Energy flow, partial equilibration, and effective temperatures in systems with slow dynamics

We show that, in nonequilibrium systems with small heal flows, there is a time-scale-dependent effective temperature that plays the same role as the thermodynamical temperature in that it controls the direction of heat flows and acts as a criterion for thermalization. We simultaneously treat the case of stationary systems with weak stirring and of glassy systems that age after cooling and show that they exhibit very similar behavior provided that time dependences are expressed in terms of the correlations of the system. We substantiate our claims with examples taken from solvable models with nontrivial low-temperature dynamics, but argue that they have a much wider range of validity. We suggest experimental checks of these ideas.