Statistical mechanics of reparametrization-invariant systems. It takes three to tango

It is notoriously difficult to apply statistical mechanics to generally covariant systems, because the notions of time, energy, and equilibrium are seriously modified in this context. We discuss the conditions under which weaker versions of these notions can be defined, sufficient for statistical mechanics. We focus on reparametrization-invariant systems without additional gauges. The key is to reconstruct statistical mechanics from the ergodic theorem. We find that a suitable split of the system into two non interacting components is sufficient for generalizing statistical mechanics. While equilibrium acquires sense only when the system admits a suitable split into three weakly interacting components—roughly: a clock and two systems among which a generalization of energy is equi-partitioned. This allows the application of statistical mechanics and thermodynamics as an additivity condition of such generalized energy. (paper)