Statistics of diffusive encounters with a small target: Three complementary approaches

TL;DR

This paper develops three methods to approximate the encounter statistics between diffusing particles and small targets within bounded domains, providing explicit formulas based on geometric features and analyzing their accuracy and applications.

Contribution

It introduces three complementary approaches for approximating encounter statistics, including a simple explicit formula depending on geometric characteristics, advancing understanding of diffusive search problems.

Findings

Derived a fully explicit approximation based on geometric features.

Compared the accuracy and limitations of three approaches.

Provided an explicit approximation for the first-crossing time distribution.

Abstract

Diffusive search for a static target is a common problem in statistical physics with numerous applications in chemistry and biology. We look at this problem from a different perspective and investigate the statistics of encounters between the diffusing particle and the target. While an exact solution of this problem was recently derived in the form of a spectral expansion over the eigenbasis of the Dirichlet-to-Neumann operator, the latter is generally difficult to access for an arbitrary target. In this paper, we present three complementary approaches to approximate the probability density of the rescaled number of encounters with a small target in a bounded confining domain. In particular, we derive a simple fully explicit approximation, which depends only on a few geometric characteristics such as the surface area and the harmonic capacity of the target, and the volume of the confining domain. We discuss the advantages and limitations of three approaches and check their accuracy. We also deduce an explicit approximation for the distribution of the first-crossing time, at which the number of encounters exceeds a prescribed threshold. Its relations to common first-passage time problems are discussed.