Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas

OpenAI

Published Oct 6, 2026• 0 citations

Abstract

We prove Bose–Einstein condensation at positive temperature in the three-dimensional dilute hard-sphere gas. For each fixed exclusion distance and every sufficiently small fixed density, there is a strictly positive temperature, independent of the volume, at which the exact canonical Gibbs state has a positive condensate fraction in the thermodynamic limit. The condensate occupies the constant orbital.

DOI: 10.83295/xfdq-re5p

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Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas

OpenAI

Published Oct 6, 2026• 0 citations

Publisher

Abstract

DOI: 10.83295/xfdq-re5p

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{\omega}

, the temperature T, and the particle number N are related by

{N \sim (T / \omega)^{3} \to\infty}

while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.

4 \pi a \left( 2 \varrho ^2 - \varrho _0^2 \right)

4πa2ϱ2-ϱ02 . Here

\varrho

ϱ denotes the density of the system and

\varrho _0

ϱ0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose–Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution.