Existence of a gas–liquid transition in the two-dimensional hard-disk model

Determine whether the two-dimensional monodisperse hard-disk model admits a gas–liquid phase transition, or prove that such a transition is excluded.

Background

The paper discusses the accepted phase behavior of the two-dimensional hard-disk system, in which the configurational partition function is generally regarded as analytic through the fluid regime up to the fluid–hexatic transition. However, the authors note that the absence of a gas–liquid transition has not been mathematically established. Resolving this issue would provide a rigorous foundation for the distinction between the low-density and high-density fluid regimes discussed in the paper and would clarify whether an additional thermodynamic phase transition exists below the fluid–hexatic transition.

References

Although there is no mathematical proof excluding a gas-liquid transition in the HD model, it is generally accepted that the configurational PF is analytic up to the fluid-hexatic transition near $\eta \simeq 0.7$ so that in this range of $\eta $ the low-density state smoothly transitions to higher-density states .

— Closed analytical form of many-body free volume and thermodynamics of monodisperse hard disks  (2610.08265 - Pergamenshchik et al., 6 Oct 2026) in Section II, subsection “Factorization of the Partition Function”