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When one thinks of chaos, then automatically the thought pops up that a very little difference in the initial conditions enlarges over time and you end up with a totally different end situation.

When I tear different pieces of an A4 paper (in the middle), no initial condition are the same, but I always end up with two half pieces. It´s a bit like throwing a little ball in the bathroom wash basin (or maybe a lightning flash). The ball always ends up in the sink.

Of course, the tearing of the paper is guided by my hands, like the little ball is guided by the surface it´s on. But are there situations, like a little ball on a flat surface , where you can´t predict (wich you more or less can in the example above where the tearing ends up always somewhere in the middle of the paper), where a small difference in the initial conditions makes the tear end up in a completely different place as in the tearing with my hands? Maybe a piece of paper that gets teared by the wind?. The wind should then be exactly the same (wich is impossible but for the sake of argument) in the two slightly different initial tears, but produces this chaotic behaviour in the tearing?

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    $\begingroup$ Your initial definition is not sufficient for chaos. One can easily find highly predictable complex maps that are not considered chaotic. $\endgroup$ – CuriousOne May 5 '16 at 5:56
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    $\begingroup$ What do you mean by complex maps? And what is my initial definition? $\endgroup$ – descheleschilder May 5 '16 at 6:07
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    $\begingroup$ The map between the initial conditions and the state at a given time. Did you look at chaos theory (which is mathematics, by the way, not physics)? $\endgroup$ – CuriousOne May 5 '16 at 6:48
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    $\begingroup$ Why the downvotes? It is a legitimate question. It is not off-topic. If the answer is no, that doesn't make it a bad question. That is the question. Please don't just downvote because it is already downvoted. $\endgroup$ – Neil May 5 '16 at 7:20
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    $\begingroup$ I don't think you've given any reason to think that the difference in initial conditions when tearing paper that leads to different tearing processes is really small. The point is your hands (and we humans in general) are really bad at yielding consistent initial conditions. Let a machine do it and you get perfectly predictable results. $\endgroup$ – ACuriousMind May 5 '16 at 10:52
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This is an interesting question.

Formally, yes, when you tear the paper, there is sensitive to initial conditions, and that explains why it is very unlikely that two pieces of paper are teared exactly in the same shape on every tear.

However, the fact that there is chaos does not imply that something can not be described macroscopically. There is nothing forbidding that macroscopically something is very likely to happen (i.e. the piece always end up in two pieces).

Let us consider a more familiar example, friction. A ball thrown to the air will suffer friction. Microscopically, particles are hitting the ball, making it to decrease its velocity. However, the trajectory of the ball can still be well approximated by Newton's law + stokes drag. Why is that?

Chaos does imply that, microscopically, the system is not integrable (and thus no long term deterministic predictions of the state are possible), but it does not imply that it can not be described statistically(1), and that its most likely outcome is so probable that the most likely outcome behaves as if it was deterministic (e.g. stoke's law).

More formally, what happens in my ball example is that the it is statistically extremely likely that the ball will follow the path described by Newton's law + stokes drag, even though the specific positions of some of the particles involved (air particles) will still be very different, due to chaos.

Likewise, in the example of yours, what happens is that it is statistically very likely that the pieces will tear in two pieces such that you end up saying "I always end up with two half pieces".

In summary, chaos implies that trajectories diverge exponentially in time and that the (microscopic) state of the system is unpredictable, but it does not imply that the evolution of an ensemble of unpredictable trajectories is unpredictable. In particular, it can happen that it is extremely likely that particles will almost always end in two separated pieces.

(1) Statistically here means over an ensemble of initial conditions. In your case, the ensemble of the different microscopic ways you decide to tear the paper.


Going a step forward in your example, you could decide to sharply fold the paper prior to tear it, for the paper to be teared along that specific fold line. What you are effectively doing is to increase the likelihood that the paper will be teared over that specific fold line.

A crucial pillar of statistical physics is that "increase likelihood to be in a specific configuration" costs you energy (e.g. to fold the paper). In physics jargon, it costs energy to decrease the entropy.

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    $\begingroup$ Does the temperature of the paper matter? $\endgroup$ – CuriousOne May 6 '16 at 8:44
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Unless I am very much mistaken, the main source of apparent randomness when tearing a paper are the inevitable inhomogeneities of the material, which are imposed on it during production.

Now, if you consider a fixed piece of paper, i.e., fixed inhomogeneities, it’s difficult to say whether we still have a high sensitivity to initial conditions. You cannot tear the very same paper twice experimentally, and thus we would need a theory of paper tearing.

But even if we suppose that paper tearing is sensitive to initial conditions, we still lack the fundamental property of topological mixing: A paper torn in two does not turn into an untorn paper. There is no perpetual dynamical flow encompassing all states.

To get a chaotic system involving paper tearing, you could consider a neverending stream of paper that is disgorged between a pair of cylinders and then split unto two pairs of cylinders and thereby torn, with the torn paper being fed back into the paper producing machine.

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