Voting and Catalytic Processes with Inhomogeneities

TL;DR

This paper investigates the effects of multiple inhomogeneities on voter and catalytic models, revealing complex steady states influenced by system dimensionality, inhomogeneity strength, and separation, with exact and numerical analyses.

Contribution

It provides exact expressions for order parameters in systems with multiple inhomogeneities and explores their dependence on system parameters, connecting to electrostatic analogies.

Findings

Steady states depend on inhomogeneity number, strength, and separation.

Explicit solutions for one and two inhomogeneities in arbitrary dimensions.

Numerical simulations for cases with more than two inhomogeneities.

Abstract

We consider the dynamics of the voter model and of the monomer-monomer catalytic process in the presence of many ``competing'' inhomogeneities and show, through exact calculations and numerical simulations, that their presence results in a nontrivial fluctuating steady state whose properties are studied and turn out to specifically depend on the dimensionality of the system, the strength of the inhomogeneities and their separating distances. In fact, in arbitrary dimensions, we obtain an exact (yet formal) expression of the order parameters (magnetization and concentration of adsorbed particles) in the presence of an arbitrary number $n$ of inhomogeneities (``zealots'' in the voter language) and formal similarities with {\it suitable electrostatic systems} are pointed out. In the nontrivial cases $n=1, 2$, we explicitly compute the static and long-time properties of the order parameters and therefore capture the generic features of the systems. When $n>2$, the problems are studied through numerical simulations. In one spatial dimension, we also compute the expressions of the stationary order parameters in the completely disordered case, where $n$ is arbitrary large. Particular attention is paid to the spatial dependence of the stationary order parameters and formal connections with electrostatics.