Dynamics of Finite Inhomogeneous Particle Systems with Exclusion Interaction

Malyshev, Vadim; Menshikov, Mikhail V.; Popov, Serguei; Wade, Andrew

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.

• Journal Article Type: Article
• Acceptance Date: Oct 24, 2023
• Online Publication Date: Nov 11, 2023
• Deposit Date: Jun 6, 2022
• Publicly Available Date: Dec 7, 2023
• Journal: Journal of Statistical Physics
• Print ISSN: 0022-4715
• Electronic ISSN: 1572-9613
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 190
• Issue: 11
• Article Number: 184
• DOI: https://doi.org/10.1007/s10955-023-03190-8
• Keywords: Lattice Atlas model, 60J27, 60K25, Jackson network, Law of large numbers, Central limit theorem, 90B22 (Secondary), Asymptotic speeds, Product-geometric distribution, Interacting particle systems, Partial stability, 60K35 (Primary), Exclusion process
• Public URL: https://durham-repository.worktribe.com/output/1201880
• Publisher URL: https://www.springer.com/journal/10955
• Related Public URLs: https://arxiv.org/abs/2205.14990