In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner's metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold's approach. The corresponding Ruppeiner's metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures.

We use a formulation of Noether’s theorem for contact Hamiltonian systems to derive a relation between the thermodynamic entropy and the Noether invariant associated with time-translational symmetry. In the particular case of thermostatted systems at equilibrium, we show that the total entropy of the system plus the reservoir are conserved as a consequence thereof. Our results contribute to understanding thermodynamic entropy from a geometric point of view.

We study the effect of the second virial coefficient on the adiabatic lapse rate of a dry atmosphere. To this end, we compute the corresponding adiabatic curves, the internal energy, and the heat capacity, among other thermodynamic parameters. We apply these results to Earth, Mars, Venus, Titan, and the exoplanet G1 851d, considering three physically relevant virial coefficients in each case: the hardsphere, van der Waals, and the square-well potential. These examples illustrate under which atmospheric conditions the effect of the second virial coefficient is relevant. Taking the latter into account yields corrections towards the experimental values of the lapse rates of Venus and Titan in some instances.

We present a rigorous mathematical treatment of Ruppeiner geometry, by considering degenerate Hessian metrics defined on radiant manifolds. A manifold M is said to be radiant if it is endowed with a symmetric, flat connection∇ and a global vector field ρ whose covariant derivative is the identity mapping. A degenerate Hessian metric on M is a degenerate metric tensor g that can locally be written as the covariant Hessian of a function, called potential. A function on M is said to be extensive if its Lie derivative with respect to ρ is the function itself. We show that the Hessian metrics appearing in equilibrium thermodynamics are necessarily degenerate, owing to the fact that their potentials are extensive (up to an additive constant). Manifolds having degenerate Hessian metrics always contain embedded Hessian submanifolds, which generalize the manifolds defined by constant volume in which Ruppeiner geometry is usually studied. By means of examples, we illustrate that linking scalar curvature to microscopic interactions within a thermodynamic system is inaccurate under this approach. In contrast, thermodynamic critical points seem to arise as geometric singularities.

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This paper presents a rigorous treatment of the concept of extensivity in equilibrium thermodynamics from a geometric point of view. This is achieved by endowing the manifold of equilibrium states of a system with a smooth atlas that is compatible with the pseudogroup of transformations on a vector space that preserve the radial vector field. The resulting geometric structure allows for accurate definitions of extensive differential forms and scaling, and the well-known relationship between both is reproduced. This structure is represented by a global vector field that is locally written as a radial one. The submanifolds that are transversal to it are embedded, and locally defined by extensive functions.