The Mathematical Structure of the Second Law of Thermodynamics
  The essence of the second law of classical thermodynamics is the `entropy principle' which asserts the existence of an additive and extensive entropy function, S, that is defined for all equilibrium states of thermodynamic systems and whose increase characterizes the possible state changes under adiabatic conditions. It is one of the few really fundamental physical laws (in the sense that no deviation, however tiny, is permitted) and its consequences are far reaching. This principle is independent of models, statistical mechanical or otherwise, and can be understood without recourse to Carnot cycles, ideal gases and other assumptions about such things as `heat', `temperature', `reversible processes', etc., as is usually done. Also the well known formula of statistical mechanics, S = -\sum p log p, is not needed for the derivation of the entropy principle.   This contribution is partly a summary of our joint work (Physics Reports, Vol. 310, 1--96 (1999)) where the existence and uniqueness of S is proved to be a consequence of certain basic properties of the relation of adiabatic accessibility among equilibrium states. We also present some open problems and suggest directions for further study. 
Elliott H. Lieb and Jakob Yngvason