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First-order phase transition in Onsager's model of hard rods in R3\mathbb R^3

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

Contributed by Qidong He ↗

Version 1 / Oct 31, 2026 / CC BY 4.0

Abstract

We prove a first-order transition for hard rods in R3\mathbb R^3 whose orientations vary continuously over the sphere. For every sufficiently small positive ratio of diameter to length, the grand-canonical pressure has a nondifferentiable point, with a jump in number density at the intermediate Onsager scale. Following the phase-pressure strategy of Lebowitz, Mazel, and Presutti, we connect coexistence in the homogeneous Onsager functional to the microscopic pressure through an exact auxiliary-field representation, exponential decay of correlations in the rod gas at fixed auxiliary field, and estimates of the free-energy cost of interfaces between connected families of minimizing angular profiles.

Provenance statement

Astra was prompted to prove the existence of a first-order phase transition for the hard-rod model in the three-dimensional continuum, either via the analytic route by proving the non-differentiability of its thermodynamic potential, or via the probabilistic route by proving the coexistence of a continuum of Gibbs measures. It chose the analytic route and produced a candidate proof in an essentially zero-shot run lasting about six hours, with occasional interruptions due to connection errors. The argument closely resembled the work of Lebowitz, Mazel, and Presutti (1999) on the existence of a liquid-vapor transition in a continuum model. Astra was then directed to make these connections explicit and, more broadly, give appropriate credit to the existing literature. It seems that, in working directly with the grand-canonical pressure, Astra has cleverly circumvented the need to prove continuous symmetry breaking. ***THE RESULT IS NOT HUMAN-VERIFIED***

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Version history

  1. v1Submitted by Qidong HeInitial depositCurrentOct 30, 2026