Statistical approach to the geometric structure of thermodynamics

We show how both the contact structure and the metric structure of the thermodynamic phase space arise in a natural way from a generalized canonical probability distribution ρ. In particular, the metric form and the contact form are found to be derived from the microscopic entropy s=-lnρ. Thus the first law and the second law of thermodynamics can be given the geometric interpretation that a thermodynamic system must possess both a contact and a compatible metric structure. We proceed to construct explicitly a new nondegenerate bilinear form on the thermodynamic phase space, whose restriction to state space yields the Weinhold-Ruppeiner metric, and whose restriction to Gibbs space can serve as an alternative to the metric proposed by Gilmore.