The microcanonical thermodynamics of finite systems: the microscopic origin of condensation and phase separations, and the conditions for heat flow from lower to higher temperatures.

Microcanonical thermodynamics [D. H. E. Gross, Microcanonical Thermodynamics, Phase Transitions in "Small" Systems (World Scientific, Singapore, 2001)] allows the application of statistical mechanics both to finite and even small systems and also to the largest, self-gravitating ones. However, one must 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 are central conditions, these fail for the systems considered here. For example, at phase separation, the entropy S(E) is necessarily convex to make e(S(E)-E/T) bimodal in E. Particularly, as inhomogeneities and surface effects cannot be scaled away, one must be careful with the standard arguments of splitting a system into two subsystems, or bringing two systems into thermal contact with energy or particle exchange. Not only the volume part of the entropy must be considered; the addition of any other external constraint [A. Wehrl, Rev. Mod. Phys. 50, 221 (1978)], such as a dividing surface, or the enforcement of gradients of the energy or particle profile, reduce the entropy. As will be shown here, when removing such constraints in regions of a negative heat capacity, the system may even relax under a flow of heat (energy) against a temperature slope. Thus the 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. However, the second law is still satisfied and the total Boltzmann entropy increases. In the final sections of this paper, the general microscopic mechanism leading to condensation and to the convexity of the microcanonical entropy at phase separation is sketched. Also the microscopic conditions for the existence (or nonexistence) of a critical end point of the phase separation are discussed. This is explained for the liquid-gas and the solid-liquid transition.