13
$\begingroup$

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?
Improve this question
$\endgroup$
2
  • $\begingroup$ Maybe the two first chapters of this: arxiv.org/abs/1509.06411 can give some hints for research and also this : arxiv.org/abs/nlin/0411062 for more info. $\endgroup$
    – Constantine Black
    Commented May 12, 2019 at 16:42
  • $\begingroup$ As an anecdote, I used to work with a mechanical test rig whose dynamics were supposed to be non-chaotic, but which in reality also had two different chaotic modes of operation, and it would flip randomly between the three states with very little provocation - e.g somebody walking around the room was often enough. We had no interest in whether it was or wasn't an ergodic system, but if the question had been asked, it would have been hard to believe that it was ergodic! $\endgroup$
    – alephzero
    Commented May 12, 2019 at 21:09

1 Answer 1

Reset to default
10
$\begingroup$

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:

enter image description here

Improve this answer
$\endgroup$
4
  • $\begingroup$ Is it usual to refer to a system as chaotic if there or invariant measures on which the maximal Lyapunov exponent is not positive? I though Lyapunov exponents (and hence chaoticity) are properties of an invariant measure, rather than the dynamical system. $\endgroup$
    – ComptonScattering
    Commented Dec 29, 2021 at 6:44
  • $\begingroup$ @ComptonScattering Yes, it's very usual, especially in physics. If a system can exhibit chaos, you can call it chaotic, as is done, e.g., with the double pendulum. $\endgroup$
    – stafusa
    Commented Dec 29, 2021 at 10:56
  • $\begingroup$ Do you know if there are examples for which every invariant manifold is individually chaotic, but the system is not ergodic? (I guess with some kind of additional stability criteria to rule out tensor product cases like the following: let $f : [0,1) \to [0,1)$ define a chaotic map, let consider $g : [0,2) \to [0,2)$ defined by $g(x) = f(x)$ for $x \in [0,1)$ and $g(x) = f(x-1)+1$ for $x \in [1,2)$. $g$ is not ergodic due to the conservation law $\lfloor x \rfloor = \lfloor g(x) \rfloor$) $\endgroup$
    – ComptonScattering
    Commented Dec 29, 2021 at 15:44
  • $\begingroup$ @ComptonScattering No, sorry, not out of the top of my head. $\endgroup$
    – stafusa
    Commented Dec 29, 2021 at 20:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.