Thursday, June 28, 2007

Near encouters and chaotic spectral statistics

Yesterday, Fritz Haake gave an interesting talk in the ASC colloquium. He explained why the observed statistics of energy levels characteristic for classically chaotic systems can be understood.

Classically, it is a characterisation of chaotic behaviour that if you start with similar initial conditions the distance will separate exponentially over time. This is measured by the Lyapunov exponent. Quantum mechanically, the situation is more complicated as the notion of paths is no longer available.

However it had been noticed already some time ago that if you quantise a classically chaotic system, the energy levels have a characteristic statistic: It's not the individual energy levels but you have to consider the difference between nearby levels. It the levels were random, the differences would be Poisson distributed (for a fixed density of states). However what one observes is a Wigner-Dyson distribution: It starts out with (E-E')^n for some small integer n (which depends on the symmetry of the system) before it falls of exponentially. This is just the same distribution that one obtains in random matrix theory (where n depends on the ensemble of matrices, orthogonal, unitary or symplectic). This distribution is supposed to be characteristic for chaos and does not depend (beyond the universality classes) on the specific system.

In the collqium now, Haake explained the connection between a positive Lyapunov exponent and level statistics.

Let us assume for simplicity that the hypersurfaces of constant energy in phase space are compact. This is for example the case for billards, the toy systems of chaos people: You draw some wall in hyperbolic space and study free motion with reflections at this wall. Now you consider very long periodic orbits (it's another property of chaotic systems that these exist). Because there is not too much room in the constant energy surface there will be a number of points where the periodic orbit nearly self-intersects (it cannot exactly self-intersect as the equation of motion in phase space is first order). You can think of the periodic orbit then as starting from this encounter point, doing some sort of loop coming back and leaving along the other loop.

Now, there is a nice fact about chaotic systems: For these self encounters there is always a nearby periodic orbit which is very similar along the loops but which connects the loops differently at the self encounter. Here is a simple proof of this fact: The strong dependence on initial conditions is just the same as stability of the boundary value problem: Let's ask what classical paths of the system are there such that x(t0)=x0 and x(t1)=x1. If you now vary x0 or x1 slightly, the solution will only very a tiny bit and the variation is exponentially small away from the endpoints x0 and x1! This is easy to see by considering a midpoint x(t) for t0<t<t1: The path has some position and velocity there. Because of the positive Lyapunov exponent, if you vary position or velocity at t, the end-points of the path will vary exponentially. Counting dimensions you see that an open set of varying position and velocity at t maps to an exponentially larger open set of x0 and x1. Thus, 'normal' variation at the end-points corresponds to exponentially small variation of mid-points.

Now you treat the point of near self encouter of the periodic orbit as boundary points of the loops and move them a bit to reconnect differnetly and you see that the change of the path in the loops is exponentially small.

Thus for a periodic orbit with n l-fold self-encounters, there are (l!)^n nearby periodic orbits that nearly differ only be reconnections at the self-encounters. This was the classical part of the argument.

On the quantum side, instead of the energy difference between adjacent levels (which is complicated to treat analytically) one should consider the two-point correlation for the density of states. This can be Fourier transformed to the time domain and for this Fourier transform there is a semiclassical expression coming from path integrals in terms of sums over periodic orbits. Now, the two point correlation receives contributions from correlations between two periodic orbits. The leading behaviour (as was known for a long time) is determined between the correlation between one periodic orbit and itself.

The new result is that the sub-leading contributions (which sum up to the Wigner Dyson distribution) can be computed by looking at the combinatorics of the a periodic orbit and its correlation with the other periodic orbits obtained by reconnecting at the near encounter points.

If you want to know the details, you have to look at the papers of Haake's group.

Another approach to these statistics is via the connection of random matrix theory to non-linear sigma models (as string theorists know). Haake claims that the combinatorics of these reconnections is in one to one correspondence to the Feynman diagrams of the NLSM perturbation theory although he didn't go into the details.

BTW, I just received a URL for the videos from Strings 07 for us Linux users which had problems with the files on the conference web page.