Nonlinear diffusion with stochastic resetting
Abstract
Resetting or restart, when applied to a stochastic process, usually brings its dynamics to a timeindependent stationary state. In turn, the optimal resetting rate makes the mean time to reach a target to be the shortest one. These and other problems have been intensively studied in the case of ordinary diffusive processes during the last decade. In this paper we consider the influence of stochastic resetting on a diffusive motion modeled in terms of the nonlinear differential equation. The reason for its nonlinearity is the powerlaw dependence of the diffusion coefficient on the probability density function or, in another context, the concentration of particles. We first derive an exact formula for the mean squared displacement and show how it attains the steadystate value under the exponential resetting. Then, we analyse the steadystate properties of the probability density function. We also explore the firstpassage properties for the nonlinear diffusion affected by the exponential resetting and find the exact expressions for the survival probability, the mean firstpassage time and the optimal resetting rate, which minimizes the mean time needed for a particle to reach a predetermined target. Finally, we find the universal property that the relative fluctuation in the mean firstpassage time of optimally restarted nonlinear diffusion is always equal to unity.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.14680
 Bibcode:
 2021arXiv210714680C
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 18 pages, 10 figures