On the Riemannian metrical structure in the classical statistical equilibrium thermodynamics

• Published in: Reports on mathematical physics 1986-08, Vol.24 (1), p.1-10
• Main Author: Janyszek, Henryk
• Format: Article
• Language: English
• Subjects: Classical ensemble theory; Classical statistical mechanics; Exact sciences and technology; Physics; Statistical physics, thermodynamics, and nonlinear dynamical systems
• ISSN: 0034-4877; 1879-0674
• DOI: 10.1016/0034-4877(86)90036-4

We consider the generalized Gibbs statistical states generated by a set of classical observables. We show that microscopical entropies (i.e. these states expressed in the logarithmic scale) form a concave hypersurface embedded in a linear space. We treat this hypersurface as a differential manifold with map given by the statistical temperatures. Next we construct the tangent space and introduce a Riemannian metric which has the sense of the well known Fisher information. Since this hypersurface is isomorphic with the hypersurface given by the logarithm of the partition function (embedded in a Euclidean space), we also adopt this metric on the latter one. As an example we take two equivalent descriptions of the ideal gas given by the Boguslavski and grand canonical distribution (each gives a two-dimensional surface). In both cases we obtain the 0-th Gaussian curvature which means that these surfaces are geometrically equivalent. In such a way the physical equivalence may be expressed in terms of the geometrical equivalence. In both cases the geodesic lines are also investigated as distinguished processes by this metrical structure.