  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.
