Riemannian geometry and stability of thermodynamical equilibrium systems

A geometrical approach to statistical thermodynamics is proposed. It is shown that any r-parameter generalised Gibbs distribution leads to a Riemannian metric of parameter space. The components of the metric tensor are represented by second moments of stochastic variables. The scalar curvature R, as a geometrical invariant, is a function of the second and third moments, so is strictly connected with fluctuations of the system. In the case of a real gas, R is positive and tends to infinity as the system approaches the critical point. In the case of an ideal gas, R=0. The obtained results, and the results of the authors previous work, suggests that for a wide class of models R tends to + infinity near the critical point. They treat R as a measure of the stability of the system. They propose some sort of statistical principle: only such models may be accepted for which R tends to infinity if the system is approaching the critical point. It is shown that, if this criterion is adopted for a class of models for which the scaling hypothesis holds, then they obtain the new inequalities for the critical indices. These inequalities are in good agreement with model calculations and experiment.