Nonlinear generalized master equations and accounting for initial correlations

By using a time-dependent operator converting a distribution function (statistical operator) of a total system under consideration into the relevant form, new exact nonlinear generalized master equations (GMEs) are derived. The inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and is suitable for obtaining both the linear and nonlinear equations describing the evolution of the (sub)system of interest. Actually, this equation provides an alternative to the BBGKY chain. To include initial correlations into consideration, this inhomogeneous nonlinear GME has been converted into the homogenous form by the method suggested earlier in [9, 10]. Obtained in such a way exact homogeneous nonlinear GME describes all stages of the (sub)system of interest evolution and influence of initial correlations at all stages of the evolution. The initial correlations are treated on the equal footing with collisions and are included in the modified memory kernel acting on the relevant part of a distribution function (statistical operator). No approximation like the Bogoliubov principle of weakening of initial correlations or RPA has been used. In contrast to homogeneous linear GMEs obtained in [9, 10], the homogeneous nonlinear GME is convenient for getting both a linear and nonlinear evolution equations. The obtained nonlinear GMEs have been tested on the space inhomogeneous dilute gas of classical particles. Particularly, a new homogeneous nonlinear equation describing an evolution of a one-particle distribution function at all times and retaining initial correlations has been obtained in the linear in the gas density approximation. This equation is closed in the sense that all two-particle correlations (collisions), including initial ones, which contribute to dissipative and nondissipative characteristics of the nonideal gas are accounted for in the memory kernel. Connection of this equation at the kinetic stage of the evolution to the Vlasov-Landau and Boltzmann equations is discussed. PACS: 05.20.Dd; 05.70.Ln