Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps

TL;DR

This paper analyzes the survival probability of a diffusing particle amidst mobile traps in various dimensions, deriving asymptotic formulas, bounds, and validating them with simulations, revealing dimension-dependent behaviors and correction effects.

Contribution

It provides new asymptotic expressions and bounds for the survival probability in different dimensions, including a novel simulation algorithm and analysis of asymmetric trap densities.

Findings

For 1<=d<2, Q(t) ~ exp(-λ_d t^{d/2}) with explicit λ_d.

For d=2, Q(t) ~ exp(-4πρ D t/ln t).

Simulations show large deviations from asymptotic predictions, indicating slow correction decay.

Abstract

The problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d. We consider the case where the traps are initially randomly distributed in space, with uniform density rho, and derive upper and lower bounds for the probability Q(t) (averaged over all particle and trap trajectories) that the particle survives up to time t. We show that, for 1<=d<2, the bounds converge asymptotically to give $Q(t) \sim exp(-\lambda_d t^{d/2})$ where $\lambda_d = (2/\pi d) sin(\pi d/2) (4\pi D)^{d/2} \rho$ and D is the diffusion constant of the traps, and that $Q(t) \sim exp(- 4\pi\rho D t/ln t)$ for d=2. For d>2 bounds can still be derived, but they no longer converge for large t. For 1<=d<=2, these asymptotic form are independent of the diffusion constant of the particle. The results are compared with simulation results obtained using a new algorithm [V. Mehra and P. Grassberger, Phys. Rev. E v65 050101 (2002)] which is described in detail. Deviations from the predicted asymptotic forms are found to be large even for very small values of Q(t), indicating slowly decaying corrections whose form is consistent with the bounds. We also present results in d=1 for the case where the trap densities on either side of the particle are different. For this case we can still obtain exact bounds but they no longer converge.