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

摘要

Microcanonical thermodynamics allows the application of statistical mechanicsboth to finite and even small systems and also to the largest, self-gravitatingones. However, one must reconsider the fundamental principles of statisticalmechanics especially its key quantity, entropy. Whereas in conventionalthermostatistics, the homogeneity and extensivity of the system and theconcavity of its entropy are central conditions, these fail for the systemsconsidered here. For example, at phase separation, the entropy, S(E), isnecessarily convex to make exp[S(E)-E/T] bimodal in E. Particularly, asinhomogeneities and surface effects cannot be scaled away, one must be carefulwith the standard arguments of splitting a system into two subsystems, orbringing two systems into thermal contact with energy or particle exchange. Notonly the volume part of the entropy must be considered. As will be shown here,when removing constraints in regions of a negative heat capacity, the systemmay even relax under a flow of heat (energy) against a temperature slope. Thusthe Clausius formulation of the second law: ``Heat always flows from hot tocold'', can be violated. Temperature is not a necessary or fundamental controlparameter of thermostatistics. However, the second law is still satisfied andthe total Boltzmann entropy increases. In the final sections of this paper, thegeneral microscopic mechanism leading to condensation and to the convexity ofthe microcanonical entropy at phase separation is sketched. Also themicroscopic conditions for the existence (or non-existence) of a criticalend-point of the phase-separation are discussed. This is explained for theliquid-gas and the solid-liquid transition.