Dissipative Time Evolution of Observables in Non-equilibrium Statistical Quantum Systems

TL;DR

This paper investigates the time evolution of observables in non-equilibrium quantum systems, proposing a non-perturbative resummation method to improve the accuracy of dynamical descriptions over long times.

Contribution

It introduces a non-perturbative resummation technique for quadratic operator correlators, ensuring energy conservation and accurate long-time dynamics in dissipative quantum systems.

Findings

Naive second order approximation breaks down at secular times.

Resummation method maintains energy conservation.

Numerical integration confirms long-time accuracy.

Abstract

We discuss differential-- versus integral--equation based methods describing out--of thermal equilibrium systems and emphasize the importance of a well defined reduction to statistical observables. Applying the projection operator approach, we investigate on the time evolution of expectation values of linear and quadratic polynomials in position and momentum for a statistical anharmonic oscillator with quartic potential. Based on the exact integro-differential equations of motion, we study the first and naive second order approximation which breaks down at secular time-scales. A method is proposed to improve the expansion by a non--perturbative resummation of all quadratic operator correlators consistent with energy conservation for all times. Motion cannot be described by an effective Hamiltonian local in time reflecting non-unitarity of the dissipative entropy generating evolution. We numerically integrate the consistently improved equations of motion for large times. We relate entropy to the uncertainty product, both being expressible in terms of the observables under consideration.