  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.
