Slashdot Mirror
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!"
Are they flying? or are they jumping ramps?
For I second I thought the title said "Holtzmann equation solved". That's probably because I was just reading Dune: The Battle of Corrin :)
You don't know what you don't know.
For the math-inclined:
http://arxiv.org/abs/0912.0888
(yes, that was from 2009)
Colorless green Cthulhu waits dreaming furiously.
please post more stories about how much we hate apple and love flash.
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.
May the Maths Be with you!
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.
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
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.
You can't handle the truth.
Navier-Stokes the next, good guys!
Colorless green Cthulhu waits dreaming furiously.
We may not understand the theory, but we'll sure love the applications!"
As long as they don't require Flash.
second link is DOWN
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.
Head on vs grazing.
This calls for a Bad Car Analogy.
Have gnu, will travel.
No, this calls for a touhou reference.
I thought I was math-inclined.
Does having a witty signature really indicate normality?
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.)
In other words, it spelt doom for Jules Verne style world of tomorrow (Steampunk)!
Who knows what surprises await in the future. Perhaps one day most or whole of contemporary Sci Fi will be regarded as "Spacepunk" or perhaps "Laserpunk", due to some revolutionary twist in Physics that will make some now popular and "futuristic" concepts bizarre.
I wanna earn $14 - the hard way!
----
WWJD...For a Klondike Bar?
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.
These posts express my own personal views, not those of my employer
How would you explain the possible practical applications of this to a non-higher-maths guy? I just don't see calculating the collisions of gaseous particles having anything to do with flying cars...
For optimal comment enjoyment, take red pill now.
Or Mothra vs. Godzilla.
Bantam Dominique roosters crow a four-note song. Once you've heard it as "Happy BIRTHday" you can't NOT hear it that way
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.
Bantam Dominique roosters crow a four-note song. Once you've heard it as "Happy BIRTHday" you can't NOT hear it that way
Did submitter actually read his own links?
No one "solved" equation. They proved existence of couple of soulutions with specific properties.
"Penn mathematicians proved the global existence of classical solutions and rapid time decay to equilibrium for the Boltzmann equation with long-range interactions. "
http://www.upenn.edu/pennnews/news/university-pennsylvania-mathematicians-solve-140-year-old-boltzmann-equation-gaseous-behaviors
"Penn mathematicians proved the global existence of classical solutions and rapid time decay to equilibrium for the Boltzmann equation with long-range interactions. "
an advance in maths that hasn't been made by Dr. Long Duk-Dong of Shanghai Jiaotong University or something. I'm amazed.
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.