Survival of a diffusing particle in an expanding cage

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 L0 and expands at a constant rate c. We calculate the joint probability density, p(r, t|r0), 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 r0 from the centre. The asymptotic (t → ∞) probability, Q, obtained by integrating over all final positions, that the particle survives, starting from the centre of the sphere, is given by Q = [4/Γ(ν + 1)λν+1]∑nbnexp [ − (αnν)2/λ], where λ = cL0/D, bn = (αnν)2ν/[Jν+1(αnν)]2, ν = (d − 2)/2 and αnν is the nth positive zero of the Bessel function Jν(z). The cases d = 1 and d = 3 are especially simple, and may be solved elegantly using backward Fokker–Planck methods.