Dynamics of finite inhomogeneous particle systems with exclusion interaction

Vadim Malyshev, Mikhail V. Menshikov, Serguei Popov, and Andrew R. Wade

Journal of Statistical Physics, 190, November 2023, article no. 184. DOI: 10.1007/s10955-023-03190-8

Supported by EPSRC award Anomalous diffusion via self-interaction and reflection (EP/W00657X/1).

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 of elementary steps, as well as by solving a finite non-linear system. The internal configuration of each stable sub-system possesses an explicit product-geometric limiting distribution, and 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 prove 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 unstable Jackson queueing networks.