# Symplectic geometry in thermodynamics

There seems to be analogues between Hamiltonian dynamics and thermodynamics given the Legendre transforms between Lagrangian and Hamiltonian functions and all of Maxwell's relations. Poincarè tried to generalise classical mechanics to the realm of statistical thermodynamics with ergodic theory yet I believe his model is not complete(?)

Therefore as my main question, does symplectic geometry underpin thermodynamics? I am currently reading about KAM theory (please see my other question regarding this) and was wondering can indeterminism in perturbation theory and chaos lead to entropy and the second law?

No answers yet? So let's take a shot at a (partial) answer:

Therefore as my main question, does symplectic geometry underpin thermodynamics?

No. In thermodynamics, we're dealing with a Legendrian submanifold of a contact manifold (cf Wikipedia). Thermodynamic variables are canonical coordinates on that manifold.

Morally speaking, in case of symplectic geometry, canonical coordinates map any symplectic manifold to the cotangent bundle $T^*\mathbb R^n$ with symplectic form $\omega = d\theta$.

In case of contact geometry, canonical coordinates map any contact manifold to the first jet bundle $J^1\mathbb R^n$ (essentially $\mathbb R\times T^*\mathbb R^n$) with contact form $\alpha = dz + \theta$ (in both cases, $\theta$ denotes the canonical 1-form of the cotangent bundle; $z$ is the coordinate of the first factor).

On the jet bundle, the submanifold in question is given by the prolongation of some state function - a thermodynamic potential expressed in its natural variables. Eg for $U = f(S, V)$, we end up with a coordinate expression $$(S, V, U, T, p) = \left(S, V, f(S, V), \frac{\partial f}{\partial S}(S, V), \frac{\partial f}{\partial V}(S, V) \right)$$

Please insert minus signs as appropriate ;)

[I] was wondering can indeterminism in perturbation theory and chaos lead to entropy and the second law?

As far as geometry is concerned, there isn't really anything special about entropy, ie this question has to be answered at the lower level of statistical mechanics; I'm happy to leave that part of the question to someone else...

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:

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

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