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?
Update: My demonstration has been updated (see here). The code has been improved, and so the presentation, due to a PhD student, Simon Tyler, that did this work. Thank you very much, Simon!
“The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.”
Taken from David Tong’s page at Cambridge (see here).
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.
ITER is our hope for our energy requirements in the future, together with NIF (see here). The success of these two experiments should grant that one of the greatest problems of humankind will be solved. ITER is a fusion reactor based on tokamak, a toroidal chamber confining a high-temperature plasma with magnetic fields. Its main aim will be just to demonstrate the feasibility of a full working reactor to satisfy our energy needs. Rumors are widespreading about delays and increasing costs of this project. I have read about this matter on an Italian newspaper (see here). General director Kaname Ikeda is nor confirmirg neither denying these rumors but presently it seems that costs are doubled and the start-up is delayed to 2025 (2018 foreseen) making a large scale available technology a century far in the future. If all this will be confirmed, it will represent a serious setback to our hope for a better future.
Update: A very informative article is appeared in Physics World. See here to read it.
Coulomb gauge is quite peculiar for the behavior of the Green functions of a Yang-Mills theory. It makes computations more involved and recent lattice computations seem to point toward some different behavior with respect to the case of Landau gauge. Indeed, the propagator is dependent on the gauge choice and could seem not so much useful. We know that things do not stay this way from our understanding of simpler theories as quantum electrodynamics. Some physics can be extracted from them and, if analytically obtained, we can build up a quantum field theory. This explains the effort of a good part of the scientific community for their understanding in the low-energy limit of Yang-Mills theory.
While in the Landau gauge the gluon propagator on the lattice is seen to reach a finite non-null value at zero momenta, in the Coulomb gauge there are strong indications that the gluon propagator is strongly suppressed bending toward zero at lower momenta. This was firstly seen by Attilio Cucchieri (see here for a more recent analysis). This situation has been somehow improved in a recent paper (see here) in a Tokyo-Berlin-Adelaide collaboration. An understanding of the underlying physics would be improved through the analysis of the ghost propagator. This should be infrared enhanced. But from the paper of Cucchieri, Mendes and Maas that I cited above, it is not seen to change from a free behavior with changing the gauge. This is a relevant indicator and should be exploited wherever one tries to see the behavior of the propagators in different gauges.
Of course, the best way to have an insight on such Green functions is through analytical means. This is overdue in the case of the Coulomb gauge. Today on arxiv appeared a notable attempt by Alkofer, Maas and Zwanziger (see here). Alkofer and Zwanziger are pioneers in the use of Dyson-Schwinger equations for gauge theories and have given important contributions to quantum field theory in this area. So, their conclusions are relevant. They show that a more demanding truncation scheme is needed for this case while it appears not trivial at all the emerging of a linear rising potential. This paper puts the foundations for further work in this direction whose understanding is essential to Yang-Mills theory in the low-energy limit.
V. Parameswaran Nair and Dimitra Karabali come out today with another paper in arxiv (see here). I am pleased to tell you about this paper as it appears really interesting. As you may know from my preceding post (see here), these authors have found a fruitful approach to Yang-Mills theory for 2+1 dimensions. The most relevant result from this is the value of the string tension given by
being the coupling constant while and denote the quadratic Casimir values for the representation R and for the adjoint representation, respectively. This is in close agreement with lattice computations, the error being of the order of 1%. Of course, one may ask how to improve such a value to make it even closer to the lattice value. This is the question the authors answer in their paper giving an expansion to compute such corrections. Indeed, they succeed to make the values closer.
Nair and Karabali approach is absolutely relevant. It should give the exact spectrum of the theory and is a serious track to be followed for our understanding of the low-energy behavior of Yang-Mills theory in our four dimensional world.