Riemannian geometric theory of critical phenomena

By combining the hypothesis that the thermodynamic curvature is the correlation volume with the hypothesis that the free energy is the inverse of the correlation volume, I propose that the thermodynamic curvature is proportional to the inverse of the free energy near the critical point. This hypothesis leads to a partial differential geometric equation for the free energy which a generalized homogeneous function reduces to a third-order nonlinear ordinary differential equation whose solution is consistent with two-scale factor universality. The resulting scaled equation of state is, overall, in very good agreement with mean-field theory, the three-dimensional Ising model, and experiment for the pure fluid. Universal ratios among the critical amplitudes are also in good agreement with known values. For the non-mean-field theory exponents, the solution considered here is not analytic in the whole one-phase region; the second derivative of the free energy suffers a discontinuity.