Gradual Diffusive Capture: Slow Death by Many Mosquito Bites

TL;DR

This paper models a diffusing particle repeatedly attacked by another diffusing particle, where each attack reduces the former's mobility until it 'dies', revealing a novel stochastic process with specific displacement and time distributions.

Contribution

It introduces a new model of gradual diffusive capture with decreasing diffusivity, analyzing the resulting displacement and lifetime distributions of the 'man' particle.

Findings

Displacement distribution follows a Cauchy form decaying as x^{-2}.

Time until death distribution decays as t^{-3/2}.

Man's diffusivity reaches zero after multiple attacks.

Abstract

We study the dynamics of a single diffusing particle (a "man") with diffusivity $D_M$ that is attacked by another diffusing particle (a "mosquito") with fixed diffusivity $D_m$. Each time the mosquito meets and bites the man, the diffusivity of the man is reduced by a fixed amount, while the diffusivity of the mosquito is unchanged. The mosquito is also displaced by a small distance $\pm a$ with respect to the man after each encounter. The man is defined as dead when $D_M$ reaches zero. At the moment when the man dies, his probability distribution of displacements $x$ is given by a Cauchy form, which asymptotically decays as $x^{-2}$, while the distribution of times $t$ when the man dies asymptotically decays as $t^{-3/2}$, which has the same form as the one-dimensional first-passage probability.