Exact Asymptotics for One-Dimensional Diffusion with Mobile Traps

TL;DR

This paper derives the precise long-time decay behavior of a diffusing particle's survival probability in a one-dimensional environment filled with mobile traps, revealing independence from the particle's diffusion constant.

Contribution

It provides the exact asymptotic form of the survival probability and first-order corrections for finite traps, advancing understanding of diffusion in trap-laden environments.

Findings

Survival probability decays as exp[-4 rho(Dt/pi)^{1/2}] asymptotically.

Asymptotic behavior is independent of the particle's diffusion constant D'.

Exact first-order results are obtained for finite numbers of traps.

Abstract

We consider a diffusing particle, with diffusion constant D', moving in one dimension in an infinite sea of noninteracting mobile traps with diffusion constant D and density rho. We show that the asymptotic behavior of the survival probability, P(t), is given by P(t) ~ exp[-4 rho(Dt/pi)^{1/2}], independent of D'. The result comes from obtaining upper and lower bounds on P(t), and showing that they coincide asymptotically. We also obtain exact results for P(t) to first order in D' for an arbitrary finite number of traps.