Large fluctuations in diffusion-controlled absorption

TL;DR

This paper analyzes the probability distribution of surviving particles in a diffusion process with an absorber, revealing different optimal histories depending on initial conditions and employing macroscopic fluctuation theory.

Contribution

It introduces a detailed analysis of particle survival probabilities and optimal system histories using macroscopic fluctuation theory for diffusion-controlled absorption.

Findings

Probability of zero survivors depends on initial distance and diffusion parameters.

Large initial distances lead to a ballistic absorption scenario.

Different histories dominate for all particles surviving versus none surviving.

Abstract

Suppose that $N_0$ independently diffusing particles, each with diffusivity $D$, are initially released at $x=\ell>0$ on the semi-infinite interval $0\leq x<\infty$ with an absorber at $x=0$. We determine the probability ${\cal P}(N)$ that $N$ particles survive until time $t=T$. We also employ macroscopic fluctuation theory to find the most likely history of the system, conditional on there being exactly $N$ survivors at time $t=T$. Depending on the basic parameter $\ell/\sqrt{4DT}$, very different histories can contribute to the extreme cases of $N=N_0$ (all particles survive) and $N=0$ (no survivors). For large values of $\ell/\sqrt{4DT}$, the leading contribution to ${\cal P}(N=0)$ comes from an effective point-like quasiparticle that contains all the $N_0$ particles and moves ballistically toward the absorber until absorption occurs.