OAI.RadialTransition.densityInterval

The theorem states that there exist a real number p, a pair potential φ : ℝ≥0 → ℝ (a function of interparticle distance), real constants B, C, η, βc, and a two-variable function f(β,ρ) such that the following hold. First, p>0 and p is the packing density, meaning that the maximal number of points in the cube [0,L]³ in ℝ³ with pairwise distances at least 1, divided by L³, tends to p as L→∞. The potential φ is bounded and continuous, with B>0 and C>0, and is stable with constant B: for every N and every configuration of N points in ℝ³, the total pair energy ∑_{i<j} φ(|xᵢ−xⱼ|) is at least −B·N. It also decays algebraically: |φ(r)| ≤ C·r^(−3−1/32) for all r≥1. Next, η>0, the interval (5p/3−η, 5p/3+η) lies in the positive reals, and βc lies in [7/8, 9/8]. Finally, for every density ρ in (5p/3−η, 5p/3+η), the function β ↦ f(β,ρ) is the canonical free energy of φ at density ρ, meaning that for every β>0 the limit as L→∞ of −log Z/(β L³) equals f(β,ρ), where Z is the partition function of N=⌊ρL³⌋ particles in the cube of side L, namely (1/N!)∫ exp(−β·energy) over the configuration cube. Moreover, β ↦ f(β,ρ) has a strict corner at βc: it has a left derivative dLeft and a right derivative dRight there with dRight<dLeft, and it is not differentiable at βc.