Quantum master equation for the microcanonical ensemble

TL;DR

This paper derives a quantum master equation for a subsystem interacting with a finite heat capacity environment, capturing microcanonical ensemble dynamics and reducing to the Redfield equation for infinite heat capacity.

Contribution

It provides a new derivation of the quantum master equation for microcanonical environments and analyzes its properties, including energy conservation and equilibrium states.

Findings

Equation describes subsystem dynamics with finite heat capacity environments.

Subsystem tends towards an equipartition equilibrium within the energy shell.

Redfield equation is recovered for infinite heat capacity environments.

Abstract

By using projection superoperators, we present a new derivation of the quantum master equation first obtained by the Authors in Phys. Rev. E {\bf 68}, 066112 (2003). We show that this equation describes the dynamics of a subsystem weakly interacting with an environment of finite heat capacity and initially described by a microcanonical distribution. After applying the rotating wave approximation to the equation, we show that the subsystem dynamics preserves the energy of the total system (subsystem plus environment) and tends towards an equilibrium state which corresponds to equipartition inside the energy shell of the total system. For infinite heat capacity environments, this equation reduces to the Redfield master equation for a subsystem interacting with a thermostat. These results should be of particular interest to describe relaxation and decoherence in nanosystems where the environment can have a finite number of degrees of freedom and the equivalence between the microcanonical and the canonical ensembles is thus not always guaranteed.