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Canonical momentum distribution and equipartition

Physical scope

Finite classical canonical model with quadratic unconstrained momenta and bounded potential on a finite-volume configuration space; excludes constraints, relativistic dispersion, and empirical equilibration.

Statement

Assume Countable Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ), as in the cotangent phase-reference convention. Let $d, N \geq 1$ and put $n = d N$ . Use the ambient smooth configuration manifold $R^{n} = (R^{d})^{N}$ and its cotangent phase space $Γ_{N} = T^{*} R^{n} ≅ R^{n} \times R^{n}$ with product Lebesgue measure $d q d p$ in canonical coordinates. Let the allowed one-particle position region $C \subset R^{d}$ be compact and Lebesgue measurable with $0 < ∣ C ∣_{d} < \infty$ , and set $C_{N} = C^{N}$ with product Lebesgue measure and volume $∣ C ∣_{d}^{N}$ . Let $U_{C} : C_{N} \to R$ be bounded and measurable, invariant under permutations of the $N$ particle labels, and interpret measurability here with respect to completed Lebesgue measure on $C_{N}$ . Choose a bounded, permutation-invariant Borel representative ${\tilde{U}}_{C}$ equal to $U_{C}$ almost everywhere, as constructed in step 1.1, and extend it to $U : R^{n} \to R \cup {+ \infty}$ by $U = {\tilde{U}}_{C}$ on $C_{N}$ and $U = + \infty$ outside $C_{N}$ . The canonical state is defined on the Borel phase reference. Its extension to the completed phase-reference sigma-algebra agrees with the density formed from the original $U_{C}$ and is independent of the Borel representative. In particular, the density vanishes outside the allowed finite-volume configuration region. The particles have common mass $m > 0$ . Adopt the canonical preparation of Canonical thermal preparation with the physical reference convention of Classical particle counting and physical observables, Hamiltonian $K (p) : = \sum_{j = 1}^{n} \frac{p_{j}^{2}}{2 m}, H (q, p) : = K (p) + U (q), p \in R^{n},$ and $β = (k_{B} T)^{- 1} > 0$ . Then the partition function is finite and positive, and the momentum components are independent centered Gaussians with variance $m / β = m k_{B} T$ . In particular, $⟨ \frac{p_{j}^{2}}{2 m} ⟩ = \frac{1}{2 β} = \frac{k_{B} T}{2}, ⟨ K ⟩ = \frac{n k_{B} T}{2} .$ These are conditional deductions of the chosen canonical model.

Facts & Assumptions

Given: Countable Choice, the canonical preparation and counting convention, positive quadratic momentum coefficients, bounded potential and finite configuration volume.

[A1]

The state has density proportional to $e^{- β H}$ relative to $d q d p / (N! a_{0}^{n})$ , with $a_{0} > 0$ the fixed action reference (Canonical thermal preparation, Classical particle counting and physical observables).

[A2]

$C_{N} = C^{N}$ is compact, Lebesgue measurable and has finite positive product volume; $U_{C}$ is bounded and completed-Lebesgue-measurable on $C_{N}$ , $β > 0$ , and $m > 0$ . The reference measure $d q d p / (N! a_{0}^{n})$ is defined on the whole ambient Borel phase space $Γ_{N}$ using the Liouville measure of Classical configuration, cotangent phase and reference measures. The canonical density, rather than this reference, vanishes outside $C_{N} \times R^{n}$ , so its integrals may be evaluated on that support.

[F1]

The supplier The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ proves $\int_{R} e^{- x^{2}} d x = \sqrt{π}$ . For each $a > 0$ , the linear diffeomorphism $y = \sqrt{a} x$ has Jacobian $\sqrt{a}$ , so one-dimensional substitution gives $\int_{R} e^{- a x^{2}} d x = a^{- 1 / 2} \int_{R} e^{- y^{2}} d y = \sqrt{π / a}$ .

[F2]

Nonnegative product integrals may be evaluated in either order (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F3]

The canonical phase coordinates are those of Classical configuration, cotangent phase and reference measures. The positive action reference is prescribed by Classical particle counting and physical observables; its SI dimension, and the position and momentum dimensions, follow Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

[F4]

The SI kelvin, Boltzmann constant and relation $β = (k_{B} T)^{- 1}$ use the published thermodynamic units convention (SI quantities and unit conventions).

[F5]

Cauchy-Schwarz inequality for $L^{2}$ gives $\int ∣ f g ∣ d μ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ for square-integrable $f, g$ .

[F6]

Under Countable Choice, $L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets identifies Lebesgue measurable sets with Borel sets modified inside Borel null sets. Countably many such modifications have a common Borel null cover.

Proof

Given: Countable Choice, [A1]–[A2] and [F1]–[F6].

1.1givenA2F2F3F6construct

Since $C_{N}$ is compact, it is Borel. Fix $M > 0$ with $∣ U_{C} ∣ \leq M$ . Approximate $U_{C}$ on $C_{N}$ by its dyadic step functions, whose errors are at most $2^{- k}$ . By [F6] and Countable Choice, replace their countably many Lebesgue-measurable level sets by Borel sets agreeing outside a common Borel null set $Z$ . Clip the resulting Borel step functions to $[- M, M]$ ; their pointwise limsup is a bounded Borel function $V$ equal to $U_{C}$ off $Z$ . Measurability of this limsup follows from countable unions and intersections of its level sets. Set ${\tilde{U}}_{C} (q) : = \frac{1}{N!} \sum_{σ \in S_{N}} V (σ q) .$ The finite average is Borel, bounded and exactly permutation-invariant. Since $U_{C}$ is invariant, the average equals it outside the Borel null set $⋃_{σ} σ^{- 1} Z$ ; these coordinate permutations preserve Lebesgue measure by [F3]. Extend by $+ \infty$ off the Borel set $C_{N}$ . The resulting $H$ is Borel on the phase space. The exceptional configuration set times $R^{n}$ is null: intersect the momentum factor with each bounded integer box, apply [F2], then take the countable union. Hence the Borel density and the completed-Lebesgue density formed from $U_{C}$ agree almost everywhere and give the same probability on the completed phase-reference sigma-algebra. The same argument compares any two representatives.

2.1A1A2F1F2step 1.1algebra

Factor $e^{- β H} = e^{- β U (q)} \prod_{j} e^{- β p_{j}^{2} / (2 m)}$ on $C_{N} \times R^{n}$ ; it is zero outside this region. By [F1]–[F2], $Z = \frac{1}{N! a_{0}^{n}} (\int_{C_{N}} e^{- β U_{C} (q)} d q) {(\frac{2 π m}{β})}^{n / 2} .$ Boundedness of $U_{C}$ , positive finite configuration volume, and positive finite Gaussian factors imply $0 < Z < \infty$ , so the adopted preparation is normalized.

3.1A1F1F3F4F5step 2.1algebra∎

The factorized density makes the momentum marginal the product of densities proportional to $e^{- β p_{j}^{2} / (2 m)}$ , independent of $q$ . Set $a = β / (2 m)$ and integrate $\frac{d}{d p} (p e^{- a p^{2}}) = e^{- a p^{2}} - 2 a p^{2} e^{- a p^{2}}$ over $R$ ; the boundary term vanishes since $p e^{- a p^{2}} \to 0$ at both ends. Dividing by the positive Gaussian integral gives $⟨ p_{j}^{2} ⟩ = 1 / (2 a) = m / β$ . Applying [F5] to $p_{j}$ and $1$ under the normalized marginal gives $⟨ ∣ p_{j} ∣ ⟩ \leq ⟨ p_{j}^{2} ⟩^{1 / 2} < \infty$ ; since the density is even its centered mean is zero. The product form proves independence. Summing the $n$ quadratic contributions proves both equipartition identities.

Remarks

The independent component calculation is on the symmetric labelled-coordinate lift specified in Classical particle counting and physical observables. A single labelled coordinate is not itself a function on the indistinguishable-particle quotient. Its marginal is the law of a uniformly selected component under the permutation-invariant lift; symmetric kinetic sums and averages, including $K / n$ , are quotient observables and retain the displayed expectations.

Canonical momentum distribution and equipartition

Statement

Facts & Assumptions

Proof

Remarks

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