Riemannian geometric theory of critical phenomena
Abstract
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 thirdorder nonlinear ordinary differential equation whose solution is consistent with twoscale factor universality. The resulting scaled equation of state is, overall, in very good agreement with meanfield theory, the threedimensional Ising model, and experiment for the pure fluid. Universal ratios among the critical amplitudes are also in good agreement with known values. For the nonmeanfield theory exponents, the solution considered here is not analytic in the whole onephase region; the second derivative of the free energy suffers a discontinuity.
 Publication:

Physical Review A
 Pub Date:
 September 1991
 DOI:
 10.1103/PhysRevA.44.3583
 Bibcode:
 1991PhRvA..44.3583R
 Keywords:

 05.70.a;
 02.40.+m;
 05.40.+j;
 64.10.+h;
 Thermodynamics;
 General theory of equations of state and phase equilibria