Ideal Bose-Einstein condensation in the canonical ensemble: exact asymptotic estimates from large deviations

What the math gives to ML

The paper provides a constructive large-deviations description of Bose-Einstein condensation in the canonical ensemble: a fixed total mass is distributed among modes, and above a critical density the excess mass concentrates into one ground-state mode while the remaining modes remain at a saturated normal density. The transferable mechanism is a sharp competition between an entropic distributed phase and a condensed phase, with a computable threshold and finite-size fluctuation law. A direct neural analogue is expert routing or memory allocation, where token assignments are constrained by a fixed batch and a controllable Gibbs energy; the BEC asymptotics predict when routing remains distributed and when one expert absorbs excess load. This yields a measurable phase diagram and a principled regularizer or capacity schedule rather than an ad hoc anti-collapse penalty.