Does there exists a symplectic formulation of statistical physics?


I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated symplectic, but I do not know anything which tries to relate them 'symplectilly'. Partial results are also welcome!



Symplectic formulation of statistical physics
You want to read Chapter IV "Statistical Mechanics" in *[Structure of Dynamical Systems](http://www.ams.org/mathscinet-getitem?mr=1461545)* (1970 French original available [here](http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm)) 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](http://www.ams.org/mathscinet-getitem?mr=2335767) 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](http://dx.doi.org/10.1007/BFb0063682)*, Lecture Notes in Math. **676** (1978), 369–397 or [scanned preprint](http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=197810025).


