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

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 $0\le \gamma \le 1$. 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 $\gamma \le 2/(d+2)$ 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 $d\le 2$ 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.

References in corpus (2)

• The target problem with evanescent subdiffusive traps
• Simulations for trapping reactions with subdiffusive traps and subdiffusive particles

Cited by in corpus (15)

• Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control
• From single-particle stochastic kinetics to macroscopic reaction rates: fastest first-passage time of $N$ random walkers
• Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
• Sample-to-sample fluctuations of power spectrum of a random motion in a periodic Sinai model
• A single predator charging a herd of prey: effects of self volume and predator-prey decision-making
• First-encounter time of two diffusing particles in confinement
• First encounters on Bethe Lattices and Cayley Trees
• Distribution of first-reaction times with target regions on boundaries of shell-like domains
• First passage time and change of entropy
• Passive advection of fractional Brownian motion by random layered flows
• First-encounter time of two diffusing particles in two- and three-dimensional confinement
• A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes
• Transport properties of diffusive particles conditioned to survive in trapping environments
• Survival probability of random walks leaping over traps
• A Monotonicity Result for the Range of a Perturbed Random Walk