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 $u-u_0\exp (-\beta t)$ and $s-\frac{u_0}{\beta [1-\exp (-\beta t\left)\right]}$ where $u_0$ is the initial velocity and $\beta$ 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 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 $\beta$ is much larger than the frequency and for values of $t\gg {\beta }^{-1\mathrm{}}$, the formula takes the form of that previously given by Smoluchowski.