Time-dependent projection operator and nonlinear generalized master equations

By introducing a special time-dependent projection operator P(t), the nonlinear version of the Nakajima-Zwanzig inhomogeneous generalized master equation (GME) for the relevant part of the N-particle distribution function containing the irrelevant initial correlation term is derived. For this projection operator the relevant part of the N-particle distribution function is a product of the distribution function of a group of s (s≤N) selected particles F_{s}(t) and the distribution function F_{N-s}(t) of N-s particles constituting the "environment" for a group of s selected particles. Thus, the linear projection operator approach results in the exact nonlinear GME for the relevant part of the distribution function, which is equivalent to the exact nonlinear GME for the distribution function of the complex of s particles of interest. This equation is further specified in the first approximation in the particle density, and then considered for one-particle and two-particle distribution functions. Given that there is no satisfactory way of dealing with the initial correlation term, the approach is suggested to rigorously include this term into consideration by converting the obtained inhomogeneous GME into homogeneous form containing initial correlations in the kernel governing the evolution of the relevant part of the distribution function. The homogeneous nonlinear GME for a one-particle distribution function is considered and the conditions for its equivalence to the nonlinear Boltzmann equation are discussed.