Is there a difference between deterministic differential equations and a probabilistic model?
Reading Max Tegmark’s terrific book, “The Mathematical Universe – My Quest for the Ultimate Nature of Reality”, inspired me to state the following fact, which I use to puzzle my friends at parties. Even in classical dynamics we cannot so easily say from observation if a system is deterministic or random. This has deep implications for the answer to the question, will the sun rise tomorrow?
Let’s look at so-called hyperbolic systems. In such a system uncertainties double at a stable rate until the cumulative unknowns destroy all specific information about the system. For such a system the so-called shadowing lemma of mathematics apply. This lemma says that for such systems one cannot tell whether a given system is deterministic or randomly perturbed (like the Saturn rings for example) at a particular scale, if you can observe it only at that scale. As you may know, the KAM Theorem explains the Saturn rings perfectly. Kolmogorov, Arnold and Moser were able to derive the stable tori in the phase space for the Saturn rings.
Hubbard gave once an intuitive example of such systems – angle doubling. “…start with a circle with its center at 0, and take a point on that circle measuring the angle that the radius to that point forms with the x-axis. Now double the angle, and double it again and again. You know the starting point exactly and you know the rules, so of course you can predict where you will be after 50 doublings, say. BUT now imagine that each time you double the angle you jiggle it by “some unmeasurably small amount”, say at the 10 to 15 decimal digit (assuming your computer cuts off numbers after the 15 digit). Now the system has become completely random in the sense that there is nothing you could measure at time zero that still gives you any information whatsoever after 50 doublings…”.
Does this disturb you?
It means that if you were presented with an arbitrarily long list of figures (each 15 decimals) produced by such an angle doubling system you could not tell whether you were looking at a randomly perturbed system or an unperturbed (deterministic) one. The data would be consistent with both explanations. The shadowing lemma tells us that (assuming we can measure angles only with 10 to 15 digits accuracy) there will always be a starting point that, under an unperturbed system will produce the same data as that produced by a randomly perturbed system. Actually, it will not be the same starting point, but you do not know the starting point, therefore you will not be able to choose between the starting point A and the deterministic system, and some unknowable starting point B and the random system.
Does this disturb you now?
There is no way of telling deterministic from random for a wide class of dynamical systems. You can choose between deterministic differential equations and a probabilistic model. The “facts” are consistent with both approaches. This is quite similar to the difference between how my compatriot Schroedinger viewed the universe in quantum mechanics and Heisenberg did. Heisenberg says that the universe is given for all time; we are just discovering more and more about it as times goes on. Schroedinger says that the world is really evolving and we are part of the changing world. Both are right – matrix mechanics and wave equation are mathematically speaking equivalent. We have known the Heisenberg-Schroedinger dichotomy for a while. It is a quite surprising fact that even in classical dynamics we cannot so easily say from observation if a system is deterministic or random.
In other words, we cannot yet answer with certainty if the sun will rise tomorrow. We could use Bayesian reasoning and use the prior observations that the sun came up every morning the last 4.5 billion years. But priors don’t mean much for non-linear systems. A small perturbation could have huge effects; our planetary system could be a chaotic systems that has been quasi-regular for many cycles and could still fall apart after a small perturbation from outer space.