Coupling and thermal equilibrium in general-covariant systems

A fully general-covariant formulation of statistical mechanics is still lacking. We take a step toward this theory by studying the meaning of statistical equilibrium for coupled, parametrized systems. We discuss how to couple parametrized systems. We express the thermalization hypothesis in a general-covariant context. This takes the form of vanishing of information flux. An interesting relation emerges between thermal equilibrium and gauge.

Figure 1

Structure of a general-covariant phase space. (a) The extended phase space ${\Gamma }_{\mathrm{ex}}$ and the constraint surface $\mathcal{C}$, which is foliated by orbits. The orbits project onto points of the physical phase space ${\Gamma }_{\mathrm{ph}}$ under the projection $\pi$. (b) The projection $\pi$ is also used to pull back forms to $\mathcal{C}$ and to push forward vector fields to the physical phase space.

Figure 2

The steps along a history can be seen as a sequence of phase-space regions, each with area $\sim h$.

Figure 3

In Ref. 12, it was shown that equilibrium between histories occurs when they transit the same number of Planck cells during the same interval—that is, when $N_1=N_2$. This figure demonstrates that the same condition holds when the thermal time derived from a statistical state is used.