TITLE: Symplectic formulation of statistical physics
QUESTION [17 upvotes]: 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!

REPLY [3 votes]: I invite you to read the following papers about "Lie group Thermodynamics" of Jean-Marie Souriau. Souriau has discovered that Gibbs equilibrium is not covariant with respect to Dynamical groups, then he has considered Gibbs equilibrium on a Symplectic Manifold with covariant model with respect to a Lie group action. Souriau has introduced a geometric (planck) temperature in the Lie Algebra of the group:
[1]Barbaresco, F. Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics. Entropy 2014, 16, 4521-4565.
http://www.mdpi.com/1099-4300/16/8/4521/pdf
[2]Barbaresco F., Koszul information geometry and Souriau Lie group thermodynamics, AIP Conf. Proc. 1641, 74 (2015)
http://djafari.free.fr/MaxEnt2014/papers/Tutorial7_paper.pdf
More information at GSI'15 conference:
www.gsi2015.org
F. Barbaresco
GSI General Chair