Poincaré recurrence theorem
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In mathematics, the Poincaré recurrence theorem states that a system having a finite amount of energy and confined to a finite spatial volume will, after a sufficiently long time, return to an arbitrarily small neighborhood of its initial state. The Poincaré recurrence time is the amount of time elapsed until the recurrence. The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics.
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 (Barreira, 2005).
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 appearent "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.
The theorem is named after Henri Poincaré, who published it in 1890.
Discussion of proof
The proof, speaking qualitatively, hinges on two premises:
- The phase trajectories of closed dynamical systems do not intersect.
- By assumption the phase volume of a finite element under dynamics is conserved.
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 harmonic oscillator. Systems that do cover all available phase volume are called ergodic.
Recurrence theorem and entropy
Recurrence theorem apparently contradicts the Second law of thermodynamics, which says that large dynamical systems evolve irreversibly towards the state with higher entropy, so that if one starts with a low-entropy state, the system will never return to it. There are many possible ways to resolve this paradox, but none of them is universally accepted. The most typical argument is that for thermodynamical systems like an ideal gas in a box, recurrence time is so large that for all practical purposes it is infinite. However this explanation is not entirely satisfactory, since there is not, in fact, any characteristic timescale in the system, compared to which the recurrence time could be said to be very large. Without a reference timescale the notion of "very large" has little meaning.
One possible way of reconciling entropy and recurrence is the following. Poincaré theorem hinges on the fact that phase trajectories don't intersect. But this premise breaks if there is environment-induced noise in the system. Roughly speaking, environment influence introduces a timescale for the duration of the period for which the system can be considered isolated. If the system is chaotic, this isolation timescale grows only logarithmically with decreasing noise level. Hence, Poincaré recurrence timescale has to be compared with this isolation timescale (and not with such an extrinsic timescale as human lifespan). The limit of large system (recurrence timescale [\rightarrow\infty]) which is perfectly isolated (isolation timescale [\rightarrow\infty]) is therefore ill-defined, and since isolation timescale grows "very slowly", while recurrence timescale grows "very quickly", physically speaking, we can't have large isolated systems. That is, "large isolated system" is a result of two idealizations, which depends on the order in which they are applied, and realistically, if the system is large, it can't be isolated for the purposes of proving the recurrence theorem.
Formal statement of the theorem
Let [(X,\Sigma,\mu)] be a 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(\ N \mbox f^n(x)\notin E \mbox n>N\}\right)=0.]
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] on PlanetMath
See also
References
- Luis Barreira, [Poincaré recurrence: old and new], XIV International Congress on Mathematical Physics (Lisboa, 2003), World Scientific, to appear, retrieved 28 November 2005.
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