Poincaré recurrence theorem

In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which energy is conserved. The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics.

The theorem is named after Henri Poincaré, who published it in 1890.

Precise formulation

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 there exist orbits that intersect the set infinitely often.

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.

Discussion of proof

The proof, speaking qualitatively, hinges on two premises:

1. The phase trajectories of closed dynamical systems do not intersect.
2. The phase volume of a finite element under dynamics is conserved.

Imagine an arbitrary small neighborhood of any point in the phase space and follow its path under dynamics of the system (usually called a "phase tube"). The volume "sweeps" points of phase space as it moves. It can never cross the regions that are already "swept", because phase trajectories do not intersect. Hence, the total volume accessible to it constantly decreases, and since the total volume is finite by assumption, in a finite time, all volume will be exhausted. At that point, the only way to continue would be for the phase tube to connect to its own starting point, which is QED.

Note that individual trajectories included in the phase tube need not connect to their respective starting points, most likely they will all be mixed up within the tube. This is why recurrence is only approximate up to the diameter of the tube. To achieve greater accuracy of recurrence, we need to take smaller initial volume, which means longer recurrence time.

Note also that nothing prevents the system from returning to its starting point before all the phase volume is exhausted. A trivial example of this is the harmonic oscillator. Systems that do cover all available phase volume are called ergodic.

Formal statement of the theorem

Let $(X,\Sigma,\mu)$ be a finite measure space and let $f\colon X\to X$ be a measure-preserving transformation. Below are two alternative statements of the theorem.

Theorem 1

For any $E\in \Sigma$, the set of those points $x$ of $E$ such that $f^n(x)\notin E$ for all $n>0$ has zero measure. That is, almost every point of $E$ returns to $E$. In fact, almost every point returns infinitely often; i.e.

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

For a proof, see proof of Poincaré recurrence theorem 1, PlanetMath.org..

Theorem 2

The following is a topological version of this theorem:

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

For a proof, see proof of Poincaré recurrence theorem 2, PlanetMath.org.

Quantum mechanical version

For quantum mechanical systems with discrete energy eigenstates, a similar theorem holds. For every $\epsilon >0$ and $T_{0}>0$ there exists a time T larger than $T_{0}$, such that $\left |\left | \psi(T)\right\rangle - \left |\psi(0)\right\rangle\right | < \epsilon$, where $\left | \psi(t)\right\rangle$ denotes the state vector of the system at time t.

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

$\left |\psi(t)\right\rangle = \sum_{n=0}^{\infty}c_{n}\exp\left(-i E_{n} t\right)\left |\phi_{n}\right\rangle$

where the $E_{n}$ are the energy eigenvalues (we use natural units, so $\hbar = 1$ ), and the $\left |\phi_{n}\right\rangle$ are the energy eigenstates. The squared norm of the difference of the state vector at time T and time zero, can be written as:

$\left |\left | \psi(T)\right\rangle - \left |\psi(0)\right\rangle\right |^{2} = 2\sum_{n=0}^{\infty}\left | c_{n}\right |^{2}\left [1-\cos\left(E_{n}T\right)\right ]$

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

$\sum_{n=N+1}^{\infty}\left | c_{n}\right |^{2}\left [1-\cos\left(E_{n}T\right)\right ] \leq \sum_{n=N+1}^{\infty}\left | c_{n}\right |^{2}$

which can be made arbitrarily small because the summation $\sum_{n=0}^{\infty}\left |c_{n}\right |^{2}$, being the squared norm of the initial state, converges to 1.

That the finite sum

$\sum_{n=0}^{N}\left | c_{n}\right |^{2}\left [1-\cos\left(E_{n}T\right)\right ]$

can be made arbitrarily small, follows from the existence of integers $k_{n}$ such that $\left |E_{n}T -2\pi k_{n}\right |<\delta$ for arbitrary $\delta>0$. This implies that there exists intervals for T on which $1-\cos\left(E_{n}T\right)<\frac{\delta^{2}}{2}$. On such intervals, we have:

$2\sum_{n=0}^{N}\left | c_{n}\right |^{2}\left [1-\cos\left(E_{n}T\right)\right ] < \delta^{2}\sum_{n=0}^{N}\left | c_{n}\right |^{2}<\delta^{2}$

The state vector thus returns arbitrarily closely to the initial state, infinitely often.

See also

• Ergodic hypothesis
• Recurrence plot
• Recurrence period density entropy
• Wandering set

Further reading

• Page, Don N. (November 25, 1994). Information Loss in Black Holes and/or Conscious Beings?. arXiv:hep-th/9411193.pdf. 

External links

• Padilla, Tony. "The Longest Time". Numberphile. Brady Haran. 

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