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The double pendulum is a simple example of a chaotic system which is extremely sensitive to tiny perturbations in its initial conditions.

If we set off two identical double pendulum systems from identical starting positions, and ignored external forces such as air resistance, friction, and vibrations, would internal quantum randomness cause these two pendulums to eventually deviate in their trajectories? And if so, on what time scale (relevant to length of pendulums and mass) would we be waiting before the difference is easily visible?

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On real physical scales quantum effects are nowhere near affecting classically chaotic systems. The point of chaos is that there is going to be some fluctuation, be it an imperfection of your model, thermal oscillations, a small whirl of air... which very early causes your trajectory to drift away from the one you predict. And there is no difference in whether the fluctuation is quantum or classical.

However, you can find microscopical systems for which classical equations of motion would exhibit chaos but they are actually quantized. The field studying the properties of such systems is called Quantum chaos. The quantum equations are linear which means that they cannot exhibit chaos and the chaos we see is only a property of the averaged "classical" picture of the dynamics. It actually turns out that the delocalization or "smearing out" of tiny deviations in initial condition typical for chaos gets often supressed in a classically-chaotic dynamical system under quantization. (But on time-scales of order $1/\hbar^2$ where $\hbar$ is the Planck constant.)

So back to your question: you should really worry more about the classical perturbations because they are much larger. If you insist on adding the "quantum" into chaos, things get much more complicated and unintuitive effects emerge which cannot be really explained by talking about quantum fluctuations.

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  • $\begingroup$ What do you mean exactly by "gets often suppressed"? Do you mean that the system is in fact not chaotic but regular? $\endgroup$ – Myridium Mar 6 '15 at 22:00
  • $\begingroup$ Yes, it is regular. As stated, quantum-mechanical equations are linear which means they cannot be dynamically unstable. However, there is the effect of "smearing out" present in the evolution of the wave-function which has an analogical effect on the probability distribution as in the classical case. On the other hand, there is a very long dynamical time (for a classical time basically infinite) after which the smearing out stops and the probability distribution is stabilised. $\endgroup$ – Void Mar 6 '15 at 22:24
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    $\begingroup$ Could you please expand a little more on how these fluctuations manifest themselves in the classical system and how it relates to the classical chaos? I don't quite understand. $\endgroup$ – Myridium Mar 6 '15 at 22:27

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