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Kinetic-plus-potential shell marginals and moments

Statement

Let integers $d, N \geq 1$ satisfy $n = d N \geq 3$ . Choose a basis $b_{1}, \dots, b_{d}$ of $R^{d}$ and the full-rank lattice $Λ : = \sum_{i = 1}^{d} Z b_{i}$ . The one-particle flat torus is $Q : = R^{d} / Λ$ with its quotient Euclidean metric and translation-invariant volume $V : = ∣ \det (b_{1}, \dots, b_{d}) ∣$ . Write $Λ^{N} : = Λ \times \dots \times Λ$ ( $N$ factors) and set $Q_{N} : = Q^{N} ≅ R^{n} / Λ^{N}$ , whose position volume is $V^{N}$ . Use the global parallel cotangent trivialization $Γ_{N} ≅ Q_{N} \times R^{n}$ and the product of quotient Lebesgue volume $d q$ with fibre Lebesgue measure $d p$ in the canonical reference $m_{N} = c_{N} d q d p$ of Classical configuration, cotangent phase and reference measures. Let $m_{j} > 0$ for $1 \leq j \leq n$ , and let $U : Q_{N} \to R$ be bounded with a smooth lift $\tilde{U} : R^{n} \to R$ satisfying $\tilde{U} (x + λ) = \tilde{U} (x)$ for all $x \in R^{n}$ and $λ \in Λ^{N}$ . Set $H (q, p) = \sum_{j = 1}^{n} \frac{p_{j}^{2}}{2 m_{j}} + U (q), α = n / 2, C_{m} = \prod_{j = 1}^{n} \sqrt{2 m_{j}} .$ Let $B_{n}$ be the Euclidean volume of the unit ball in $R^{n}$ . Fix an energy $E$ that is a regular value of $H$ and whose shell $Σ_{E} = H^{- 1} ({E})$ is nonempty. For every continuous $g : Q_{N} \to R$ , the coarea shell measure from Coarea and invariant probability on compact regular phase shells satisfies $μ_{E} (g \circ π_{N}) = c_{N} C_{m} α B_{n} \int_{Q_{N}} g (q) (E - U (q))_{+}^{α - 1} d q,$ where $x_{+} = \max (x, 0)$ . Thus the position marginal of $P_{E}$ has density proportional to $(E - U)_{+}^{α - 1}$ . In the scaled momentum coordinates $r_{j} = p_{j} / \sqrt{2 m_{j}}$ , the coarea shell measure over each $q$ with $E > U (q)$ has fiber factor $d A_{r} / (2 ∣ r ∣)$ on the sphere $∣ r ∣ = \sqrt{E - U (q)}$ . Define $σ_{E, q}$ as Euclidean surface area on that sphere divided by its total area. It satisfies $\int \frac{p_{j}^{2}}{2 m_{j}} d σ_{E, q} = \frac{E - U (q)}{n}, \int \sum_{j} \frac{p_{j}^{2}}{2 m_{j}} d P_{E} = \int (E - U (q)) d P_{E} (q) .$ The result concerns the full kinetic-plus-potential shell; it does not replace it by a shell of potential energy alone.

Facts & Assumptions

Given: The flat compact configuration torus, positive masses, bounded smooth potential, regular nonempty full shell, and full AC for the coarea supplier.

[A1]

Assume the Axiom of Choice required by Coarea and invariant probability on compact regular phase shells and its published Euclidean supplier (The Axiom of Choice).

[A2]

$Q_{N}$ is compact, $U$ is smooth, and $E$ is a regular value; therefore $Σ_{E}$ is a compact smooth shell and a compact energy band exists around $E$ .

[F1]

The phase reference is $c_{N} d q d p$ in canonical coordinates (Classical configuration, cotangent phase and reference measures).

[F2]

The compact-band shell measure disintegrates phase volume over energy and equals the coarea density (Coarea and invariant probability on compact regular phase shells).

[F3]

A nonnegative $C^{1}$ change of variables transforms Euclidean integrals by the absolute Jacobian under the stated choice premise (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions, The Axiom of Choice).

[F4]

Dominated convergence permits differentiation limits with one integrable majorant (Dominated convergence).

[F5]

Finite measures agreeing on a generating $π$ -system agree on its generated sigma-algebra (Finite measures agreeing on a generating pi-system and on the whole space are equal).

[F6]

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

Proof

Given: [A1]–[A2] and the phase-volume, coarea, change-of-variables, convergence, uniqueness and product-integration facts [F1]–[F6].

1.1A1A2F1F3F6algebra

Put $r_{j} = p_{j} / \sqrt{2 m_{j}}$ , so $d p = C_{m} d r$ and the kinetic energy is $∣ r ∣^{2}$ . For $t \geq 0$ , scalar dilation $r = \sqrt{t} u$ has Jacobian $t^{n / 2}$ , so the $r$ -volume of ${∣ r ∣^{2} \leq t}$ is $B_{n} t^{α}$ . Integrating over $q$ therefore gives the sublevel volume $m_{N} {H < E} = c_{N} C_{m} B_{n} \int_{Q_{N}} (E - U (q))_{+}^{α} d q .$ The level boundary has zero phase volume: in each finite regular graph chart, every one-dimensional vertical section is empty or a singleton, so [F6] gives zero ambient measure; finitely many charts cover the shell.

2.1A2F2F4step 1.1

Choose $E_{-} < E$ in $J$ and vary the upper endpoint $E$ in a compact subinterval of $J$ . The coarea disintegration [F2], applied to the compact band $H^{- 1} ([E_{-}, E])$ , gives the volume of ${E_{-} < H < E}$ as $\int_{E_{-}}^{E} μ_{s} (1) d s$ . The explicit sublevel formula in step 1.1 makes this band volume $F (E) - F (E_{-})$ , where $F (E) = c_{N} C_{m} B_{n} \int (E - U)_{+}^{α} d q$ . Its difference quotients are bounded on compact $E$ intervals by a finite constant because $Q_{N}$ is compact and $U$ bounded; [F4] gives $F^{'} (E) = c_{N} C_{m} α B_{n} \int (E - U)_{+}^{α - 1} d q$ . The finite graph-chart formula in [F2], with $∣ \partial_{i} H ∣$ bounded below on smaller compact chart supports, makes $s \mapsto μ_{s} (1)$ continuous by [F4]. The fundamental theorem of calculus therefore gives $F^{'} (E) = μ_{E} (1)$ at every regular $E$ .

3.1A2F2F4F5step 2.1algebra

Repeating the radial calculation in step 1.1 and the local energy-derivative argument in step 2.1 with the bounded factor $g (q)$ gives the displayed identity for every continuous $g$ . The two finite Borel measures on $Q_{N}$ consequently agree on continuous test functions; approximating indicators of closed sets by $q \mapsto \max (0, 1 - k d i s t (q, F))$ and applying dominated convergence shows they agree on closed sets, and [F5] extends equality to all Borel sets. Dividing by the positive total shell mass proves the position-marginal assertion.

4.1A2F2F3step 3.1algebra∎

Fix $q$ with $E > U (q)$ and put $R = \sqrt{E - U (q)}$ . On a sphere chart where $r_{j} \neq 0$ , solve $r_{j} = h (q, r^{'}, E)$ with $∣ r ∣ = R$ . Implicit differentiation gives $∣ \partial_{E} h ∣ = 1 / (2 ∣ r_{j} ∣)$ and $\nabla_{(q, r^{'})} h = (- \nabla U / (2 r_{j}), - r^{'} / r_{j})$ . Consequently the full graph-area factor is $\sqrt{1 + ∣ \nabla_{(q, r^{'})} h ∣^{2}} = ∣ \nabla_{q, r} H ∣ / (2 ∣ r_{j} ∣)$ , while its fixed- $q$ sphere-area factor is $\sqrt{1 + ∣ \nabla_{r^{'}} h ∣^{2}} = R / ∣ r_{j} ∣$ . Thus $d A_{f u l l} / ∣ \nabla H ∣ = d q d r^{'} / (2 ∣ r_{j} ∣) = d q d A_{r} / (2 R)$ , so the local chart coarea expression [F2], up to the common factor $c_{N} C_{m}$ , has the asserted fiber factor. Orthogonal coordinate permutations preserve Euclidean surface area by their unit tangent Jacobian, so all $r_{j}^{2}$ have equal averages under normalized area $σ_{E, q}$ . Their sum is the constant $R^{2}$ , hence each average is $R^{2} / n = (E - U (q)) / n$ . Since $p_{j}^{2} / (2 m_{j}) = r_{j}^{2}$ , this proves the first fiber identity. The total kinetic energy equals $E - U (q)$ pointwise on $Σ_{E}$ , and integration against $P_{E}$ proves the last identity.

Kinetic-plus-potential shell marginals and moments

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources