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| In [[mathematics]], the '''Poincaré recurrence theorem''' states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. The '''Poincaré recurrence time''' is the length of time elapsed until the recurrence (this time may vary greatly depending on the exact initial state and required degree of closeness). The result applies to isolated mechanical systems subject to some constraints, e.g., all particles must be bound to a finite volume. The theorem is commonly discussed in the context of [[ergodic theory]], [[dynamical systems]] and [[statistical mechanics]].
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| The theorem is named after [[Henri Poincaré]], who published it in 1890.
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| == Precise formulation ==
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| Any [[dynamical system]] defined by an [[ordinary differential equation]] determines a [[flow (mathematics)|flow map]] ''f''<sup> ''t''</sup> 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 system]]s are volume-preserving because of [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]. The theorem is then: If a [[flow (mathematics)|flow]] preserves volume and has only bounded orbits, then for each open set there exist orbits that intersect the set infinitely often.<ref name="Barreira 2005">{{Cite book | last1=Barreira | first1=Luis | editor1-last=Zambrini | editor1-first=Jean-Claude | title=XIVth International Congress on Mathematical Physics | publisher=[[World Scientific]] | isbn=978-981-256-201-2 | doi=10.1142/9789812704016_0039 | year=2006 | chapter=Poincaré recurrence: old and new | pages=415–422 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref>
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| As an example, the deterministic [[baker's map]] exhibits Poincaré recurrence which can be demonstrated in a particularly dramatic fashion when acting on 2D images. A given image, when sliced and squashed hundreds of times, turns into a snow of apparent "random noise". However, when the process is repeated thousands of times, the image reappears, although at times marred with greater or lesser bits of noise.
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| == Discussion of proof == | |
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| The proof, speaking qualitatively, hinges on two premises:<ref name="gibbs">Chapter X of {{cite book |last=Gibbs |first=Josiah Willard |authorlink=Josiah Willard Gibbs |title=[[Elementary Principles in Statistical Mechanics]] |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York}}</ref>
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| # A finite upper bound can be set on the total potentially accessible phase space volume. For a mechanical system, this bound can be provided by requiring that the system is contained in a bounded ''physical'' region of space (so that it cannot, for example, [[escape velocity|eject particles that never return]]) — combined with the conservation of energy, this locks the system into a finite region in phase space.
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| # The phase volume of a finite element under dynamics is conserved. (for a mechanical system, this is ensured by [[Liouville's theorem (Hamiltonian)|Liouville's theorem]])
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| Imagine any finite starting volume of [[phase space]] and follow its path under dynamics of the system. The volume "sweeps" points of phase space as it evolves, and the "front" of this sweeping has a constant size. Over time the explored phase volume (known as a "phase tube") grows linearly, at least at first. But, because the accessible phase volume is finite, the phase tube volume must eventually saturate because it cannot grow larger than the accessible volume. This means that the phase tube must intersect itself. In order to intersect itself, however, it must do so by first passing through the starting volume. Therefore, at least a finite fraction of the starting volume is recurring.
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| Now, consider the size of the non-returning portion 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 of the non-returning portion must return. But that would be a contradiction, since any part of the non-returning portion that returns, also returns to the original starting volume. Thus, the non-returning portion of the starting volume cannot be finite and must be infinitely smaller than the starting volume itself. QED.
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| The theorem does not comment on certain aspects of recurrence which this proof cannot guarantee:
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| * 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.
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| * 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.
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| * 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 [[harmonic oscillator]]. Systems that do cover all accessible phase volume are called [[ergodic hypothesis|ergodic]] (this of course depends on the definition of "accessible volume").
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| * 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.
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| * 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.
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| == Formal statement of the theorem ==
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| Let <math>(X,\Sigma,\mu)</math> be a finite [[measure space]] and let <math>f\colon X\to X</math>
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| be a [[measure-preserving transformation]]. Below are two alternative statements of the theorem.
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| ===Theorem 1===
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| For any <math>E\in \Sigma</math>, the set of those points <math>x</math> of <math>E</math> such that <math>f^n(x)\notin E</math> for all <math>n>0</math> has zero measure. That is, almost every point of <math>E</math> returns to <math>E</math>. In fact, almost every point returns infinitely often; ''i.e.''
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| :<math>\mu\left(\{x\in E:\mbox{ there exists } N \mbox{ such that }
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| f^n(x)\notin E \mbox{ for all } n>N\}\right)=0.</math>
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| For a proof, see {{planetmath reference|id=6035|title=proof of Poincaré recurrence theorem 1}}.
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| ===Theorem 2===
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| The following is a topological version of this theorem:
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| If <math>X</math> is a [[second-countable]] [[Hausdorff space]] and <math>\Sigma</math> contains the [[Borel sigma-algebra]], then the set of recurrent points of <math>f</math> has full measure. That is, almost every point is recurrent.
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| For a proof, see {{planetmath reference|id=6036|title=proof of Poincaré recurrence theorem 2}}
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| == Quantum mechanical version ==
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| For quantum mechanical systems with discrete energy eigenstates, a similar theorem holds. For every <math>\epsilon >0</math> and <math>T_{0}>0</math> there exists a time T larger than <math> T_{0}</math>, such that <math>\left |\left | \psi(T)\right\rangle - \left |\psi(0)\right\rangle\right | < \epsilon</math>, where <math>\left | \psi(t)\right\rangle</math> denotes the state vector of the system at time t.<ref>{{cite journal |first=P. |last=Bocchieri |first2=A. |last2=Loinger |title=Quantum Recurrence Theorem |journal=[[Physical Review|Phys. Rev.]] |volume=107 |issue=2 |pages=337–338 |year=1957 |doi=10.1103/PhysRev.107.337 |bibcode = 1957PhRv..107..337B }}</ref><ref>{{cite journal |first=I. C. |last=Percival |title=Almost Periodicity and the Quantal H theorem |journal=[[Journal of Mathematical Physics|J Math. Phys.]] |volume=2 |issue= 2|pages=235–239 |year=1961 |doi=10.1063/1.1703705 |bibcode = 1961JMP.....2..235P }}</ref><ref>{{cite journal |first=L. S. |last=Schulman |title=Note on the quantum recurrence theorem |journal=Phys. Rev. |volume=A 18 |issue=5 |pages=2379–2380 |year=1978 |doi=10.1103/PhysRevA.18.2379 |bibcode = 1978PhRvA..18.2379S }}</ref>
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| The essential elements of the proof are as follows. The system evolves in time according to:
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| :<math>\left |\psi(t)\right\rangle = \sum_{n=0}^{\infty}c_{n}\exp\left(-i E_{n} t\right)\left |\phi_{n}\right\rangle</math>
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| where the <math>E_{n}</math> are the energy eigenvalues (we use natural units, so <math>\hbar = 1</math> ), and the <math>\left |\phi_{n}\right\rangle</math> are the energy eigenstates. The squared norm of the difference of the state vector at time T and time zero, can be written as:
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| :<math>\left |\left | \psi(T)\right\rangle - \left |\psi(0)\right\rangle\right |^{2} = 2\sum_{n=0}^{\infty}\left | c_{n}\right |^{2}\left [1-\cos\left(E_{n}T\right)\right ]</math>
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| We can truncate the summation at some n = N independent of T, because
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| <math>\sum_{n=N+1}^{\infty}\left | c_{n}\right |^{2}\left [1-\cos\left(E_{n}T\right)\right ] \leq \sum_{n=N+1}^{\infty}\left | c_{n}\right |^{2}</math>
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| which can be made arbitrarily small because the summation <math>\sum_{n=0}^{\infty}\left |c_{n}\right |^{2}</math>, being the squared norm of the initial state, converges to 1.
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| That the finite sum
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| :<math>\sum_{n=0}^{N}\left | c_{n}\right |^{2}\left [1-\cos\left(E_{n}T\right)\right ]</math>
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| can be made arbitrarily small, follows from the existence of integers <math>k_{n}</math> such that <math>\left |E_{n}T -2\pi k_{n}\right |<\delta</math> for arbitrary <math>\delta>0</math>. This implies that there exists intervals for T on which <math>1-\cos\left(E_{n}T\right)<\frac{\delta^{2}}{2}</math>. On such intervals, we have:
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| :<math>2\sum_{n=0}^{N}\left | c_{n}\right |^{2}\left [1-\cos\left(E_{n}T\right)\right ] < \delta^{2}\sum_{n=0}^{N}\left | c_{n}\right |^{2}<\delta^{2}</math>
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| The state vector thus returns arbitrarily closely to the initial state, infinitely often.
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| When the state is spanned by the arbitrarily large but finite number of eigenstates the quantum Poincaré recurrence time is therefore equal to the
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| full [[quantum revival]] time and can be calculated exactly when the energies of the system are rational fractions of a constant.
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| It is giant in years already for systems as simple as hydrogen atom and states consisting of few eigenstates.
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| ==See also==
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| * [[Ergodic hypothesis]]
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| * [[Recurrence plot]]
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| * [[Recurrence period density entropy]]
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| * [[Wandering set]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * {{cite journal|last=Page|first=Don N.|title=Information Loss in Black Holes and/or Conscious Beings?|date=November 25, 1994|arxiv=hep-th/9411193.pdf|bibcode = 1994hep.th...11193P }}
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| ==External links==
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| * {{cite web|last=Padilla|first=Tony|title=The Longest Time|url=http://www.numberphile.com/videos/longest_time.html|work=Numberphile|publisher=[[Brady Haran]]}}
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| {{PlanetMath attribution|id=6033|title=Poincaré recurrence theorem}}
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| {{DEFAULTSORT:Poincare recurrence theorem}}
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| [[Category:Theorems in dynamical systems]]
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| [[Category:Ergodic theory]]
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| [[Category:Statistical mechanics]]
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