The paper deals with a diffusing particle that escapes from a cavity to the outer world through a narrow cylindrical tunnel. We derive expressions for the Laplace transforms of the particle survival probability, its lifetime probability density, and the mean lifetime. These results show how the quantities of interest depend on the geometric parameters (the cavity volume and the tunnel length and radius) and the particle diffusion coefficients in the cavity and in the tunnel. Earlier suggested expressions for the mean lifetime, which correspond to different escape scenarios, are contained in our result as special cases. In contrast to these expressions, our formula predicts correct asymptotic behavior of the mean lifetime in the absence of the cavity or tunnel. To test the accuracy of our approximate theory we compare the mean lifetime, the lifetime probability density, and the survival probability (the latter two are obtained by inverting their Laplace transforms numerically) with corresponding quantities found by solving numerically the three-dimensional diffusion equation, assuming that the cavity is a sphere and that the particle has the same diffusion coefficient in the cavity and in the tunnel. Comparison shows excellent agreement between the analytical and numerical results over a broad range of the geometric parameters of the problem.
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28 June 2009
Research Article|
June 30 2009
Escape from cavity through narrow tunnel
Alexander M. Berezhkovskii;
Alexander M. Berezhkovskii
a)
1Mathematical and Statistical Computing Laboratory, Division of Computational Bioscience, Center for Information Technology,
National Institutes of Health
, Bethesda, Maryland 20892, USA
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Alexander V. Barzykin;
Alexander V. Barzykin
2
The Royal Bank of Scotland
, 250 Bishopsgate, London EC2M 4AA, United Kingdom
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Vladimir Yu. Zitserman
Vladimir Yu. Zitserman
3Joint Institute for High Temperatures,
Russian Academy of Sciences
, Izhorskaya 13, Bldg. 2, Moscow 125412, Russia
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a)
Electronic mail: [email protected].
J. Chem. Phys. 130, 245104 (2009)
Article history
Received:
February 11 2009
Accepted:
June 08 2009
Citation
Alexander M. Berezhkovskii, Alexander V. Barzykin, Vladimir Yu. Zitserman; Escape from cavity through narrow tunnel. J. Chem. Phys. 28 June 2009; 130 (24): 245104. https://doi.org/10.1063/1.3160546
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