Survival probability of a particle in a sea of mobile traps: A tale of tails

TL;DR

This paper investigates the long-time decay of a diffusing particle's survival probability amidst subdiffusive traps, deriving asymptotically exact bounds and revealing regimes where traps or the particle appear effectively immobile.

Contribution

It provides asymptotically exact decay laws for survival probability in systems with subdiffusive traps, extending understanding of trapping dynamics in various dimensions.

Findings

For $oldsymbol{oldsymbol{ ext{γ} extless 2/(d+2)}}$, decay matches immobile trap case.

For $oldsymbol{ ext{γ} extgreater 2/(d+2)}$ and $oldsymbol{d extless 3}$, decay matches stationary particle case.

In dimensions greater than 2, the decay law remains ambiguous due to non-coinciding bounds.

Abstract

We study the long-time tails of the survival probability $P(t)$ of an $A$ particle diffusing in $d$-dimensional media in the presence of a concentration $\rho$ of traps $B$ that move sub-diffusively, such that the mean square displacement of each trap grows as $t^{\gamma}$ with $0\leq \gamma \leq 1$. Starting from a continuous time random walk (CTRW) description of the motion of the particle and of the traps, we derive lower and upper bounds for $P(t)$ and show that for $\gamma \leq 2/(d+2)$ these bounds coincide asymptotically, thus determining asymptotically exact results. The asymptotic decay law in this regime is exactly that obtained for immobile traps. This means that for sufficiently subdiffusive traps, the moving $A$ particle sees the traps as essentially immobile, and Lifshitz or trapping tails remain unchanged. For $\gamma > 2/(d+2)$ and $d\leq 2$ the upper and lower bounds again coincide, leading to a decay law equal to that of a stationary particle. Thus, in this regime the moving traps see the particle as essentially immobile. For $d>2$, however, the upper and lower bounds in this $\gamma$ regime no longer coincide and the decay law for the survival probability of the $A$ particle remains ambiguous.