Optimal potentials for diffusive search strategies

TL;DR

This paper finds optimal external potentials for one-dimensional diffusive search strategies to minimize mean first passage time, relating the results to entropy measures and comparing with resetting models.

Contribution

It introduces a method to determine the optimal potential minimizing search time for arbitrary target distributions, linking it to Rényi entropy and comparing with resetting strategies.

Findings

Optimal potential minimizes MFPT for given target distribution.

Minimal MFPT relates to cumulative Rényi entropy.

Comparison shows differences with resetting-based search models.

Abstract

We consider one dimensional diffusive search strategies subjected to external potentials. The location of a single target is drawn from a given probability density function (PDF) $f_G(x)$ and is fixed for each stochastic realization of the process. We optimize the quality of the search strategy as measured by the mean first passage time (MFPT) to the position of the target. For a symmetric but otherwise arbitrary distribution $f_G(x)$ we find the optimal potential that minimizes the MFPT. The minimal MFPT is given by a nonstandard measure of the dispersion, which can be related to the cumulative R\'enyi entropy. We compare optimal times in this model with optimal times obtained for the model of diffusion with stochastic resetting, in which the diffusive motion is interrupted by intermittent jumps (resets) to the initial position. Additionally, we discuss an analogy between our results and a so-called square-root principle.