Fibonacci Chaos And Time's Arrow
    By Johannes Koelman | August 2nd 2009 08:58 PM | 14 comments | Print | E-mail | Track Comments
    About Johannes

    I am a Dutchman, currently living in India. Following a PhD in theoretical physics (spin-polarized quantum systems*) I entered a Global Fortune


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    Call it irreversibility, call it time's arrow, call it the second law of thermodynamics. Fact is that everything evolves in such a way that things get more messy. Disorder rises. Entropy increases. We do not observe the opposite happening. Heat flows from from hot to cold, not the other way around. Fluids mix but don't unmix. Shattered pieces of crystal don't reassemble into a vase.

    Yet the laws of physics tell us that when studying the evolution of physical systems in all its microscopic details, there is no preferred 'direction of time'. The fundamental laws of physics obey time symmetry. For every forward evolution there exists a time reversed evolution. Both evolutions are equally valid and equally likely. Reverse all velocities in Newton's dynamics and the system traces back its history.

    The laws of physics tell us that forward and reverse directions of time are absolutely indistinguishable down to the minute and microscopic details. Yet, we all know that when we are watching a movie we can quickly tell whether the movie is displayed in reverse or not. We seem to have manouvred ourselves in a paradoxial situation.

    Enter Ludwig Boltzmann, the posthumously famous Austrian physicist who was tragically ahead of his time and - following many years of struggle to get his ideas accepted by the scientific establishment - died by suicide in 1906. Boltzmann left a legacy that is now much better understood. He provided us with the guidance and the tools to marry the time-reversible laws of physics with the time-irreversible behavior we observe in our everyday world.

    At the heart of Boltzmann's theories is a statistical approach towards the description of nature. In particular, Boltzmann showed that the thermodynamic entropy of a system, the amount of energy no longer available to do physical work, can be defined rigorously in a statistical way. In simple terms, Boltzmann's entropy S is the minimum length of the encoding you need to fully describe the state of the system. It is the amount of info (number of bits or digits) you need to trace back the history of the system.

    The following 'Fibonacci fleas model' should make this more concrete.

    Consider a dozen fleas feeding on nine cats (targets labeled '1-9') and a human (target '0'). At discrete intervals, the fleas jump from one target to the other according to a sequence of 12 digit integers being generated. These integers represent the states of our discrete dynamic model by enumerating the locations of the fleas. For instance,


    means flea number 1 is located on cat 5, flea 2 can be found on human 0, flea three on cat 4, and so on up till flea 12 who is located on cat 9. If the next state is


    we observe a synchronized jump with the first flea jump from cat 5 to cat 4, the second flea from the human to cat 3, etc.

    The generation of these integers in this discrete model follows a simple time-symmetric mechanics: based on the last two numbers generated, the next number is three times the last minus the one-but-last. If the number computed exceeds 999,999,999,999 or becomes negative, it is mapped into a 12-digit positive integer by adding or subtracting the appropriate multiples of 1,000,000,000,000. So referring to the number generated at timestep t as W(t), we have:

      W(t+1) = 3 W(t) – W(t-1) (mod 1,000,000,000,000)

    It is important to stress at this point that the dynamics is indeed time-symmetric as this equation can also be written as:

      W(t-1) = 3 W(t) – W(t+1) (mod 1,000,000,000,000).

    Starting from W(0) = 0 and W(1) = 1, this reversible arithmetic in forward mode yields the even Fibonacci numbers:

      0, 1, 3, 8, 21, 55, ...

    whilst the odd fibonacci numbers are generated when starting from W(0) = 1 and W(1) = 1:

      1, 1, 2, 5, 13, 34, 89, ...

    Let's start with the latter initial condition, and see what happens to the distribution of fleas. We write the full 12 digit numbers so as to make explicit the location of each flea. We observe:

      W(0) = 000,000,000,001
      W(1) = 000,000,000,001
      W(2) = 000,000,000,002
      W(3) = 000,000,000,005
      W(4) = 000,000,000,013
      W(5) = 000,000,000,034
      W(6) = 000,000,000,089
      W(7) = 000,000,000,233

    and so on. We can now track the total number of fleas on the human (the number of zeros in the 12-digit numbers) Obviously, starting at t = 0 with all the fleas on the human (twelve zero's), the number of fleas on the human gradually decreases. This trend continues untill after 28 steps a state is reached that has no fleas on the human:

      W(28) = 139,583,862,445

    From that moment on the number of fleas on the human fluctuates around values close to zero:

      W(29) = 365,435,296,162 (no fleas bothering the human)
      W(30) = 956,722,026,041 (two fleas)
      W(31) = 504,730,781,961 (two fleas)


    Let us now focus on the entropy of this flea dynamics. Recall that we defined entropy as the minimum number of digits needed to re-trace the system's history. To be able to do this, one needs just the last two flea-state numbers. Using these two numbers we compute all preceding numbers simply by applying the same dynamics in reverse direction (second equation above).

    So what does this tell us about the entropy S?

    Having reached state 29, retracing the history requires the two full 12-digit numbers W(29) and W(28) or an entropy of 24 digits. The same is true once the dynamics has brought us beyond t = 29. However, when the dynamics has evolved no further than some earlier time, this is no longer true. As an example, consider the evolution to t = 7. We need the two numbers W(6) and W(7) to retrace the full history. So that is again two times twelve digits, right? Wrong. You only need a total of 6 digits (S = 6) as ignoring the leading zero's (which are irrelevant for computing the dynamics) both numbers have only three digits.

    Carefully counting the number of digits in subsequent pairs of W-numbers whilst ignoring the leading zero's, we observe the entropy to increase to its maximum value S = 24 digits in about 28 timesteps (see figure). Now this is interesting. The irreversibility paradox manifests itself in its full glory in this simple fleas model. Using nothing more than a very simple fully time symmetric Fibonacci-type dynamics, we observe an 'arrow of time' in the form of an entropy that increases in one direction, and decreases in the reverse direction.

    It should now be clear what is causing the entropy increase with time. Entropy increases simply because te system started in a specially prepared low-entropy starting state. There is no paradox related to the fact that the dynamics is reversible. This can be observed more clearly by continuing the reverse dynamics into negative times. This results into a perfectly time-symmetrical plot of entropy (red curve) and number of fleas on the human (blue curve). 

    In his superb book “The Emperor's New MindRoger Penrose takes this reasoning further and applies it to the universe and its origin. He stresses the fact that the universe started off at what he refers to as a 'ridiculously tiny entropy' state. Based on a closed universe model (a universe that eventually collapses into a big crunch) and some holographic considerations, he comes to the conclusion that the creator had to select some 10^123 digits each to a unique value such as to create a universe with the known low entropy as we witness it today.

    In other words, only one out of 10^(10^123) possible initial universes had the right properties for the universe to evolve with a second law of thermodynamics as powerful in effect as in our universe. These are no mind boggling figures. These numbers are simply incomprehensible. In a future blog I will revisit this issue and present you with a simple model for an expanding universe that in terms of entropy behaves in a remarkably different way. 


    The thing is, all the other [big number] possible universes with less than the right properties as you describe, probably did have their go at existing, or those that haven't, will eventually. However, nothing came of it, each time a universe is run. And other universes that come after ours. Nothing might come of theirs either. But every once in a while, one comes along that fits the requirements.

    If there are truly an infinite number of Universes, there must be an infinite number of Universes that are completely or slightly different from our own Universe, a infinite number of Universes that are exactly the same as our own (with exact copies of everthing and everybody) and an infinite numbers of Universes that are exactly the same as our own but with time running (slightly or a lot) ahead or behind time in our own Universe.

    Maybe there isn't infinite. Maybe there's actually only "almost infinite" or a nominal infinite, which for most purposes will do. Thus, there will be a boundary or horizon of possibility. All the things that are possible are bounded by this possibility envelope. All possibilities are effectively (or at least, might as well be) executed. Whether they actually are executed or not is immaterial, it's the fact that they are possible or not, that denotes that they might possibly be executed. Therefore, all possibilities might as well be executable, and among all the possibilities, some will work out, some won't, but all were at least possible.

    Johannes Koelman
    Yes, the anthropic principle could be in effect here, and it might have taken some 10^(10^123) attempts before we emerged.

    Having said that, I am a strong believer in Occam's razor (the opening sentence in the Wikipedia article seems particularly appropriate in this context!).

    A much simpler explanation is possible. More about that soon...
    There probably only needs to be three universes. A "higher than" universe (for any given observable parameter), a "lower than" universe, and our "just right" universe.

    In fact, those three could be embodied in just the one, if you allow the "higher than" and "lower than" possibilities to represent themselves as mere possibilities, not actual manifestations in our present reality. At least, not today. Not on Tuesdays.

    If the big bang is not a supernatural event but a mere natural event, it probably has happened an infinite number of times already, it is probably happening an infinite number of times right now (whatever "now" means), and it will probably continue to happen an infinite number of times...

    Interesting article. From your reference to Boltzmann, I think you would like:

    Gadflies and Geniuses in the History of Gas Theory
    Stephen G. Brush
    Synthese Vol 119 Issue 1-2 pp 11-43 (1999)

    Robert H. Olley / Quondam Physics Department / University of Reading / England
    Johannes Koelman
    Thanks for the reference.

    And.. nice summary! 
    wow this is so scientific

    Your fibonacci & sequence articles are so interesting that bookmarked your page to my blogiste.

    In the case you interested some work how they are applied in the financial market (which after all is less science) take a peek.

    Any work of time series with fibonacci sequence would be interesting to read.

    Time is the presence of motion and forces. Time is due to expansion of space. Time is slow where expansion of space is slow like around large masses. As total motion and forces within a mass is a constant therefore when linear motion is increased then internal motion as well forces within that object slow which is then percieved as slowing of time. If we think of two objects in orbit around each other and imagine time slowing down then we should motion slow down as well as forces get weaker. Forces are part of time. Forces determine the arrow of time. Without forces time will be perfectly symetrical and lack any direction.

    If given enough time, the universe can and will return to a state of low entropy as it once was.

    Of course you and I can never perceive such an event for it supersedes the capability of all mortals.

    I, on the other hand, am immortal.

    So these things come naturally for me

    >If given enough time, the universe can and will return to a state of low entropy as it once was.

    That's know as Poncare recurrence, given enough (and these are figures in the googleplex range) time, any
    state of the universe will happen again though random combinations of particles reforming that state, so by
    Poncare recurrence will all immortal. And the average observer is an Boltzmann Brain, randomly formed out of

    In practice though cosmic expansion and especially cosmic acceleration, will dilute the universe, meaning that
    there won't be any chance for these random recombinations to happen, the universe will continue to fall in
    density, producing an cosmic arrow of time, and making random recombinations to rare in practice to reform any
    particular state of the universe.

    Dear Johannes,

    many sincere thanks for this very nice article !

    Interestingly, with your Fibonacci model you have come to this same conclusion, about what drives the entropy to increase, as Arieh ben Naim in his books about the topic.

    Still, the entropy is well known to be "anthropomorphic", in the sense that you may define it different ways depending on the problem at hand. Thus, how your Fibonacci model would solve the problem of entropy's "anthropomorphicity" ?

    Respectfully yours,

    Evgeni B Starikov