Thermodynamic Geometry

The geometric study of thermodynamics began with Gibbs’s reformulation of the theory in terms of equilibrium states rather than processes [1]. The surface of the set of equilibrium states was Gibbs’s primary object of study and foreshadowed much of the modern differential geometric theory manifolds. Gibbs surfaces is where the geometric theory stood when I became a graduate student. The story that I relate in these lectures begins here. This is when Frank Weinhold published a series of five papers [2] based on the following observation: If we interpret differentials as vector quantities, and use the second derivative matrix of the internal energy as a metric, then many of the thermodynamic identities can be interpreted as well known geometric equations, e.g., the Pythagorean theorem, the law of sines, etc. At the time, I was working on another aspect of the geometrical structure related to thermodynamic potentials (briefly mentioned in the last lecture) and reacted to these papers with great skepticism. The geometry of Gibbs seemed to