TITLE:
Absorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere
AUTHORS:
Hongyun Wang, Hong Zhou
KEYWORDS:
Diffusion Equation, Brownian Diffusion, Asymptotic Solutions, Absorption Rate
JOURNAL NAME:
Applied Mathematics,
Vol.7 No.7,
April
28,
2016
ABSTRACT: We study the problem of a diffusing particle confined in a large
sphere in the n-dimensional space being absorbed into a small sphere at the
center. We first non-dimensionalize the problem using the radius of large
confining sphere as the spatial scale and the square of the spatial scale
divided by the diffusion coefficient as the time scale. The non-dimensional
normalized absorption rate is the product of the physical absorption rate and
the time scale. We derive asymptotic expansions for the normalized absorption
rate using the inverse iteration method. The small parameter in the asymptotic
expansions is the ratio of the small sphere radius to the large sphere radius.
In particular, we observe that, to the leading order, the normalized absorption
rate is proportional to the (n － 2)-th power of the small parameter for .