Modified Fokker-Planck equations

Abstract

The exact Fokker-Planck-like equation of motion for the singlet phase space density derived by Lebowitz and Resibois has been investigated with the objective of improving on the usual replacement of the exact translational collision operator by the Fokker-Planck operator. The exact collision operator is expressed in terms of a projected force correlation operator which acts on the position and momentum of the tagged particle. The lowest order operator dependence of this correlation function is exposed, resulting in an improved FP collision operator that contains third and fourth-order momentum derivatives and a streaming term, which, roughly speaking, allows free translation during the correlation time of the fluctuating forces. The improved FP operator is used to obtain a short time expansion (up to t ⁴) for the velocity correlation function. Through a cumulant expansion technique, the K-space singlet density is also obtained and a criteria, based on force correlation functions, is devised for the existence of the super Burnett coefficient.