Heat can flow from cold to hot in Microcanonical Thermodynamics of finite systems. The microscopic origin of condensation and phase separations

摘要

Microcanonical Thermodynamics allows the application of Statistical Mechanicson one hand to closed finite and even small systems and on the other to thelargest,self-gravitating ones. However, one has to reconsider the fundamentalprinciples of Statistical Mechanics especially its key quantity, entropy.Whereas in conventional Thermostatistics the homogeneity and extensivity of thesystem and the concavity of its entropy S(E) are central conditions, these failfor the systems considered here. E.g. at phase separation the entropy S(E) isnecessarily convex to make e^{S(E)-E/T} bimodal in E (the two coexistingphases). This is so even for normal macroscopic systems with short-rangecoupling. As inhomogeneities and surface effects in particular cannot be scaledaway,one has to be careful with the standard arguments of splitting a systeminto two or bringing two systems into thermal contact. Not only the volume partof the entropy must be considered. When removing an external constraint inregions of a negative heat capacity, the system may even relax under a flow ofheat (energy) against the temperature slope. Thus Clausius formulation of theSecond Law: "Heat always flows from hot to cold" can be violated. Temperatureis not a necessary or fundamental control parameter of Thermostatistics. In thefinal sections of this paper the general microscopic mechanism leading tocondensation and to the convexity of the microcanonical entropy S(E) at phaseseparation is sketched. Also the microscopic conditions for the existence ornon-existence of a critical end-point of the phase-separation are discussed.This is explained for the liquid--gas and the solid--liquid transition.