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A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the high-friction case

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  • Titulaer, U.M.

Abstract

The motion of a Brownian particle in an external field can be described on two levels: by a Fokker-Planck equation for the joint probability distribution of position and velocity, and by a Smoluchowski equation for the distribution in position space only. We derive the second description, with corrections, from the first by means of a systematic expansion procedure of the Chapman-Enskog type in terms of the inverse friction coefficient. We also derive equations describing the initial period, when the Smoluchowski description is not yet valid; in particular we find formulae connecting the initial value to be used for the Smoluchowski equation with that of the full Fokker-Planck equation. The special case of an harmonically bound Brownian particle can be solved exactly; the results are used to check and to illustrate our expressions for general potential.

Suggested Citation

  • Titulaer, U.M., 1978. "A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the high-friction case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 91(3), pages 321-344.
    • Handle: RePEc:eee:phsmap:v:91:y:1978:i:3:p:321-344
    DOI: 10.1016/0378-4371(78)90182-6
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    Cited by:

    1. Subramanian, G. & Brady, J.F., 2004. "A Chapman–Enskog formalism for inertial suspensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 334(3), pages 385-416.
    2. Faure, Dimitri, 2022. "Averaging of semigroups associated to diffusion processes on a simplex," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 323-357.

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