Category: Chaos theory

Quantum Chaos

This is a discussion of the work that I did over the summer of 2012-13 at the Quantum Science department of ANU as part of the Summer Research Scholarships program there. I did a project on the computational modelling of a chaotic quantum mechanical system, focusing on how the chaos in the system changes as a function of its energy scale – from the near-classical to the fully quantum. We intend to publish our results, so I can’t post them here; however I will give you a run-down on the ideas behind it.

A little bit of chaos theory

Chaos is a ‘sexy’  kind of word in mathematics and physics alike. It is easy to vigorously hand-wave about the butterfly effect and sensitivity to initial conditions and other such things but the fact is that there’s no easy way to quantify or measure ‘unpredictability’ – due to, not surprisingly, its unpredictable nature. What we need instead is some rigorous handwaving.

So how do we decide just what is a chaotic system? All of our good quantitative descriptions of chaotic systems come from studying phase space behaviour – the vector space spanned by \left\{x_1, x_2, \dots, \frac{dx_1}{dt}, \frac{dx_2}{dt}, \dots \right\}, where each x_i is a physical degree of freedom. We consider points in phase space, each corresponding to a state of the system, and their time evolution. These produce trajectories through phase space that reflect the dynamics of the system.

Imagine a simple harmonic oscillator, the paragon of an elementary dynamical system -and something which is definitely not chaotic. This system has one variable, the position x, which is governed by the differential equation \frac{d^2x}{dt^2} + k^2x = 0 for some constant k. Now, if we pull the oscillator out to some particular distance d_0 and then let it go (i.e. set the system up with some particular set of initial conditions), the trajectory through phase space is simply an ellipse in the plane, centered on the origin. The aspect ratio of the ellipse is given by the ratio of the physical parameters of the system, the mass of the oscillator and the strength of the spring. For any given parameter set, with appropriate change of variables we can always make this ellipse a circle. The square of the radius gives the total energy in the system, which for us would be proportional to d_0^2).

Now suppose we were to repeat the experiment with the simple harmonic oscillator – we begin by pulling our oscillator out to distance d_0, but instead of letting go here we pull an additional amount \delta further. This means that our initial states for this oscillator are separated in phase space (under the Euclidean metric) by an amount \delta and that our initial energies differ by a constant proportional to (d_0 + \delta)^2 - d_0^2 = 2d_0\delta + \delta^2. But given that we know the energy gives the radius of the circular trajectory through the appropriately scaled phase space, which is constant in our lossless system, we know that at all times t, these two trajectories are separated by a constant amount.

We see that the simple harmonic oscillator is predictable in the sense that, given a small difference \delta between the initial conditions, the trajectories of two points follow essentially same curve. It is this which separates chaotic systems from their non-chaotic brethren: chaotic systems are extremely sensitive to small changes of initial state and as such become dynamically unpredictable given small perturbations. We may state:

A system is considered chaotic if each pair of phase space trajectories, initially separated by a differential amount, eventually become dissimilar.

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