  A general-covariant statistical framework capable of describing classical
fluctuations of the gravitational field is a thorny open problem in theoretical
physics, yet ultimately necessary to understand the nature of the gravitational
interaction and a key to quantum gravity. Inspired by Souriau's symplectic
generalization of the Maxwell-Boltzmann-Gibbs equilibrium in Lie group
thermodynamics, we investigate a spacetime-covariant formulation of statistical
mechanics for parametrized first-order field theories, as a simplified model
sharing essential general covariant features with canonical general relativity.
Starting from a covariant multi-symplectic phase space formulation, we define a
general-covariant notion of Gibbs state in terms of the covariant momentum map
associated with the lifted action of the diffeomorphisms group on the extended
phase space. We show how such a covariant notion of equilibrium encodes the
whole information about symmetry, gauge and dynamics carried by the theory,
associated to a canonical spacetime foliation, where the covariant choice of a
reference frame reflects in a Lie algebra-valued notion of local temperature.
We investigate how physical equilibrium, hence time evolution, emerges from
such a state and the role of the gauge symmetry in the thermodynamic
description.
