Survival of a diffusing particle in an expanding cage

TL;DR

This paper analyzes the survival probability of a diffusing particle within an expanding spherical boundary, deriving the joint probability density of its position and survival time in a d-dimensional setting.

Contribution

It provides a novel analytical calculation of the joint probability density for a Brownian particle in an expanding absorbing sphere.

Findings

Derived explicit formulas for survival probabilities.

Extended analysis to d-dimensional spheres.

Applicable to diffusion in expanding domains.

Abstract

We consider a Brownian particle, with diffusion constant D, moving inside an expanding d-dimensional sphere whose surface is an absorbing boundary for the particle. The sphere has initial radius L_0 and expands at a constant rate c. We calculate the joint probability density, p(r,t|r_0), that the particle survives until time t, and is at a distance r from the centre of the sphere, given that it started at a distance r_0 from the centre.