Asymptotic behavior of a stochastic particle system of 5 neighbors

TL;DR

This paper investigates the long-term behavior of a stochastic particle system with five neighbors, proposing a conjecture linking local patterns to asymptotic distribution and deriving mean flux both theoretically and in the deterministic limit.

Contribution

It introduces a conjecture connecting local pattern counts to asymptotic distribution and derives mean flux formulas for stochastic and deterministic cases.

Findings

Conjecture relating local patterns to asymptotic distribution.

Derived mean flux depending on conserved quantities.

Established mean flux in the deterministic limit.

Abstract

We analyze a stochastic particle system of 5 neighbors. Considering eigenvalue problem of transition matrix, we propose a conjecture that asymptotic distribution of the system is determined by the number of specific local patterns in the asymptotic solution. Based on the conjecture, mean flux which depends of a pair of the conserved quantities is derived theoretically. Moreover, we obtain mean flux in the deterministic case through the limit of stochastic parameter.