For each hard-sphere radius a>0 and every sufficiently small fixed density ρ>0, there is a temperature T(a,ρ)>0 independent of volume at which the exact three-dimensional canonical Gibbs state has a positive condensate fraction in the constant orbital as N/L³ tends to ρ.
For each hard-sphere radius a>0 and every sufficiently small fixed density ρ>0, there is a temperature T(a,ρ)>0 independent of volume at which the exact three-dimensional canonical Gibbs state has a positive condensate fraction in the constant orbital as N/L³ tends to ρ.
For each hard-sphere radius a>0 and every sufficiently small fixed density ρ>0, there is a temperature T(a,ρ)>0 independent of volume at which the exact three-dimensional canonical Gibbs state has a positive condensate fraction in the constant orbital as N/L³ tends to ρ.
If correct, establishes macroscopic coherence at a fixed positive temperature in the exact dilute hard-sphere model.
Does the dilute three-dimensional hard-sphere gas condense at a volume-independent positive temperature?
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OpenAI reports positive-temperature condensation for dilute hard spheres
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