Differential Invariants in Thermodynamics

Abstract

Due to the first and second law of thermodynamics, the state of a thermodynamic system is described by a Legendrian manifold. This Legendrian manifold is locally determined by the information gain function. We describe the algebra of rational differential invariants of the information gain function under the action of two different Lie groups appearing naturally as a result of measuring random vectors, and we discuss our results in the context of ideal and van der Waals gases.