Effect of Inertia on the Brownian Motion of Rigid Particles in a Viscous Fluid

From Boussinesq's work it is known that the frictional resistance of a particle in a viscous, inert fluid, depends on its history. This plays an important part in Brownian motion. The general theory of fluctuations in fluid dynamics leads to explicit expressions for the autocorrelation function G(t) for the random force acting on a Brownian particle and the autocorrelation function φ(t) for its velocity. It is demonstrated that G(t) contains an inertial term which depends on the velocity distribution in the liquid surrounding the particle. The results obtained lead to the correct value of the diffusivity.