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.

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.