Dynamics of finite inhomogeneous particle systems with exclusion interaction

TL;DR

This paper analyzes finite inhomogeneous particle systems with exclusion interactions on a 1D lattice, providing explicit partitions, distributions, and fluctuation results based on particle jump rates.

Contribution

It introduces a linear-time algorithm to partition the system into stable sub-systems and characterizes their distributions and asymptotic behaviors, extending classical queueing network results.

Findings

Explicit partition of the system into stable sub-systems based on jump rates

Product-geometric limiting distribution for each stable sub-system

Central limit theorem for fluctuations in the fully stable system

Abstract

We study finite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses jumps that would lead to more than one particle occupying any site. We show that the particle jump rates determine explicitly a unique partition of the system into maximal stable sub-systems, and that this partition can be obtained by a linear-time algorithm using only elementary arithmetic. The internal configuration of each stable sub-system possesses an explicit product-geometric limiting distribution, and the location of each stable sub-system obeys a strong law of large numbers with an explicit speed; the characteristic parameters of each stable sub-system are simple functions of the rate parameters for the corresponding particles. For the case where the entire system is stable, we provide a central limit theorem describing the fluctuations around the law of large numbers. Our approach draws on ramifications, in the exclusion context, of classical work of Goodman and Massey on partially-stable Jackson queueing networks.