Boltzmann Equation Solved, the New Way 104
xt writes "The Boltzmann equation is old news. What's news is that the 140-year-old equation has been solved, using mathematical techniques from the fields of partial differential equations and harmonic analysis, some as new as five years old. This solution provides a new understanding of the effects due to grazing collisions, when neighboring molecules just glance off one another rather than collide head on. We may not understand the theory, but we'll sure love the applications!"
Research paper here: (Score:5, Informative)
For the math-inclined:
http://arxiv.org/abs/0912.0888 [arxiv.org]
(yes, that was from 2009)
Re:Research paper here: (Score:2, Informative)
Well apparently it only recently passed peer-review, if you consider 3 months "recent". That's not unusual for a research paper anyway.
http://www.pnas.org/content/107/13/5744.short [pnas.org]
(behind a paywall)
Meaning of "Solved" (Score:5, Informative)
It's worth noting that someone says that an equation has been "solved" in modern mathematics, they typically don't mean that you plug in the initial conditions and then get a formulae for your answer. Generally what they mean is that you can apply some other--probably numerical or approximate--techniques in an effort to solve the equation, and as long as you are careful, use enough computational resources, and don't go to far out, your solutions will be reasonably accurate.
This appears to be more or less what the team has done. They've proven the "the global existence of classical solutions and rapid time decay to equilibrium for the Boltzmann equation with long-range interactions". In other words, they've proven that the equation has "well behaved" solutions and not solutions for which something goes horribly wrong at some distance from your starting point.
While it doesn't sound like much, this is actually a very big deal. If the proof had gone the other way, it would mean that the equation would produce something akin to "ultraviolet catastrophes" under certain conditions, which means that the equation did not properly describe physical systems. With this proof, that's not an issue anymore and we now know that the equation will always produce reasonable solutions when given reasonable (i.e. physical) initial conditions.
Perhaps they've gone farther than just existence proofs and also provided a formula or technique for obtaining or approximating solutions. However, the Proceedings of the National Academy of Sciences journal is a closed publisher and the article is locked behind a paywall, so I guess the vast majority of us will never know.
Re:Meaning of "Solved" (Score:5, Informative)
No, they just showed that there *is* a solution, and the solution behaves "well".
Mathematically speaking, it makes little sense to say the "correctness" of the Boltzmann equation. It is Just Another Equation (TM). Physically speaking, the application of said equation to physical bodies has been established in physical ways.
Re:Meaning of "Solved" (Score:1, Informative)
Providing a link to ultraviolet catastrophe [wikipedia.org]. Posting as AC to avoid Karma whoring.
Re:Meaning of "Solved" (Score:2, Informative)
What they really proved, at long last, is that gaseous systems are stable for small perturbations.
In layman's terms: the Butterfly Effect is bogus. It takes a very large perturbation to convert a stable portion of atmosphere into a storm, and the flutter of a butterfly's wings is not significant to tipping the balance.
Re:Meaning of "Solved" (Score:3, Informative)
Is an ultraviolet catastrophe a math term, or a physics one?
Physics, the Boltzmann Equation describes the behavior of gas. "Ultraviolet catastrophes" don't happen, so if the equation allows for them (or similar bad behavior), then the equation is wrong.
What they've shown is that it still accurately describes gassy behaviors that were hitherto unknown until a few years ago.
In other words, it still works. :)
Re:Meaning of "Solved" (Score:5, Informative)
"Ultraviolet catastrophe" is a physics term, talking about a time when math that had seemed to work out well produced some puzzling answers. The solution was that they had to scrap the old math and replace it with something radically different. Equivalent to somebody accidentally proving that there was no such thing as molecules, and having to re-do chemistry from scratch.
In the case of the "ultraviolet catastrophe", the old math said that a hot object should emit photons at every wavelength. Fewer at shorter, higher-energy wavelengths, but some nonetheless. The math worked for longer wavelengths, but for shorter ones (say, ultraviolet) it got worse. For ultra-short wavelengths, any body hotter than absolute zero should be emitting photons of near-zero wavelength with arbitrarily large amounts of energy. Infinite, in fact. Quite a catastrophe.
The solution turned out to be to say that the energy had to come in discrete packets. The new theory is perplexing, but more accurate and way more useful. (Computers, lasers, etc etc etc.)
Ultimately it turned out well, but nobody at the time really wanted to have to throw out everything they knew about energy. In this case, it's unsurprising that the new solutions should confirm that we're not looking at another similar revolution. I don't think anybody was looking forward to scrapping what we think we know about gases.
Re:Research paper here: (Score:3, Informative)
Given that hard maths can take months to work through, three months is actually quite impressive.
Re:Meaning of "Solved" (Score:3, Informative)
Is an ultraviolet catastrophe a math term, or a physics one?
It's a physics term, but math and physics are pretty intertwined at that point.
The basic idea is that random populations of things tend to follow a normal distribution, or bell curve. If you have a bunch of molecules bouncing about then some will be moving fast, some slow, but most will be at a moderate speed. All things being equal the percentage of slow vs fast should be roughly similar, producing a graph that looks like a bell - round peak in the middle, the sides falling off and leveling out.
According to classical physics a "black body" (an ideal object at a certain temperature) should emit some photons of higher energy, and some photons of lower energy, with most photons of a moderate energy. The graph of these should follow a bell curve, if everything else was equal. At lower temperatures the curve was approximately a bell curve, centered around the infrared wavelengths. However, as the temperature is raised there is a shortfall of higher energy photons. The graph starts to develop a "lean", it looks like it has a fat tail on the lower energy photon side and a long, thin tail on the higher energy photon side. Because many of those high energy photons are in the ultraviolet range it was called the "ultraviolet catastrophe" - it was a highly unexpected result which turned the physics community on its head.
Ultimately quantum theory explained the reason for this. Quantum energy levels for the electrons in atoms results in the lower energy transitions being more likely than higher energy transitions, thus tending to produce a higher amount lower energy photons and a lower amount of higher energy photons than classical physics predicted.
Re:Meaning of "Solved" (Score:4, Informative)
Some equations can be proven, eg:
a+b+c-a-b = c
(for number systems that have fully associative addition and subtraction)
However, the Boltzmann equation is more like your example:
a+b=c
That can never be proven correct or incorrect, because it depends on a, b, and c. However, given that equation, and the values for two of the variables, you can solve for the value of the third. Or given that equation and just one variable's value, you can solve for a new equation that shows a relationship between the other two variables. But asking whether "a+b=c" is correct has little meaning. It's correct when a=b=c=0, and incorrect when a=b=c=1, and the Boltzmann equation is similar.
Re:Meaning of "Solved" (Score:4, Informative)
I think most people have the wrong idea about the "Butterfly Effect." IIRC, the weather scientists were talking about the precision with which they would need to know air movement to make longer term predictions. i.e. the longer the forecast the more digits of precision are needed in your measurement. They were referring to the level of precision and not to butterflies causing a tornado or other such nonsense.
I think this paper [csuchico.edu] says that the butterfly/tornado link came directly from Edward Norton Lorenz, an American mathematician and meteorologist, and a pioneer of chaos theory:
In the title of a talk given by Lorenz at the 139th meeting of the American Association for the Advancement of Science in December, 1972, the butterfly made its first appearance: ''Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?'' In this talk, Lorenz raised the fundamental issue: ''The question which really interests us is whether they (the butterflies) can do even this--whether, for example, two particular weather situations differing by as little as the immediate influence of a single butterfly will generally after sufficient time evolve into two situations differing by as much as the presence of a tornado. In more technical language, is the behavior of the atmosphere unstable with respect to perturbations of small amplitude?''
Re:Meaning of "Solved" (Score:2, Informative)
What they really proved, at long last, is that gaseous systems are stable for small perturbations.
In layman's terms: the Butterfly Effect is bogus. It takes a very large perturbation to convert a stable portion of atmosphere into a storm, and the flutter of a butterfly's wings is not significant to tipping the balance.
What they really proved, at long last, is that gaseous systems are stable for small perturbations.
In layman's terms: the Butterfly Effect is bogus. It takes a very large perturbation to convert a stable portion of atmosphere into a storm, and the flutter of a butterfly's wings is not significant to tipping the balance.
Uhm, no. You do not understand what systems are modelled by the Boltzmann equation, what Lyapunov exponents are nor what "global in time solutions" actually are. Lets pretend that the Boltzmann equation is a good model, on it's own, for atmospheric dynamics. This paper proves global existence of various norms of the solution, so that says that there is no time T less than infinity at which those norms become unbounded. Solution trajectories that start arbitrarily close are allowed to diverge exponentially in time. That is, there can be exponential sensitivity to initial conditions (ie. butterfly effect) with no violation of these results. Someone above mentioned the example y' = y^2, y(0) = 1 which has solutions which become unbounded as t approaches 1 from below. These results rule out that behaviour but not behaviour like y' = y. Solutions here are bounded as t goes to infinity but are bounded for all bounded times. Please, go back to grad school for a few years before claiming to understand what you clearly don't.
Re:Meaning of "Solved" (Score:1, Informative)
Some equations can be proven, eg:
a+b+c-a-b = c
(for number systems that have fully associative addition and subtraction)
Actually, to prove this you also need the addition to be commutative. From associativity, only
a+b+c=c+b+a
follows, and that is not a trivial equation in some systems.
Re:Meaning of "Solved" (Score:4, Informative)
I won't try to explain the specifics, but ...
Some calculation indicated that 'black bodies' (which I think means "radiates heat" or something) would emit infinite energy. However, this didn't correspond to the reality that those things don't, in fact, radiate infinite energy.
The solution these guys got for this equation showed that the equation (which describes particle collisions in a gas I think) doesn't spiral out of control and emit infinite energy. It's still an exceedingly complex equation that we can't solve, but this tells them they're on the right track.
Which is good, because that very complex equation has been shown to at least usable. Which I think lets us do better CGI of water for Avatar 2, plus the real science that comes from being able to model fluids accurately and look at the wacky physics there. ;-)
Any actual physicists can now pillory me and my lame attempt to explain this. :-P
Re:Meaning of "Solved" (Score:3, Informative)
Ask a chemist.
Well, ask a physical chemist, they're all in the ground floor labs with the heavy equipment pretending to be physicists (while all the physicists are off pretending to be mathematicians).
For anyone who doesn't get the reference:
http://xkcd.com/435/ [xkcd.com]
Re:Meaning of "Solved" (Score:1, Informative)
Not just associativity! This requires commutativity, or you can't rearrange your equation to have the a and -a next to each other. /.ers make mistakes like this all the time; if you want to talk about math and pretend you know stuff, at least do a better job of it.