Equilibration in two chambers connected by a capillary

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Abstract
A problem common to biophysics, chemical engineering, physical chemistry, and physiology relates to a description of the kinetics of particle transport between two or more chambers. In this paper we analyze the case of two chambers connected by a cylindrical capillary. We derive general solutions for the Laplace transforms of the relaxation functions describing the equilibration of particles between the two chambers and the capillary. These solutions show how the relaxation functions depend on geometric parameters (volumes of the two chambers, the length and radius of the capillary) as well as diffusion coefficients in the three compartments. The general solutions are used to analyze the reduction to single-exponential kinetics which describes equilibration of the particles when the capillary is not too long. When all of the diffusion constants are equal we derive simple expressions for the average relaxation times. Brownian dynamics simulations were run to check the accuracy of approximations used to derive the results. We found excellent agreement between the theoretical predictions and numerical results.