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Dynamics of Finite Inhomogeneous Particle Systems with Exclusion Interaction

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

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Vadim Malyshev

Serguei Popov


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
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
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