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 $\Omega$ with a reflecting boundary $\partial \Omega$, that contains a thin reactive filament. The filament is a tube of cross-sectional diameter $\epsilon \ll 1$ around a smooth closed curve $\Gamma$, and its surface is partially reactive with reactivity $b$. As $\epsilon \to 0$ 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$(-\log [\epsilon d_c])$. This contrasts with the $O({\epsilon }^{-1})$ scaling for spatially localized traps of radius $\epsilon$. Using matched asymptotic expansions, we derive a Fredholm integral equation on $\Gamma$ for the line density of the reactive flux. It sums all powers of $\nu \equiv -1/\log [\epsilon d_c]$ and involves two geometric quantities: the regular part $R$ of the Neumann Green's function, which encodes the interaction of $\Gamma$ with $\partial \Omega$, and an intrinsic self-energy of $\Gamma$ 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.