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Exact relative-shell canonical normalization and response

Statement

Use the relative Kepler Hamiltonian and shell $Q = {q \in R^{3} : r_{0} \leq ∣ q ∣ \leq R}$ from Regulated relative Kepler phase volume and shell law, with $μ, A, r_{0} > 0$ and $R > 16 r_{0}$ . For every $β > 0$ , define $Z_{r e l} (β) = \frac{1}{h_{*}^{3}} \int_{Q} \int_{R^{3}} e^{- β h (q, p)} d^{3} p d^{3} q .$ Then $0 < \frac{∣ Q ∣ e^{β A / R}}{h_{*}^{3}} {(\frac{2 π μ}{β})}^{3 / 2} \leq Z_{r e l} (β) \leq \frac{∣ Q ∣ e^{β A / r_{0}}}{h_{*}^{3}} {(\frac{2 π μ}{β})}^{3 / 2} < \infty .$ All energy moments are finite, and on every compact positive $β$ interval one integrable majorant justifies all derivatives of $Z_{r e l}$ . Put $d ν_{r e l} = h_{*}^{- 3} d^{3} q d^{3} p$ on $Q \times R^{3}$ and $p_{β} = Z_{r e l} (β)^{- 1} e^{- β h}$ . For a measurable observable $A : Q \times R^{3} \to R$ with finite second moment, define $⟨ A ⟩_{β} : = \int A p_{β} d ν_{r e l}$ and ${Var}_{β} (A) : = \int (A - ⟨ A ⟩_{β})^{2} p_{β} d ν_{r e l}$ . The normalized canonical density has $⟨ h ⟩_{β} = - \partial_{β} \log Z_{r e l}, \frac{d ⟨ h ⟩_{β}}{d β} = - {Var}_{β} (h), C_{K} : = \frac{d ⟨ h ⟩_{β}}{d T_{K}} = k_{B} β^{2} {Var}_{β} (h) \geq 0, T_{K} = (k_{B} β)^{- 1} .$ This is the exact six-dimensional relative canonical preparation; center-of-mass degrees are absent.

Facts & Assumptions

Given: The relative phase shell and Hamiltonian, $β > 0$ , $μ, A, r_{0} > 0$ , and $R > 16 r_{0}$ .

[F1]

The relative shell is $Q = {q : r_{0} \leq ∣ q ∣ \leq R}$ and $h (q, p) = ∣ p ∣^{2} / (2 μ) - A / ∣ q ∣$ , with measure $d^{3} q d^{3} p / h_{*}^{3}$ . (Regulated relative Kepler phase volume and shell law)

[F2]

Radial integration gives $∣ Q ∣ = 4 π \int_{r_{0}}^{R} r^{2} d r = \frac{4 π}{3} (R^{3} - r_{0}^{3})$ . (Radial integration over a bounded Euclidean shell)

[F3]

Gaussian momentum moments are finite and uniformly bounded on compact positive inverse-temperature intervals. (Finite-dimensional Gaussian normalization and moment envelope)

[F4]

Differentiation under an integral is valid when the parameter derivative has an integrable uniform majorant. (Differentiation under the integral sign)

[F5]

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

Proof

1.1F1F2F3F5algebra

On $Q$ , $- A / r_{0} \leq - A / ∣ q ∣ \leq - A / R$ . Therefore $e^{β A / R} e^{- β ∣ p ∣^{2} / (2 μ)} \leq e^{- β h (q, p)} \leq e^{β A / r_{0}} e^{- β ∣ p ∣^{2} / (2 μ)} .$ The position volume is positive and finite by [F2]. Tonelli [F5] and the Gaussian formula [F3] integrate the momentum factor as $(2 π μ / β)^{3 / 2}$ , proving both strict positivity and the stated finite bounds.

1.2F1F2F3F4algebra

Fix $[β_{0}, β_{1}] \subset (0, \infty)$ . Since $∣ - A / ∣ q ∣ ∣ \leq A / r_{0}$ , for every integer $j \geq 0$ a finite constant $B_{j}$ gives $∣ h (q, p) ∣^{j} e^{- β h (q, p)} \leq B_{j} (1 + ∣ p ∣^{2 j}) e^{- β_{0} ∣ p ∣^{2} / (2 μ)} .$ This is integrable on $Q \times R^{3}$ by [F2] and [F3], independently of $β$ in the compact interval. The integrand derivatives are $(- h)^{j} e^{- β h}$ , so [F4] applied successively proves all moment and derivative formulas.

2.1F3F4step 1.1step 1.2algebra∎

The first two derivatives are $Z_{r e l}^{'} = - \int h e^{- β h} d ν$ and $Z_{r e l}^{''} = \int h^{2} e^{- β h} d ν$ . Dividing by $Z_{r e l} > 0$ and differentiating the quotient gives $d ⟨ h ⟩_{β} / d β = - ⟨ h^{2} ⟩_{β} + ⟨ h ⟩_{β}^{2} = - {Var}_{β} (h)$ . Since $d T_{K} / d β = - 1 / (k_{B} β^{2})$ , the chain rule yields the displayed $C_{K} \geq 0$ .

Exact relative-shell canonical normalization and response

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources