# Symplectic formulation of statistical physics

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!

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arxiv.org/abs/1108.3472 "Thermodynamics and the moment map" Mikhail Kapranov Abstract: We give a thermodynamical interpretation of the moment map for toric varieties... I put this paper on my table since usually Kapranov's papers are very insightful, unfortunately I did not have time to read it... –  Alexander Chervov Jul 19 '12 at 12:33
You cannot go wrong with a classic, Mathematical foundations of statistical mechanics by A. I. Khinchin, Dover. The word symplectic is never used in the paper, but one of the first fact he proves is Liouville'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. –  Liviu Nicolaescu Jul 19 '12 at 14:30
in a similar vein, one might ask for a symplectic formulation of quantum mechanics: arxiv.org/abs/1208.5969 –  Carlo Beenakker Aug 30 '12 at 7:31

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.

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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