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Absorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere

DOI: 10.4236/am.2016.77065, PP. 709-720

Keywords: Diffusion Equation, Brownian Diffusion, Asymptotic Solutions, Absorption Rate

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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 \"\".

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