Factorised Steady States in Mass Transport Models on an Arbitrary Graph

TL;DR

This paper investigates a broad class of mass transport models on arbitrary graphs, deriving conditions under which the steady state distribution factorizes, encompassing many known models and extending previous results to general graph structures.

Contribution

It provides a general criterion for factorization of steady states in mass transport models on arbitrary graphs, unifying and extending prior specific cases.

Findings

Derived a sufficient condition for factorisable steady states.

Included special cases like Zero-range process and asymmetric random average process.

Extended previous results to arbitrary graph structures.

Abstract

We study a general mass transport model on an arbitrary graph consisting of $L$ nodes each carrying a continuous mass. The graph also has a set of directed links between pairs of nodes through which a stochastic portion of mass, chosen from a site-dependent distribution, is transported between the nodes at each time step. The dynamics conserves the total mass and the system eventually reaches a steady state. This general model includes as special cases various previously studied models such as the Zero-range process and the Asymmetric random average process. We derive a general condition on the stochastic mass transport rules, valid for arbitrary graph and for both parallel and random sequential dynamics, that is sufficient to guarantee that the steady state is factorisable. We demonstrate how this condition can be achieved in several examples. We show that our generalized result contains as a special case the recent results derived by Greenblatt and Lebowitz for $d$-dimensional hypercubic lattices with random sequential dynamics.