Continuum phase transitions for radial pair potentials
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
A continuum temperature singularity for a radial pair potential
We construct a stable radial pair potential in three dimensions whose canonical free energy has a strict downward derivative jump at one common finite positive inverse temperature throughout an open interval of positive densities. The potential has a divergent repulsive core, a nontrivial attractive interval, and an integrable…
Full source abstract
We construct a stable radial pair potential in three dimensions whose canonical free energy has a strict downward derivative jump at one common finite positive inverse temperature throughout an open interval of positive densities. The potential has a divergent repulsive core, a nontrivial attractive interval, and an integrable tail satisfying \phi(r)=o(r^{-3}). Its free-cube canonical free energy is finite at every positive inverse temperature and density.
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A radial continuum phase transition with algebraic decay
We construct a bounded, continuous, stable radial pair potential in three dimensions with |\phi(r)|≤ 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…
Full source abstract
We construct a bounded, continuous, stable radial pair potential in three dimensions with |\phi(r)|≤ 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.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
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