Non-equilibrium steady state in the hydro regime

We study the existence and properties of the non-equilibrium steady state which arises by putting two copies of systems at different temperatures into a thermal contact. we solve the problem for the relativistic systems that are described by the energy-momentum of a perfect hydro with general equation of state (eos). in particular, we examine several simple examples: a hydro with a linear eos, a holographic cft perturbed by a relevant operator and a barotropic fluid, i.e., $$ p=pleft(mathrm{mathcal{e}}right) $$ . our studies suggest that the formation of steady state is a universal result of the hydro regime regardless of the kind of fluid.