Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

Abstract

We apply the generalized method of separation of variables (GMSV) to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (i.e., an arbitrary configuration of non-overlapping partially reactive spherical sinks or obstacles). We consider both exterior and interior problems and all most common boundary conditions: Dirichlet, Neumann, Robin, and conjugate one. Using the translational addition theorems for solid harmonics to switch between the local spherical coordinates, we obtain a semi-analytical expression of the Green function as a linear combination of partial solutions whose coefficients are fixed by boundary conditions. Although the numerical computation of the coefficients involves series truncation and matrix inversion, the use of the solid harmonics as basis functions naturally adapted to the intrinsic symmetries of the problem makes the GMSV particularly efficient, especially for exterior problems. The obtained Green function is the key ingredient to solve boundary value problems and to determine various characteristics of stationary diffusion such as reaction rate, escape probability, harmonic measure, residence time, and mean first passage time, to name but a few. The relevant aspects of the numerical implementation and potential applications in chemical physics, heat transfer, electrostatics, and hydrodynamics are discussed.

Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

“…However, the proposed approach is not limited to the spherical boundary. For instance, the case of a hyperplane was partly studied in [34,101]; apart from straightforward extensions to disks and cylinders, one can consider more complicated catalytic surfaces formed by multiple non-overlapping spheres, for which the Dirichlet propagator in the steadystate regime was recently investigated in [26]. In general, the eigenbasis of the Dirichlet-to-Neumann operator M p can be constructed numerically; since M p is independent of the surface reactivity, this construction has to be performed only once for a given catalytic surface.…”

“…The reactivity is thus related to the probability of reaction event at the encounter [12][13][14]. Robin boundary condition with homogeneous (constant) reactivity κ was often employed to describe many chemical and biochemical reactions and permeation processes [15][16][17][18][19][20][21][22][23][24][25][26], to model stochastic gating [27][28][29][30], or to approximate the effect of microscopic heterogeneities in a random distribution of reactive sites [31][32][33] (see a recent overview in [34]). In spite of its practical importance, diffusion-controlled reactions with heterogeneous surface reactivity κ(s) remain much less studied.…”

Full distribution of first exit times in the narrow escape problem

“…In turn, when the starting point lies outside this solid angle (i.e. q e > ), the distance to the target is larger than q -R r H r p , , ; ( )should decay faster, the contributions in (C. 18) and (C. 19) are thus cancelled, and one needs more refined asymptotic analysis. Without dwelling further on this analysis, we present below probabilistic arguments to get a reasonable approximation.…”

Imperfect Diffusion-Controlled Reactions

“…In turn, the intricate exploration of the partially reactive surface via diffusionmediated jumps in complicated structures such as multiscale porous media or domains with irregular or fractal boundaries, remains poorly understood. In this light, efficient numerical techniques such as fast Monte Carlo methods [82,67,68,83] or semi-analytical solutions [84,85] become particularly important.…”