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Finite gravitational Gibbs normalization and differentiable moments

Statement

Use the box, mass $m$ , integer $N \geq 1$ , regulator $a > 0$ , and Hamiltonian $H_{N, a} = K_{N} + U_{a}$ of Finite softened gravitational Hamiltonian and domain, with $K_{N} = \sum_{i} ∣ p_{i} ∣^{2} / (2 m)$ . Put $C_{N} = G m^{2} N (N - 1) / (2 a)$ and use the reference measure of Finite gravitational phase and entropy reference. For every $β > 0$ , $0 < \frac{∣ B ∣^{N}}{N! h_{*}^{3 N}} {(\frac{2 π m}{β})}^{3 N / 2} \leq Z_{N} (β) \leq e^{β C_{N}} \frac{∣ B ∣^{N}}{N! h_{*}^{3 N}} {(\frac{2 π m}{β})}^{3 N / 2} < \infty .$ The density $p_{β} = Z_{N}^{- 1} e^{- β H_{N, a}}$ is a positive probability density. For each integer $j \geq 0$ , $\int_{Γ_{N}} ∣ H_{N, a} ∣^{j} e^{- β H_{N, a}} d ν_{N}$ is finite, and $Z_{N}$ is $C^{\infty}$ on $(0, \infty)$ with $Z_{N}^{(j)} (β) = \int_{Γ_{N}} (- H_{N, a})^{j} e^{- β H_{N, a}} d ν_{N} .$ On every compact positive $β$ interval, one integrable majorant works for each fixed derivative order.

Facts & Assumptions

Given: A bounded measurable box $B \subseteq R^{3}$ with $0 < ∣ B ∣ < \infty$ , $N \geq 1$ , $m, G, a, β > 0$ , and the finite phase model.

[F1]

The softened model has $H_{N, a} = K_{N} + U_{a}$ on $B^{N} \times R^{3 N}$ , where each pair term in $U_{a}$ is $- G m^{2} / \sqrt{∣ q_{i} - q_{j} ∣^{2} + a^{2}}$ . (Finite softened gravitational Hamiltonian and domain)

[F2]

For $d \geq 1$ , $m > 0$ , and a compact interval $[β_{0}, β_{1}] \subset (0, \infty)$ , all Gaussian momentum moments $\int_{R^{d}} (1 + ∣ p ∣^{2 j}) e^{- β ∣ p ∣^{2} / (2 m)} d p$ have a common finite bound. (Finite-dimensional Gaussian normalization and moment envelope)

[F3]

If an integrable section is differentiable in a parameter and its derivative is dominated by one integrable majorant on an open interval, the derivative passes through the integral. (Differentiation under the integral sign)

[F4]

Tonelli factors nonnegative measurable integrals on finite products of Euclidean Lebesgue spaces. (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product)

Proof

1.1givenF1F2F4

Each pair denominator is at least $a$ , so $- C_{N} \leq U_{a} \leq 0$ and $K_{N} \geq 0$ . Consequently $e^{- β K_{N}} \leq e^{- β H_{N, a}} \leq e^{β C_{N}} e^{- β K_{N}}$ . Tonelli [F4] and [F2] evaluate the momentum integral as $(2 π m / β)^{3 N / 2}$ ; the position integral is $∣ B ∣^{N}$ , so the stated positive lower and finite upper bounds follow after division by $N! h_{*}^{3 N}$ .

2.1F1F2step 1.1algebra

Fix a compact interval $[β_{0}, β_{1}] \subset (0, \infty)$ . Since $∣ U_{a} ∣ \leq C_{N}$ and $K_{N} = ∣ p ∣^{2} / (2 m)$ in $d = 3 N$ momentum coordinates, for every fixed $j$ there is a finite $A_{j}$ such that $∣ H_{N, a} ∣^{j} e^{- β H_{N, a}} \leq A_{j} (1 + ∣ p ∣^{2 j}) e^{- β_{0} ∣ p ∣^{2} / (2 m)}$ for every $β \in [β_{0}, β_{1}]$ and every phase point. The right side is integrable by [F2] times the finite position volume $∣ B ∣^{N}$ . This proves all absolute energy moments and supplies an integrable compact-parameter envelope.

3.1F2F3step 2.1algebra∎

For each $j$ , the parameter derivative of $(- H_{N, a})^{j} e^{- β H_{N, a}}$ is $(- H_{N, a})^{j + 1} e^{- β H_{N, a}}$ ; its measurability and an integrable uniform bound are provided by step 2.1 at order $j + 1$ . Apply [F3] successively on compact subintervals of $(0, \infty)$ , starting with $j = 0$ , to differentiate under the integral any finite number of times. The formula for $Z_{N}^{(j)}$ follows by induction, and the moment of order zero gives the positive normalized density.

Finite gravitational Gibbs normalization and differentiable moments

Statement

Facts & Assumptions

Proof

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