On the Theory of the Brownian Motion
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 u-u0exp(-βt) and s-u0β[1-exp(-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 t>>β-1, the formula takes the form of that previously given by Smoluchowski.
- Publication:
-
Physical Review
- Pub Date:
- September 1930
- DOI:
- 10.1103/PhysRev.36.823
- Bibcode:
- 1930PhRv...36..823U