Thermodynamical Formalism

In this first chapter we review the thermodynamical formalism in the abstract form convenient for applications to Statistical Mechanics. To emphasize its connections to other subjects in Theoretical Physics, we often use the language of symplectic and contact geometry. After reviewing the basic notions and Laws of Thermodynamics, we discuss the thermodynamical notion of entropy and its physical consequences. The major part of the chapter is devoted to the definition, properties, physical interpretation, and mutual relations between the several thermodynamical potentials. We introduce and study a number of them, posing particular emphasis on the ones associated to the equilibrium ensembles to be introduced in Chap. 2 . The various thermal susceptibilities (response functions) are expressed as derivatives of the thermodynamical potentials and connected via the Maxwell relations. In the last section we reproduce Boltzmann’s thermodynamic proof of the Stefan law as a testament of the “unreasonable effectiveness” of Thermodynamics in Physics.