Billiards, Chaos and Portfolio Optimization

First of all I have to excuse that the physics friday has become a physics saturday night. Reason for the delay has been a business trip to the UK.

The branch of physics studying how chaotic classical dynamical systems can be described in terms of quantum theory is called quantum chaos. The famous correspondence principle states that classical mechanics is the classical limit of quantum mechanics. If this is true then there must be quantum mechanisms underlying classical chaos. 

I think a good starting point here would be to give you a small example of classical chaos - this brings us to play around with billiards. A billiard can be described as a number of particles moving around in a region confined by hard walls.

Assume a 2d rectangle billiard with an aspect ratio β in a magnetic field perpendicular to the motion plane. Fixing the strength of the magnetic field one can tune the cyclotron radius Rc by varying the velocity of the particle. In the single-particle case, we focus on the dynamics of the system as a function of β and  μ=Rc/Lx where Lx is the choosable side of the rectangle. In both limits  (μ going to zero and  μ going to infinity) the motion is regular and between these two limits the dynamics is generally mixed except at particular values of μ when the system is completely chaotic.

The picture shows a phase space map for a rectangular billiard with aspect ratio β=2 and μ=6The color map indicates the phase-space distance ∆  (as a measure whether an initial point in the phase space leads to chaotic or regular motion) between two orbits having a small initial perturbation. (b) Ordered phase-space distances for all the cells plotted in (a). The dashed line shows the threshold (∆ = 0.1) between chaotic and regular motion.

To see how this classical picture relates to a quantum mechanical one and how one can use the concepts applied there for improving portfolio optimization you should mark next friday red in your calendar.