The Statistical Mechanics of Indistinguishable Energy States and the Glass Transition

The statistical mechanics of particles that populate indistinguishable energy sub-states is explored. In particular, the mathematical treatment of the microstates differs from conventional statistical mechanics where for a given degeneracy, the energy sub-levels or sub-states are universally treated as distinguishable, and differentiated by unique quantum numbers, or addressed by distinct spatial locations. Results from combinatorial counting problems are adapted to derive exact distribution functions for both classical and quantum particles at a high degeneracy limit. Quantum particles obey a non-extensive entropy $\mathcal{S}\propto \sqrt{N}$, that satisfies an Area Law: $\mathcal{S}\propto A$ in $d=2$ bulk spatial dimensions. Classical particles exhibit a definitive glass transition, similar to supercooled liquids where the configurational entropy vanishes below a finite temperature $T_K$.