On the escape of the diffusing particle from the cavity

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Abstract

The problem of escape of a Brownian particle from a cylindrical cavity through a hole on the surface of one of the cylinder ends is considered. Using the method of surface homogenization, a one-dimensional description of the process is proposed. The solution obtained with its help allows finding the average lifetime of a particle in such a cavity with any size of the hole. Its qualitative difference from the well-known solution for the mean lifetime of a particle diffusing in an isometric (sphere-like) cavity is that the previously obtained result depends only on the volume of the cavity while the solution found in this work depends both on the volume and on the length of the cylinder.

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About the authors

V. Yu. Zitserman

Joint Institute for High Temperatures, Russian Academy of Sciences

Author for correspondence.
Email: vz1941@mail.ru
Russian Federation, Moscow, 125412

Yu. A. Makhnovskii

A. V. Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences

Email: vz1941@mail.ru
Russian Federation, Moscow, 119991

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2. Fig. 1. Illustration of formula (9). Dependences of the relative contribution of the time to reach the right boundary, x = L, during the lifetime of a particle diffusing in the interval (0, L), on the dimensionless length L/R for different values ​​of the parameter a/R, indicated in the legend to the figure.

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3. Fig. 2. Dependences of the dimensionless lifetime of a particle, formula (7), (solid curves), the second term in this formula (dashed curves) and this term without taking into account the scale factor F(s) (dotted curves) on the dimensionless length L/R for different values ​​of the parameter a/R. The values ​​a/R=0.15 correspond to “thick” curves, and the values ​​a/R=0.3 correspond to “thin” curves.

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