Hierarchy Bloch Equations for the Reduced Statistical Density Operators in Canonical and Grand canonical Ensembles

TL;DR

This paper derives hierarchy equations for reduced statistical density operators in canonical and grand canonical ensembles, providing a framework to study temperature dependence in many-body quantum systems.

Contribution

It introduces a new set of hierarchy equations for reduced density operators in both ensembles, extending Bloch equations to interacting many-body systems.

Findings

Derived hierarchy equations for reduced density operators.

Established temperature-dependent laws for reduced density operators.

Discussed potential solutions and applications of the equations.

Abstract

Starting from Bloch equation for a canonical ensemble, we deduce a set of hierarchy equations for the reduced statistical density operator for an identical many-body system with two-body interaction. They provide a law according to which the reduced density operator varies in temperature. By definition of the reduced density operator in Fock space for a grand canonical ensemble, we also obtain the analogous Bloch equation and the corresponding hierarchy reduced equations for the identical interacting many-body system. We discuss their possible solutions and applications.