A Problem in Brownian Motion

Abstract

Gerlach investigated the rotatorial Brownian motion of a small mirror suspended on a fine wire. It follows from the theorem of equipartition that the average square deviation of the mirror will depend on the temperature alone of the surrounding gas. Gerlach verified this for a large range of pressures (1 to ${10}^{\ensuremath{-}6}$ atm). The analogy which we found that exists between this problem and the well-known treatment of the shot effect by Schottky enables us to give a more detailed theory of this phenomenon. If the displacement, registered during a time, long compared with the characteristic period of the mirror, is developed into a Fourier series, we find the square of the amplitude of each Fourier component to be a function of the pressure and molecular weight of the surrounding gas as well as of its temperature, (formula 18). The sum of the squares, however, is a function of the temperature alone (proved in section 4). This explains why the curves registered by Gerlach at different pressures, though all giving the same mean square deviation, are quite different in appearance. To get the fluctuating torque on the mirror, the expression: $\stackrel{-}{{{\ensuremath{\delta}}_{p}}^{2}}=\frac{16}{\ensuremath{\pi}}\ifmmode\cdot\else\textperiodcentered\fi{}\frac{1}{n}\ifmmode\cdot\else\textperiodcentered\fi{}\frac{1}{\overline{c}\ensuremath{\Delta}t\ensuremath{\Delta}o}$ is obtained for the fluctuation in time of the pressure of a gas on the wall (section 5). In this $n$ represents the number of molecules per cc, $\overline{c}$ is the mean velocity and $\ensuremath{\Delta}o$ is the surface of the wall.

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Affiliated Institutions

• University of Michigan US

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