Asymptotic analysis of target fluxes in the three-dimensional narrow capture problem
File(s)
Published version
Author(s)
Bressloff, Paul C
Type
Journal Article
Abstract
We develop an asymptotic analysis of target fluxes in the three-dimensional (3D)
narrow capture problem. The latter concerns a diffusive search process in which the targets are much
smaller than the size of the search domain. The small target assumption allows us to use matched
asymptotic expansions and Green's functions to solve the diffusion equation in Laplace space. In
particular, we derive an asymptotic expansion of the Laplace transformed flux into each target in
powers of the nondimensionalized target size \epsilon . One major advantage of working directly with fluxes
is that one can generate statistical quantities, such as splitting probabilities and conditional first
passage time moments, without having to solve a separate boundary value problem in each case.
However, in order to derive asymptotic expansions of these quantities, it is necessary to eliminate
Green's function singularities that arise in the limit s \rightarrow 0, where s is the Laplace variable. We
achieve this by considering a triple expansion in \epsilon , s, and \Lambda \sim \epsilon /s. This allows us to perform
partial summations over infinite power series in \Lambda , which leads to multiplicative factors of the form
\Lambda n/(1 + \Lambda )n. Since \Lambda n/(1 + \Lambda )n \rightarrow 1 as s \rightarrow 0, the singularities in s are eliminated. We then
show how corresponding asymptotic expansions of the splitting probabilities and conditional mean
first passage times (MFPTs) can be derived in the small-s limit. The resulting expressions agree
with previous asymptotic expansions derived by solving a separate boundary value problem for each
statistical quantity, although each expansion was only carried out to second order in the expansions.
Here we also determine the third order contributions, which are O(\epsilon
2
) and O(\epsilon ) in the cases of the
splitting probabilities and conditional (MFPTs), respectively. Finally, we illustrate the theory by
considering a pair of targets in a spherical search domain, for which the Green's functions can be
calculated explicitly.
narrow capture problem. The latter concerns a diffusive search process in which the targets are much
smaller than the size of the search domain. The small target assumption allows us to use matched
asymptotic expansions and Green's functions to solve the diffusion equation in Laplace space. In
particular, we derive an asymptotic expansion of the Laplace transformed flux into each target in
powers of the nondimensionalized target size \epsilon . One major advantage of working directly with fluxes
is that one can generate statistical quantities, such as splitting probabilities and conditional first
passage time moments, without having to solve a separate boundary value problem in each case.
However, in order to derive asymptotic expansions of these quantities, it is necessary to eliminate
Green's function singularities that arise in the limit s \rightarrow 0, where s is the Laplace variable. We
achieve this by considering a triple expansion in \epsilon , s, and \Lambda \sim \epsilon /s. This allows us to perform
partial summations over infinite power series in \Lambda , which leads to multiplicative factors of the form
\Lambda n/(1 + \Lambda )n. Since \Lambda n/(1 + \Lambda )n \rightarrow 1 as s \rightarrow 0, the singularities in s are eliminated. We then
show how corresponding asymptotic expansions of the splitting probabilities and conditional mean
first passage times (MFPTs) can be derived in the small-s limit. The resulting expressions agree
with previous asymptotic expansions derived by solving a separate boundary value problem for each
statistical quantity, although each expansion was only carried out to second order in the expansions.
Here we also determine the third order contributions, which are O(\epsilon
2
) and O(\epsilon ) in the cases of the
splitting probabilities and conditional (MFPTs), respectively. Finally, we illustrate the theory by
considering a pair of targets in a spherical search domain, for which the Green's functions can be
calculated explicitly.
Date Issued
2021-01
Date Acceptance
2021-02-02
Citation
SIAM: Multiscale Modeling and Simulation, 2021, 19 (2), pp.612-632
ISSN
1540-3459
Publisher
Society for Industrial and Applied Mathematics
Start Page
612
End Page
632
Journal / Book Title
SIAM: Multiscale Modeling and Simulation
Volume
19
Issue
2
Copyright Statement
© 2021 Society for Industrial and Applied Mathematics
Identifier
http://dx.doi.org/10.1137/20m1380326
Publication Status
Published
Date Publish Online
2021-04-07