpha }$ on ${\{ 0,1\} ^s}$ with density $0 \leqslant \alpha = \mu [\eta (x) = 1] \leqslant 1$. Ergodic theorems are proved concerning the convergence of the system to one of the ${\mu _\alpha }$.

A Characterization of the Invariant Measures for an Infinite Particle System with Interactions. II

A Characterization of the Invariant Measures for an Infinite Particle System with Interactions. II

Random Stirring of the Real Line

Random stirring of the real line $R_1$ is defined. This notion is derived from a generalization of the nearest-neighbor simple exclusion model on the one-dimensional lattices discussed by Spitzer and by Harris. Under the random stirring, the motion of an infinite particle system is Markovian and has a Poisson process as an invariant probability measure. An ergodic theorem is established concerning the convergence of a system to a Poisson process.

Ergodic properties of an infinite system of particles moving independently in a periodic field

We investigate the ergodic properties of a general class of infinite systems of independent particles which undergo nontrivial “collisions” with an external field, e.g. fixed convex barriers (the Lorentz gas). We relate the ergodic properties of these systems to the ergodic properties for a single particle moving in a finite box (with periodic boundary conditions) with the same dynamics. We prove that when the one particle system is mixing or aK-system for a sequence of boxes approaching infinity so is the infinite particle system with an equilibrium measure obtained as a Poisson construction over the one particle phase space.

Recurrent random walk of an infinite particle system

Let $p(x,y)$ be the transition function for a symmetric irreducible recurrent Markov chain on a countable set $S$. Let ${\eta _t}$ be the infinite particle system on $S$ moving according to simple exclusion interaction with the one particle motion determined by $p$. Assume that $p$ is such that any two particles moving independently on $S$ will sooner or later meet. Then it is shown that every invariant measure for ${\eta _t}$ is a convex combination of Bernoulli product measures ${\mu _\alpha }$ on ${\{ 0,1\} ^s}$ with density $0 \leqslant \alpha = \mu [\eta (x) = 1] \leqslant 1$. Ergodic theorems are proved concerning the convergence of the system to one of the ${\mu _\alpha }$.

A characterization of the invariant measures for an infinite particle system with interactions. II

Let $p(x,y)$ be the transition function for a symmetric, irreducible Markov chain on the countable set $S$. Let $\eta (t)$ be the infinite particle system on $S$ with the simple exclusion interaction and one-particle motion determined by $p$. The present author and Spitzer have determined all of the invariant measures of $\eta (t)$, and have obtained ergodic theorems for $\eta (t)$, under two different sets of assumptions. In this paper, these problems are solved in the remaining case.

Some results on the limiting behaviour of infinite particle systems

We conclude by mentioning two counter-examples. An example may be constructed in which the initial configuration has an average γ and the velocity distribution although symmetric is such that if it is truncated there will be convergence to P(γ) but without truncation there will not be convergence. It is also possible to show that if Z ∋ W there may be convergence to P(γ) without the initial configuration having an average.

A Characterization of the Invariant Measures for an Infinite Particle System with Interactions

Let $p(x,y)$ be the transition function for a symmetric, irreducible, transient Markov chain on the countable set S. Let ${\eta _t}$ be the infinite particle system on S with the simple exclusion interaction and one-particle motion determined by p. A characterization is obtained of all the invariant measures for ${\eta _t}$ in terms of the bounded functions on S which are harmonic with respect to $p(x,y)$. Ergodic theorems are proved concerning the convergence of the system to an invariant measure.

An Infinite Particle System with Zero Range Interactions

Existence and uniqueness is proved for Spitzer's zero range interaction process. It is then shown that a certain class of measures on the configuration space is invariant for this process.

Asymptotic orbits in a free Fermi gas

We consider the time evolution of local observables and physical states in an infinite system of non-interacting Fermi particles. The orbit of an observable in theC*-algebra of the canonical anticommutation relations is proved to be asymptotic to a set of observables consisting of sums of products of elements of grade two and lower with support in a family of separated cells in ℝ3 (alacunary paving of ℝ3) under time evolution. A space-factorization (“clustering”) property for primary, even, locally Fock states is established. A class of such states whose space-correlations decay as (logd)−(1+a) witha positive andd the (space-) separation is, then, proved to be time-asymptotic to their associated quasi-free states.

Infinite particle systems of recurrent random walks

Infinite particle systems of recurrent random walks

Infinite particle systems

We consider a system of denumerably many particles that are distributed at random according to a stationary distribution P on some closed subgroup X of Euclidean space. We assume that the expected number of particles in any compact set is finite. We investigate the relationship between P and the distribution Q of particles as viewed from a particle selected “at random” from some set. The distribution Q is called the tagged particle distribution. We give formulas for computing P in terms of Q and Q in terms of P and show that, with the appropriate notion of convergences, P n → P {P_n} \to P implies Q n → Q {Q_n} \to Q and vice versa. The particles are allowed to move in an appropriate translation invariant manner and we show that the tagged particle distribution Q’ at a later time 1 is the same as the distribution of particles at time 1 as viewed from a particle selected “at random” from those initially in some set. We also show that Q’ is the same as the distribution of particles at time 1 as viewed from a particle selected at random from those at the origin, when initially the particles are distributed according to Q. The one-dimensional case is treated in more detail. With appropriate topologies, we show that in this case there is a homeomorphism between the collection of stationary distributions P and tagged particle distributions Q. A stationary spacings distribution Q 0 {Q_0} related to Q is introduced, and we show that with the appropriate topology the map taking Q to Q 0 {Q_0} is a homeomorphism. Explicit expressions are found for all these maps and their inverses. The paper concludes by using the one-dimensional results to find stationary distributions for a class of motions of denumerably many unit intervals and to establish criteria for convergence to one of these distributions.

A characterization of the invariant measures for an infinite particle system with interactions

Let p ( x , y ) p(x,y) be the transition function for a symmetric, irreducible, transient Markov chain on the countable set S. Let η t {\eta _t} be the infinite particle system on S with the simple exclusion interaction and one-particle motion determined by p. A characterization is obtained of all the invariant measures for η t {\eta _t} in terms of the bounded functions on S which are harmonic with respect to p ( x , y ) p(x,y) . Ergodic theorems are proved concerning the convergence of the system to an invariant measure.

Markovian Interaction Processes with Finite Range Interactions

An elementary proof of an existence theorem for infinite particle systems interacting through a finite range interaction is given. The results are then used to prove that the system preserves ergodicity of the initial measure.

Existence Theorems for Infinite Particle Systems

Sufficient conditions are given for a countable sum of bounded generators of semigroups of contractions on a Banach space to be a generator. This result is then applied to obtain existence theorems for two classes of models of infinite particle systems. The first is a model of a dynamic lattice gas, while the second describes a lattice spin system.

Existence theorems for infinite particle systems

Sufficient conditions are given for a countable sum of bounded generators of semigroups of contractions on a Banach space to be a generator. This result is then applied to obtain existence theorems for two classes of models of infinite particle systems. The first is a model of a dynamic lattice gas, while the second describes a lattice spin system.