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!"
That's quite interesting (Score:5, Interesting)
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.
OK Boltzmann down (Score:2, Interesting)
Navier-Stokes the next, good guys!
ultraviolet catastrophe (Score:3, Interesting)
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.
Re:Meaning of "Solved" (Score:3, Interesting)
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 [abstrusegoose.com] these days.
Both. (Score:3, Interesting)
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.