On the derivation of an N-particle analogue of the Fokker-Planck equation

An analogue Fokker-Planck equation also referred to as the Fokker-Planck-Klein-Kramers equation for N interacting Brownian particles in a fluid is derived using the matrix representation of the Liouville equation in the general theory of irreversible processes developed by the Brussels school. It is shown that the equation is valid when the mass and the density of a Brownian particle are large as compared to the mass and density of the fluid. Also, a fast decay of the correlations between the fluid and the Brownian particles as well as of the internal correlations of the fluid has to be assumed. On the other hand, no hypothesis about the correlations among the Brownian particles is needed.