First-encounter time of two diffusing particles in confinement

TL;DR

This paper analyzes how confinement influences the first-encounter time of two diffusing particles in one-dimensional settings, providing exact results for equal diffusivities and exploring long-time behavior for unequal diffusivities.

Contribution

It offers analytical solutions for the survival probability and first-encounter time density in confined one-dimensional systems, highlighting boundary effects and challenges when diffusivities differ.

Findings

Exact results for equal diffusivities in a half-line and interval.

Boundary effects significantly impact diffusion-controlled kinetics.

Long-time behavior analyzed for unequal diffusivities.

Abstract

We investigate how confinement may drastically change both the probability density of the first-encounter time and the related survival probability in the case of two diffusing particles. To obtain analytical insights into this problem, we focus on two one-dimensional settings: a half-line and an interval. We first consider the case with equal particle diffusivities, for which exact results can be obtained for the survival probability and the associated first-encounter time density over the full time domain. We also evaluate the moments of the first-encounter time when they exist. We then turn to the case when the diffusivities are not equal, and focus on the long-time behavior of the survival probability. Our results highlight the great impact of boundary effects in diffusion-controlled kinetics even for simple one-dimensional settings, as well as the difficulty of obtaining analytic results as soon as translational invariance of such systems is broken.