Riemannian geometry of thermodynamics and systems with repulsive power-law interactions

A Riemannian geometric theory of thermodynamics based on the postulate that the curvature scalar R is proportional to the inverse free energy density is used to investigate three-dimensional fluid systems of identical classical point particles interacting with each other via a power-law potential energy γr^-α. Such systems are useful in modeling melting transitions. The limit α→∞ corresponds to the hard sphere gas. A thermodynamic limit exists only for short-range (α>3) and repulsive (γ>0) interactions. The geometric theory solutions for given α>3, γ>0, and any constant temperature T have the following properties: (1) the thermodynamics follows from a single function b(ρT^-3∕α), where ρ is the density; (2) all solutions are equivalent up to a single scaling constant for ρT^-3∕α, related to γ via the virial theorem; (3) at low density, solutions correspond to the ideal gas; (4) at high density there are solutions with pressure and energy depending on density as expected from solid state physics, though not with a Dulong-Petit heat capacity limit; (5) for 3<α<3.7913, the solution goes from the low to the expected high density limit smoothly; (6) for α>3.7913 a phase transition is required to go between these regimes; (7) for any α>3 we may include a first-order phase transition, which is expected from computer simulations; and (8) if α→∞, the density approaches a finite value as the pressure increases to infinity, with the pressure diverging logarithmically in the density difference.