The statistical mechanics of particles that populate indistinguishable energy substates is explored. In particular, the mathematical treatment of microstates differs from conventional statistical mechanics, where, for a given degeneracy, the energy sublevels or substates 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 nonextensive entropy $\mathcal{S}\propto \sqrt{N}$ that satisfies the area law $\mathcal{S}\propto A$ in $d=2$ bulk spatial dimensions. Classical particles exhibit a definitive glass transition similar to supercooled liquids, for which the configurational entropy vanishes below a finite temperature $T_K$.

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• Classical statistical mechanics
• Critical phenomena
• Eigenstate thermalization
• Exotic phases of matter
• Kinetic theory
• Nucleation
• Phase transitions
• Quantum statistical mechanics