General Mechanism for a Positive Temperature Entropy Crisis in Stationary Metastable States: Thermodynamic Necessity and Confirmation by Exact calculations

We introduce the concept of stationary metastable states (SMS’s), and give a prescription to study it using a restricted partition function formalism. This requires introducing a continuous entropy function S(E) even for a finite system, a standard practice in the literature though never clearly stated, so that it can be differentiated. The formalism ensures that SMS free energy exists all the way to T = 0, and remains stable. We introduce the concept of the reality condition, according to which the entropy S(T ) of a set of coupled degrees of freedom must be non-negative.The entropy crisis, which does not affect stability, is identified as the violation of the reality condition. We identify and validate rigorously, using general thermodynamic arguments, the following general thermodynamic mechanism behind the entropy crisis in SMS. The free energy Fdis(T ) of any SMS must be equal to the T = 0 crystal free energy E0 at two different temperatures T = 0, and T = Teq > 0. Thus, the stability requires Fdis(T ) to possess a maximum at an intermediate but a strictly positive temperature TK, where the energy is E = EK. The SMS branch below TK gives the entropy crisis and must be replaced by hand by an ideal glass free energy of constant energy EK, and vanishing entropy. Hence, TK > 0 represents the Kauzmann temperature. The ideal glass energy EK is higher than the crystal energy E0 at absolute zero, which is in agreement with the experimenatal fact that the extrapolated energy of a real glass at T = 0 is higher than its T = 0 crystal energy. We confirm the general predictions by two exact calculations, one of which is not meanfield. The calculations clearly show that the notion of SMS is not only not vaccuous, but also not a consequence of a mean-field analysis. They also show that certain folklore cannot be substantiated.