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

TL;DR

This paper studies a diffusion model with stochastic resetting where the diffusion rate depends on the resetting rate, revealing phase transitions between different diffusive regimes, relevant for soft matter physics.

Contribution

It introduces a generalized model linking diffusivity to resetting rate, uncovering both continuous and discontinuous phase transitions in the system.

Findings

Identified a continuous phase transition between linear and non-linear response regimes.

Discovered a discontinuous phase transition with a sudden change in diffusivity.

Linked the model to physical systems like soft matter under shear.

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 H\'ebraud-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.