“God plays dice with the Universe. But they’re loaded dice. And the main objective of physics now is to ﬁnd out what rules were they and how we can use them for our own ends.” -Joe Ford

Joe Ford was one of the greats of “Chaos Theory.” He is largely responsible for turning this into a topic of interest in the West (the Soviets invented much of it independently) through his founding of the journal Physica D. It is one of the indignities of physics history that he isn’t more widely recognized for his contributions. I never met the guy, as he died around the time I began studying his ideas, but my former colleagues sing his praises as a great scientist and a fine man. One of his lost ideas, working with student Matthias Ilg and coworker Giorgio Mantica, is the “Ford paradox.” The Ford paradox is so obscure, a google search on it only turns up comments by me. This is a bloody shame, as it is extremely interesting.

Definitions: In dynamical systems theory, we call the motion of a constrained system an “orbit.” No need to think of planets here; they are associated with the word “orbit” because they were the first orbital systems formally studied. It’s obvious what an orbit is if you look at the Hamiltonian, but for now, just consider an orbit to be some kind of constrained motion.

In most nontrivial dynamical systems theory, we also define something called the “phase space.” The phase space is that which fully defines the dynamical state of the system. In mechanics, the general convention is to define it by position and momentum of the objects under study. If the object is constrained to travel in a plane and its mass doesn’t change, like, say, a pendulum, you only have two variables; angular position, and its time derivative, and you can easily visualize the phase space:

For my last definition, I will define the spectrum for the purposes of this exposition. The spectrum is the Fourier transform with respect to time of the orbits. Effectively, it is the energy levels of the dynamical system. If you know the energy and the structure of the phase space, classically speaking, you know what the motion is.

Consider a chaotic system, such as the double pendulum. Double pendulums, as you might expect, have two moving parts, so the phase space is four dimensional, but we can just look at the angle of the bottom most pendulum with respect to the upper pendulum:

If you break down the phase space into regions, and assign a string to each region, one can characterize chaos by the length of the string in bits. If it is a repeated string, the system is non-chaotic. Chaotic systems are random number generators. They generate random strings. This is one of the fundamental results of modern dynamical systems theory. A periodic orbit can be reduced to simple sequences, like: {1 0 1 0 1 0}, {1 1 0 1 1 0 1 1 0}. Effectively, periodic orbits are integers. Chaotic orbits have no simple repeating sequences. Chaotic orbits look like real numbers. Not floats which can be represented in a couple of bytes: actual real numbers, like base of the natural log or or the golden ratio . In a very real sense, chaotic orbits generate new information. Chaotic randomness sounds like the opposite of information, but noisy signals contain lots of information. Otherwise, qua information theory, you could represent the noise with a simple string, identify it, and remove it. People have invented mechanical computers that work on this principle. This fact also underlies the workings of many machine learning algorithms. Joe Ford had an extremely witty quoteable about this: “Evolution is chaos with feedback.”



This is all immediately obvious when you view the phase space for a chaotic system, versus a non-chaotic system. Here is a phase space for the end pendulum of a double pendulum at a non-chaotic set of parameters: it behaves more or less like a simple pendulum. My plots are in radians (unlike the above one for a normal pendulum, which I found somewhere else), but otherwise, you should see some familiar features:

It looks squished because, well, it is a bipendulum. The bottom which looks like lines instead of distorted ellipses is where the lower pendulum flips over the upper pendulum. The important thing to notice is, the orbits are all closed paths. If you divided the phase space into two regions, the path defined string would reduce to something like {1 0 1 0 1 0…} (or in the lower case { 0 0 0 0…}) forever.

Next, we examine a partially chaotic regime. The chaotic parts of the phase space look like fuzz, because we don’t know where the pendulum will be on the phase space at any given instant. There are still some periodic orbits here. Some look reminiscent of the non-chaotic orbits. Others would require longer strings to describe fully. What you should get from this; the orbits in the chaotic regions are random. Maybe the next point in time will be a 1. Maybe a 0. So, we’re generating new information here. The chaotic parts and not so chaotic parts are defined on a manifold. Studying the geometry of these manifolds is much of the business of dynamical systems theory. Non-chaotic systems always fall on a torus shaped manifold. You can see in the phase space that they even look like slices of a torus. Chaotic systems are, by definition, not on a torus. They’re on a really weird manifold.

Finally: a really chaotic double pendulum. There are almost no periodic orbits left here; it’s all motion in chaotic, and the path the double pendulum follows generates random bits on virtually any path available to it in the phase space:

Now, consider quantum mechanics. In QM, we can’t observe the position and momentum of an object with infinite precision, so the phase space is “fuzzy.” I don’t feel like plotting this out using Husimi functions, but the ultimate result of it is the chaotic regions are smoothed over. Since the universe can’t know the exact trajectory of the object, it must remain agnostic as to the path taken. The spectrum of a quantum mechanical orbital system looks like … a bunch of periodic orbits. The quantum spectrum vaguely resembles the parts of the classical phase space that look like slices of a torus. I believe it was W.P. Reinhardt who waggishly called this the “vague tori.” He also said, “the vague tori, being of too indistinct a character to object, are then heavily exploited…” Quantum chaologists are damn funny.

This may seem subtle, but according to quantum mechanics, the “motion” is completely defined by periodic orbits. There are no chaotic orbits in quantum mechanics. In other words, you have a small set of periodic orbits which completely define the quantum system. If the orbits are all periodic, there is less information content than orbits which are chaotic. If this sort of thing is true in general, it indicates that classical physics could be a more fundamental theory than quantum mechanics.

As an interesting aside: we can see neat things in the statistics of the quantum spectrum when the classical equivalent is chaotic; the spectrum looks like the eigenvalues of a random matrix. Since quantum mechanics can be studied as matrix theory, this was a somewhat expected result. Eigenvalues of a random matrix were studied at great length by people interested in the spectra of nuclei, though the nuclear randomness comes from the complexity of the nucleus (aka, all the many protons and neutrons), rather than the complexity of the underlying classical dynamics. Still, it was pretty interesting when folks first noticed it in simple atomic systems with classically chaotic dynamics. The quantum spectra of a classically non-chaotic system are more or less near neighbor Poisson distributed. Quantum spectra repulse one another. You know something is up when near neighbor spectral distribution starts to look like this:

Random matrix theory is now used by folks in acoustics. Since sound is wave mechanics, and since wave mechanics can be approximated in the short wavelength regime by particles, the same spectral properties apply. One can design better concert hall acoustics by making the “short wavelength” regime chaotic. This way there are no dead spots or resonances in the concert hall. Same thing applies to acoustically invisible submarines. I may expand upon this, and its relationship to financial and machine learning problems in a later blog post. Spectral analysis is important everywhere.

Returning from the aside to the Ford paradox. Our chaotic pendulum is happily chugging along producing random bits we can use to, I dunno, encrypt stuff or otherwise perform computations. But, QM orbits behave like classical periodic orbits, albeit ones that don’t like standing too close to one another. If quantum mechanics is the ultimate theory of the universe: where do the long strings of random bits come from in a classically chaotic system? Since people believe that QM is the ultimate law of the universe, somehow we must be able to recover all of classical physics from quantum mechanics. This includes information generating systems like the paths of chaotic orbits. If we can’t derive such chaotic orbits from a QM model, that indicates that QM might not be the ultimate law of nature. Either that, or our understanding of QM is incomplete. Is there a point where the fuzzy QM picture turn into the classical bit generating picture? If so, what does it look like in the transition?

I’ve had physicists tell me that this is “trivial,” and that the “correspondence principle” handles this case. The problem is, classically chaotic systems egregiously violate the correspondence principle. Classically chaotic systems generate information over time. Quantum mechanical systems are completely defined by stationary periodic orbits. To say the “correspondence principle handles this” is to merely assert that we’ll always get the correct answer, when, in fact, there are two different answers. The Ford paradox is asking the question: if QM is the ultimate theory of nature, where do the long bit strings in a classically chaotic dynamical system come from? How is the classical chaotic manifold constructed from quantum mechanical fundamentals?

Joe Ford was a scientist’s scientist who understood that “the true method of knowledge is experiment.” He suggested we go build one of these crazy things and see what happens, rather than simply yakking about it. Why not build a set of small and precise double pendulums and see what happens? The double pendulum is pretty good, in that its classical mechanics has been exhaustively studied. If you make a small enough one, and study it on the right time scales, quantum mechanics should apply. In principle, you can make a bunch of them of various sizes, excite them to the chaotic manifold, and watch the dynamics unfold. You should also do this in simulation, of course. My pal Luca made some steps in that direction. This experiment could also be done with other kinds of classically chaotic systems; perhaps the stadium problem is the right approach. Nobody, to my knowledge, is thinking of doing this experiment, though there are many potential ways to do it.

It’s possible Joe Ford and I have misunderstood things. It is possible that spectral theory and the idea of the “quantum break time” answers the question sufficiently. But the question has not to my knowledge been rigorously answered. It seems to me much a more interesting question than the ones posed by cosmology and high energy physics. For one thing, it is an answerable question with available experimental tests. For another, it probably has real-world consequences in all kinds of places. Finally, it is probably a productive approach to unifying information theory with quantum mechanics, which many people agree is worth doing. More so than playing games postulating quantum computers. Even if you are a quantum computing enthusiast, this should be an interesting question. Do the bits in the long chaotic string exist in a superposition of states, only made actual by observation? If that is so, does the measurement produce the randomness? What if I measure differently?

But alas, until someone answers the question, I’ll have to ponder it myself.

Edit add:

For people with a background in physics who want to understand the information theory behind this idea, the following paper is useful:

“The Arnol’d Cat: Failure of the Correspondence Principle” J. Ford, G. Mantica, G. H. Ristow, Physica D, Volume 50, Issue 3, July 1991, Pages 493–520