Homogeneous and nonlinear generalized master equations accounting for initial correlations

TL;DR

This paper develops an exact homogeneous nonlinear generalized master equation that incorporates initial correlations in many-particle systems, extending previous linear approaches and enabling accurate modeling of system dynamics across all stages.

Contribution

It introduces a new method to derive exact nonlinear GMEs that account for initial correlations without time-scale restrictions, advancing the theoretical framework for many-particle dynamics.

Findings

Derived exact homogeneous nonlinear GME including initial correlations.

Established a method to convert inhomogeneous equations into homogeneous form.

Provided a framework for modeling all evolution stages of many-particle systems.

Abstract

To take initial correlations into account, a method, based on the time-independent projection operator technique, that allows converting the conventional linear inhomogeneous (containing a source caused by initial correlations) time-convolution generalized master equation (TC-GME) and time-convolutionless GME (TCL-GME) into the homogeneous form exactly, is proposed. This approach results in the exact linear time-convolution and time-convolutionless homogeneous generalized master equations (TC-HGME and TCL-HGME) which take the dynamics of initial correlations into account via modified memory kernels governing the evolution of the relevant part of a distribution function of a many-particle system. However, to derive the desired nonlinear equations (the Boltzmann equation in particular) from this actually linear equation, we should make an additional approximation neglecting the time-retardation of a one-particle distribution function which restricts the time scale by times much smaller than the relaxation time for a one-particle distribution function. To obtain the actually nonlinear evolution equations and avoid any restrictions on the time scales, we develop a new method, based on using a time-dependent operator converting a distribution function of a total system into the relevant form, that allows deriving the new exact nonlinear generalized master equations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogeneous form by the method which we used for conventional linear GMEs. The obtained exact homogeneous nonlinear GME describes all evolution stages of the system of interest and treats initial correlations on an equal footing with collisions by means of the modified memory kernel.