Inmathematics andphysics, thePoincaré recurrence theorem states that certaindynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state.

ThePoincaré 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 ofergodic theory,dynamical systems andstatistical mechanics. Systems to which the Poincaré recurrence theorem applies are calledconservative systems.

The theorem is named afterHenri Poincaré, who discussed it in 1890.^[1][2] A proof was presented byConstantin Carathéodory usingmeasure theory in 1919.^[3][4]

Anydynamical system defined by anordinary differential equation determines aflow mapf ^t mappingphase space on itself. The system is said to bevolume-preserving if the volume of a set in phase space is invariant under the flow. For instance, allHamiltonian systems are volume-preserving because ofLiouville's theorem. The theorem is then: If aflow preserves volume and has only bounded orbits, then, for eachopen set, any orbit that intersects this open set intersects it infinitely often.^[5]

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

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 boundedphysical 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 inphase space.
2. The phase volume of a finite element under dynamics is conserved (for a mechanical system, this is ensured byLiouville's theorem).

Imagine any finite starting volume${\displaystyle D_1}$ of thephase 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${\displaystyle k_1}$ the phase tube must intersect itself. This means that at least a finite fraction${\displaystyle R_1}$ of the starting volume is recurring.Now, consider the size of the non-returning portion${\displaystyle D_2}$ 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${\displaystyle R_2}$ of it must return after${\displaystyle k_2}$ steps. But that would be a contradiction, since in a number${\displaystyle k_3=}$lcm${\displaystyle (k_1,k_2)}$ of step, both${\displaystyle R_1}$ and${\displaystyle R_2}$ would be returning, against the hypothesis that only${\displaystyle R_1}$ was. Thus, the non-returning portion of the starting volume cannot be theempty set, i.e. all${\displaystyle D_1}$ 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 theharmonic oscillator. Systems that do cover all accessible phase volume are calledergodic (this of course depends on the definition of "accessible volume").
• Whatcan 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

${\displaystyle (X,\mathrm{\Sigma },\mu )}$

be a finitemeasure space and let

${\displaystyle f:X\to X}$

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

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

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

${\displaystyle \mu \left(\{x\in E:\text{ there exists }N\text{ such that }{f}^n(x)\notin E\text{ for all }n>N\}\right)=0.}$

The following is a topological version of this theorem:

If${\displaystyle X}$ is asecond-countableHausdorff space and${\displaystyle \mathrm{\Sigma }}$ contains theBorel sigma-algebra, then the set ofrecurrent points of${\displaystyle f}$ has full measure. That is, almost every point is recurrent.

More generally, the theorem applies toconservative 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${\displaystyle \epsilon >0}$ and${\displaystyle T_0>0}$ there exists a timeT larger than${\displaystyle T_0}$, such that, where${\displaystyle |\psi (t)⟩}$ denotes the state vector of the system at time t.^[7][8][9]

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

${\displaystyle |\psi (t)⟩=\sum _{n=0}^{\mathrm{\infty }}{c}_n\exp (-i{E}_{n}t)|{\varphi }_n⟩}$

where the${\displaystyle E_n}$ are the energy eigenvalues (we usenatural units, so${\displaystyle \hslash =1}$ ), and the${\displaystyle |{\varphi }_n⟩}$ are the energyeigenstates. The squared norm of the difference of the state vector at time${\displaystyle T}$ and time zero, can be written as:

${\displaystyle ||\psi (T)⟩-|\psi (0)⟩{|}^2=2\sum _{n=0}^{\mathrm{\infty }}|c_n{|}^2[1-\cos (E_{n}T)]}$

We can truncate the summation at somen = N independent ofT, because

${\displaystyle \sum _{n=N+1}^{\mathrm{\infty }}|c_n{|}^2[1-\cos (E_{n}T)]\le 2\sum _{n=N+1}^{\mathrm{\infty }}|c_n{|}^2}$

which can be made arbitrarily small by increasingN, as the summation${\displaystyle \sum _{n=0}^{\mathrm{\infty }}|c_n{|}^2}$, being the squared norm of the initial state, converges to 1.

The finite sum

${\displaystyle \sum _{n=0}^N|c_n{|}^2[1-\cos (E_{n}T)]}$

can be made arbitrarily small for specific choices of the timeT, according to the following construction. Choose an arbitrary${\displaystyle \delta >0}$, and then chooseT such that there are integers${\displaystyle k_n}$ that satisfies

,

for all numbers${\displaystyle 0\le n\le N}$. For this specific choice ofT,

As such, we have:

.

The state vector${\displaystyle |\psi (T)⟩}$ thus returns arbitrarily close to the initial state${\displaystyle |\psi (0)⟩}$.

• 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?".arXiv:hep-th/9411193.

• Padilla, Tony."The Longest Time".Numberphile.Brady Haran. Archived fromthe original on 2013-11-27. Retrieved2013-04-08.Recent version of the original site.
• "Arnold's Cat Map: An interactive graphical illustration of the recurrence theorem of Poincaré".

This article incorporates material from Poincaré recurrence theorem onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

• Ergodic theory
• Henri Poincaré
• Statistical mechanics
• Theorems in dynamical systems

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