Static- and dynamical-phase transition in multidimensional voting models on continua

TL;DR

This paper investigates static and dynamical phase transitions in a multidimensional continuum voting model, revealing how bulk reactions and boundary conditions influence system behavior and relaxation dynamics.

Contribution

It introduces a continuum version of a multidimensional voting model and analyzes the distinct static and dynamical phase transitions within it.

Findings

Static phase transition driven by bulk reactions.

Dynamical phase transition influenced by bulk reactions and boundary conditions.

Different mechanisms govern stationary and relaxation behaviors.

Abstract

A voting model (or a generalization of the Glauber model at zero temperature) on a multidimensional lattice is defined as a system composed of a lattice each site of which is either empty or occupied by a single particle. The reactions of the system are such that two adjacent sites, one empty the other occupied, may evolve to a state where both of these sites are either empty or occupied. The continuum version of this model in a Ddimensional region with boundary is studied, and two general behaviors of such systems are investigated. The stationary behavior of the system, and the dominant way of the relaxation of the system toward its stationary state. Based on the first behavior, the static phase transition (discontinuous changes in the stationary profiles of the system) is studied. Based on the second behavior, the dynamical phase transition (discontinuous changes in the relaxation-times of the system) is studied. It is shown that the static phase transition is induced by the bulk reactions only, while the dynamical phase transition is a result of both bulk reactions and boundary conditions.