On the Theory of the Brownian Motion

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 $u\ensuremath{-}{u}_{0}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}t)$ and $s\ensuremath{-}\frac{{u}_{0}}{\ensuremath{\beta}[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}t)]}$ where ${u}_{0}$ is the initial velocity and $\ensuremath{\beta}$ 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 ${s}^{2}$ of Ornstein and F\"urth. 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 $\ensuremath{\beta}$ is much larger than the frequency and for values of $t\ensuremath{\gg}{\ensuremath{\beta}}^{\ensuremath{-}1}$, the formula takes the form of that previously given by Smoluchowski.

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