## KAM theorem and ergodicity

Integrable Hamiltonian systems have the property that, when they are slightly perturbed, their behavior gets only slightly modified. One can state this result in a more technical way through a beautiful theorem due to Kolmogorov, Arnold (see here) and Moser (KAM). In phase space, these systems move on tori and the effect of a small perturbation is to produce a small deformation of these tori. A condition on resonances must hold for KAM theorem to apply. Indeed, for a small set of initial conditions the motion is no more bounded. So, when the perturbation increases, invariant tori get progressively destroyed and chaos sets in. The system goes to occupy a large part of phase space and we are in a condition for ergodicity to be true.

One may ask what can happen when the perturbation becomes increasingly large. A first idea is that ergodicity is maintained and we keep on being in a situation of fully developed chaos. Indeed, this idea is plainly wrong. For an infinitely large perturbation, a dual KAM theorem holds and again we get invariant tori and bounded motion. I proved this in my recent paper (see here). Increasing the perturbation makes tori reform and we lose ergodicity again. Indeed, ergodicity appears to be there only for a limited range of parameters of the Hamiltonian system. This can make us think that this property, that appears to be essential to our understanding of thermodynamics and, more generally, of statistical mechanics, is not ubiquitous.

So, one may ask why all systems appear to behave as if ergodicity holds. The answer to this question is quite straightforward. What makes Hamiltonian systems behave ergodically is the fact that they are composed by a very large number of particles. It is this that provokes the correct working of our statistical approach and produces everyday reality we observe. This conclusion is quite important as makes clear that we do not need ergodicity at a very fundamental level but just at a macroscopic one. This intuition was already present in Boltzmann‘s Stosszahlansatz hypothesis. The existence of a dual KAM theorem makes all this very clear. Indeed, thermodynamic limit can make quantum system quite unstable with respect to coherent evolution producing a classical ergodic system.

### 5 Responses to KAM theorem and ergodicity

1. […] haya estudiado mecánica analítica puede comprenderla). Marco nos lo resume en su blog ”KAM theorem and ergodicity,” The Gauge Connection, June 25th, 2009. El artículo técnico es de fácil lectura para […]

2. avalgoma says:

First of all, I like very much this blog, congratulations!!!

I see that all systems appear to behave as if ergodicity holds because they are composed by a very large number of particles (N>>1). However, is there any mathematical proof of this?

Many thanks,

— Agus

3. mfrasca says:

Hi avalgoma,

Thank you a lot for appreciating my blog.

The reason for this, that increasing the number of particles Boltzmann’s hypothesis holds and so a large system can appear ergodic, comes from quantum mechanics. There is a theorem by Lieb and Simon that grants that such a system is unstable with respect to quantum behavior, in most cases, so that, at a classical level, one has that Stosszahlansatz hypothesis can be taken to hold and the H-theorem is really a theorem.

You are welcome.

Marco

• avalgoma says:

Thanks a lot for your response.

Imagine that I accept that Lieb and Simon’s theorem grants that such a system is unstable with respect to quantum behavior. Then, how do you know that the degree of unstability is an ergodic one?

I mean, who ensures you that we are in between a small perturbation and a large perturbation system just to ensure ergodic or quasi-ergodic behavior?

Thanks.

• mfrasca says:

This is a really interesting question and the answer is at the boundary between classical and quantum behavior. Let me give you the sequence I follow with my reasoning: You will have a quantum kinetic equation describing your system. Increasing the number of particles can make the system more and more classical (see this). Your classical limit will hit, in most cases, the classical kinetic equation. But then, all the correlations will have decayed in time because this is the classical limit for quantum correlations and the Boltzmann equation does hold. H-theorem holds and the equilibrium distribution is the Boltzmann’s one. Turning back the reasoning, you get an ergodic system. But this limit, even if is quite generic, for some systems fails and so, there exist several systems retaining their quantum properties macroscopically.

This irreversible behavior of quantum large systems has been experimentally observed. You should check the works by Horacio Pastawski and his group and all the studies on Loschmidt echo. So, ergodicity should not be looked for in classical systems, as also my dual KAM theorem is there to say, but starting from quantum many body systems. This implies that irreversibility is an unavoidable consequence for large quantum systems.