On the Mathematics of Thermodynamics
  We show that the mathematical structure of Gibbsian thermodynamics flows from the following simple elements: the state space of a thermodynamical substance is a measure space together with two orderings (corresponding to "warmer than" and "adiabatically accessible from") which satisfy certain plausible physical axioms and an area condition which was introduced by Paul Samuelson. We show how the basic identities of thermodynamics, in particular the Maxwell relations, follow and so the existence of energy, free energy, enthalpy and the Gibbs potential function. We also discuss some questions which we have not found dealt with in the literature, such as the amount of information required to reconstruct the equations of state of a substance and a systematic approach to thermodynamical identities. 
J.B. Cooper, T. Russell
math-ph math.MP