The Mean First-Reaction Time on a Thin Filament in a 3-D Domain
Abstract
We study the mean first-reaction time (MFRT) of a Brownian particle in a bounded three-dimensional domain Ω with a reflecting boundary ∂Ω, that contains a thin reactive filament. The filament is a tube of cross-sectional diameter ε≪ 1 around a smooth closed curve Γ, and its surface is partially reactive with reactivity b. As ε→ 0 the filament reduces to a line source in the outer problem. Its cross-section enters only through a logarithmic reactive capacitance dc(b), so that the MFRT is 𝒪(−log[ε dc]). This contrasts with the 𝒪(eps−1) scaling for spatially localized traps of radius eps. Using matched asymptotic expansions, we derive a Fredholm integral equation on Γ for the line density of the reactive flux. It sums all powers of ν≡ −1/log[ε dc] and involves two geometric quantities: the regular part R of the Neumann Green's function, which encodes the interaction of Γ with ∂Ω, and an intrinsic self-energy of Γ that depends only on its shape. The framework extends to multiple filaments, splitting probabilities, a spatially heterogeneous diffusivity, and the principal eigenvalue of the Laplacian. For the unit sphere we solve the integral equation with a simple trapezoidal Nyström scheme, and we validate the predictions for rings, linked rings and torus knots against full 3-D finite-element solutions.
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BibTeX
@misc{grebenkov2026mean,
title = {{The Mean First-Reaction Time on a Thin Filament in a 3-D Domain}},
author = {Denis S. Grebenkov and Theodore Kolokolnikov and Alan E. Lindsay and Michael J. Ward},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38715},
eprint = {2610.38715},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38715},
}
RIS
TY - UNPB
TI - The Mean First-Reaction Time on a Thin Filament in a 3-D Domain
AU - Denis S. Grebenkov
AU - Theodore Kolokolnikov
AU - Alan E. Lindsay
AU - Michael J. Ward
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38715
UR - https://arxiv.org/abs/2610.38715
AB - We study the mean first-reaction time (MFRT) of a Brownian particle in a bounded three-dimensional domain Ω with a reflecting boundary ∂Ω, that contains a thin reactive filament. The filament is a tube of cross-sectional diameter ε≪ 1 around a smooth closed curve Γ, and its surface is partially reactive with reactivity b. As ε→ 0 the filament reduces to a line source in the outer problem. Its cross-section enters only through a logarithmic reactive capacitance d_c(b), so that the MFRT is 𝒪(−log[ε d_c]). This contrasts with the 𝒪(eps^−1) scaling for spatially localized traps of radius eps. Using matched asymptotic expansions, we derive a Fredholm integral equation on Γ for the line density of the reactive flux. It sums all powers of ν≡ −1/log[ε d_c] and involves two geometric quantities: the regular part R of the Neumann Green's function, which encodes the interaction of Γ with ∂Ω, and an intrinsic self-energy of Γ that depends only on its shape. The framework extends to multiple filaments, splitting probabilities, a spatially heterogeneous diffusivity, and the principal eigenvalue of the Laplacian. For the unit sphere we solve the integral equation with a simple trapezoidal Nyström scheme, and we validate the predictions for rings, linked rings and torus knots against full 3-D finite-element solutions.
ER -