Heat can flow from cold to hot in Microcanonical Thermodynamics of   finite systems. The microscopic origin of condensation and phase separations
  Microcanonical Thermodynamics allows the application of Statistical Mechanics on one hand to closed finite and even small systems and on the other to the largest,self-gravitating ones. However, one has to reconsider the fundamental principles of Statistical Mechanics especially its key quantity, entropy. Whereas in conventional Thermostatistics the homogeneity and extensivity of the system and the concavity of its entropy S(E) are central conditions, these fail for the systems considered here. E.g. at phase separation the entropy S(E) is necessarily convex to make e^{S(E)-E/T} bimodal in E (the two coexisting phases). This is so even for normal macroscopic systems with short-range coupling. As inhomogeneities and surface effects in particular cannot be scaled away,one has to be careful with the standard arguments of splitting a system into two or bringing two systems into thermal contact. Not only the volume part of the entropy must be considered. When removing an external constraint in regions of a negative heat capacity, the system may even relax under a flow of heat (energy) against the temperature slope. Thus Clausius formulation of the Second Law: "Heat always flows from hot to cold" can be violated. Temperature is not a necessary or fundamental control parameter of Thermostatistics. In the final sections of this paper the general microscopic mechanism leading to condensation and to the convexity of the microcanonical entropy S(E) at phase separation is sketched. Also the microscopic conditions for the existence or non-existence of a critical end-point of the phase-separation are discussed. This is explained for the liquid--gas and the solid--liquid transition. 
D.H.E.Gross