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Derivation of Smoluchowski equations with corrections for Fokker-Planck and BGK collision models

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  • Skinner, James L.
  • Wolynes, Peter G.

Abstract

The time evolution of the phase space distribution function for a classical particle in contact with a heat bath and in an external force field can be described by a kinetic equation. From this starting point, for either Fokker-Planck or BGK (Bhatnagar-Gross-Krook) collision models, we derive, with a projection operator technique, Smoluchowski equations for the configuration space density with corrections in reciprocal powers of the friction constant. For the Fokker-Planck model our results in Laplace space agree with Brinkman, and in the time domain, with Wilemski and Titulaer. For the BGK model, we find that the leading term is the familiar Smoluchowski equation, but the first correction term differs from the Fokker-Planck case primarily by the inclusion of a fourth order space derivative or super Burnett term. Finally, from the corrected Smoluchowski equations for both collision models, in the spirit of Kramers, we calculate the escape rate over a barrier to fifth order in the reciprocal friction constant, for a particle initially in a potential well.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:96:y:1979:i:3:p:561-572
    DOI: 10.1016/0378-4371(79)90013-X
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    Citations

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    Cited by:

    1. Widder, M.E. & Titulaer, U.M., 1989. "Brownian motion in a medium with inhomogeneous temperature," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 154(3), pages 452-466.
    2. Titulaer, U.M., 1980. "Corrections to the Smoluchowski equation in the presence of hydrodynamic interactions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 100(2), pages 251-265.
    3. Stewart, Glen R., 1982. "Long-time behavior of a non-Markovian Brownian oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 115(3), pages 519-530.
    4. Marchesoni, Fabio & Grigolini, Paolo, 1983. "The Kramers model of chemical relaxation in the presence of a radiation field," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 121(1), pages 269-285.
    5. Vollmer, H.D. & Risken, H., 1982. "Eigenvalues and eigenfunctions of the Kramers equation. Application to the Brownian motion of a pendulum," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 110(1), pages 106-127.
    6. Janssen, J.A.M., 1988. "Solution of Kramers' problem," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 152(1), pages 145-176.
    7. Bunde, Armin & Gouyet, Jean-François, 1985. "Brownian motion in the bistable potential at intermediate and high friction: Relaxation from the instability point," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 132(2), pages 357-374.
    8. Felderhof, B.U., 1983. "Reciprocity in electrohydrodynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 122(3), pages 383-396.

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