When Chaos Goes Quantum

Not many Mondays ago we had a Departmental Colloquium here at USC entitled “Quantum Chaos and the Foundations of Statistical Mechanics”, by Mark Srednicki, of UCSB.

This was a double treat for me, since I’ve known Mark since my days in Santa Barbara, and remember many happy lunchtimes sitting at lunch with him overlooking the lagoon talking about everything from physics to Bablyon 5. That was during those truly amazing days of being a postdoc in string theory at the time when D-brane technology was turning the field upside down, and a lot of the torque needed for this was being generated right there in Santa Barbara, sometimes in lunchtime conversations. I was reminiscing about those days just a week before in Cambridge, having run into Karl Landsteiner and Roberto Emparan, two other postdocs from those fantastic times. The reason for us all being in Cambridge was to attend the Andrew Chamblin memorial conference, which I told you about in an earlier post. Andrew was also a postdoc there, around the same time as us, and we rapidly forged the good friendships that you’ve read about in a number of earlier posts linked from the previous link.

Mark used to tell me a bit about Quantum Chaos back then too, and I found it interesting, but always wanted to hear the story laid out properly, and to hear what he was doing with it, once I understood the basics. It also sounded like a perfect colloquium topic, since my mission is to expose the department to a variety of things going on in physics, and we’ve not had anything from this area of theoretical physics for quite some time. I don’t even know what area you’d call this. It’s just interesting stuff.

He gave us a reminder of a number of classically chaotic systems of interest, including some of the famous billiards examples that you may have seen elsewhere. (Here is a nice website with some billiard simulation examples (Java applets) you can play with yourself. I also found a website by Matthew Trump, at the university of Texas introducing chaos to the layperson. It is a tad annoying in that you have to keep clicking to move the sentences along, but seems good. There’s an extensive Wikipedia article here.) A simple one I’d not heard of before which was fascinating is the following: Take a slope at some angle, at the bottom of which is a vertical wall. Bounce a ball down the slope. Follow its trajectory. If the angle of the slope is less than or equal to 45 degrees to the horizontal, the system is in fact non-chaotic. Above that, it is chaotic. I find this remarkable.

Classical chaos, phrased in terms of trajectories, is one thing. Then we come to quantum chaos, another thing altogether. What is it exactly? One really can not be talking about trajectories at all in this context, given the central tenets of quantum mechanics that remove the whole concept of a well-defined trajectory from the physics, via the uncertainty principle. One must try to talk about chaos in quantum systems using the language of quantum mechanics. You might try several ways of sneaking up on the definition of what chaos actually might mean in the quantum context, and I won’t go into all the different ways right now. It suffices to say that it is interesting enough to focus on aspects of the spectrum of a system to learn something interesting about the difference between chaotic and non-chaotic in this context. One might start out by quantizing classical systems that are non-chaotic on the one hand and chaotic on the other, comparing the properties of the resulting quantum systems.

So indeed, Mark talked the contrast between the energy spectrum of a chaotic system vs that of a non-chaotic system, and the contrasts in behaviour of the associated wavefunctions as well. For example, the characteristic behaviour of the statistical properties of the eigenvalues (how they are distributed) are quite different in the two cases. Rather than getting a classic p(s)=e^-s behaviour for the probability density, where s is the energy spacing in suitable units and p(s) is telling you the probability of finding an eigenvalue a “distance” s away, (this follows essentially from the Poisson statistics underlying the fact that the eigenvalues just don’t care about each other) you get behaviour like p(s)=sⁿ e^-s2, where n is 1 or 2 depending upon the system (time reversible or not). This is of course familiar to people who know their history as the type of statistics that underlies distribution of energy levels in nuclei, as uncovered by the giants like Wigner, and Dyson. They modeled these systems by random Hamiltonians…. and so it is as if chaotic systems behave as though their underlying Hamiltonian is an ensemble of random ones. Put another way, energy levels for these systems do care about each other, and they repel each other with a particular interaction (they actually form what is known as a Dyson gas, which is one dimensional and has a Coulombic repulsion)… By now, I’m hooked, as this is all the starting point of stuff I worked on for years, but in a totally different context – Matrix models and string theory. Wonderful the way things are all so connected (which is why I never try to separate out the “fundamental” from the rest of physics as many in our field are so ready to do – it is so arrogant, naive, and so often wrong). But I’ll stop here.

A quick search seems to show that there does not seem to be a huge amount out there in the way of nice online resources describing Quantum Chaos. However, I found a site written by someone called Todd Timberlake that seems to go into some of it at a general physics level. Pop there if you want to hear more. There is a bibliography to some of the classics there. There’s also a bit of a bibliography at a woefully incomplete (at date of writing) Wikipedia article, linked here. I imagine a more focused search can turn up some more sources.

I also found a Physics Today article by Eric Heller reviewing a 2002 textbook by Hans-Jürgen Stöckmann entitled “Quantum Chaos: an Introduction”. The review itself is informative in a general way, and also talks a little about the classic work. A number of the results of this area discussed have had applications to experiment, particularly in the context of quantum dots and other mesoscopic systems. Here’s a website (from the group at Washington State University) talking about some of this.

The next part of Mark’s talk was all about the origins of Statistical Mechanics. The issue is simple to state: At the heart of the physics of thermodynamics is really microscopic quantum mechanical behaviour. That box of a gas of particles is a quantum mechanical system of very many particles, but yet we get from that to a simple thermal description in terms of a thermal state (all data about initial conditions lost) at a sharply defined temperature. How does one really get from the one to the other? Can you derive the expressions we use for the descriptions of such thermal states starting from the microscopic? How does a closed system thermalise at all? This is the onset of chaos, in a specific sense. In fact, if the system is integrable (same number of degrees of freedom as conserved quantities), one might not expect it to thermalise at all, and therefore, one expects non-integrability, and indeed chaos, to play a role. I’m not going to do even a half decent job of explaining to you the details of what he does -it is a long time since the talk, for a start- but I hope you see that this is a really interesting question.

As he said, you won’t find the answer in any textbook, in fact (even though as physicists we are all taught statistical mechanics, and there are some words about thermalisation) and it was a frustration with not knowing the explicit answer to this question that led Mark to do his work in this area. He used some of the results of quantum chaos that he taught us earlier in the colloquium to show how to derive a thermalised description of the right variables from the microscopic. I used the emphasis in the previous sentence to drive home the point that there is no particular reason to expect that, for a given system, all variables settle down to a nice thermalised description. This is not unreasonable, after a bit of thought, and the conditions emerge in the derivation. There are some references to the detailed papers given on our colloquium bibliography page (generously constructed and maintained by our librarian Sara Tompson) here – scroll down to October 23rd.

I’ll stop at this point, no doubt leaving you annoyed that I did not describe more detail, but in the hope that I’ve at least whet your appetite to go away and read up a bit on this stuff, especially if it is outside your physics comfort zone. It is/was a large gap in my own education, and so I was happy to see it spelled out. In fact, there’s a connection to things I’ve been working on that I’d not thought about before, and I got a nice slap-in-the-face of a crazy idea I shall have to think about and see if it is worth working up into a sharp statement of physics at some point.

-cvj