Boltzmann Equation Solved, the New Way

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!"

Comments

Holtzmann equation

[weicco]

For I second I thought the title said "Holtzmann equation solved". That's probably because I was just reading Dune: The Battle of Corrin :)

Re:Holtzmann equation

[jd]

I'd have preferred a solution to Block Transfer Computation and the Skasis Paradigm.

Re:Holtzmann equation

[Profane MuthaFucka]

I thought it said MY PENIS equation and I wondered just how hard it could be to count to two and a half.

Research paper here:

[gzipped_tar]

For the math-inclined:

http://arxiv.org/abs/0912.0888

(yes, that was from 2009)

Re:Research paper here:

[gzipped_tar]

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

(behind a paywall)

Re:Research paper here:

[jd]

Given that hard maths can take months to work through, three months is actually quite impressive.

this sounds like science

[Anonymous Coward]

please post more stories about how much we hate apple and love flash.

Meaning of "Solved"

[ObsessiveMathsFreak]

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"

[gzipped_tar]

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"

[blair1q]

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"

[Bigjeff5]

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"

[jfengel]

"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:Meaning of "Solved"

[ObsessiveMathsFreak]

Is an ultraviolet catastrophe a math term, or a physics one?

A mathematical one. The simplest example would be something like the solution to the equation dy/dx=y^2, with y=1 at x=0. This has the solution 1/(1-x), which "blows up" at x=1. Technically, you would say the solution has a singularity at x=1. The singularity is characteristic of the differential equation itself, and not really of the initial conditions or the methods used to solve it. Inherently, you're going to face this problem when attempting to solve the differential equation.

A "blow" up or pole is just one kind of singularity. There are many others. In the context of physical equations, their presence suggests that the assumptions of your equation break down as the solution approaches the singularity, or that your assumptions were flawed to begin with. In the context of mathematics, it means that any numerical system you use to solve the equation is going to break down horribly as it gets close to the singularity. This is a huge problem as if you're using a numerical solver, you typically have no idea where the singularities are anyway. What's worse, most numerical methods will actually continue along happily after they have passed the singularity, the only problem being every number they return after that point is more than likely totally wrong.

It depends on what the authors mean by "classical solutions", but my reading of it is that they mean solutions without singularities and which decay quickly, both of which are reasonable solutions for the equation in question. Since the paper (at the arXiv) is 50 pages long, I'm not entirely sure. Their solution might allow certain other types of singularity or govern the propagation of singularities. I honestly have no idea, which I understand is fairly common these days.

Re:Meaning of "Solved"

[Graff]

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"

[Your.Master]

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"

[TheSync]

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 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"

[haderytn]

after sufficient time evolve into two situations differing by as much as the presence of a tornado derrr derrr yerpppp

Re:Meaning of "Solved"

[grep -v '.*' *]

... the flutter of a butterfly's wings is not significant...

It all depends on the size of the butterfly. Haven't you ever heard of Miracle-Gro, or seen the movie: Monsters vs. Aliens??

Re:Meaning of "Solved"

[jfw]

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"

[jo_ham]

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).

Re:Meaning of "Solved"

[gstoddart]

Somebody please explain the ultraviolet catastrophe to me. What is the underlying reason for the old model providing incorrect predictions? I am very science-minded but cannot understand this. The wikipedia page could not help much.

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"

[Samah]

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/

That's quite interesting

[jd]

I'm not sure of any direct uses (flying cars won't be one), but it has implications in other areas of mathematics.

One of the big problems for computational fluid dynamics is that the equations evolved are a real pain. So much so that most of the engineers who need CFD often don't trust the results as better than a first approximation. The new solutions found to the Boltzman equations doesn't really help directly, as CFD uses customized versions of the Navier-Stokes equations for specific types of conditions, but the tools developed to find those new solutions may be useful in producing more generic CFD solutions and may result in analytics techniques that produce far more valid results than current CFD methods.

(A gas can often be treated as a compressible fluid in CFD, so if you can model a gas better, or even just sanity-check intermediate calculations, you can improve CFD for those types of calculations.)

The actual article (as opposed to the blog posting) mentions that the system is 7-dimensional. In maths, this has a different meaning than in physics. It doesn't mean 7 spacial dimensions, it means that in order to define anything you have to have 7 parameters. So, no, boiling water and turning it into a gas won't open a portal to a parallel universe. (If it were that easy, you think I'd still be here?)

For those interested in actually doing the maths, rather than talking about it, there are a great many open source PDE solvers. I've listed a few on Freshmeat, but you could spend the rest of your life collecting them. Might make for a unique hobby, but applying them to this sort of problem seems much more interesting.

Re:That's quite interesting

[Warlord88]

A gas can often be treated as a compressible fluid in CFD,

A gas is a compressible fluid.

there are a great many open source PDE solvers

The most general solvers are the most handicapped. Even the ridiculously costly commercial solvers (Ansys, Fluent, etc.) solve a limited number of problems. I was working on a project that attempted to numerically simulate the effect of electromagnetic waves on the brain. Obviously, you need to solve the Maxwell's equations in horrible medium that is your brain. That's when I realized how woefully indadequate the commercial solvers (that claim to simulate the problem) are.

love the application?

[roman_mir]

We may not understand the theory, but we'll sure love the applications!"

- Yeah, apparently the application suspends notquitewrong.com accounts. I think it's a winner.

OK Boltzmann down

[gzipped_tar]

Navier-Stokes the next, good guys!

ultraviolet catastrophe

[Crispy Critters]

I believe the term came from blackbody radiation (this is just the electromagnetic radiation of anything with a finite temperature; think infrared night-vision goggles.). Originally, the measured and theoretical emission spectra increased with frequency like the frequency squared. If you try to add up the total emission, though, you get something infinite (because you can't integrate x^2 from 0 to infinity). This is nonsense, because warm bodies, such as yourself, are not radiating infinite power. This is the ultraviolet catastrophe.

The answer to the problem is that quantum mechanical effects cause the spectrum to turn around again and head toward zero at high frequency, giving you something with a finite integral.

So the original poster meant that sometimes you can prove that what you have isn't the whole picture, but that is not the case here.

Thanks!

[hoytak]

I thought I was math-inclined.

Re:I don't get the comic

[fishexe]

Are they flying? or are they jumping ramps?

Does it matter? The text below the comic says "I’m guessing any guys that can solve that might as well be hooking up a flying car," but even if you assume it's jumping ramps, it still makes sense because it looks like the car would make a "grazing collision" on one of those obstacles, and they now have a solved equation for that. So just take your pick and enjoy it either way.

Ideal gas is not chaotic, weather is

[syousef]

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.

Um since when was weather on this planet equivalent to an ideal gas? An ideal gas is not a chaotic system. So the Boltzmann equation has nothing to do with the butterfly effect. You're talking out of your arsehole, and slashdot is collectively too ignorant to call you for it, hence you've been modded informative. Fucking sad.

Both.

[Ungrounded Lightning]

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.

No, they were referring to both.

One of the issues with chaotic systems is that there are regions in the regime where a small perturbation DOES expand without limit and small changes produce large effects. Weather is such a system.

On one hand, it means that current instrumentation can only measure things down to the point where the models track the actual weather for 3 to 5 or so days (depending on conditions) before they diverge into uselessness. On the other it means that there are literally situations where a landing plane makes the difference between a foggy and a clear morning, a contrail grows into a storm system, or a butterfly taking off makes the difference, weeks later, of whether a hurricane hits Cuba or Texas - or even forms at all.

Not EVERY butterfly takeoff creates or destroys the next month's hurricanes. But some do. Go out far enough and the details of the recoil of a molecule can make the difference between El Nino and La Nina.

Which does not necessarily mean that weather doesn't converge into predictable climate. Many chaotic systems still follow a predictable set of tracks.

Mugs

[csrster]

These guys must be crazy. Why waste your career proving the regularity of solutions to the Boltzmann equation when you could get a million bucks for doing the same for the Navier-Stokes equations.