Stochastic resetting of a population of random walks with resetting-rate-dependent diffusivity

Abstract

We consider the problem of diffusion with stochastic resetting in a population of random walks where the diffusion coefficient is not constant, but behaves as a power-law of the average resetting rate of the population. Resetting occurs only beyond a threshold distance from the origin. This problem is motivated by physical realizations like soft matter under shear, where diffusion of a walk is induced by resetting events of other walks. We first reformulate in the broader context of diffusion with stochastic resetting the so-called Hébraud-Lequeux model for plasticity in dense soft matter, in which diffusivity is proportional to the average resetting rate. Depending on parameter values, the response to a weak external field may be either linear or non-linear with a non-zero average position for a vanishing applied field, and the transition between these two regimes may be interpreted as a continuous phase transition. Extending the model by considering a general power-law relation between diffusivity and average resetting rate, we notably find a discontinuous phase transition between a finite diffusivity and a vanishing diffusivity in the small field limit.

Capture of a diffusing lamb by a diffusing lion when both return home

“…This study was taken further for a system of Brownian particles where interactions are relevant, for example, in population genetics [32]. Inclusion of interactions further allows to consider more nontrivial forms of resetting mechanisms such as those which are driven by the interactions between the constituent particles [33] or space dependent resetting in interacting Brownian particles [34]. One of the most important examples which involves interacting random walks is the well known prey predator model which culminates when the prey is captured by the predator [35].…”