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This question can be asked about any chaotic dynamical system, but hydrodynamics of the atmosphere makes it more concrete. Arnold describes his 1966 result as follows:

I have calculated the curvature of this group [diffeomorphism group in hydrodynamics] and even used it to show that weather prediction is impossible for periods longer than two weeks. In a month you lose 3 digits in the prediction, just because of the curvature. This instability is not the Euler instability, it’s not describing a chaotic attractor of Euler equations – but it comes from the same line of ideas.

How final are the two weeks? One could imagine collecting data of greater precision and getting a meaningful longer term forecast. But there seems to be a theoretical limit to increasing precision, as with the diffraction limit to optical resolution. At a too fine enough precision, positions and momenta can not both be specified even theoretically, and the classical description breaks down. Quantum effects are usually negligible at classical scales, but does this apply to chaotic classical systems? In them initial discrepancies quickly magnify. Does this mean that quantum effects become classically relevant and long-term prediction of such systems is impossible in principle? Is there a theoretical time limit on weather forecasts for example?

Practical limits are discussed in How to calculate the upper limit on the number of days weather can be forecast reliably? Apparently, 15 days come up as well.

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  • $\begingroup$ Search phrase of some relevance: "Sensitive dependence on initial conditions". $\endgroup$ – dmckee --- ex-moderator kitten Oct 16 '15 at 20:27
  • $\begingroup$ @dmckee That's the definition of chaos, which the OP seems to already know about. $\endgroup$ – PyRulez Oct 16 '15 at 23:28
  • $\begingroup$ there are plenty of other impredictibility long before quantum effects, starting with thermals noise. Besides, you would need to scan the whole initiate state of whole atmosphere + ocean + vegetation, and neglect no action of anything (comprising animals moving, at some point). $\endgroup$ – Fabrice NEYRET Oct 17 '15 at 20:12
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I am not exactly sure what your question is (and which of sentences ending with a question mark are rethorical questions), so I try to answer them one at a time (and skip some questions that I consider to be answered by the answers to others):

Quantum effects are usually negligible at classical scales, but does this apply to chaotic classical systems?

There is no such thing as a purely classical system in reality, so quantum effects apply to all chaotic systems. Whether these effects are the predominant sources of inaccuracy depends on what exactly you regard as the system, more specifically whether there are any bigger influences (such as humans) that you do not consider part of the system.

On the other hand, in a theoretical purely classical system, there are no quantum effects by definition, but I suppose that’s not what you wanted to know.

Does this mean that quantum effects become classically relevant and long term prediction of such systems is impossible in principle?

Yes. Suppose, your system is the entire universe and you can measure everything as exactly as permitted by quantum effects. Further suppose that you are “outside” the universe, i.e., you are not part of the system yourself and your measurements and prediction efforts do not influence the system. Finally suppose that you have sufficent knowledge of physics to run a simulation as precise as permitted by your measurements. Then eventually quantum effects will affect macroscopic predictions by means of your simulation.

The reason for this is that the universe almost certainly has a positive Lyapunov exponent: It certainly contains a myriad of subsystems that have a positive Lyapunov exponent when isolated and there is no reason to assume that “coupling” them with the rest of the universe will change this in all cases.

Is there a theoretical time limit on weather forecasts for example?

Yes, though its exact value depends on what you consider to be part of your theory/system, e.g., are human behaviour or solar fluctuations part of your theory? Is your forecasting happening outside the system? Another factor is what exactly you consider a successful wheather forecast. If all those things are known, then there is an upper limit, but giving any reasonable value for it will probably be a huge (largely pointless) piece of work.

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  • $\begingroup$ "....but giving any reasonable value for it will probably be a huge (largely pointless) piece of work." But from the known classical chaotic behavior of the system, the known Lyapunov exponents describing how fast perturbations grow as a function of time, one should be able to come up with a reasonable estimate for the time scale after which quantum fluctuations in the initial state will start to become important. $\endgroup$ – Count Iblis Oct 17 '15 at 19:40
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    $\begingroup$ @CountIblis: The known classical behaviour of which system? Planet earth? Weather? Weather under very idealistic assumptions? — I agree with you that once we have a (computationally feasible) model of a system, it’s easy. But the systems for which we have such models are so far from reality that it’s safe to say that there are bigger butterflies to consider than quantum effects. $\endgroup$ – Wrzlprmft Oct 17 '15 at 19:52
  • $\begingroup$ I'm not an expert in dynamical systems, but I was under the impression that you can extract the values of Lyapunov exponents for real systems like the weather, and from that you can estimate the time scale beyond which prediction is fundamentally impossible due to quantum fluctuations. $\endgroup$ – Count Iblis Oct 17 '15 at 20:22
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    $\begingroup$ @CountIblis: you can extract the values of Lyapunov exponents for real systems – Yes – like the weather – No. Weather is far too high-dimensional for this. To estimate the Lyapunov exponents from data, you have to have a reasonable amount of observations that are close-by in phase space (without being close in time). For the weather this means that you have to have several instances, where the weather is roughly the same everywhere on the whole planet. $\endgroup$ – Wrzlprmft Oct 17 '15 at 20:57
  • $\begingroup$ And this is not the only problem. Estimating the Lyapunov exponent requires that the dynamics can be reasonably approximated as linear on the temporal and geometrical scales of your measurements (unless you want to do a phase-space reconstruction, which requires much more data given the number of dimensions we are talking about here). This does simply not apply to weather, e.g., because of human beings. $\endgroup$ – Wrzlprmft Oct 17 '15 at 20:57

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