Infinite Particle Systems Research Articles

Symmetric Two-Particle Exclusion-Eating Processes

We consider infinite particle systems on a countable set $S$ with two-particle exclusion-eating motion determined by a symmetric transition function $p(x, y)$. This is, in a certain sense, a mixture of the exclusion process and the voter model. We discuss the dual process of this process and use the dual process to give a description of the set of invariant measures and to prove an ergodic theorem.

Solution of Kirkwood-Salsburg equations for a one-dimensional lattice gas

Expressions are obtained for the lowest correlation functions directly from the Kirkwood-Salsburg equations for an infinite system of particles on a one-dimensional lattice with two-body nearest-neighbor interaction in certain external fields. The problem of finding the external field that makes the density oscillation near the wall uniform is considered.

A coupling of infinite particle systems

A coupling of infinite particle systems

On a variational problem for an infinite particle system in a random medium Part II: The local growth rate

This paper solves the second of two variational problems arising in the study of an infinite system of particles that branch and migrate in a random medium. This variational problem involves a non-linear functional on a subset of the stationary probability measures on [ℕ×ℝ+]ℤ, describing the interplay between particles and medium. It is shown that the variational problem can be solved in terms of the Lyapunov exponent of a product of random ℕ×ℕ matrices. This Lyapunov exponent is calculated via a random continued fraction. By analyzing the latter we are able to compute the maximum and the maximizer in the variational problem. It is found that these quantities exhibit interesting non-analyticities and changes of sign as a function of model parameters, which correspond to phase transitions in the infinite particle system. By combining with results from Part I we obtain a complete picture of the phase diagram.

On a variational problem for an infinite particle system in a random medium.

Article On a variational problem for an infinite particle system in a random medium. was published on January 1, 1994 in the journal Journal für die reine und angewandte Mathematik (volume 1994, issue 454).

Occupation times in systems of null recurrent Markov processes

We study asymptotic properties of differences of occupation times for infinite systems of noninteracting Markovian particles. Under a suitable normalisation we prove convergence in law to a nondegerate Gaussian field. We also obtain large deviations properties. These results generalise previous results obtained separately by both authors.

Ergodic Theorems for Infinite Systems of Locally Interacting Diffusions

Let $x(t) = \{x_i(t), i \in \mathbb{Z}^d\}$ be the solution of the system of stochastic differential equations $dx_i(t) = \bigg(\sum_{j\in\mathbb{Z}^d}a(i,j)x_j(t) - x_i(t)\bigg) dt + \sqrt{2g (x_i(t))} dw_i(t), \quad i \in \mathbb{Z}^d.$ Here $g: \lbrack 0, 1\rbrack \rightarrow \mathbb{R}^+$ satisfies $g > 0$ on (0, 1), $g(0) = g(1) = 0, g$ is Lipschitz, $a(i,j)$ is an irreducible random walk kernel on $\mathbb{Z}^d$ and $\{w_i(t), i \in \mathbb{Z}^d\}$ is a family of standard, independent Brownian motions on $\mathbb{R}; x(t)$ is a Markov process on $X = \lbrack 0, 1\rbrack^{\mathbb{Z}^d}$. This class of processes was studied by Notohara and Shiga; the special case $g(v) = v(1 - v)$ has been studied extensively by Shiga. We show that the long term behavior of $x(t)$ depends only on $\hat{a}(i,j) = (a(i,j) + a(j, i))/2$ and is universal for the entire class of $g$ considered. If $\hat{a}(i,j)$ is transient, then there exists a family $\{\nu_\theta, \theta \in \lbrack 0, 1\rbrack\}$ of extremal, translation invariant equilibria. Each $\nu_\theta$ is mixing and has density $\theta = \int x_0 d\nu_\theta$. If $\hat{a}(i,j)$, is recurrent, then the set of extremal translation invariant equilibria consists of the point masses $\{\delta_0, \delta_1\}$. The process starting in a translation invariant, shift ergodic measure $\mu$ on $X$ with $\int x_0 d\mu = \theta$ converges weakly as $t \rightarrow \infty$ to $\nu_\theta$ if $\hat{a}(i,j)$ is transient, and to $(1 - \theta)\delta_0 + \theta\delta_1$ if $\hat{a}(i,j)$ is recurrent. (Our results in the recurrent case remove a mild assumption on $g$ imposed by Notohara and Shiga.) For the case $\hat{a}(i,j)$ transient we use methods developed for infinite particle systems by Liggett and Spitzer. For the case $\hat{a}(i,j)$, recurrent we use a duality comparison argument.

Quantum oscillator in an infinite-particle harmonic system

The behavior of a quantum oscillator in an infinite-particle system is studied for the case of linear interaction. The relation between the spectrum of the dynamic matrix of a complete system and oscillator damping is established. The dependence of the spectrum on the parameters of interaction is determined.

A random field approach to weak convergence of processes

Random fields in L ′( R d+1 ) are associated to processes with paths in D([0, 1], L ′( R d )). This embedding provides a way to analyze weak convergence for such processes. The approach is also useful for real valued processes. The idea is to regard processes as time random fields. The method is illustrated with the fluctuations of an infinite particle system.

Diffusion in a one-dimensional gas of hard point particles

We simulate the classical diffusion of a particle of massM in an infinite one-dimensional system of hard point particles of massm in equilibrium. Each computer run corresponds to about 108 collisions of the diffusive particle. We find that 〈νν(t)〉 ∼ 1/t δ fort large enough, and a crossover from an M ≠ m regime whereδ=2 toδ=3 forM=m. The diffusion constant has a sharp maximum atM=m. We study moments 〈x(t)2〉 and 〈x(t)4〉, and examine the behavior ofq 2 (t)=〈x(t)4〉/3〈x(t)2〉2. We find thatq(t)→1 (consistent with a normal distribution) in theM →∞ limit (for all timest) and in the t→∞ limit for allM.

Central Limit Theorem for a Random Walk with Random Obstacles in $\mathrm{R}^d$

A random walk with obstacles in $\mathbf{R}^d, d \geq 2$, is considered. A probability measure is put on a space of obstacles, giving a random walk with random obstacles. A central limit theorem is then proven for this process when the obstacles are distributed by a Gibbs state with sufficiently low activity. The same problem is treated for a tagged particle of an infinite hard core particle system.

Hard particles in narrow pores. Transfer-matrix solution and the periodic narrow box

We derive an exact transfer-matrix solution for an infinite system of hard particles confined in a manner that precludes non-nearest-neighbor interactions. The solution takes the form of a functional eigenvalue equation which may be solved numerically for the thermodynamic and structural properties of the confined fluid. Barker [Aust. J. Phys. 15, 127 (1962)] originally derived this solution by a different route, and we apply it in a number of new ways. We present the first calculations based on this solution for systems of hard disks between parallel lines, and for hard spheres in a cylindrical pore. Through comparison with Monte Carlo simulations, we examine the range of validity of the solution when applied to systems in which non-nearest-neighbor interactions may occur. We find that the transfer-matrix approach provides acceptable results for pore widths up to two particle diameters, and that the approximation becomes quite poor as the pore is widened further, particularly at high density. This solution may be used to test the narrow-pore limit of more versatile theories of confined fluids. We further apply the solution to the so-called ‘‘periodic narrow box,’’ which has previously been solved in only two dimensions. We reproduce the two-dimensional result, and present the solution (which may be formulated explicitly rather than as an eigenvalue equation) for the three-dimensional, hard sphere version of the model. This simple model provides a remarkably accurate description of the thermodynamic behavior of a bulk hard sphere crystal.

Fleming–Viot Processes in Population Genetics

Fleming and Viot [Indiana Univ. Math. J., 28 (1979), pp. 817–843] introduced a class of probability-measure-valued diffusion processes that has attracted the interest of both pure and applied probabilists. This paper surveys the subject of Fleming–Viot processes as it relates to population genetics. Topics include:1. Introduction.2. Some measure-valued Markov chains.2.1. A diploid model. 2.2. The Wright–Fisher model.2.3. A Moran model.3. The Fleming–Viot process: characterization.4. Convergence.5. Ergodicity.6. An infinite particle system.7. Bounded mutation operators.8. Reversibility.9. Examples.9.1. Continuous-state stepwise-mutation model.9.2. Infinitely-many-neutral-alleles model.9.3. Infinitely-many-neutral-alleles model with ages.9.4. Two-locus model with recombination.9.5. n-locus model with gene conversion.9.6. Infinitely-many-sites model without recombination.9.7. Infinitely-many-neutral-alleles model with allelic genealogies.

Space‐time large deviations for interacting particle systems

AbstractWe prove a process‐level large deviation principle for the space‐time empirical averages of continuous‐time systems on an infinite lattice. Our methods rely on the Donsker‐Varadhan large deviation theory for Markov processes, and allow us to express the rate function rather explicitly in terms of the Markov generator of the infinite particle system. We can prove our principle for a large class of spin systems with no particle exchange, as well as for infinite diffusion processes whose drift is the gradient of a finite range Hamiltonian. © 1993 John Wiley & Sons, Inc.

A Variational Approach to Branching Random Walk in Random Environment

This paper considers an infinite system of particles on the integers $\mathbb{Z}$ that: (1) step to the right with a random delay, and (2) split or die along the way according to a random law depending on their position. The exponential growth rate of the particle density is computed in the long time limit in the form of a variational formula that can be solved explicitly. The result reveals two phase transitions associated with localization vs. delocalization and survival vs. extinction. In addition, the system exhibits an intermittency effect. Greven and den Hollander considered the more difficult situation where the particles may step both to the left and right, but the analysis of the phase diagram was less complete.

Branching random walk in random environment: phase transitions for local and global growth rates

Let (η n ) be the infinite particle system on ℤ whose evolution is as follows. At each unit of time each particle independently is replaced by a new generation. The size of a new generation descending from a particle at sitex has distributionF x and each of its members independently jumps to sitex±1 with probability (1±h)/2,h∈[0, 1]. The sequence {F x } is i.i.d. with uniformly bounded second moment and is kept fixed during the evolution. The initial configurationη 0 is shift invariant and ergodic. Two quantities are considered:

Non-Markovian fluctuation limits of infinite particle systems

S′-valued Gaussian processes of a certain class are shown to have the semimartingale decomposition similar to that one given by the Ornstein-Uhlenbeck equation. There is given a simple sufficient condition for the family of fluctuations to be tight.The non–Markov property for this class of processes is investigated

Population growth in random media. I. Variational formula and phase diagram

We consider an infinite system of particles on the integer lattice Z that: (1) migrate to the right with a random delay, (2) branch along the way according to a random law depending on their position (random medium). In Part I, the first part of a two-part presentation, the initial configuration has one particle at each site. The long-time limit exponential growth rate of the expected number of particles at site 0 (local particle density) does not depend on the realization of the random medium, but only on the law. It is computed in the form of a variational formula that can be solved explicitly. The result reveals two phase transitions associated with localization vs. delocalization and survival vs. extinction. In earlier work the exponential growth rate of the Cesaro limit of the number of particles per site (global particle density) was studied and a different variational formula was found, but with similar structure, solution, and phases. Combination of the two results reveals an intermediate phase where the population globally survives but locally becomes extinct (i.e., dies out on any fixed finite set of sites).

Infinite reversible nearest particle systems in inhomogeneous and random environments

In this paper we prove that an infinite reversible nearest particle system in a random environment determined by i.i.d. random variables λ(x), x ∈ Z , survives almost surely if E log λ (0) > 0 and dies out almost surely if Eλ(0) < 1. Both a.s. survival and a.s. extinction may happen if E log λ(0) < 0 and Eλ(0) > 1.

ERGODIC PROPERTIES OF SYSTEMS WITH AN EXTERNAL POTENTIAL IN CLASSICAL STATISTICAL MECHANICS

For infinite-particle systems of classical statistical mechanics, a method of describing the dynamics of a charged one-dimensional Brownian particle is obtained. Bibliography: 7 titles.