Invalid Microstate Densities for Model Systems Lead to Apparent Violation of Thermodynamic Law

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 systems which can, in principle, reach thermodynamic equilibrium since restrictions arise from the underlying equilibrium statistical mechanic axioms of independence and a priori equal probability of microstates. Here we derive a concise criterion specifying the condition on Ω which must be met in order for a system to be able, in principle, to reach thermodynamic equilibrium. Natural quantum systems obey this criterion and therefore can, in principle, reach thermodynamic equilibrium. However, models which do not respect this criterion will present inconsistencies when treated under equilibrium thermodynamic formalism. This has relevance to a number of recent models in which negative heat capacity and other violations of fundamental thermodynamic law have been reported.