Thermodynamic geometry and the metrics of Weinhold and Gilmore

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Abstract
The thermodynamic geometries of Gibbs, Weinhold, and Gilmore are compared and the benefits of each are pointed out along with the structures which must be abandoned in order to reap the benefits. While the measurement of distances is not required (or even meaningful) in a traditional Gibbsian picture, Weinhold’s metric can be used to measure distances in the equation-of-state surface. Using Weinhold’s metric for more than a single state of equilibrium necessitates abandoning a Gibbsian picture of convex surfaces of thermodynamic states. Gilmore’s metric, on the other hand, is compatible with standard Gibbsian thermodynamics. This metric measures distance in the potential surface of statistical mechanics rather than the equation-of-state surface of equilibrium thermodynamics.