Latest of fathers of KAM theorem, Vladimir Arnold, passed away yesterday. He was 72. He has been one of the most influential mathematical-physicists of XX century. I studied analytical mechanics on his book and I have learned a lot about differential equations from other fine texts of him. It is with great sadness for me to apprehend this piece of news that is going around all the World, giving the exact stature of this man. The reason for my deep sadness relies on the fact that I built on his very strong shoulders, being him a giant (see here). KAM theorem has been a key result in dynamical systems opening the way toward a deep understanding of chaos in conservative systems and extending beyond to more general classes of Hamiltonian systems. Personally, I lived the struggling to understand plasma heating under the spell of this theorem but applications range everywhere in all fields of physics. I think that the best way to remember a scientist is through his works and I think that a simple post is not enough to embrace all the contributions due to Arnold. He is also remembered for being a pioneer in catastrophe theory and solved XIII Hilbert problem. He has been awarded a Fields medal but Soviet government impeded him to accept the prize. He will live forever in the work of people of our community and a lot more will be the fruits coming from his ideas.
Today, the Editor of Journal of Mathematical Physics communicated to me that my paper on KAM tori reforming has been accepted for publication. This is one of the most prestigious journals in mathematical physics and the result is really important and I am very happy for this very good news. Thank you very much, folks!
This question is indeed crucial for ergodicity and the arrow of time so much discussed in these days. See here for a post about this matter.
Some days ago I received an email from Wolfram Research asking to me to produce a demonstration for their demonstrations project based on my last proved theorem about KAM tori reforming (see here). Being aware of the power of Mathematica I have found the invitation quite stimulating. The idea behind these demonstrations is to use Mathematica’s command Manipulate that permits to have interactive presentations. A typical application is exactly in the area of differential equations where you can have some varying parameters. But the possibilities are huge for this method and, indeed, you can find almost 5000 demonstrations at that site. Indeed, in this way you are able to explore the behavior of mathematical models interactively and this appears as a really helpful tool. If you mean to send a demo of yours, be advised that it will undergo peer-review. So, such a publication has exactly the same value of other academic titles.
In a few days I prepared the demo and I have sent it to Wolfram. It was accepted for publication last friday. You can find it here. You can check by yourself the truthfulness of my theorem. The advantage to work in classical mechanics is that you can have immediately an idea of what is going on by numerics. Manipulate of Mathematica is a powerful tool in your hands to accomplish such an aim.
Finally, you can download the source code and modify it by yourself changing ranges, equations and so on. I tried the original Duffing oscillator without dissipation and, granted the validity of KAM theorem, one can verify an identical behavior with tori reforming for a very large perturbation. A shocking evidence without experiments, isn’t it?
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.