Non-equilibrium temperatures in systems with internal variables

We discuss different definitions of non-equilibrium temperature in systems with internal variables. In equilibrium states, all the different definitions of temperature (caloric, entropic, kinetic, vibrational, configurational, fluctuational, and so on) lead to the same value, but in non-equilibrium they lead to different values. Out of equilibrium, equipartition is not to be expected and therefore the "temperatures" of the different degrees of freedom will be different from each other. Here, we focus our attention on the caloric temperature, based on the internal energy corresponding to the internal variable, the entropic temperature, based on the entropy, and several thermometric or empirical temperatures, based on the zeroth law. We illustrate the difference and the connections between them in some simple systems with a heat flux (two-level and three-level systems, ideal gases), and we also consider a solid system with internal variables. These variables may be measured, but not controlled, and they appear in the Gibbs equation like the classical thermodynamic variables. As an example, we consider a crystal with dislocations submitted to an external flux of energy (like a wall of a fusion reactor bombarded by energetic particles).