Talk:Poincaré recurrence theorem

Orbit definition

I would appreciate if you'd define the term "orbit" before you use it. What orbit is this? An orbit in phase space? An orbit in the real space? Are we talking about some bound systems, like an atom? If not, what are the mathematical properties of an "orbit"? Is it a curve? Is it a surface? Does it have to be continuous? Does it have to be differentiable? Does it have to be closed? 92.52.23.13 (talk) 16:48, 3 February 2017 (UTC)

Precise formulation

There is no precise formulation in section "precise formulation". I would suggest it to be excluded. In it's place there could be some section giving motivation to what follows. Motivation, by the way, need not be too much complicated. There is no need to mention ODE, flow maps, Hamiltonian systems, Liouville what ever... this could go in an "optional" section at the end of the article. André Caldas (talk) 01:43, 13 December 2010 (UTC)

I completely disagree. The article does not have any careful statement of the theorem anywhere. I think the least an article can do is say what it is talking about.Daqu (talk) 07:51, 17 May 2015 (UTC)

is this topological or measurable dynamics?

I vaguely recall that the poincare recurrence theorem is a statement in topological dynamics, not measurable dynamics. But I might be wrong. Haven't thought about this stuff for a while. Dmharvey File:User dmharvey sig.png Talk 01:42, 27 August 2005 (UTC)

I vaguely recall a defintion based on measure. I don't think they had topological dynamics in Poincare's time. But I dunno. I'll keep my eye out for a reference. linas 00:56, 30 August 2005 (UTC)

And BTW, have you seen the baker's map images? They're stunning, I remember the first time I saw them. First umpteen-ten-or-hundred-thousand iterations, you have pure, unadulterated white noise. And then, all of a sudden, wham, the original image re-emerges (although its tainted by speckles.) Very dramatic. The speckles are clearly show the recurrance is "measure based". linas 01:01, 30 August 2005 (UTC)

Expanding the article

Hi, all

I took the liberty to start expanding this article. It definitely needs much more. On the other hand, the last paragraph contains material which is a result of my own thinking about the subject, so it may not be appropriate for inclusion in WP. I hope there are people here who can judge it better.

Dmanin 07:53, 26 November 2005 (UTC)

I think it is appropriate to have a formal statement of the theorem. I don't know the formal statement -- I did once upon a time -- and I'd like to see it there. Dmharvey 14:21, 27 November 2005 (UTC)

I agree completely. Dmanin 23:40, 27 November 2005 (UTC)

This doesn't seem right

Hi.

I noticed this:

"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."

But this does not seem to work. For example, consider the simple system given by iterating the function f(z) = z/2, in, say, the unit disk of the complex plane. (This can be expressed as a function of time as f(z, t) = z/(2^t), which generalizes it to continuous time as well.) This is confined to a finite amount of (complex) space, and yet all points inevitably tend toward zero, and except for zero itself, never return to an "arbitrarily small neighborhood" of the initial state. The unit disk shrinks inexorably to zero under the iteration. But Poincaré's theorem is true (otherwise it would not be a theorem), so I must be missing something here. Does the "energy" of the system need to be conserved in order for this to work, and so does my system above not have a constant "energy" (what is the general, mathematical definition of "energy" for a dynamical system, anyway?) with respect to time t? If so, then should not the statement of the theorem in the opener be modified? mike4ty4 20:37, 11 September 2007 (UTC)

The first line is a rather imprecise formulation of the theorem, meant to be accessible to as many people as possible. The precise statement, later on, explains that the map has to be volume-preserving. In your example, the map shrinks sets and thus Poincaré's theorem does not apply.

I understand that the first sentence of the article, that you quoted above, is misleading, so I rewrote it. I hope that helps. -- Jitse Niesen (talk) 07:51, 15 September 2007 (UTC)

questionable paragraph

This paragraph:

One possible way of reconciling entropy and recurrence is the following. Poincaré's 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.

Seems to me to be questionable, and in any case is not sourced to any citation, and so unless a citation is given we'd have to treat it as a piece of OR. So I will remove it (someone can add it back in if they have a proper cite). --SJK (talk) 06:38, 20 April 2008 (UTC)

I don't think it's OR, I've seen this discussed in the literature, sorry I have no references. it's interesting, dunno if it's "true". linas (talk) 19:39, 21 November 2010 (UTC)

Section: "Recurrence theorem and entropy" Needs editing

I'm not happy with this section. It doesn't cite any references, and I'm not satisfied that its claim regarding the fact that their are no times to compare the recurrence time to are true, principally because there is certainly a time scale associated with the mean free path, velocity and the density of the gas, which can be shown to be very small compared to the recurrence time. In terms of experiment one can also claim the recurrence times are very large compared to the constraints of any possible experiments.

Actually, the recurrence time for any macroscopical system is going to be longer than most of the times listed in 1 E19 s and more, and there is little doubt that, for example, the half-life of the proton is a characteristic time scale of a system made of hydrogen gas in a box. --A r m y 1 9 8 7 ! ! ! 10:39, 8 September 2008 (UTC)

Yeah, that sentence was hogwash; OR, I guess. I removed it. It's reproduced below.

However this explanation is not entirely satisfactory, since there is not, in fact, any characteristic timescale in the system, <x! -- yes there is! the radius of the box divided by the root-mean-square velocity of a particle in the box gives a characteristic timescale -- x> 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.

I removed above. linas (talk) 19:44, 21 November 2010 (UTC)

Deleting incorrect example

I am deleting the following text which was found in the section "Precise formulation"

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.

The phenomenon discussed here is not an example of Poincare recurrence. Although Poincare recurrence predicts that eventually the image would return to a state close to the original state, this is in a "monkeys on a typewriter eventually producing the works of Shakespeare" sense, not within "thousands of times". The editor who added this may have been thinking of certain number-theoretic phenomena which cause this kind of behavior for certain transformations (though as far as I know not for the baker's transformation), see e.g. the article of Freeman Dyson and Harold Falk in the American Mathematical Monthly, "Period of a discrete cat mapping". David9550 (talk) 19:06, 30 June 2016 (UTC)

Links with Nietzsche's Eternal Return

What about adding a section of 'Links with philosophy' regarding the possible connection to the Eternal Return by Nietzsche? James Dioniso (talk) 14:32, 19 September 2020 (UTC)

Interesting. I think it's relevant.

-Mildare — Preceding unsigned comment added by 2806:2F0:9260:5A55:B875:5ADB:7B8E:C22F (talk) 05:23, 23 April 2021 (UTC)