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A Lagrangian Description of Thermodynamics

The fact that a temperature and an entropy may be associated with horizons in semi-classical general relativity has led many to suspect that spacetime has microstructure. If this is indeed the case then its description via Riemannian geometry must be regarded as an effective theory of the aggregate behavior of some more fundamental degrees of freedom that remain unknown, in many ways similar to the treatment of fluid dynamics via the Navier-Stokes equations. This led us to ask how a geometric structure may naturally arise in thermodynamics or statistical mechanics and what evolution may mean in this context. In this article we argue that it is possible to view thermodynamic processes as the evolution of a dynamical system, described by a quadratic Lagrangian and a metric on the thermodynamic configuration space. The Lagrangian is an invariant distance between equilibrium thermodynamic states as defined by the metric, which is straightforwardly obtained from a complete set of equations of state.