Macroscopic fluctuation theory and the absorption of Brownian particles by partially reactive targets
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918106460323840 |
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| author | Bressloff, Paul C |
| author_facet | Bressloff, Paul C |
| contents | 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 $Ω\subset \R^d$. We proceed by coarse-graining a generalised Dean-Kawasaki equation with Robin boundary conditions at the target surfaces. The exterior surface $\partial Ω$ 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 $κ$. Finally, we extend our analysis to multiple partially absorbing targets. First, we obtain the general result that, in the case of partially absorbing targets ($0<κ<\infty$), 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 $κ\rightarrow \infty$.) 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 $Ω$ and applying methods from singular perturbation theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Macroscopic fluctuation theory and the absorption of Brownian particles by partially reactive targets Bressloff, Paul C Statistical Mechanics 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 $Ω\subset \R^d$. We proceed by coarse-graining a generalised Dean-Kawasaki equation with Robin boundary conditions at the target surfaces. The exterior surface $\partial Ω$ 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 $κ$. Finally, we extend our analysis to multiple partially absorbing targets. First, we obtain the general result that, in the case of partially absorbing targets ($0<κ<\infty$), 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 $κ\rightarrow \infty$.) 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 $Ω$ and applying methods from singular perturbation theory. |
| title | Macroscopic fluctuation theory and the absorption of Brownian particles by partially reactive targets |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2507.19833 |