Survival probability of an immobile target surrounded by mobile traps

TL;DR

This paper derives exact asymptotic formulas for the survival probability of an immobile target surrounded by traps performing general continuous-time random walks, encompassing diffusive, subdiffusive, superdiffusive, and anomalous behaviors.

Contribution

It establishes a universal relation between survival probability and maximum trap displacement, providing exact asymptotics for a broad class of CTRWs, including superdiffusive and anomalous traps.

Findings

Derived exact asymptotic form of survival probability for large time.

Unified framework for diffusive, subdiffusive, superdiffusive, and anomalous traps.

Explicit expressions for stretching exponent and constants in survival probability.

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 $f(\eta)$ and arbitrary waiting time distribution $\psi(\tau)$. The traps are initially distributed uniformly in space with density $\rho$. 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 $f(\eta)$ and $\psi(\tau)$, 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, $P_s(t)\sim a\, \exp[-b\, t^{\theta}]$, for large $t$, with exact expressions for the stretching exponent $\theta$ and the constants $a$ and $b$ for arbitrary CTRW. By choosing appropriate $f(\eta)$ and $\psi(\tau)$, 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.