Peoples Friendship University of Russia,

6 Mikluho-Maklaya Str.,

117198 Moscow,

RUSSIA

E-mail: rudikar@mail.ru, ogol@oldi.ru

**The geometric representations in classical and statistical
equilibrium thermodynamics (contact, Riemann and information geometries)**

**Abstract:**

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*(2*r*+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)’s
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.