You want to read Chapter IV "Statistical Mechanics" in *Structure of Dynamical Systems* (1970 French original available here) by J.-M. Souriau, one of the pioneers of symplectic mechanics.

Given a symplectic manifold $X$ on which a Lie group $G$ acts with moment map $\Psi$, Souriau defines the *Gibbs states* as the probability measures on $X$ that maximize entropy for a given mean value of $\Psi$. He shows (thm 16.219) that they have the form

$$
\text{const}\times e^{-\langle\Psi(\cdot),\beta\rangle}\lambda,
\qquad
(\lambda=\text{Liouville measure})
$$
for some $\beta\in\mathfrak g$ which generalizes the "inverse temperature" when $G=\mathbf R=$ {time translations}. The rest of the Chapter is devoted to the study of these states; in particular when $G=$ SO(3), $\beta$ can be interpreted as a rotation vector, and the fact that planets revolve in a common plane as the equality of their equilibrium $\beta$s. See also *Définition covariante des équilibres thermodynamiques*, Suppl. Nuovo Cimento **1** (1966), 203–216.

Later Souriau developed a general-relativistic viewpoint on dissipative processes which explains why they preserve the mean value of $\Psi(x)$. Thus, quoting from this summary to whet your appetite: "the first principle of thermodynamics [loses] its primitive status and [becomes] a necessary consequence of the invariance of the symplectic structure in gravitational gauge transformations." For more details, see *Thermodynamique et géométrie*, Lecture Notes in Math. **676** (1978), 369–397 or scanned preprint.

Mathematical foundations of statistical mechanicsby A. I. Khinchin, Dover. The word symplectic is never used in the paper, but one of the first fact he proves isLiouville's theorem: a Hamiltonian flow on the standard symplectic $\mathbb{R}^{2n}$ preserves the symplectic volume. In any case, it is an excellent read for a mathematician. $\endgroup$ – Liviu Nicolaescu Jul 19 '12 at 14:30