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Hamiltonian volume, recurrence, and ergodic scope

Statement

Assume the Axiom of Choice. Let $(M, ω)$ be a smooth $2 n$ -dimensional symplectic manifold and $H \in C^{\infty} (M, R)$ , with Hamiltonian field defined by $ι_{X_{H}} ω = d H$ . Wherever its local flow $ϕ_{t}$ is defined, it preserves the symplectic form and the volume measure $m$ determined by $ω^{n} / n!$ . No completeness is asserted. Let $R$ be a measurable region on which a nonzero-time map $T = ϕ_{τ}$ is defined and is forward invariant, $T (R) \subseteq R$ . The restriction $T : R \to R$ preserves $m ∣_{R}$ if $T (R) = R$ , or more generally if $m (R ∖ T (R)) = 0$ . This latter condition holds automatically when $m (R) < \infty$ . Thus for every finite-measure forward-invariant $R$ and every measurable $E \subseteq R$ , almost every point of $E$ returns to $E$ infinitely often under forward iteration. Mere forward invariance of an infinite-volume region does not imply relative measure preservation.

If $m (R) = 1$ and $T$ is ergodic, then for every $f \in L^{1} (R, m)$ the averages $N^{- 1} \sum_{j = 0}^{N - 1} f (T^{j} x)$ converge almost everywhere and in $L^{1}$ to $\int_{R} f d m$ . Without ergodicity, the limit is the invariant conditional expectation supplied by Birkhoff's theorem, not necessarily the space mean. Finally, fix any real scalar $k_{B}$ . If each stated time map $ϕ_{t}$ is a diffeomorphism of a finite invariant region $R$ , a density $p_{0} \geq 0$ with $\int_{R} p_{0} d m = 1$ and $\int_{R} ∣ p_{0} \log p_{0} ∣ d m < \infty$ transports by $p_{t} = p_{0} \circ ϕ_{- t}$ with conserved fine-grained entropy $- k_{B} \int_{R} p_{t} \log p_{t} d m$ , using $0 \log 0 = 0$ . Volume preservation alone does not imply ergodicity or a time-average/ensemble identity.

Facts & Assumptions

Given: The symplectic/Hamiltonian definitions, the local flow domain, and each separately stated finite invariant probability system.

[A1]

AC is adopted; $(M, ω)$ is smooth symplectic of dimension $2 n$ , $H$ is smooth, and $X_{H}$ follows $ι_{X_{H}} ω = d H$ .

[A2]

The local flow is used only where defined. The sampled region is measurable and forward invariant; for an infinite-measure region the restricted-measure assertion additionally requires $m (R ∖ T (R)) = 0$ (in particular, onto invariance suffices). For entropy transport, each stated $ϕ_{t}$ is a diffeomorphism of $R$ with inverse $ϕ_{- t}$ . For recurrence and Birkhoff, $R$ is forward invariant, $T = ϕ_{τ}$ and every iterate exist on $R$ , and $m (R) < \infty$ ; for the space-mean conclusion $m (R) = 1$ and $T$ is ergodic in the sense of [F5].

[A3]

The entropy scale $k_{B}$ is a fixed real mathematical parameter, with no physical value or unit adopted here. The transported density is measurable, nonnegative, normalized, and has finite absolute entropy integral, with $0 \log 0 = 0$ .

[F1]

The Axiom of Choice, Symplectic form and symplectic manifold, Hamiltonian vector field and Hamiltonian function, and Symplectic manifolds have a canonical orientation and volume form: AC is the inherited measure-theorem convention; $ω$ is closed and nondegenerate, $ι_{X_{H}} ω = d H$ , and $ω^{n} / n!$ determines the canonical positive volume measure.

[F2]

Hamiltonian flows preserve the symplectic form: every local Hamiltonian flow preserves $ω$ wherever defined. Liouville volume preservation then gives preservation of the symplectic volume on the same domain.

[F3]

Measure-preserving transformations and systems and Integral invariance under measure-preserving maps: the stated invariant flow map is measure preserving, and integrals of integrable functions are unchanged by its measure-preserving substitution.

[F4]

Poincare recurrence for finite measure-preserving systems: on a finite-measure system, almost every point of each measurable set returns to that set infinitely often.

[F5]

Ergodicity relative to an invariant measure, Birkhoff pointwise ergodic theorem, and Birkhoff ergodic theorem for ergodic finite-measure systems: ergodicity means every invariant event is null or conull; for an integrable observable on a finite measure-preserving probability system, Birkhoff supplies almost-everywhere convergence. Ergodic averages converge in Lp on finite-measure spaces at $p = 1$ supplies $L^{1}$ convergence on finite measure spaces; Finite-measure identification of the Birkhoff limit under AC identifies the limit as the unique integrable function measurable for $I = {E : T^{- 1} E = E}$ with $\int_{E} f^{*} d m = \int_{E} f d m$ for every $E \in I$ . This is the invariant conditional expectation meant here; the ergodic case has constant limit equal to the space integral.

[F6]

Probability measures and probability spaces and Integrable real and complex functions, and their integrals: normalized densities and the stated absolute-integrability condition define the probability and entropy integrals.

Proof

Given: [A1]–[A3] and the inspected supplier interfaces [F1]–[F6].

1.1A1A2F1F2F3algebra

By [F1], $ω$ is closed and $ι_{X_{H}} ω = d H$ . The Hamiltonian-flow theorem [F2], under its local existence domain, gives $ϕ_{t}^{*} ω = ω$ and hence $ϕ_{t}^{*} (ω^{n} / n!) = ω^{n} / n!$ . The volume measure defined by this positive form is therefore preserved by each local flow diffeomorphism on its domain. For a measurable $E \subseteq R$ , injectivity and volume preservation give $m ((T ∣_{R})^{- 1} E) = m (E \cap T (R))$ . If $m (R ∖ T (R)) = 0$ , this equals $m (E)$ , so the restriction is the measure-preserving system of [F3]. When $m (R) < \infty$ , volume preservation gives $m (T (R)) = m (R)$ ; because $T (R) \subseteq R$ , finite additivity gives $m (R ∖ T (R)) = 0$ . Thus the finite forward-invariant case follows without an onto hypothesis. Also $d H (X_{H}) = ω (X_{H}, X_{H}) = 0$ because a two-form is alternating, so each defined orbit stays on its energy level. This argument grants no flow outside its domain.

2.1A2A3F3F6step 1.1

Apply [F3] to $p_{t} = p_{0} \circ ϕ_{- t}$ . It preserves $\int p_{t} d m = \int p_{0} d m = 1$ . The scalar function $u \mapsto u \log u$ , defined to be $0$ at $u = 0$ by [A3], is Borel; its absolute integrability is preserved because the same measure-preserving substitution gives $\int_{R} ∣ p_{t} \log p_{t} ∣ d m = \int_{R} ∣ p_{0} \log p_{0} ∣ d m < \infty .$ Applying integral invariance to its positive and negative parts proves $\int_{R} p_{t} \log p_{t} d m = \int_{R} p_{0} \log p_{0} d m$ . Multiplication by the fixed real scalar $- k_{B}$ gives the stated scaled entropy conservation.

2.2A2F3F4step 1.1

The map $T$ on $R$ is measure preserving by 1.1 and [A2]. Its total measure is finite, so [F4] gives infinite returns for almost every point in each measurable $E \subseteq R$ . The conclusion is explicitly almost everywhere and applies to the sampled map at the specified $τ$ .

3.1A2F5step 1.1∎

Under the additional probability and ergodicity assumptions in [A2], the hypotheses of [F5] hold for each $f \in L^{1} (R, m)$ . Birkhoff therefore gives almost-everywhere and $L^{1}$ convergence to the invariant conditional expectation, and its ergodic corollary identifies that limit with $\int_{R} f d m$ . If ergodicity is removed, only the invariant conditional expectation conclusion remains. Neither [F2] nor recurrence [F4] supplies ergodicity.

Hamiltonian volume, recurrence, and ergodic scope

Statement

Facts & Assumptions

Proof

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