ASYMPTOTIC ANALYSIS OF TARGET FLUXES IN THE THREE-DIMENSIONAL NARROW CAPTURE PROBLEM

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 -> 0, where s is the Laplace variable. We achieve this by considering a triple expansion in epsilon, s, and Lambda similar to 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) -> 1 as s -> 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.