Precise formulation[edit]

Any dynamical system defined by an ordinary differential equation determines a flow map f^t mapping phase space on itself. The system is said to be volume-preserving if the volume of a set in phase space is invariant under the flow. For instance, all Hamiltonian systems are volume-preserving because of Liouville's theorem. The theorem is then: If a flow preserves volume and has only bounded orbits, then, for each open set, any orbit that intersects this open set intersects it infinitely often.^[5]

There may be some special phases that never return to the starting phase volume, or that only return to the starting volume a finite number of times then never return again. These however are extremely "rare", making up an infinitesimal part of any starting volume.

Not all parts of the phase volume need to return at the same time. Some will "miss" the starting volume on the first pass, only to make their return at a later time.

Nothing prevents the phase tube from returning completely to its starting volume before all the possible phase volume is exhausted. A trivial example of this is the . Systems that do cover all accessible phase volume are called ergodic (this of course depends on the definition of "accessible volume").

harmonic oscillator

What can be said is that for "almost any" starting phase, a system will eventually return arbitrarily close to that starting phase. The recurrence time depends on the required degree of closeness (the size of the phase volume). To achieve greater accuracy of recurrence, we need to take smaller initial volume, which means longer recurrence time.

For a given phase in a volume, the recurrence is not necessarily a periodic recurrence. The second recurrence time does not need to be double the first recurrence time.

The proof, speaking qualitatively, hinges on two premises:^[6]

Imagine any finite starting volume $D_{1}$ of the phase space and to follow its path under the dynamics of the system. The volume evolves through a "phase tube" in the phase space, keeping its size constant. Assuming a finite phase space, after some number of steps $k_{1}$ the phase tube must intersect itself. This means that at least a finite fraction $R_{1}$ of the starting volume is recurring. Now, consider the size of the non-returning portion $D_{2}$ of the starting phase volume – that portion that never returns to the starting volume. Using the principle just discussed in the last paragraph, we know that if the non-returning portion is finite, then a finite part $R_{2}$ of it must return after $k_{2}$ steps. But that would be a contradiction, since in a number $k_{3}=$ lcm $(k_{1},k_{2})$ of step, both $R_{1}$ and $R_{2}$ would be returning, against the hypothesis that only $R_{1}$ was. Thus, the non-returning portion of the starting volume cannot be the empty set, i.e. all $D_{1}$ is recurring after some number of steps.

The theorem does not comment on certain aspects of recurrence which this proof cannot guarantee:

Arnold's cat map

Ergodic hypothesis

Quantum revival

Recurrence period density entropy

Recurrence plot

Wandering set

Page, Don N. (25 November 1994). "Information loss in black holes and/or conscious beings?". :hep-th/9411193.

arXiv

Padilla, Tony. . Numberphile. Brady Haran. Archived from the original on 2013-11-27. Retrieved 2013-04-08. Recent version of the original site.

"The Longest Time"

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"Arnold's Cat Map: An interactive graphical illustration of the recurrence theorem of Poincaré"

This article incorporates material from Poincaré recurrence theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.