Talk:Fermi–Pasta–Ulam–Tsingou problem

common sense in typesetting

Please: Use some common sense. I understand that some editors may not know that a period set _outside_ the "math" tags in "displayed" TeX may fail to get properly aligned (I've fixed the ones that were in this article). But the following offends common sense:

:<math>\begin{align} \left(\frac{x_{j+1}+x_{j-1}-2x_j}{h^2}\right)&=\frac{u(x+h,t)+u(x-h,t)-2u(x,t)}{h^2}\\ &=u_{xx}(x,t)+\left(\frac{h^2}{12}\right)u_{xxxx}(x,t)+O(h^4)\end{align}</math>.

Where was the period at the end of this sentences supposed to appear to be?? Look at the result:

${\begin{aligned}\left({\frac {x_{j+1}+x_{j-1}-2x_{j}}{h^{2}}}\right)&={\frac {u(x+h,t)+u(x-h,t)-2u(x,t)}{h^{2}}}\\&=u_{xx}(x,t)+\left({\frac {h^{2}}{12}}\right)u_{xxxx}(x,t)+O(h^{4})\end{aligned}}$.

I changed it to read as follows:

${\begin{aligned}\left({\frac {x_{j+1}+x_{j-1}-2x_{j}}{h^{2}}}\right)&={\frac {u(x+h,t)+u(x-h,t)-2u(x,t)}{h^{2}}}\\&=u_{xx}(x,t)+\left({\frac {h^{2}}{12}}\right)u_{xxxx}(x,t)+O(h^{4}).\end{aligned}}$

Michael Hardy (talk) 19:15, 26 December 2012 (UTC)

"Solved" ?

http://phys.org/news/2015-03-mathematicians-year-old-problem.html

Keith McClary (talk) 03:58, 24 March 2015 (UTC)

Yes, interesting, and some kind of synopsis should be placed here. I quote: for greater lengths of time, ... energy within the system did eventually (thermalize) ... how precisely was this happening? ... the key lies in a gradual transfer of energy during coincidences of six modes in the system. When precisely six modes interact, the energy is transferred in a nonreversible way. ... interactions of triads, quartets, and quintets are reversible; in other words, they do not bring the FPU system closer to thermal equilibrium. However, the interaction of waves in sixtets does lead to irreversible transfer of energy. My gut instinct is that this is the tip of some huge iceberg, since solitons and KdV-like phenomena are wide-spread, and "six" is one of those numbers that smells like some kind of elliptic equation. "Watch this space", methinks, routes to thermalization are very poorly understood in physics, so this seems like a big deal, to me. The preprint is here: A route to thermalization in the α-Fermi-Pasta-Ulam system, Miguel Onorato, Lara Vozella, Davide Proment, Yuri V. Lvov (arXiv)

Draft outline of their results, and some comments and some more quotes:

• They rewrite the FPUT equations in terms of fourier modes, and find a three-wave interaction. i.e. "scattering" in particle-physics terms, a three-point feynman diagram. But it has no resonant interactions. (oops - currently a WP redlink!) They are looking for these, as resonances are the ones that spread energy around. BTW, the three-wave interaction is of order ε in the interaction, where ε << 1 is the FPUT non-linear interaction.
• Such resonant interactions are the base for the so called wave turbulence theory - currently a WP redlink, but apparently "its a thing" (There some chapter in some book on it...)
• The Fourier modes in the FPU system can be divided into free and bound modes. ... To illustrate the concept of the bound modes ... higher harmonics are bound to the primary free sinusoidal mode and they do not obey to the linear dispersion relation. Cnoidal waves and solitons are similar objects: they are characterized by a large number of harmonics that are not free to interact with each other. ... the equipartition phenomenon is ... only for those that are free to interact. The rest of the modes in the spectrum do not have an independent dynamics and are phase-locked to the free ones. Very nice! This is an elegant insight into the problem!
• Build a spectrum characterized only by free modes. ...remove via canonical transformation ... all interactions that are not resonant. After one round of these certain canonical transformations, they get something that is more-or-less the Zakharov equation (up to sums->integrals, yada yadda).
• After one round of canonical transformations, they get a system with 4-wave interactions (and the 3-wave interaction is now gone). This system does have resonant interactions. Thus, quartets of four waves will thermalize between the four waves. However, these quartets are non-interacting: if you excite one quartet, it does not mix with the others. They are isolated from one another! Put differently, any given mode belongs to only one quartet. BTW, the four-wave interaction is of order ε² in the interaction.
• Fivewave interactions are non resonant. I'm not clear if this is supposed to be a generic statement, or is specific to FPUT. I think it's FPUT-only, but somehow obvious in the same way that the 3-wave interaction wasn't resonant.
• Working out higher orders, they find a six-wave interaction, of strength ε⁴. It is resonant, and the resonance mix together all of the different modes. That is, it each mode appears in more than one (resonant) six-wave interaction, and they chain together, so all modes are inter-connected. This is the process by which thermalization eventually occurs!
• The time to thermalize is given by the kinetic equation of wave turbulence theory .. it is apparently widely and intensely studied. Time to thermalize involves correllators, obtained via the BBGKY hierarchy. The final result is that time to thermalize is proportional to ε⁸ which I think has something to do with a correllator between two 6-wave interactions. The EOM Springer article on Bogolyubov–Born–Green–Kirkwood–Yvon hierarchy says more than the WP article.
• There is some quite remarkable commentary about the Toda lattice!! Apparently, FPUT looks like Toda with an extra perturbation. Why is Toda exactly integrable, but not FPUT? Well, they work out the 4-wave interaction term and find that it vanishes identically! This is where the remarkable claim shows up: one can now do another canonical transformation, to once again remove the "bound modes"; then repeating, ad infinitum one is able to linearize the Toda lattice (after an infinite number of steps). The suggestion seems to be that all such integrable systems are linearizable via repeated canonical transformations. I think that is pretty cool! This was not an afternoon wasted!

Anyway, that's it; they do some numerical work to confirm the theoretical work. Above is meant to be a rough draft for any new section that explains the results. Seems pretty darn notable to me, but I'm an outsider looking in, so what do I know... The mathematician inside of me wonders what it's like, when one chains together an infinite number of canonical transformations, moving in some direction. I can only presume all the usual suspects enter into the picture. Fascinating!

67.198.37.16 (talk) 19:36, 12 September 2020 (UTC)

Requested move 20 April 2017

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: Moved.(non-admin closure)All recent sources use the proposed name. Winged Blades ^Godric 11:42, 7 May 2017 (UTC)

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Fermi–Pasta–Ulam problem → Fermi–Pasta–Ulam–Tsingou problem – The 2008 article in Physics Today cited in the Wikipedia article called for the FPU problem to be renamed to the FPUT problem. While it has been slow to retroactively rename the problem to reflect the contributions of Tsingou, current sources call it by its new name. I'm requesting that we update the page to reflect the name used in recent literature. Here are some contemporary sources that reflect the name change:

• Vibrational resonances in 1D Morse and FPU lattices (2008 - within the intro paragraph the authors reference Tsingou and use FPUT to refer to the problem for the remainder of the paper)
• Coupled Oscillator Models with no Scale Separation (2008)
• Caltech physics PhD thesis (2009)
• Advances in Chemical Physics, Vol 153 (2013)
• Scaling perspective on intramolecular vibrational energy flow: analogies, insights, and challenges (2013)
• Holographic Thermalization, stability of AdS, and the Fermi–Pasta–Ulam–Tsingou paradox (2014)
• Perimeter Institute of Theoretical Physics talk (2015)
• Leeds University lecture notes on FPUT (2016)
• University of Trieste computational physics lecture notes mentioning FPUT (2016)
• Early time evolution of a localized nonlinear excitation in the β-FPUT chain (2017)
• Study unravels long-held Fermi puzzle tied to nonlinear systems (2017 - this happened last week)

While it is true that the old literature refers to the FPU problem, the current community has embraced the name change and Wikipedia should reflect this. Blueclaw (talk) 15:08, 20 April 2017 (UTC)--Relisting. TonyBallioni (talk) 21:51, 28 April 2017 (UTC)

• Oppose, based on my research to this point. For one thing, the existing name better satisfies Conciseness, one of the Five Virtues of article titles. And the other four are pretty much a wash either way. So WP:AT says not to do this if we can avoid it. Nominator has assembled a good list of examples, but what about counter-examples? Clicking on the link in the thread above this one takes me to This 2015 article at Physics.org (whatever that is) where they say "Yuri Lvov, has found an elegant explanation for the long-standing Fermi-Pasta-Ulam (FPU) problem..." And that's very recent. Google Ngrams can't help us here. Google brings up links to many instances of just "Fermi-Pasta-Ulam problem" though. Here's one at random. It's from 2008 so it's "old", but 2008 is not that old.

We don't want to get ahead of the trend, here. If you imagine a Google Ngram, the area under each curve matters, as well as the trend and current position of the lines... Google hits are fairly worthless as indication of anything, what with all the mirrors and all, but for what it is worth the exact string "Fermi–Pasta–Ulam problem" gets 9,000 hits while "Fermi–Pasta–Ulam–Tsingou problem" gets 600.

I understand that Tsingou got the shaft here, and that there's a movement to redress this (which I approve of). It is likely (and good) that in future going forward the Fermi–Pasta–Ulam–Tsingou version will gain the ascendency. I'm not convinced that we are there yet. Also, Conciseness. Herostratus (talk) 03:32, 27 April 2017 (UTC)

• Support per nom and per the present wording of the article and its sources. This is one of those "women in the kitchen" old-timey historical oddities which are being corrected one-by-one, and the Wikipedia title likely should reflect the actual events, the wording of the page, and even the sources used. Randy Kryn 14:30, 30 April 2017 (UTC)
• Support despite this google ngram search, or 15,700 vs 802 ghits, and NOTADVOCACY and the normal rule that Wikipedia should not get ahead of the trend. There is sufficient evidence for the trend to correct an historic injustice. --SmokeyJoe (talk) 06:17, 5 May 2017 (UTC)
• Support. All recent sources use the newer name. Andrewa (talk) 07:04, 6 May 2017 (UTC)

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The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Why not full names for all authors?

My good-faith edits on this page were reverted by Headbomb, and I don't understand why. All I did was add first names to some of the authors identified by only last names (and sometimes also an initial). In many cases, such incomplete information is far from enough to identify a person. For instance, who is Kruskal? Coming from outside the subject, and knowing several scientists with this name, I had no way to know at first glance who this particular Kruskal the article referred to; only clicking on the link, and then getting back, could I figure it out. Then again, why refer to Enrico Fermi and not simply to Fermi, for uniformity? Surely there must be some kind of reason, but sorry, the whole naming scheme does not look adequate to me as it stands. Turgidson (talk) 23:38, 20 June 2018 (UTC)

See WP:CITEVAR. Headbomb {t · c · p · b} 23:42, 20 June 2018 (UTC)

That does not tell me anything new besides the curt invocation of policy made in the revert. I still have not received a clear, logical, convincing explanation for this unjustified-on-the-merits revert. Just proof by intimidation type of argument. Turgidson (talk) 23:48, 20 June 2018 (UTC)

The reason is WP:CITEVAR. Do what you want in the main text, but don't mess with referencing style. Headbomb {t · c · p · b} 23:50, 20 June 2018 (UTC)