# Are there necessary and sufficient conditions for ergodicity?

What are the necessary and sufficient conditions (if any) for ergodicity (or non-ergodicity)?

I see for instance that some integrable systems are not ergodic. For instance a linear chain of harmonic oscillators put to oscillate in a determinate normal mode will remain in that mode. If I add non-linearity it breaks down integrability and (perhaps) the system becomes ergodic. Thus I suspect that concepts such as integrability, chaos and non-linearity are intimately related to ergodicity. Is there any set of conditions relating these concepts with ergodicity?

• If the system is Hamiltonian, then, the phase-space volume is conserved, by Louiville's theorem. By Poincare's recurrence theorem, such a system is ergodic: for each open set there exist orbits that intersect the set infinitely many times. – Dr. Ikjyot Singh Kohli Oct 25 '17 at 21:59
• @Dr.IkjyotSinghKohli Poincare's recurrence theorem is weaker than ergodicity. More precisely: the assumptions of Poincare's theorem are not enough to prove ergodicity. – lcv Oct 26 '17 at 5:17
• – honeste_vivere Oct 26 '17 at 13:59
• @lcv yes. I am aware. I was just giving an example off the top of my head! :) – Dr. Ikjyot Singh Kohli Oct 26 '17 at 14:22

## 1 Answer

integrability, chaos and non-linearity are intimately related to ergodicity. Is there any set of conditions relating these concepts with ergodicity?

Yes.

In order to exhibit chaos, a system must be nonlinear: either through an explicitly nonlinear term, or indirectly, such as when the nonlinearity arises from partial differentiation or a time-delay.

Now, in general, chaos and unpredictability are actually a matter of degree, rather than a sharp distinction between chaotic and non-chaotic systems. That's well expressed by the ergodic hierarchy:

$$\text{Bernoulli} \supset \text{Kolmogorov} \supset \text{Mixing} \supset \text{Ergodic}$$

where I omit the distinct degrees of mixing. Bernoulli systems are the most chaotic, equivalent to shift maps. Kolmogorov systems (often simply K-systems) have positive Lyapunov exponents and correspond to what is most often considered a chaotic system. (Strongly) mixing systems intuitively have the behavior implied by their name and, while they don't necessarily have exponentially divergent trajectories, there is a degree of unpredictability which can justify calling them weakly chaotic. Ergodic systems, on the other hand, have time correlations that don't necessarily decay at all, so are clearly not chaotic.

Therefore, if a system is chaotic, it's necessarily ergodic. But an important remark must be made: while most systems are chaotic, they are seldom so in their entire phase space. Thus, we usually restrict our attention to the accessible regions. When that is not necessary, it's usually indicated by the qualifier fully (chaotic, ergodic, etc.).

As for the title question:

Are there necessary and sufficient conditions for ergodicity?

I'm afraid that's a math question and unfortunately there doesn't seem to exist any condition besides those of the different equivalent definitions of ergodicity. Concrete examples of proofs of ergodicity can be found in ergodic theory textbooks (Charles Walkden's notes on ergodic theory are available on-line (pdf1, pdf2)).