Microscopic Theory of Brownian Motion in an Oscillating Field; Connection with Macroscopic Theory

Recently, Lebowitz and Rubin, and Résibois and Davis, showed that the Fokker-Planck equation for the distribution function of a Brownian particle ($B$ particle) of mass $M$, in a fluid of particles of mass $m$, may be derived directly from the Liouville equation for the joint distribution of fluid and $B$ particle. It is the lowest order term in a ${\left(\frac{m}{M}\right)}^{\frac{1}{2}}$ expansion of the effect of the fluid on the distribution of the $B$ particle. These authors studied in particular the steady-state distribution function of $B$ particles acted on by a small constant external field E, which results from a balance between the effects of the driving force and those of the fluid. In this paper we extend these studies to the case where the $B$ particle is acted on by a time-dependent field $e^{i\omega t}$. We find that the effect of the fluid on the distribution function of the $B$ particle is again given, to lowest order in ${\left(\frac{m}{M}\right)}^{\frac{1}{2}}$, by a Fokker-Planck term, albeit one with a frequency-dependent friction constant, $M\zeta \left(\omega \right)\sim \int 0^{\infty }〈\mathcal{F}\mathrm{}\left(0\right)\mathrm{}·\mathcal{F}\mathrm{}\left(t\right)\mathrm{}〉e^{i\omega t}\mathrm{dt}$. Here $\mathcal{F}$ is the microscopic, $N$-body force acting on a stationary $B$ particle and the average is over the equilibrium distribution function of the fluid in the presence of this fixed $B$ particle. We further show that $-M\zeta \left(\omega \right){\mathrm{V}}_0{e}^{i\omega t}$ is equal to the force acting on a $B$ particle moving through the fluid with a prescribed small velocity ${\mathrm{V}}_0{e}^{i\omega t}$, Under appropriate circumstances this latter force may be computed from kinetic theory or from hydrodynamics. We thus have complete agreement between our microscopic theory and that obtained from stochastic considerations. We also clarify the relation between the different formalisms used by Lebowitz and Rubin and by Résibois and Davis.