Geometrical aspects of statistical mechanics

Abstract

Investigation of the geometry of thermodynamic state space, based upon the differential geometric approach to parametric statistics developed by Chentsov [Statistical Decision Rules and Optimal Inference (Nauka, Moscow, 1972)], Efron [Ann. Stat. 3, 1189 (1975)], Amari [Ann. Stat. 10, 357 (1982)], and others, provides a deeper understanding of the mathematical structure of statistical thermodynamics. In the present paper, the Riemannian geometrical approach to statistical mechanical systems due to Janyszek [J. Phys. A 23, 477 (1990)] is applied to various models including the van der Waals gas and magnetic models. The scalar curvature for these models is shown to diverge not only at the critical points but also along the entire spinodal curve. The critical behavior of the curvature derived from the Fisher information metric turns out to coincide with that derived from the entropy differential metric by Ruppeiner [Phys. Rev. A 20, 1608 (1979)].