Survival probability of an immobile target in a sea of evanescent diffusive or subdiffusive traps: a fractional equation approach

TL;DR

This paper derives a fractional reaction-subdiffusion equation to analyze the survival probability of an immobile target amidst evanescent traps, revealing how anomalous transport and trap decay influence survival across dimensions.

Contribution

The authors develop a dimension-independent fractional equation for target survival with evanescent traps, extending previous one-dimensional models and clarifying the effects of anomalous diffusion and trap decay.

Findings

Finite survival probability with exponential trap decay in all dimensions.

Eternal survival depends on decay and diffusion exponents in power-law trap decay.

Explicit survival probabilities are obtained for different trap decay regimes.

Abstract

We calculate the survival probability of an immobile target surrounded by a sea of uncorrelated diffusive or subdiffusive evanescent traps, i.e., traps that disappear in the course of their motion. Our calculation is based on a fractional reaction-subdiffusion equation derived from a continuous time random walk model of the system. Contrary to an earlier method valid only in one dimension (d=1), the equation is applicable in any Euclidean dimension d and elucidates the interplay between anomalous subdiffusive transport, the irreversible evanescence reaction and the dimension in which both the traps and the target are embedded. Explicit results for the survival probability of the target are obtained for a density \rho(t) of traps which decays (i) exponentially and (ii) as a power law. In the former case, the target has a finite asymptotic survival probability in all integer dimensions, whereas in the latter case there are several regimes where the values of the decay exponent for \rho(t) and the anomalous diffusion exponent of the traps determine whether or not the target has a chance of eternal survival in one, two and three dimensions.