Pressure fluctuations, viscosity, and Brownian motion

Frank Munley

Published 2020-01-02Version 1

Brownian motion occurs in a variety of fluids, from rare gases to liquids. The Langevin equation, describing friction and agitation forces in statistical balance, is one of the most successful ways to treat the phenomenon. In relatively dense fluids, such as water and air at standard temperature and pressure, friction is normally treated as a mesoscopic effect arising from the coordinated viscous action of fluid surrounding the particle. It is often assumed that the friction in denser fluids obeys Stokes' law. The appropriateness of this assumption involves a discussion of recent experimental research in the ballistic or "coasting" phase of motion occurring at a very short time scale. Given the mesoscopic nature of the friction force for relatively dense fluids, we should expect the agitation force to also be mesoscopic. It has been suggested occasionally that pressure fluctuations with a well-defined minimum time scale are an appropriate mesoscopic agitation force for denser fluids. The purpose of this paper is to do that in the simplest possible way. To accomplish the goal, the simple random walk will be used to approximate the time and space scales below which ballistic motion begins and diffusive motion ends. Following that, pressure fluctuations and the associated time scale will be introduced and shown to be consistent with the fluctuation-dissipation theorem describing the statistical balance between agitation and friction. As successful as the pressure fluctuation model is, it fails for fluids like glycerin that have viscosities a thousand times and more that of water. A simple phenomenological model of collective fluid motion will be presented to explain Brownian motion in such fluids.