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In a recent lecture on ergodicity and many-body localization, the presenter, Dmitry Abanin, mentioned that it is possible for a classical dynamical system to be chaotic but still fail to obey the ergodic hypothesis, which is frankly a pretty remarkable combination of properties. Unfortunately, the lecture had a lot of ground to cover and Abanin did not elaborate.

So:

  • Are any explicit examples known that have been shown to be both chaotic and non-ergodic?
  • Is there some clear explanation for what properties of those systems allow them to show this behaviour?
Emilio Pisanty
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1 Answers1

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A trivial example of a non-ergodic, chaotic system is a 2D conservative system that is not fully chaotic, i.e., with a mix of regular and chaotic regions in its phase space: each individual chaotic region is ergodic in itself, but since trajectories cannot cross the regular, invariant barriers between those regions, the systems as a whole is not ergodic.

An example of such a system is Chirikov's Standard Map:

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stafusa
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