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Solution of Kramers' problem

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  • Janssen, J.A.M.

Abstract

The escape of a Brownian particle across a potential barrier was introduced by Kramers. He derived expressions for the escape or transition rate in the limits of large and small damping. Several expressions that are supposed to be valid for the whole damping regime were put forward some years ago. It seemed that the solution of Matkowsky, Schuss and Tier settled the bridging issue. In this article, however, it will be argued why the mean-first-passage-time approach followed by Matkowsky et al. is not suited to formulate Kramers' problem. An alternative treatment is given based on the Kramers equation which is solved separately in two regions of phase space (corresponding to well and barrier). The escape rate follows from the matching conditions of both solutions. The new expressions resembles the one of Matkowsky et al., but is simpler, and clarifies the connection with unimolecular chemical reactions.

Suggested Citation

  • Janssen, J.A.M., 1988. "Solution of Kramers' problem," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 152(1), pages 145-176.
    • Handle: RePEc:eee:phsmap:v:152:y:1988:i:1:p:145-176
    DOI: 10.1016/0378-4371(88)90069-6
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    References listed on IDEAS

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    1. Van Kampen, N.G. & Oppenheim, I., 1986. "Brownian motion as a problem of eliminating fast variables," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 138(1), pages 231-248.
    2. Skinner, James L. & Wolynes, Peter G., 1979. "Derivation of Smoluchowski equations with corrections for Fokker-Planck and BGK collision models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 96(3), pages 561-572.
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