Dynamical Study of Brownian Motion

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We study the motion of a Brownian particle in a fluid from a dynamical point of view, i.e., without the a priori introduction of purely stochastic elements. The Brownian particle is distinguished primarily by having a mass $M$ which is much greater than the mass of the fluid particles $m$. Our method consists of rewriting the Liouville equation for $\ensuremath{\mu}$, the joint distribution of fluid and Brownian particle, as a pair of coupled equations for the distribution function of the Brownian particle $f$ and the conditional distribution of the fluid $P=(\frac{\ensuremath{\mu}}{f})$. The equation for $P$ is then solved formally in a perturbation series in the square root of the mass ratio ($\frac{m}{M}$), which is then substituted in the equation for $f$ to obtain a collision term $\ensuremath{\delta}f$ representing the effect of the fluid on $f$. We consider two situations: (1) A constant external force acts on the Brownian particle and $f$ is stationary, the external force being balanced by $\ensuremath{\delta}f$, and (2) a general time-dependent $f$. We find in both cases, as expected, that to lowest order $\ensuremath{\delta}f$ has the form of a Fokker-Planck type collision term, though in the second case this only holds for times much larger than the fluid relaxation time after an initial time at which $\ensuremath{\mu}$ is arbitrary. The next order terms in $\ensuremath{\delta}f$ differ for the two cases. Furthermore, because of the limitations on the times at which $\ensuremath{\delta}f$ is valid in the second case, $f(t)$ does not really obey a Markoffian equation to this order when the initial state is arbitrary. In the Appendixes we consider the formal structure of $\ensuremath{\delta}f$, the form of $f$ in the stationary case, a quasistochastic model of Brownian motion, the motion of a composite Brownian particle, and the motion of a Brownian particle in a crystal. The latter makes contact with the work of Hemmer and Rubin.