Emergence of macroscopic temperatures in systems that are not thermodynamical microscopically: towards a thermodynamical description of slow granular rheology

A scenario for systems with slow dynamics is characterized by stating that there are several temperatures coexisting for the sample, with a single temperature shared by all observables at each (widely separate) timescale. In preparation for the study of granular rheology, we show within this framework that glassy systems with driving and friction that are generic and do not correspond to a thermal bath-and whose microscopic 'fast' motion is hence not thermal-have a well-defined macroscopic temperature associated with the slow degrees of freedom. This temperature is what a thermometer coupled to the system will measure if tuned to respond to low frequencies, and, since it can be related to the number of stationary configurations, it is the formalization of Edwards' 'compactivity' ideas.