Cumulant expansion of a Fokker–Planck equation: Rotational and translational motion in dense fluids

Abstract

A frequency generalized cumulant method is applied to the rotational (R) and translational (T) Fokker–Planck equation derived by Hwang and Freed. The lowest order nonvanishing cumulant is second order in the streaming part of the Liouville operator and generates the R–T coupled Smoluchowski equation. The structure of the fourth and sixth order cumulants for uncoupled R–T motion of a spherical top allows an approximate summation of the cumulant series which is accuate for dense fluids. The orientational time correlation functions for lth rank spherical harmonics calculated using the summed cumulants are in agreement with the high density Fixman and Rider solutions to the rotational Fokker–Planck equation. The zero frequency part of the orientational correlation functions gives an improved Hubbard relation relating the orientational and angular momentum correlation times. For the translational case, the summed cumulants lead to a wave vector and frequency dependent single partice diffusion constant. The effects of R–T coupling on the single particle orientational correlation functions are calculated using the Happel and Brenner model for the hydrodynamic part of the torque–force correlation functions. The effects are concluded to be small for a Brownian particle, even if the particle has strongly coupled orientational and translational modes.