Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On Brownian motion, Boltzmann's equation, and the Fokker-Planck equation


Authors: Julian Keilson and James E. Storer
Journal: Quart. Appl. Math. 10 (1952), 243-253
MSC: Primary 60.0X
DOI: https://doi.org/10.1090/qam/50216
MathSciNet review: 50216
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Abstract: In order to describe Brownian motion rigorously, Boltzmann's integral equation must be used. The Fokker-Planck type of equation is only an approximation to the Boltzmann equation and its domain of validity is worth examining.

A treatment of the Brownian motion in velocity space of a particle with known initial velocity based on Boltzmann's integral equation is given. The integral equation, which employs a suitable scattering kernel, is solved and its solution compared with that of the corresponding Fokker-Planck equation. It is seen that when $ M/m$, the mass ratio of the particles involved, is sufficiently high and the dispersion of the velocity distribution sufficiently great, the Fokker-Planck equation is an excellent description. Even when the dispersion is small, the first and second moments of the Fokker-Planck solution are reliable. The higher moments, however, are then in considerable error--an error which becomes negligible as the dispersion increases.


References [Enhancements On Off] (What's this?)

  • [1] Ming Chen Wang and G. E. Uhlenbeck, On the theory of the Brownian motion. II, Rev. Modern Phys. 17 (1945), 323–342. MR 0013266, https://doi.org/10.1103/RevModPhys.17.323
  • [2] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
  • [3] Lawson and Uhlenbeck, Threshold signals, (MIT) Radiation Laboratory Series, 24, Chap. III, McGraw-Hill (1950).
  • [4] J. Keilson, The statistical nature of inverse Brownian Motion in velocity space, Technical Report No. 127, Cruft Laboratory, Harvard University, May 10, 1951.

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DOI: https://doi.org/10.1090/qam/50216
Article copyright: © Copyright 1952 American Mathematical Society


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