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

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Fixed softening does not guarantee finite extensive free energy

Statement refuted

Fix $β > 0$ . For each box $B_{N}$ , let $V_{N} = ∣ B_{N} ∣$ , let $Z_{N} (β)$ be its canonical partition function, and set $F_{N} (β) = - β^{- 1} \log Z_{N} (β)$ . 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. The claim refuted is that a fixed positive softening length guarantees such a finite limit for every sequence of unscaled self-gravitating boxes whose volumes are bounded above and below by positive multiples of $N$ . The counterexample below has both $f_{N} \to - \infty$ and $g_{N} \to - \infty$ .

Facts & Assumptions

Given: The softened Newtonian pair potential with fixed $a > 0$ and the growing-box hypotheses in Point collapse and unscaled gravity nonextensivity.

[F1]

For boxes containing a fixed positive-volume cube, the softened unscaled model has $\log Z_{N} / N \to + \infty$ along the concentration sequence, so its free energy per particle tends to $- \infty$ . (Point collapse and unscaled gravity nonextensivity)

Counterexample

1.1givenF1construct

Fix $L > 0$ , take $C = [0, L]^{3}$ , and set $B_{N} = [0, L N^{1 / 3}]^{3}$ for every integer $N \geq 1$ . Then $C \subseteq B_{N}$ , $∣ C ∣ = L^{3} > 0$ , and $V_{N} = L^{3} N$ , so this is a bounded-box sequence satisfying the softened concentration hypotheses in [F1].

2.1F1step 1.1algebra∎

Clause 2 of [F1] gives $\log Z_{N} / N \geq β c (N - 1) - \log N + K_{β} \to + \infty$ , hence $f_{N} = - β^{- 1} \log Z_{N} / N \to - \infty$ . Since $V_{N} = L^{3} N$ , also $g_{N} = f_{N} / L^{3} \to - \infty$ . Neither normalization has a finite real thermodynamic-limit value along this sequence, contradicting the claimed guarantee.

Depends on

Point collapse and unscaled gravity nonextensivity

Fixed softening does not guarantee finite extensive free energy

Statement refuted

Facts & Assumptions

Counterexample

Depends on

Dependency tree · 2 levels

Sources