Survival of a static target in a gas of diffusing particles with exclusion

TL;DR

This paper uses macroscopic fluctuation theory to analyze the probability that a spherical target remains unhit by particles in a diffusive lattice gas, revealing different behaviors based on system parameters and initial conditions.

Contribution

It provides a detailed analysis of the survival probability of a target in a lattice gas using MFT, including stationary and non-stationary solutions, and extends the approach to interacting particle systems.

Findings

For small R/√(D_0 T), the probability is determined by a stationary MFT solution.

For large R/√(D_0 T), non-stationary solutions dominate, with different scaling behaviors.

Initial conditions influence the survival probability in the non-stationary regime.

Abstract

Let a lattice gas of constant density, described by the symmetric simple exclusion process, be brought in contact with a "target": a spherical absorber of radius $R$. Employing the macroscopic fluctuation theory (MFT), we evaluate the probability ${\mathcal P}(T)$ that no gas particle hits the target until a long but finite time $T$. We also find the most likely gas density history conditional on the non-hitting. The results depend on the dimension of space $d$ and on the rescaled parameter $\ell=R/\sqrt{D_0T}$, where $D_0$ is the gas diffusivity. For small $\ell$ and $d>2$, ${\mathcal P}(T)$ is determined by an exact stationary solution of the MFT equations that we find. For large $\ell$, and for any $\ell$ in one dimension, the relevant MFT solutions are non-stationary. In this case $\ln {\mathcal P}(T)$ scales differently with relevant parameters, and it also depends on whether the initial condition is random or deterministic. The latter effects also occur if the lattice gas is composed of non-interacting random walkers. Finally, we extend the formalism to a whole class of diffusive gases of interacting particles.