On the Theory of the Brownian Motion

G. E. Uhlenbeck and L. S. Ornstein
Phys. Rev. 36, 823 – Published 1 September 1930
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Abstract

With a method first indicated by Ornstein the mean values of all the powers of the velocity u and the displacement s of a free particle in Brownian motion are calculated. It is shown that uu0exp(βt) and su0β[1exp(βt)] where u0 is the initial velocity and β the friction coefficient divided by the mass of the particle, follow the normal Gaussian distribution law. For s this gives the exact frequency distribution corresponding to the exact formula for s2 of Ornstein and Fürth. Discussion is given of the connection with the Fokker-Planck partial differential equation. By the same method exact expressions are obtained for the square of the deviation of a harmonically bound particle in Brownian motion as a function of the time and the initial deviation. Here the periodic, aperiodic and overdamped cases have to be treated separately. In the last case, when β is much larger than the frequency and for values of tβ1, the formula takes the form of that previously given by Smoluchowski.

  • Received 7 July 1930

DOI:https://doi.org/10.1103/PhysRev.36.823

©1930 American Physical Society

Authors & Affiliations

G. E. Uhlenbeck and L. S. Ornstein

  • University of Michigan, Ann Arbor and Physisch Laboratorium der R. U. Utrecht, Holland

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Issue

Vol. 36, Iss. 5 — September 1930

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