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 :)

Research paper here:

[gzipped_tar]

For the math-inclined:

http://arxiv.org/abs/0912.0888

(yes, that was from 2009)

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"

[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"

[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"

[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

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.

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.