Survival probability of an immobile target surrounded by mobile traps

Abstract

We study analytically, in one dimension, the survival probability $P_{s}(t)$ up to time $t$ of an immobile target surrounded by mutually noninteracting traps each performing a continuous-time random walk (CTRW) in continuous space. We consider a general CTRW with symmetric and continuous (but otherwise arbitrary) jump length distribution $f(η)$ and arbitrary waiting time distribution $ψ(τ)$. The traps are initially distributed uniformly in space with density $ρ$. We prove an exact relation, valid for all time $t$, between $P_s(t)$ and the expected maximum $E[M(t)]$ of the trap process up to time $t$, for rather general stochastic motion $x_{\rm trap}(t)$ of each trap. When $x_{\rm trap}(t)$ represents a general CTRW with arbitrary $f(η)$ and $ψ(τ)$, we are able to compute exactly the first two leading terms in the asymptotic behavior of $E[M(t)]$ for large $t$. This allows us subsequently to compute the precise asymptotic behavior, $P_s(t)\sim a\, \exp[-b\, t^θ]$, for large $t$, with exact expressions for the stretching exponent $θ$ and the constants $a$ and $b$ for arbitrary CTRW. By choosing appropriate $f(η)$ and $ψ(τ)$, we recover the previously known results for diffusive and subdiffusive traps. However, our result is more general and includes, in particular, the superdiffusive traps as well as totally anomalous traps.