Poincaré recurrence theorem

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.{{cite conference |last1=Barreira |first1=Luis |editor1-last=Zambrini |editor1-first=Jean-Claude |conference=XIVth International Congress on Mathematical Physics |publisher=World Scientific |isbn=978-981-256-201-2 |doi=10.1142/9789812704016_0039 |year=2006 |title=Poincaré recurrence: Old and new |pages=415–422}}

The proof, speaking qualitatively, hinges on two premises:{{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=Charles Scribner's Sons |location=New York, NY |at=Chapter X|title-link=Elementary Principles in Statistical Mechanics }}

1. 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, eject particles that never return) – combined with the conservation of energy, this locks the system into a finite region in phase space.
2. The phase volume of a finite element under dynamics is conserved (for a mechanical system, this is ensured by Liouville's theorem).

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:

• 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 harmonic oscillator. Systems that do cover all accessible phase volume are called ergodic (this of course depends on the definition of "accessible volume").
• 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.

Let

:$(X,\backslash Sigma,\backslash mu)$

be a finite measure space and let

:$f\backslash colon\; X\backslash to\; X$

be a measure-preserving transformation. Below are two alternative statements of the theorem.

For any $E\backslash in\; \backslash Sigma$, the set of those points $x$ of $E$ for which there exists $N\backslash in\backslash mathbb\{N\}$ such that $f^n(x)\backslash notin\; E$ for all $n>N$ has zero measure.

In other words, almost every point of $E$ returns to $E$. In fact, almost every point returns infinitely often; i.e.

:$\backslash mu\backslash left(\backslash \{x\backslash in\; E:\backslash text\{\; there\; exists\; \}\; N\; \backslash text\{\; such\; that\; \}$

f^n(x)\notin E \text{ for all } n>N\}\right)=0.

The following is a topological version of this theorem:

If $X$ is a second-countable Hausdorff space and $\backslash Sigma$ contains the Borel sigma-algebra, then the set of recurrent points of $f$ has full measure. That is, almost every point is recurrent.

More generally, the theorem applies to conservative systems, and not just to measure-preserving dynamical systems. Roughly speaking, one can say that conservative systems are precisely those to which the recurrence theorem applies.

For time-independent quantum mechanical systems with discrete energy eigenstates, a similar theorem holds. For every $\backslash varepsilon\; >0$ and $T\_0>0$ there exists a time T larger than $T\_0$, such that , where $|\; \backslash psi(t)\backslash rangle$ denotes the state vector of the system at time t.{{cite journal |first1=P. |last1=Bocchieri |first2=A. |last2=Loinger |title=Quantum Recurrence Theorem |journal=Phys. Rev. |volume=107 |issue=2 |pages=337–338 |year=1957 |doi=10.1103/PhysRev.107.337 |bibcode=1957PhRv..107..337B}}{{cite journal |first=I.C. |last=Percival |title=Almost Periodicity and the Quantal H theorem |journal=J. Math. Phys. |volume=2 |issue=2 |pages=235–239 |year=1961 |doi=10.1063/1.1703705 |bibcode=1961JMP.....2..235P}}{{cite journal |first=L. S. |last=Schulman |title=Note on the quantum recurrence theorem |journal=Phys. Rev. A |volume=18 |issue=5 |pages=2379–2380 |year=1978 |doi=10.1103/PhysRevA.18.2379 |bibcode=1978PhRvA..18.2379S}}

The essential elements of the proof are as follows. The system evolves in time according to:

:$|\backslash psi(t)\backslash rangle\; =\; \backslash sum\_\{n=0\}^\backslash infty\; c\_n\; \backslash exp(-i\; E\_n\; t)|\backslash phi\_n\backslash rangle$

where the $E\_n$ are the energy eigenvalues (we use natural units, so $\backslash hbar\; =\; 1$ ), and the $|\backslash phi\_n\; \backslash rangle$ are the energy eigenstates. The squared norm of the difference of the state vector at time $T$ and time zero, can be written as:

:$|\; |\backslash psi(T)\; \backslash rangle\; -\; |\backslash psi(0)\backslash rangle|^2\; =\; 2\backslash sum\_\{n=0\}^\backslash infty\; |c\_n|^2\; [1-\backslash cos(E\_n\; T)]$

We can truncate the summation at some n = N independent of T, because

$\backslash sum\_\{n=N+1\}^\backslash infty\; |c\_n|^2\; [1-\backslash cos(E\_n\; T)]\; \backslash leq\; 2\backslash sum\_\{n=N+1\}^\backslash infty\; |c\_n|^2$

which can be made arbitrarily small by increasing N, as the summation $\backslash sum\_\{n=0\}^\backslash infty\; |c\_n|^2$, being the squared norm of the initial state, converges to 1.

The finite sum

:$\backslash sum\_\{n=0\}^N\; |c\_n|^2\; [1-\backslash cos(E\_n\; T)]$

can be made arbitrarily small for specific choices of the time T, according to the following construction. Choose an arbitrary $\backslash delta>0$, and then choose T such that there are integers $k\_n$ that satisfies

:,

for all numbers $0\; \backslash leq\; n\; \backslash leq\; N$. For this specific choice of T,

:

As such, we have:

:.

The state vector $|\backslash psi(T)\backslash rangle$ thus returns arbitrarily close to the initial state $|\backslash psi(0)\backslash rangle$.

• Arnold's cat map
• Ergodic hypothesis
• Quantum revival
• Recurrence period density entropy
• Recurrence plot
• Wandering set

{{Reflist}}

• {{cite arXiv |last=Page |first=Don N. |title=Information loss in black holes and/or conscious beings? |eprint=hep-th/9411193 |date=November 25, 1994 |df=dmy-all}}

• {{cite web |last=Padilla |first=Tony |title=The Longest Time |url=http://www.numberphile.com/videos/longest_time.html |work=Numberphile |publisher=Brady Haran |access-date=2013-04-08 |archive-url=https://web.archive.org/web/20131127054243/http://numberphile.com/videos/longest_time.html |archive-date=2013-11-27 |url-status=dead }} [https://www.numberphile.com/videos/the-longest-time Recent version of the original site.]
• {{cite web |url=http://gerdbreitenbach.de/arnold_cat/cat.html |title=Arnold's Cat Map: An interactive graphical illustration of the recurrence theorem of Poincaré}}

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Category:Ergodic theory

Recurrence theorem

Category:Statistical mechanics

Category:Theorems in dynamical systems