A generalized Dean–Kawasaki equation for an interacting Brownian gas in a partially absorbing medium
Author(s)
Bressloff, Paul
Type
Journal Article
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.
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.
Date Issued
2024-08
Date Acceptance
2024-07-03
Citation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2024, 480 (2296)
ISSN
1364-5021
Publisher
The Royal Society
Journal / Book Title
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume
480
Issue
2296
Copyright Statement
© 2024 The Author(s). Published by the Royal Society under the terms of the
Creative Commons Attribution License http://creativecommons.org/licenses/
by/4.0/, which permits unrestricted use, provided the original author and
source are credited.
Creative Commons Attribution License http://creativecommons.org/licenses/
by/4.0/, which permits unrestricted use, provided the original author and
source are credited.
License URL
Identifier
https://royalsocietypublishing.org/doi/10.1098/rspa.2023.0915
Publication Status
Published
Article Number
20230915
Date Publish Online
2024-08-28