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Point collapse and unscaled gravity nonextensivity

Statement

Use the dimensionless phase reference of Finite gravitational phase and entropy reference and the softened Hamiltonian of Finite softened gravitational Hamiltonian and domain.

For two equal positive masses in a bounded three-dimensional box with nonempty interior, the distinct point Hamiltonian $H_{2}^{p t} (q, p) = \frac{∣ p_{1} ∣^{2} + ∣ p_{2} ∣^{2}}{2 m} - \frac{G m^{2}}{∣ q_{1} - q_{2} ∣}$ makes the canonical partition integral infinite for every $β > 0$ .
For fixed $a > 0$ , let $B_{N}$ be a sequence of bounded boxes containing one fixed cube $C$ of volume $v > 0$ and diameter $D < \infty$ , with constants $b_{1}, b_{2} > 0$ such that $b_{1} N \leq ∣ B_{N} ∣ \leq b_{2} N$ . The softened canonical partition functions satisfy $\frac{\log Z_{N} (β)}{N} \geq β c (N - 1) - \log N + K_{β}, c = \frac{G m^{2}}{2 \sqrt{D^{2} + a^{2}}} > 0,$ where $K_{β}$ is independent of $N$ . Hence $\log Z_{N} / N \to + \infty$ and the free energy per particle $- β^{- 1} \log Z_{N} / N$ tends to $- \infty$ .
For the unscaled point attraction $- G m^{2} / r$ restricted to the hard-core position domain $D_{N} = {q \in B_{n}^{N} : ∣ q_{i} - q_{j} ∣ > a for i \neq j}$ with fixed $a > 0$ , there is a fixed-density sequence of integers $n \geq 2$ , $N = n^{3}$ , boxes $B_{n} = (0, n d)^{3}$ of side $O (n)$ for fixed $d > a$ , and positive-volume admissible position sets on which the interaction energy is at most $- c_{0} N^{5 / 3}$ for some $c_{0} > 0$ . Its canonical partition functions therefore also satisfy $\log Z_{N} / N \to + \infty$ .

For the sequences in clauses 2 and 3, write $F_{N} (β) = - β^{- 1} \log Z_{N} (β)$ . In clause 2, the free energy per particle $F_{N} / N$ tends to $- \infty$ . Since $b_{1} \leq ∣ B_{N} ∣ / N \leq b_{2}$ , the free energy per volume $F_{N} / ∣ B_{N} ∣$ also tends to $- \infty$ . Clause 3 gives the same per-particle limit along its fixed-density sequence, and there $∣ B_{n} ∣ / N = d^{3}$ , so the per-volume free energy also tends to $- \infty$ . Thus finite boxes at each particle count, fixed softening in clause 2, and hard-core exclusion with unscaled attraction in clause 3 do not ensure a finite real thermodynamic-limit value for either normalized free energy in these sequences. These results do not address differently scaled interactions or every possible gravitational thermodynamic limit.

Facts & Assumptions

Given: The phase reference, the softened Hamiltonian, $G, m, a, β > 0$ , and the box or packing hypotheses in the Statement.

[F1]

The phase reference is $d ν_{N} = d^{3 N} q d^{3 N} p / (N! h_{*}^{3 N})$ . (Finite gravitational phase and entropy reference)

[F2]

The softened model has kinetic energy $\sum_{i} ∣ p_{i} ∣^{2} / (2 m)$ and pair attraction $- G m^{2} / \sqrt{∣ q_{i} - q_{j} ∣^{2} + a^{2}}$ . (Finite softened gravitational Hamiltonian and domain)

[F3]

For $d = 3 N$ , the momentum Gaussian integral is $(2 π m / β)^{3 N / 2}$ and its polynomial moments are finite. (Finite-dimensional Gaussian normalization and moment envelope)

[F4]

Tonelli's theorem applies to the nonnegative Boltzmann integrand on product Lebesgue spaces. (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product)

Proof

1.1givenF1F3F4algebra

Choose an interior point $q_{0}$ of the box and a length $ℓ > 0$ small enough that the cubes below lie in the box for all sufficiently large integers $n$ . Let $Q_{1, n} = q_{0} + (0, ℓ / n)^{3}$ and $Q_{2, n} = q_{0} + (2 ℓ / n, 0, 0) + (0, ℓ / n)^{3}$ . They are disjoint, have volumes $(ℓ / n)^{3}$ , and every pair satisfies $0 < ∣ q_{2} - q_{1} ∣ \leq \sqrt{11} ℓ / n$ . On their product the point-potential Boltzmann weight is at least $\exp (β G m^{2} n / (\sqrt{11} ℓ))$ , so its positional integral is at least $(ℓ / n)^{6} \exp (β G m^{2} n / (\sqrt{11} ℓ))$ . This lower bound tends to infinity, proving divergence of the positional integral. The independent momentum Gaussian integral is positive and finite by [F3]; Tonelli [F4] therefore makes the full partition integral infinite.

1.2F1F2F3givenalgebra

In the softened model, when every $q_{i} \in C$ , all pair distances are at most $D$ and hence $U_{a} (q) \leq - G m^{2} N (N - 1) / (2 \sqrt{D^{2} + a^{2}}) = - c N (N - 1)$ . Restricting the nonnegative partition integral to $C^{N} \times R^{3 N}$ and integrating momenta gives $Z_{N} (β) \geq \frac{v^{N}}{N! h_{*}^{3 N}} e^{β c N (N - 1)} {(\frac{2 π m}{β})}^{3 N / 2} .$ Use $N! \leq N^{N}$ , divide logarithms by $N$ , and gather all $N$ -independent terms into $K_{β}$ to obtain the claimed bound. Since $β c N - \log N \to + \infty$ , the free-energy conclusion follows.

2.1F1F3givenalgebra∎

For each fixed $N = n^{3}$ , the hard-core condition bounds each pair attraction below by $- G m^{2} / a$ ; the finite box and Gaussian momentum integral therefore make its partition function finite. Fix $d > a + 2 δ$ with $0 < δ < d$ . In $B_{n} = (0, n d)^{3}$ , place centers at $d (k_{1} + \frac{1}{2}, k_{2} + \frac{1}{2}, k_{3} + \frac{1}{2})$ for $k_{i} \in {0, \dots, n - 1}$ and choose the open cube of side $δ$ centered at each center. These cubes lie in $B_{n}$ and have positive volume. Distinct grid centers are at least $d$ apart, and two points in their cubes differ from their centers by at most $\sqrt{3} δ$ ; hence every particle separation exceeds $d - \sqrt{3} δ > a$ , since $2 > \sqrt{3}$ . The product cubes are admissible. Each pair distance is at most $\sqrt{3} (n - 1) d + \sqrt{3} δ \leq c_{r} n d$ with $c_{r} = \sqrt{3}$ . Since $∣ B_{n} ∣ = n^{3} d^{3} = N d^{3}$ , this sequence has fixed number density $d^{- 3}$ . There are $N (N - 1) / 2$ pairs; for $N \geq 2$ , $U_{N} \leq - \frac{G m^{2}}{c_{r} n d} \frac{N (N - 1)}{2} \leq - c_{0} N^{5 / 3}, c_{0} = \frac{G m^{2}}{4 c_{r} d} > 0,$ because $n = N^{1 / 3}$ . Restricting the finite partition integral to these product cubes, integrating momenta, and using $N! \leq N^{N}$ gives $\log Z_{N} / N \geq β c_{0} N^{2 / 3} - \log N + K_{β} \to + \infty$ .

Point collapse and unscaled gravity nonextensivity

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources