Notes on the third law of thermodynamics: I

We analyse some aspects of the third law of thermodynamics. We first review both the entropic version ( N ) and the unattainability version ( U ) and the relation occurring between them. Then, we heuristically interpret ( N ) as a continuity boundary condition for thermodynamics at the boundary T = 0 of the thermodynamic domain. On a rigorous mathematical footing, we discuss the third law both in Carathéodory's approach and in Gibbs' one. Carathéodory's approach is fundamental in order to understand the nature of the surface T = 0. In fact, in this approach, under suitable mathematical conditions, T = 0 appears as a leaf of the foliation of the thermodynamic manifold associated with the non-singular integrable Pfaffian form δ Q rev . Being a leaf, it cannot intersect any other leaf S = const of the foliation. We show that ( N ) is equivalent to the requirement that T = 0 is a leaf. In Gibbs' approach, the peculiar nature of T = 0 appears to be less evident because the existence of the entropy is a postulate; nevertheless, it is still possible to conclude that the lowest value of the entropy S has to be attained at the boundary of the convex set where S is defined.