Browse › Probability and statistical mechanics › 228 Continuum phase transitions for radial pair potentials
A radial continuum phase transition with algebraic decay
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We construct a bounded, continuous, stable radial pair potential in three dimensions with $|\phi(r)|\le Cr^{-3-1/32}$ for r ≥ 1. Throughout an open interval of positive densities, its canonical thermodynamic free energy is finite at every positive inverse temperature and has a strict downward derivative jump at one common finite positive inverse temperature.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Continuum phase transitions for radial pair potentials (Lean scope notes) (lean/docs/228.md, 2026). Release's scope note: it describes the formalized statement of this paper's theorem (bounded continuous potential, $|\phi(r)|\le Cr^{-3-1/32}$, a common $\beta_c\in[7/8,9/8]$ on a density interval around $5p/3$) plus the earlier fixed-density statement, with comparator files
RadialTransition.leanandRadialDensityInterval.lean. [family-level] - OpenAI, RadialTransition/DensityInterval.lean (openai/math, commit adc7f12). The release's Lean module with theorem
densityInterval, the formal counterpart of this paper's Theorem 1.1.
Possibly relevant literature (found by the agent, not checked)
- Israel, Existence of phase transitions for long-range interactions (Comm. Math. Phys. 43 (1975), 59–68; entry in the author's publication list). This is a convex tangent-functional method for producing coexistence, related to this paper's argument that preserves pressure supporting slopes; the companion paper cites it, but this paper does not.
- Ruelle, On manifolds of phase coexistence (Theor. Math. Phys. 30 (1977), 24–29). Builds on Israel's convexity theorem to show that coexistence points are not isolated in the space of interactions, which is related to the paper's persistence of a ball of supporting slopes under successive attractions; the companion paper cites it, but this paper does not.
- Göbel, Jenssen, Michelen, Pappik, Perkins, Schiller, Uniqueness, analyticity and mixing for Gibbs point processes via spectral gaps (arXiv:2606.28009, June 2026). Shows that for certain repulsive radial pair potentials there is no phase transition at any activity; this contrasts with the paper's construction, which adds attractive radial terms to a bounded repulsive core.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 228
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026,
author = {{OpenAI}},
title = {{A radial continuum phase transition with algebraic decay}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf}{OAI:A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/228.md. The Lean statements have not been checked by people against the paper's claims.
Family 228: Continuum phase transitions for radial pair potentials
Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval. One potential has a divergent repulsive core; another is bounded and continuous with an integrable power-law tail, realizing the type of transition sought in Simon's continuum problem.
Other papers in this family:
- A continuum temperature singularity for a radial pair potential · September 24, 2026