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

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 meanfield 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-Barabási 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.

Run-and-tumble particles on a line with a fertile site

“…This stationary density profiledoesnot depend on the initial conditions. From a formal perspective, the rate of exponential growth of the density of right-movers at the origin was obtained from an integral equation, which resembles the renewal equations used to extract the steady state of systems under resetting (see [9][10][11][18][19][20][21][22][23][24][25][26][27][28] for examples, as well as [29] and references therein for a review). The Laplace transform of the equations of motion was also observed to yield the stationary probability density of a single runand-tumble particle subjected to resetting in [9].…”

Susceptibility to disorder of the optimal resetting rate in the Larkin model of directed polymers

“…The quantities L(Y, r, 0) and L(0, r, 0) are just Laplace transforms in time of the diffusive propagator of Eq. (18).…”

“…Stochastic resetting has since become a source of developments in out-of-equilibrium statistical physics [3][4][5][6], with applications including RNA polymerisation processes [7,8], active matter [9][10][11], randomised searching problems [12] and lifting of entropy barriers [13]. The corresponding renewal arguments [1,2,10] have been applied to models of active matter [10,14], predator-prey dynamics [15,16], population dynamics [17,18], and stochastic processes [19][20][21][22] (see [23] for a review, and references therein for more applications). For experimental realisations, see [24].…”

Winding number of a Brownian particle on a ring under stochastic resetting

“…Intuitively, stochastic resetting decreases the mean first-passage time because the amplitude of excursions in the wrong direction is unbounded. Stochastic resetting has found applications in a variety of fields, including active matter [3,4], population dynamics [5][6][7][8], reaction-diffusion systems [9,10] and stochastic processes [11][12][13][14][15]. For a review, see [16] and references therein.…”