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Poincare recurrence for the model's Hamiltonian flow on a bounded invariant set of finite phase volume: almost every state of any Borel subset returns to it infinitely often

Physical scope

Poincare recurrence for the Hamiltonian flow of a declared autonomous potential presentation: if a bounded Borel set of phase space with compact closure in the phase domain is invariant under a complete flow on that set, then the flow preserves phase volume on its Borel subsets by a finite local-cover and time-subdivision argument, and almost every state of each such subset returns to it infinitely often at integer multiples of the time unit. The statement assumes the Axiom of Countable Choice inherited from the local phase-volume theorem and asserts nothing about return times, which individual states return, infinite phase volume or incomplete flows outside the invariant set, and nothing empirical.

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), inherited from the phase-volume preservation theorem Liouville's theorem for a potential presentation: the phase-volume form $ω^{3 N} / (3 N)!$ and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains (its measure-preservation clause uses the change-of-variables theorem under that assumption). Adopt the force model with potential presentation and the Hamiltonian data of Liouville's theorem for a potential presentation: the phase-volume form $ω^{3 N} / (3 N)!$ and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains: time-independent potential $U : Q \to R$ with residual forces $d \equiv 0$ , Cartesian phase domain $W = Q \times R^{3 N}$ (The canonical one-form and the two-form of a potential presentation: $λ = p \cdot d x$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart), Hamiltonian $H : I_{0} \times W \to R, H (t, x, p) = \frac{1}{2} p \cdot M^{- 1} p + U (x),$ phase-volume form $Ω = ω^{\land 3 N} / (3 N)!$ . Let $A \subseteq W$ be a bounded Borel set (The Borel sigma-algebra of a topological space) whose closure is contained in the open phase domain $W$ . Let $Φ_{t}$ denote the potential-branch flow on its maximal time-slice domain, and suppose it is defined at every point of $A$ for all real times and maps $A$ onto itself: $Φ_{t} (A) = A (t \in R),$ so that $A$ is invariant in the sense of The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets clause (iv). Then $λ_{6 N} (A) < \infty$ , and the flow preserves $λ_{6 N}$ on every Borel subset of $A$ , by the finite-cover and time-subdivision argument in the proof below. For every Borel set $E \subseteq A$ almost every $x \in E$ with respect to the Lebesgue measure $λ_{6 N}$ of the chart (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ ) has $Φ_{n} (x) \in E for infinitely many n \geq 1.$ The time unit is the frame time $[t] = τ$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), so "infinitely many $n$ " means infinitely many returns at unit time steps.

Facts & Assumptions

Given: ${A C}_{ω}$ ; the potential presentation and Hamiltonian data with $d \equiv 0$ ; the invariant bounded Borel set $A$ with compact closure $K : = \overline{A} \subset W$ ; and the potential-branch flow along $A$ for every real time.

[L1]

Local Liouville preservation (The Hamiltonian evolution of a potential presentation preserves the two-form, the bracket and the determinant on its actual time-slice domains, also when the Hamiltonian depends on time and Liouville's theorem for a potential presentation: the phase-volume form $ω^{3 N} / (3 N)!$ and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains clauses 1, 2 and 4): at each phase point of the autonomous $C^{2}$ Hamiltonian model there is an open neighborhood $U_{i}$ and a time $h_{i} > 0$ on which the local flow is a $C^{1}$ diffeomorphism with unit Jacobian determinant; for every $∣ s ∣ \leq h_{i}$ and bounded Borel $B \subseteq U_{i}$ with $\overline{B} \subseteq U_{i}$ , its image is Borel and $λ_{6 N} (Φ_{s} (B)) = λ_{6 N} (B)$ . These are local-in-time statements on actual solution domains.

[L2]

Recurrence in finite measure (Poincare recurrence for finite measure-preserving systems): in a measure-preserving system $(X, A, μ, T)$ with $μ (X) < \infty$ , for every $E \in A$ almost every $x \in E$ has $T^{n} x \in E$ for infinitely many positive integers $n$ .

[L3]

Measurability of the time-one map (The flow domain of a C¹ autonomous field is open and the flow is C¹ on it): the Hamiltonian field is $C^{1}$ because $H$ is $C^{2}$ ; its maximal-flow domain is open and its flow map is $C^{1}$ there. Every point of $A$ has a complete orbit by hypothesis, so the time-one domain is an open neighborhood of $A$ and $Φ_{1}$ is continuous, hence Borel measurable, on that neighborhood.

[L4]

Heine-Borel in Euclidean space (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line): a closed and bounded subset of $R^{6 N}$ is compact.

Proof

technique · direct

1.1givenL1L4algebra

If $A = \emptyset$ , every Borel subset of $A$ is empty and the almost-everywhere recurrence assertion is vacuous. Otherwise $K = \overline{A}$ is nonempty. Compact phase set and local cover: $K = \overline{A}$ is closed and bounded in $R^{6 N}$ , so [L4] makes it compact; by hypothesis it lies in the open phase domain $W$ . For each $z \in K$ , [L1] supplies a local-flow neighborhood $U_{z}$ and $h_{z} > 0$ . Choose an open neighborhood $V_{z}$ of $z$ with $\overline{V_{z}} \subset U_{z}$ , and by compactness select finitely many $V_{1}, \dots, V_{m}$ covering $K$ , with corresponding local neighborhoods $U_{i}$ and times $h_{i}$ . Set $h : = \min_{i} h_{i} > 0$ . Since $A$ is bounded, $λ_{6 N} (A) < \infty$ .

2.1step 1.1givenL1algebra

One short time slice preserves volume on every Borel subset of $A$ : fix $∣ s ∣ \leq h$ and a Borel set $B \subseteq A$ . Partition $B$ into disjoint Borel pieces $B_{i}$ by assigning each point to the first $V_{i}$ containing it. Then $\overline{B_{i}} \subseteq K \cap \overline{V_{i}} \subset U_{i}$ , so [L1] applies to each piece and gives a Borel image with $λ_{6 N} (Φ_{s} (B_{i})) = λ_{6 N} (B_{i})$ . The images are pairwise disjoint because $Φ_{s}$ is injective on $A$ , with inverse $Φ_{- s} ∣_{A}$ ; they lie in $A$ by invariance. Finite additivity therefore yields $λ_{6 N} (Φ_{s} (B)) = λ_{6 N} (B)$ .

3.1step 2.1givenalgebra

All-time volume preservation by subdivision: for $t \neq 0$ , choose an integer $M \geq 1$ with $∣ t ∣ / M \leq h$ and put $s : = t / M$ . Starting from any Borel $B \subseteq A$ , apply step 2.1 successively to $B, Φ_{s} (B), \dots, Φ_{(M - 1) s} (B)$ ; each is a Borel subset of $A$ , so every step preserves measure and the final set is $Φ_{t} (B)$ . The case $t = 0$ is immediate. Thus $λ_{6 N} (Φ_{t} (B)) = λ_{6 N} (B)$ for every real $t$ and every Borel $B \subseteq A$ .

4.1step 3.1givenL3algebra

The measure-preserving system: put $X : = A$ , let $A$ be the Borel $σ$ -algebra of $X$ (The Borel sigma-algebra of a topological space), $μ : = λ_{6 N} ∣_{A}$ and $T : = Φ_{1} ∣_{A}$ . By step 1.1, $μ (X) < \infty$ . The map $T$ maps $X$ onto itself, is a bijection with inverse $Φ_{- 1} ∣_{A}$ , and is $A$ -measurable by [L3]. For every $E \in A$ , $T^{- 1} (E) = Φ_{- 1} (E)$ and step 3.1 gives $μ (T^{- 1} (E)) = μ (E)$ ; hence $(X, A, μ, T)$ is a measure-preserving system with finite total measure.

5.1step 4.1L2∎

Conclusion: apply [L2] to the system of step 4.1: for every $E \in A$ , almost every $x \in E$ satisfies $T^{n} x \in E$ for infinitely many $n \geq 1$ , that is $Φ_{n} (x) \in E$ for infinitely many $n \geq 1$ , which is the assertion of the Statement.

Remarks

What the hypothesis buys. The finite-volume invariant set is the essential hypothesis: free or unconfined models have infinite phase volume (their energy sublevels are unbounded), and the theorem deliberately says nothing about them. A mechanical realization is a compact energy sublevel of a trapped model, where energy conservation and compact continuation supply the invariance; the page's three-dimensional Cartesian quartic example works this out explicitly.
Choice and qualifications. The countable-choice assumption is inherited from Liouville's theorem for a potential presentation: the phase-volume form $ω^{3 N} / (3 N)!$ and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains; the recurrence theorem Poincare recurrence for finite measure-preserving systems is itself proved without further choice (its proof uses countable subadditivity of the measure). No return-time estimate, no rate and no statement about individual states is asserted: the conclusion is almost-everywhere with respect to the phase volume.
No empirical content. Recurrence is a property of the flow on an invariant finite-measure set; it is not a statement about observed gases, thermalization times or the second law, and it identifies no measured quantity.

Poincare recurrence for the model's Hamiltonian flow on a bounded invariant set of finite phase volume: almost every state of any Borel subset returns to it infinitely often

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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