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We develop a Liouville perturbation theory for weakly driven and weakly openquantum systems in situations when the unperturbed system has a number ofconservations laws. If the perturbation violates the conservation laws, itdrives the system to a new steady state which can be approximately butefficiently described by a (generalized) Gibbs ensemble characterized by oneLagrange parameter for each conservation law. The value of those has to bedetermined from rate equations for conserved quantities. Remarkably, even weakperturbations can lead to large responses of conserved quantities. We present aperturbative expansion of the steady state density matrix; first we give thecondition that fixes the zeroth order expression (Lagrange parameters) and thendetermine the higher order corrections via projections of the Liouvillian. Theformalism can be applied to a wide range of problems including two-temperaturemodels for electron-phonon systems, Bose condensates of excitons or photons orweakly perturbed integrable models. We test our formalism by studyinginteracting fermions coupled to non-thermal reservoirs, approximately describedby a Boltzmann equation.

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