The question of negative temperatures in thermodynamics and statistical mechanics

We show that both positive and negative absolute temperatures and monotonically increasing and decreasing entropy in adiabatic processes are consistent with Carathéodory’s version of the second law and we explore the modifications of the Kelvin–Planck and Clausius versions which are needed to accommodate these possibilities.We show, in part by using the equivalence of distributions and the canonical distribution, that the correct microcanonical entropy, is the surface (Boltzmann) form rather than the bulk (Gibbs) form thereby providing for the possibility of negative temperatures and we counter the contention on the part of a number of authors that the surface entropy fails to satisfy fundamental thermodynamic relationships.