The geometric representations in classical and statistical equilibrium thermodynamics (contact, Riemann and information geometries)
By the aid of the formalism of the differential 1- and 2-forms in the framework of the contact geometry the analysis is given of the family of r-dimensional Gibbs surfaces G(r), parametrized by the Euclidean r-dimensional (r>=2) base space R(r) of the thermodynamical parameters for the macroscopic system in the state of thermal equilibrium. It is shown, that G(r)=L(r), where L(r) is the family of Legendre (sub)manifolds, which describe the solutions of the Pfaff equations (up to contactomorphisms) in the Euclidean space R(2r+1); Darboux theorem in this case corresponds to the thermodynamic Gibbs- Helmholtz relations.
In the scope of the classical thermodynamics (CTD) the surfaces G(r)
are considered at first as imbedded in R(r+1) Riemannian
manifolds. It is shown, that the conditions of the thermodynamic stability
of the system (in the form of Gibbs inequalities) imposed by the Second
Law relative to spontaneous equilibrium fluctuations demand all the G(r)ís
to be strictly convex. This, in turn, means the strict positivity of the
external Gauss curvature K, which is proportional to Det
h, where h is the Hessian of the Massieu -Planck potential for the system under consideration.
Alternatively, G(r) are considered as Riemannian manifolds without any imbedding, but with h chosen as the metric matrix g. We consider the most interesting case r=2, which is not subjected to the restrictions of the Liouville theorem and allows a great variety of conformal, or "isothermic" base space coordinate ransformations. In this case the Gauss' "theorema egregium" expresses K (now as the internal curvature) through the derivatives of g, and in this way we obtain some conditions of the thermodynamic stability additional to the standard conditions which include only the positive definiteness of matrix h.
In the framework of the statistical thermodynamics (STD) the information geometry, or geometrostatistics (the last term is due to Kolmogorov) is studied using the fact that the global quasi-metric in the topological space P of all parametric r-dimensional (r>=2) probability density functions (pdf) is locally equivalent to the Riemannian metric with the metric matrix equal to the Fisher information matrix gF, which is (by definition) always positive definite.
In the case of strictly additive base space variables due to the theorem of Koopman-Pitman-Dynkin-Jeffries P is the space of all exponential pdf, which were obtained in STD much earlier and are known as Gibbs-Einstein-Szilard pdf. For this class of pdf it is shown that gF equals to h and therefore the Fisher-Kramer-Rao theorem implies Einstein thermodynamic "uncertainties relation". Moreover, the space P, considered as the embedded Riemannian manifold parametrized by the base space R(r), should be, just as G(r), be strictly convex, because in this case gF equals to g. Finally, the expressions are obtained for gF and K (at r=2) for the q-deformed Gibbs-Einstein-Szilard pdf, i.e. non-exponential pdf of the Renyi-Tsallis type.