Survival Probability of a Ballistic Tracer Particle in the Presence of Diffusing Traps

TL;DR

This paper derives exact and asymptotic expressions for the survival probability of a ballistic tracer particle amidst diffusing traps in various dimensions, revealing exponential decay characterized by explicit decay exponents.

Contribution

It provides the first exact formula for survival probability in all times for d<2 and asymptotic behavior for d≥2, including explicit decay exponents in key dimensions.

Findings

Survival probability decays exponentially at large times.

Explicit decay exponents are derived for dimensions d≤2 and d=3.

Exact solutions are obtained for all times in d<2.

Abstract

We calculate the survival probability P_S(t) up to time t of a tracer particle moving along a deterministic trajectory in a continuous d-dimensional space in the presence of diffusing but mutually noninteracting traps. In particular, for a tracer particle moving ballistically with a constant velocity c, we obtain an exact expression for P_S(t), valid for all t, for d<2. For d \geq 2, we obtain the leading asymptotic behavior of P_S(t) for large t. In all cases, P_S(t) decays exponentially for large t, P_S(t) \sim \exp(-\theta t). We provide an explicit exact expression for the exponent \theta in dimensions d \leq 2, and for the physically relevant case, d=3, as a function of the system parameters.