Ruppeiner, George
(1992)
*Riemannian geometry of the thermodynamics and critical phenomena.*
Accademia Peloritana dei Pericolanti, Classe di Scienze FF. MM. NN., LXX .
pp. 101-133.

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## Abstract

This paper describes a Riemannian geometry of thermodynamics with a metric based on thermodynamic fluctuation theory. The resulting Riemannian thermodynamic curvature has been interpreted as the volume where classical thermodynamic fluctuation theory breaks down. Near the critical point, this volume is the correlation volume. Combining this interpretation with the well known proportionality between the free energy per volume and the inverse of the correlation volume yields the following hypothesis: the thermodynamic curvature is proportional to the inverse of the free energy. This thermodynamic hypothesis may be expressed as a partial differential equation for the free energy near the critical point. Its solution yields an equation of state in very good agreement with mean field theory, the three-dimensional Ising model, and experiment for the pure fluid. For the non-mean field theory exponents, the solution considered is not analytic in the whole one-phase region; the second derivative of the free energy suffers a discontinuity.

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