Homogeneous generalized master equation retaining initial correlations
Victor F. Los · Journal of Physics A Mathematical and General · 2001
Using the projection operator technique, the exact homogeneous generalized master equation (HGME) for the relevant part of a distribution function (statistical operator) is derived. The exact (mass) operator governing the evolution of the relevant part of a distribution function and comprising arbitrary initial correlations is found. Neither the Bogolyubov principle of weakening of initial correlations with time nor any other approximation such as random phase approximation has been used to obtain the HGME. These approximations are usually used to derive the approximate homogeneous equation for the relevant part of a distribution function from the conventional exact generalized master equation (GME), which has a source containing the irrelevant part (initial correlations). The HGME does not have a source and contains only the linear, relative to the relevant part of a distribution function, terms of the GME modified by the dynamics of initial correlations. The obtained equation is valid on any timescale, for any initial moment of time and any initial correlations. In particular, it describes the short-time behaviour and allows for treating the influence of initial correlations consistently. As an example, we have considered a dilute gas of classical particles. By selecting the appropriate projection operator, we have derived the homogeneous equation for a one-particle distribution function retaining initial correlations in the linear approximation on the small density parameter and for the space homogeneous case. This equation allows for considering all stages of the time evolution. It converts into the conventional Boltzmann equation on the appropriate timescale if the contribution of all initial correlations vanishes on this timescale.