Nonlinear generalized master equations and accounting for initial correlations

TL;DR

This paper derives new exact nonlinear generalized master equations that incorporate initial correlations and describe the full evolution of a subsystem, tested on dilute gases and connecting to classical kinetic equations.

Contribution

It introduces a homogeneous nonlinear GME framework that accounts for initial correlations at all evolution stages, extending previous linear models.

Findings

Derived a new homogeneous nonlinear GME including initial correlations.

Applied the equation to dilute gases, capturing all two-particle correlations.

Connected the new kinetic equation to classical Boltzmann and Vlasov-Landau equations.

Abstract

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 evolution equations. 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 homogeneous nonlinear GME describes all stages of the (sub)system of interest evolution and influence of initial correlations at all stages thereof. 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. Connection of this equation at the kinetic stage of the evolution to the Vlasov-Landau and Boltzmann equations is discussed.