CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Hard-core exclusion does not guarantee a finite unscaled-gravity limit

Statement refuted

Fix $β > 0$ and consider the fixed-density sequence indexed by $N = n^{3}$ . Let $V_{N}$ be the box volume, $Z_{N} (β)$ the canonical partition function, and $F_{N} (β) = - β^{- 1} \log Z_{N} (β)$ . The per-particle free-energy density is $f_{N} = F_{N} / N$ (in joules), and the per-volume free-energy density is $g_{N} = F_{N} / V_{N}$ (in joules per cubic metre). Here a “finite extensive free-energy density” means that the chosen normalized sequence has a finite real thermodynamic-limit value as $n \to \infty$ . The claim refuted is that a fixed hard-core exclusion distance and fixed particle density guarantee this for the unscaled Newtonian long-range attraction.

Facts & Assumptions

Given: The unscaled point attraction $- G m^{2} / r$ , a positive hard-core separation, and boxes with side proportional to $N^{1 / 3}$ .

[F1]

For $N = n^{3}$ , a cubic packing with fixed minimum separation still has interaction energy at most $- c_{0} N^{5 / 3}$ on positive-volume position sets, and its canonical free energy per particle tends to $- \infty$ . (Point collapse and unscaled gravity nonextensivity)

Counterexample

1.1givenF1construct

Fix a grid spacing larger than the hard-core radius and take $N = n^{3}$ particles, one in each small cube around the cubic grid sites. The enclosing box has side proportional to $n = N^{1 / 3}$ and the product of the small cubes is an admissible positive-volume position set, matching [F1].

2.1F1step 1.1algebra∎

By [F1], $f_{N} = F_{N} / N = - β^{- 1} \log Z_{N} / N \to - \infty$ . Here $V_{N} = ∣ B_{n} ∣ = d^{3} N$ , so $g_{N} = F_{N} / V_{N} = d^{- 3} f_{N} \to - \infty$ . Neither normalization has a finite real thermodynamic-limit value, contradicting the claimed guarantee.

Depends on

Point collapse and unscaled gravity nonextensivity

Hard-core exclusion does not guarantee a finite unscaled-gravity limit

Statement refuted

Facts & Assumptions

Counterexample

Depends on

Dependency tree · 2 levels

Sources