# Perturbative approach to weakly driven many-particle systems in the presence of approximate conservation laws

## TL;DR

This paper introduces a perturbative method to analyze weakly driven quantum many-particle systems with approximate conservation laws, enabling efficient description of their steady states via generalized Gibbs ensembles.

## Contribution

It develops a Liouville perturbation theory that accounts for conservation laws and their violations, providing a systematic way to find steady states in weakly driven quantum systems.

## Key findings

- Perturbative expansion of steady states using Lagrange multipliers.
- Application to models like electron-phonon systems and Bose condensates.
- Validation through study of interacting fermions with non-thermal reservoirs.

## Abstract

We develop a Liouville perturbation theory for weakly driven and weakly open quantum systems in situations when the unperturbed system has a number of conservations laws. If the perturbation violates the conservation laws, it drives the system to a new steady state which can be approximately but efficiently described by a (generalized) Gibbs ensemble characterized by one Lagrange parameter for each conservation law. The value of those has to be determined from rate equations for conserved quantities. Remarkably, even weak perturbations can lead to large responses of conserved quantities. We present a perturbative expansion of the steady state density matrix; first we give the condition that fixes the zeroth order expression (Lagrange parameters) and then determine the higher order corrections via projections of the Liouvillian. The formalism can be applied to a wide range of problems including two-temperature models for electron-phonon systems, Bose condensates of excitons or photons or weakly perturbed integrable models. We test our formalism by studying interacting fermions coupled to non-thermal reservoirs, approximately described by a Boltzmann equation.