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Violation of Fundamental Thermodynamic Law or Invalid Microstate Densities?

It is often incorrectly assumed that the number of microstates Ω(E,V,N,...) available to an isolated system can have arbitrary dependence on the extensive variables E,V,N, .... However, this is not the case for natural systems which can reach thermodynamic equilibrium since restrictions exist arising from the underlying equilibrium axioms of independence and \it{a priori} equal probability of microstate, and the fundamental constants of Nature. Here we derive a concise formula specifying the condition on Ω which must be met for real systems. Models which do not respect this condition will present inconsistencies when treated under equilibrium thermodynamic formalism. This has relevance to a number of recent models in which negative heat capacity and violation of fundamental thermodynamic law have been reported. Natural quantum systems obey the axioms and abide by the fundamental constants, and thus natural systems, in the absence of infinite range forces, can, in principle, attain thermodynamic equilibrium.