Enhancing search efficiency through diffusive echo

TL;DR

This paper investigates a diffusive process in an annular domain, revealing a 'diffusive echo' phenomenon that can enhance search efficiency by creating a focused flux to a target location.

Contribution

It provides analytical expressions for first passage distributions in annular geometries and identifies the diffusive echo effect, a novel insight into search dynamics.

Findings

Identification of the diffusive echo phenomenon in certain configurations.

Analytic expressions for joint distribution of hitting time and angle.

Potential application to improve search strategies in constrained environments.

Abstract

Despite having been studied for decades, first passage processes remain an active area of research. In this contribution we examine a particle diffusing in an annulus with an inner absorbing boundary and an outer reflective boundary. We obtain analytic expressions for the joint distribution of the hitting time and the hitting angle in two and three dimensions. For certain configurations we observe a 'diffusive echo', i.e. two well-defined maxima in the first passage time distribution to a targeted position on the absorbing boundary. This effect, which results from the interplay between the starting location and the environmental constraints, may help to increase the efficiency of the random search by generating a sustained flux to the targeted position over a short period. Finally, we examine the corresponding one-dimensional system for which there is no well-defined echo.