Non-conserving zero-range processes with extensive rates under resetting

TL;DR

This paper analyzes a non-conserving zero-range process with extensive rates, providing exact solutions for occupation numbers and steady states, especially under resetting, with applications to population dynamics and network models.

Contribution

It introduces a functional method to solve the master equation, deriving integral equations and exact Laplace domain solutions, extending the house-of-cards model with resetting effects.

Findings

Occupation numbers forget initial conditions exponentially fast.

Exact Laplace domain solutions for steady states with resetting.

Steady state modified by initial conditions and resetting rate.

Abstract

We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and vanish with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the mean-field geometry leads to a negative binomial law for the number of particles at each site, with parameters depending on the hopping, creation and annihilation rates. This model can be mapped to population dynamics (if the creation rates are reproductive fitnesses in a haploid population, and the hopping rate is the mutation rate). It can also be mapped to a Bianconi--Barab\'asi model of a growing network with random rewiring of links (if creation rates are the rates of acquisition of links by nodes, and the hopping rate is the rewiring rate). The steady state has recently been worked out and gives rise to occupation numbers that reproduce Kingman's house-of-cards model of selection and mutation. In this paper we solve the master equation using a functional method, which yields integral equations satisfied by the occupation numbers. The occupation numbers are shown to forget initial conditions at an exponential rate that decreases linearly with the fitness level. Moreover, they can be computed exactly in the Laplace domain, which allows to obtain the steady state of the system under resetting. The result modifies the house-of-cards result by simply adding a skewed version of the initial conditions, and by adding the resetting rate to the hopping rate.