Non-linear diffusion with stochastic resetting

Abstract

Resetting or restart, when applied to a stochastic process, usually brings its dynamics to a time-independent stationary state. In turn, the optimal resetting rate makes the mean time to reach a target to be the shortest one. These and other intriguing problems have been intensively studied in the case of ordinary diffusive processes over the last decade. In this paper we consider the influence of stochastic resetting on a diffusive motion modeled in terms of the non-linear differential equation. The reason for its non-linearity is the power-law dependence of the diffusion coefficient on the probability density function or, in another context, the concentration of particles. We briefly outline this issue at first to prepare the foundations for our further considerations. Then, we derive an exact formula for the mean squared displacement and demonstrate how it attains the steady-state value under the influence of exponential resetting. This mechanism brings also about that the spatial support of the probability density function, which for the free non-linear diffusion is confined to the set of a finite measure, tends to span the entire domain of real numbers. In addition, we explore the first-passage properties for the non-linear diffusion intermittent by the exponential resetting and find analytical expressions for the mean first-passage time and determine numerically the optimal resetting rate which minimizes the mean time needed for a particle to reach a pre-determined target. Finally, we test and confirm the universal property that the relative fluctuation in the mean first-passage time of optimally restarted non-linear diffusion is equal to unity.

arXiv comment22 pages, 11 figures, The current paper is the essential modification of the previous version. The manuscript captures new results, especially for the mean hitting time where the absorbing boundary condition for the probability density function has been rejected. As we will show in a separate paper, it can not be used in the case of the non-linear diffusion equation

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