Monotonicity and Condensation in Homogeneous Stochastic Particle Systems

TL;DR

This paper investigates stochastic particle systems with condensation, proving that finite-critical-density condensing systems are necessarily non-monotone, and explores the monotonicity in systems with infinite critical density.

Contribution

It establishes a fundamental link between condensation and non-monotonicity in homogeneous particle systems with finite critical density.

Findings

Condensing systems with finite critical density are necessarily non-monotone.

An example of a condensing process with infinite critical density may be monotone.

Numerical evidence and partial proofs suggest some infinite-critical-density systems are monotone.

Abstract

We study stochastic particle systems that conserve the particle density and exhibit a condensation transition due to particle interactions. We restrict our analysis to spatially homogeneous systems on finite lattices with stationary product measures, which includes previously studied zero-range or misanthrope processes. All known examples of such condensing processes are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space and the canonical measures (with a fixed number of particles) are not monotonically ordered. For our main result we prove that condensing homogeneous particle systems with finite critical density are necessarily non-monotone. On finite lattices condensation can occur even when the critical density is infinite, in this case we give an example of a condensing process that numerical evidence suggests is monotone, and give a partial proof of its monotonicity.