Asymmetric particle systems on R

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.