Inertial Effects in Motion Driven by Hydrodynamic Fluctuations

Abstract

The motion of a bounding surface in a fluid is considered stochastically in the presence of a volume source of thermal fluctuations. The two problems studied are the Brownian motion of a sphere and the thermally excited displacement of a free surface. The stochastic equation of motion for the velocity of the Brownian particle, and for the displacement of the surface, is written as a generalized Langevin equation. The systematic and random forces are identified with the solutions to explicitly defined boundary value problems. It is shown that a fluctuation-dissipation theorem applies. Recent papers which calculate time correlation functions for these problems from the systematic force alone are thus shown to be consistent with the physical origin of the source of fluctuations. The physical similarity of the two problems studied is emphasized. The long time t−3/2 behavior of the velocity correlation function for a Brownian particle is closely related to the modified propagation of surface waves predicted recently by Bouchiat and Meunier. Experimental evidence for the latter phenomenon is discussed briefly.