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 Ω ⊂ Rd. We proceed by coarse-graining a generalised Dean–Kawasaki equation with Robin boundary conditions at the target
surfaces. The exterior surface ∂Ω is maintained at a constant density ℘ by contact with a particle bath. We first derive Euler–Lagrange or 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 two simple geometric configurations, namely, the finite interval and a circular annulus. In both cases we determine how the variance of the current depends on the rate of absorption (reactivity constant) κ. 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 < κ < ∞), 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 κ → ∞.) 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 Ω and applying methods from singular perturbation theory. This yields a non-perturbative expression for the covariance matrix with respect to the small parameter ν = −lnϵ, assuming that the target radii are O(ϵL) with 0 < ϵ ≪ 1. The leading order contribution is a diagonal matrix whose entries
are consistent with the result for a circular annulus, whereas higher-order terms generate cross-correlations.