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Is there order in chaos

  1. Aug 13, 2013 #1
    I was reading this post on livescience web site

    It's saying that the #1 unsolved mystery in physics is that we are not sure if there is order in chaos

    I was confused since I thought that chaos only exist due to the lake of infinitely precise calculations, but if there is an all seeing eye that can track every single particle in the universe with infinitely precise accuracy it can predict the state of the system at any given time.
    Do we have any other reason to think that there is no order in chaos except quantum theory which introduce wave functions that randomly collapse upon their observation / measurement ?
     
  2. jcsd
  3. Aug 13, 2013 #2
    " Do we have any other reason to think that there is no order in chaos except quantum theory which introduce wave functions that randomly collapse upon their observation / measurement ?' - TeCNoYoTTa

    In my opinion this is a very wrong statement, why would wave functions randomly collapse?? The quantum theory gives us probablity for each state!!!

    Let us look at it from statistical point of view, the most probable macro state is that which has maximum number of micro states!!!
    The second law of thermodynamics - Entropy of the universe always increases!!! We are moving to a state of increasing chaos!!!
     
  4. Aug 13, 2013 #3
    Thanks for your reply but what I meant is if there is anything other than quantum theory that contradict with the deterministic view of the world.
     
  5. Aug 14, 2013 #4
    Well that`s a very interesting question, sorry if I got you wrong.

    In my understanding, even the string theory is non-deterministic!!
     
  6. Aug 14, 2013 #5

    Filip Larsen

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    Chaos does not in itself imply non-determinism, if that is what you mean.

    If a system is chaotic then by its usual definition ([1] or any good textbook on the topic) it holds that no matter how precise you know your initial conditions (for instance as an open ball around a particular state) the trajectories of those initial states will in some sense diverge (that is separate from the trajectory of the exact initial state) after some time, and hence you will not be able to predict trajectories with arbitrary precision for arbitrary lengths of time.

    If one, as you now ask, know the exact initial condition and if the equations governing the dynamics are deterministic, then the trajectory from that state will indeed be deterministic and in some meta-physical sense predicable, but this does not mean there are no chaos (as it is defined).

    The interesting question I suspect the post is referring to, is whether the precise trajectories in a chaotic system can be described in a (mathematical) way that do not require one to "solve the actual system". Or in other words, if we link chaos with predictability, whether or not a deterministic chaotic system can be predicted with arbitrary precision without "running" the actually system, that is, simulate the full system with infinite precision (which we can't due to truncation error [2]).

    Personally, as an engineer, I do not think such a mathematical "order in chaos" is possible in general, and even if it were it would not matter much in "real world". Searching for order in natural occurring chaos would be like searching for perfect geometric structures (like circles) in nature; there may well be an abundance of approximations of such perfect structures in nature, but this does not necessarily imply that nature is really "built" using these perfect structures or that a single example of such a perfect structure even exists. Add quantum mechanics to the models and equations and you can forget any hope of finding order in natural occurring chaos.

    I will leave it to the mathematicians to evaluate whether there is any chance to find order in some (yet unknown) subset of chaotic motion. I am currently under the impression that it can be mathematical proven that no such description of order can exist, but if "order in chaos" is classified as an unsolved mystery by people in the know then I may very well be mistaking.


    [1] http://en.wikipedia.org/wiki/Chaos_theory
    [2] http://en.wikipedia.org/wiki/Truncation_error
     
  7. Aug 15, 2013 #6

    Pythagorean

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    It depends on what you call order. Generally, people speak of order as opposed to chaos. I.e. in chaos, you have neighboring states diverging (thus, a positive Lyapunov exponent). In order, you have neighboring states converging to some manifold (thus, a negative Lyapunov exponent).

    One outcome I observe in the special case of systems with multiple attractors is that in ordered systems your basins of attraction are well separated: you can define particular regions with clear boundaries for which initial conditions will always go to a particular state in the steady-state solution.

    Whereas in chaos, the basins are (usually?) riddled. The boundaries between basins are not continuous lines, but are more fractal-like:

    http://www.scholarpedia.org/article/Basin_of_attraction

    Of course, I don't claim that this defines order, just that non-riddled basins seem to be a characteristic of it (I've never seen a formal definition of order... usually authors talk about a transition form order to chaos though, so I naturally assume it means not chaos... thus negative Lyapunov exponent). In a system with just one attractor that is chaotic, the riddled basin definition isn't very meaningful since the whole system is one basin, but you can still talk about the sign of the Lyapunov exponent.
     
  8. Aug 15, 2013 #7

    atyy

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    The livescience post linked in the OP seems to be specifically about the Navier Stokes equations: "In fact, it isn't known whether a general solution of the so-called Navier-Stokes equations even exists, or, if there is a solution, whether it describes fluids everywhere, or contains inherently unknowable points called singularities."

    Perhaps it's related to the Clay Millenium problem http://www.claymath.org/millennium/Navier-Stokes_Equations/ [Broken].
     
    Last edited by a moderator: May 6, 2017
  9. Aug 15, 2013 #8

    Filip Larsen

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    I think you are right, atyy. So, it is not a new form of order in general chaotic system they refer to, but "merely" a closed, or at least partial solution, to a specific subset of chaotic systems, namely those described by the Navier-Stokes equations. Makes sense to me.
     
  10. Aug 16, 2013 #9

    Claude Bile

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    The linked article contains (or at least, implies) a false equivalence; namely that equations without well-defined solutions are chaotic.

    Claude.
     
  11. Aug 16, 2013 #10

    Pythagorean

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    The linked article is a reminder that pagination must die. Here's the one relevant paragraph, bolding the one reference to chaos.

    "Physicists can't exactly solve the set of equations that describes the behavior of fluids, from water to air to all other liquids and gases. In fact, it isn't known whether a general solution of the so-called Navier-Stokes equations even exists, or, if there is a solution, whether it describes fluids everywhere, or contains inherently unknowable points called singularities. As a consequence, the nature of chaos is not well understood. Physicists and mathematicians wonder, is the weather merely difficult to predict, or inherently unpredictable? Does turbulence transcend mathematical description, or does it all make sense when you tackle it with the right math?"

    I really have no idea what the author is saying at all. The subheading was "Is there order in chaos?"
     
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