Asymmetric particle systems on R

TL;DR

This paper investigates asymmetric interacting particle systems on the real line, identifying conditions for Poisson invariant distributions and analyzing the structure of particle spacings, including explicit distributions and diffusion properties.

Contribution

It introduces a class of particle systems with Poisson invariant measures and derives explicit spacing distributions and diffusion coefficients for various dynamics.

Findings

Invariant particle distribution is Poisson under certain conditions.

Spacing distribution for uniform jump fraction is a gamma distribution.

Explicit diffusion coefficients are computed for different dynamics.

Abstract

We study interacting particle systems on the real line which generalize the Hammersley process [D. Aldous and P. Diaconis, Prob. Theory Relat. Fields 103, 199-213 (1995)]. Particles jump to the right to a randomly chosen point between their previous position and that of the forward neighbor, at a rate which may depend on the distance to the neighbor. A class of models is identified for which the invariant particle distribution is Poisson. The bulk of the paper is devoted to a model where the jump rate is constant and the jump length is a random fraction $r$ of the distance to the forward neighbor, drawn from a probability density $\phi(r)$ on the unit interval. This is a special case of the random average process of Ferrari and Fontes [El. J. Prob. 3, Paper no. 6 (1998)]. The discrete time version of the model has been considered previously in the context of force propagation in granular media [S.N. Coppersmith et al., Phys. Rev. E 53, 4673 (1996)]. We show that the stationary two-point function of particle spacings factorizes for any choice of $\phi(r)$. Under the assumption that this implies pairwise independence, the invariant density of interparticle spacings for the case of uniform $\phi(r)$ is found to be a gamma distribution with parameter $\nu$, where $\nu = 1/2$, 1 and 2 for continuous time, backward sequential and discrete time dynamics, respectively. A heuristic derivation of a nonlinear diffusion equation is presented, and the tracer diffusion coefficient is computed for arbitrary $\phi(r)$ and different types of dynamics.