Diffusion with Optimal Resetting

TL;DR

This paper investigates how various resetting strategies affect the mean time for a diffusive particle to be absorbed by a target, introducing generalizations like space-dependent resetting rates and random resetting positions, and explores optimal resetting policies.

Contribution

It extends previous models by analyzing generalized resetting mechanisms and identifies optimal resetting strategies for different target distributions.

Findings

Non-resetting windows can reduce absorption time when initial position is far from target.

Optimal resetting distribution undergoes a transition as target distribution narrows.

Introducing space-dependent resetting can optimize search efficiency.

Abstract

We consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate $r$. We consider several generalisations of the model of M. R. Evans and S. N. Majumdar (2011), Diffusion with stochastic resetting, Phys. Rev. Lett. 106, 160601: (i) a space dependent resetting rate $r(x)$ ii) resetting to a random position $z$ drawn from a resetting distribution ${\cal P}(z)$ iii) a spatial distribution for the absorbing target $P_T(x)$. As an example of (i) we show that the introduction of a non-resetting window around the initial position can reduce the mean time to absorption provided that the initial position is sufficiently far from the target. We address the problem of optimal resetting, that is, minimising the mean time to absorption for a given target distribution. For an exponentially decaying target distribution centred at the origin we show that a transition in the optimal resetting distribution occurs as the target distribution narrows.