Fluid turbulence is a highly complex and non-linear chaotic phenomenon. Great difficulties and complications are encountered when trying to accurately and robustly calculate or simulate fluid flows, due to the chaotic, multi-scale effects of turbulence.

I am just wondering if there is anything known to Physics that is more chaotic than fluid turbulence (i.e. more non-linear and harder to simulate)? Are analogous situations to fluid turbulence encountered in other branches of Physics? For example, is there any parallel to turbulence in quantum systems?

Note: I am aware that turbulent effects also occur in plasmas; however, that is probably a result of their close relationship with continuum fluids.


Prompted by some of the comments, I also asked this linked question, about to what extent it is possible to quantify how chaotic a dynamical system is:

Is it possible to quantify how chaotic a system is?

The answers there suggest that the main method is by comparing Lyapunov Exponents. So, in light of that, I think this question can be reduced to:

  1. Are there any known physical systems that have larger Lyapunov Exponents than fluid turbulence?
  2. What physical system has the largest known Lyapunov Exponents?
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    $\begingroup$ How is something more chaotic/non-linear? Isn't it an either-or? $\endgroup$ – Kyle Kanos Nov 1 '19 at 15:08
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    $\begingroup$ The UK House of Commons was certainly very turbulent and chaotic until this week... probably more so than any known fluid. The election results are also certainly highly non-linear and harder to simulate than any known fluid. $\endgroup$ – ZeroTheHero Nov 1 '19 at 15:12
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    $\begingroup$ There is quantum turbulence. But I'm not sure how you propose to measure how much non-linearity there is exactly. I suppose what you're really after is a problem with the largest Lyapunov exponent? But that doesn't mean it is "hard" to simulate, at least depending on your definition of "hard"... $\endgroup$ – tpg2114 Nov 1 '19 at 15:15
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    $\begingroup$ Spacetime inside the event horizon of a blackhole merger, perhaps. $\endgroup$ – JEB Nov 1 '19 at 15:30
  • $\begingroup$ Do you mean more chaotic in the sense of having a higher Lyapunov exponent? Chaotic means two systems with similar starting state separate exponentially as time progresses. The Lyaypunov exponent controls how fast. - Edit I guess I should read all the comments first. I see that @tpg2114 already said this. $\endgroup$ – mmesser314 Nov 1 '19 at 15:52

An interesting question that unfortunately might not have a clear answer and/or be off topic (see this discussion on big list questions).

Why is it hard to answer?

First, Lyapunov exponents $\lambda$ have unit of frequency, i.e., 1/time and systems have intrinsic time scales $\tau$, so a normalized Lyapunov exponent $$\lambda_\tau=\lambda/\tau$$ would probably make more sense for the purpose of making comparisons. Otherwise we'll have that the system with the fastest dynamics will present the largest exponent. This makes finding an answer more challenging, since attributing a time scale is not necessarily trivial, for instance, what's the typical time scale of the Solar system?

Besides that, also the example mentioned in the original post might not be appropriate. Turbulence is not yet fully understood and it's actually more complicated than chaos: it can be modeled by spatio-temporal chaos (in that measuring both at a fixed spot or along a fluid point reveals chaos), but probably even that fails to capture the full complexity of turbulence.

Therefore comparing the Lyapunov exponent estimated for a turbulent system with that of, say, a mechanical system, might not be very meaningful.

At any rate, in order to include at least one experimental value for a system's Lyapunov exponent, let's mention that Parlitz reports in their 1993 paper an exponent of $2000$ bits/second for a realization of the Chua's circuit.

If we consider the strong connection between the Kolmogorov-Sinai entropy and Lyapunov exponents, we might speculate that systems that generate entropy the fastest may correspond to the most chaotic processes. If this is the case, then relativistic heavy-ion collisions, which should bring entropy from $0$ to $5000$ in $2\times 10^{−24}$ seconds, must be among the front-runners.

  • $\begingroup$ Thanks! You make some very good points here. $\endgroup$ – Time4Tea Nov 6 '19 at 13:42
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    $\begingroup$ or 2 yoctoseconds (ys) :) $\endgroup$ – The_Sympathizer Nov 18 '19 at 2:47

Taking specifically the case of the chaotic "scrambling" of quantum information, there is a famous conjecture, currently the subject of much study, that the fastest possible scrambler is a black hole. As an amateur on this subject, perhaps the best I can do is to quote the title and abstract of one of the relevant papers:

A bound on chaos

We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent $\lambda_L \leq 2\pi k_B T/ \hbar$. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

In other words, this conjecture, if correct, does not establish an absolute fastest timescale for chaos, but rather a bound at any given temperature, which is saturated by a black hole with the associated black hole temperature.

Plugging in numbers, this ends up being 823 GHz/K , or if you take the inverse to get a time constant, $1/\lambda_L=$ $1.22*10^{-12}$ s*K. For a sense of scale, a solar-mass black hole has a Hawking temperature of around $10^{-8}$ K, while for a supermassive black hole it might be a billion times lower.


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