Quantum-Mechanical, Microscopic Brownian Motion

Recently, the Fokker-Planck (F-P) equation, satisfied by the phase-space distribution function of a Brownian (B) particle, has been derived from the Liouville equation, satisfied by the phase-space distribution function of the composite system, B particle plus fluid. We have started from a strictly quantum-mechanical formulation of the problem, using a density-matrix description of the composite system, and have shown that in an appropriate classical limit the F-P equation results. To lowest order in $\hslash$, our technique gives the same expression for the friction coefficient obtained earlier from the Liouville equation. In principle it yields corrections to the F-P equation to all orders in $\hslash$. We have shown that for a uniform fluid, the quantum corrections to the F-P equation to first order in $\hslash$ appear only as a shift in the friction coefficient. If the classical friction coefficient ${\zeta }_0$ is replaced by ${\zeta }_0+\hslash {\zeta }_1$, with an appropriate ${\zeta }_1$, then the F-P equation is given correctly to order $\hslash$. There is a difference in the mass dependence of ${\zeta }_0$ and ${\zeta }_1$ which suggests the existence of an isotope effect in the mobility of a particle moving through a quantum fluid. The technique employed in the derivation is to consider the expectation value of the density matrix of the system in a state in which the position and momentum of the particles are known with maximum accuracy (Kennard packet). This expectation value $\mu$ is an appropriate, positive, semiclassical phase-space distribution function. An appropriate equation is derived for $\mu$, which to lowest order in $\hslash$ is the Liouville equation. The remainder of the derivation is similar to the derivation in the classical case (a suitable projection technique is used).