A generalized Dean–Kawasaki equation for an interacting Brownian gas in a partially absorbing medium

Abstract

The Dean–Kawasaki (DK) equation is a stochastic partial differential equation (SPDE) for the global density ρ = N −1 ∑ j = 1 N δ(x − Xj (t)) of a gas of N overdamped Brownian particles, where Xj(t) is the position of the jth particle. In the thermodynamic limit N → ∞ with weak pairwise interactions, the expectation ⟨ρ⟩ with respect to the white noise processes converges in distribution to the solution of a McKean–Vlasov (MV) equation. In this article, we use an encounter-based approach to derive a generalized DK equation for an interacting Brownian gas on the half-line with a partially absorbing boundary at x = 0. Each particle is independently absorbed when its local time Lj(t) at x = 0 exceeds a random threshold ℓj . The global density is now summed over the set of particles that have not yet been absorbed, and expectations are taken with respect to the Gaussian noise and the random thresholds ℓj . Assuming the DK equation has a well-defined mean-field limit, we derive the corresponding MV equation on the half-line. We illustrate the theory by (i) analysing stationary solutions for a Curie–Weiss (quadratic) interaction potential and a totally reflecting boundary; and (ii) calculating the effective rate of particle loss in the weak absorption limit. Extensions to finite intervals and partially absorbing traps are also considered.