Macroscopic fluctuation theory and the absorption of Brownian particles by partially reactive targets

TL;DR

This paper applies macroscopic fluctuation theory to analyze current fluctuations of Brownian particles near partially absorbing targets, deriving equations for optimal paths and fluctuations, and exploring effects of multiple targets and absorption rates.

Contribution

It develops a comprehensive MFT framework for Brownian particles with partially absorbing targets, including multiple targets and boundary conditions, extending previous models.

Findings

Derived MFT equations for single and multiple targets.

Obtained Gaussian distribution for small current fluctuations.

Calculated covariance matrices showing cross correlations in non-interacting particles.

Abstract

We use macroscopic fluctuation theory (MFT) to analyse current fluctuations in a non-interacting Brownian gas with one or more partially absorbing targets within a bounded domain $\Omega \subset \R^d$. We proceed by coarse-graining a generalised Dean-Kawasaki equation with Robin boundary conditions at the target surfaces. The exterior surface $\partial \Omega$ is maintained at a constant density $\owp$. We first derive MFT equations for the optimal noise-induced path for a single target under a saddle-point approximation of the associated path integral action. We then obtain the Gaussian distribution characterising small current fluctuations by linearising the MFT equations about the corresponding deterministic or noise-averaged system and solving the resulting stationary equations. The Robin boundary conditions are handled using the spectrum of a Dirichlet-to-Neumann operator defined on the target surface. We illustrate the theory by considering the finite interval and a circular annulus. In both cases we determine how the variance of the current depends on the rate of absorption $\kappa$. Finally, we extend our analysis to multiple partially absorbing targets. First, we obtain the general result that, in the case of partially absorbing targets ($0<\kappa<\infty$), the covariance matrix for current fluctuations supports cross correlations even in the absence of particle interactions. (These cross-correlations vanish in the totally absorbing limit $\kappa\rightarrow \infty$.) We then explicitly calculate the covariance matrix for circular targets in a 2D domain by assuming that the targets are much smaller than the characteristic size $L$ of the domain $\Omega$ and applying methods from singular perturbation theory.