Bose-Einstein condensation of the dilute hard-sphere Bose gas in the thermodynamic limit, at zero and small positive temperature

Lineage and corrections

This event attributed to Unreleased internal OpenAI model

Companion: A density-uniform condensate bound for dilute Bose gases (September 27, 2026) evidence for this event

Companion: Ground-state condensation in the dilute hard-sphere gas (September 24, 2026) evidence for this event

Solovej, Mathematical physics of dilute Bose gases (C. R. Physique 2025), Section 5 evidence for this event

Related: Bose-Einstein condensation at positive temperature in the dilute hard-sphere gas evidence for this event

Bose-Einstein condensation of the dilute hard-sphere Bose gas in the thermodynamic limit, at zero and small positive temperature parent of this event

Positive temperature (October 5): for every $a>0$ there is $\rho_*(a)>0$ such that for each fixed $0<\rho<\rho_*$ some fixed $T(a,\rho)>0$ gives $\liminf N^{-1}\langle u_{0,L},\gamma^{(1)}_{N,L,T}u_{0,L}\rangle>0$ for the exact canonical hard-sphere Gibbs state as $L\to\infty$, $N/L^3\to\rho$. Ground state (September 24 companion): absolute $\varepsilon_0,c_0>0$ with condensate fraction at least $c_0$ in every ground state when $\rho a^3<\varepsilon_0$. A further companion gives a density-uniform bound for bounded nonnegative finite-range potentials at temperatures up to $\rho^2$. Not shown: the transition temperature, condensation at moderate density, two dimensions, or potentials with attractive parts. parent of this event

VibeMathed record: Bose-Einstein condensation of the dilute hard-sphere Bose gas in the thermodynamic limit, at zero and small positive temperature evidence for this event

Lean proof of the ground-state companion: OAI/Analysis/HardSphere/Main.lean evidence for this event