The Statistical Mechanics of Indistinguishable Energy States and the Glass Transition

TL;DR

This paper develops a statistical mechanics framework for indistinguishable particles in high degeneracy states, revealing a glass transition in classical particles and a non-extensive entropy in quantum particles.

Contribution

It introduces a novel treatment of microstates for indistinguishable particles, deriving exact distribution functions and uncovering a glass transition and area law entropy.

Findings

Classical particles exhibit a glass transition with vanishing configurational entropy below a finite temperature.

Quantum particles obey a non-extensive entropy proportional to the square root of particle number, satisfying an Area Law.

Derived exact distribution functions for particles in high degeneracy energy states.

Abstract

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 $T_K$.