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Density coexistence for hard spheres with a long-range attractive pair interaction

Primarily AI-generated textHuman understanding: no partsmath-ph — Mathematical Physicsmath.PR — Probability

Contributed by Qidong He ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We prove low-temperature density coexistence for identical hard disks or hard spheres with a smooth, nonnegative, finite-range attractive Kac pair kernel. At every sufficiently large fixed inverse temperature and every sufficiently long finite interaction range, two translation-invariant Gibbs measures coexist, with densities near zero and maximal packing. The pressure is therefore nondifferentiable in chemical potential. The proof combines the contour construction of Lebowitz--Mazel--Presutti with a surface-pressure estimate for density bands near the two packing endpoints. A packing bound controls the fine density profile; centering the interaction makes its residual boundary contribution quadratic in the band width. Compatible near-packed configurations and conditional entropy then control the hard-core boundary contribution. These estimates allow the dense restricted ensemble to retain positional order and long-range correlations.

Provenance statement

Astra was prompted to prove the existence of a first-order phase transition in a continuum system of hard disks or spheres with a long-range attractive two-body Kac interaction, either via the analytic route by proving the non-differentiability of its thermodynamic potential, or via the probabilistic route by proving the coexistence of two Gibbs measures distinguished by their particle densities. It chose the analytic route and, voluntarily restricting its attention to the low-temperature regime, produced a candidate proof in about 1.5 hours. It seems that Astra has managed to completely sidestep the need to understand the mixing properties and near-optimal packings of the continuum hard-core model. ***THE RESULT IS NOT HUMAN-VERIFIED***

Tools used

OpenAI
ChatGPTVersion 6 Astra
Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1Submitted by Qidong HeInitial depositCurrentOct 31, 2026