Monday, March 19, 2012

QCD With Monte Carlo Methods

For the last fifty years we have know pretty much exactly the laws of the universe that govern how atoms and their subcomponents quarks, are held together. But, the equations have been too hard to produce accurate answers in even moderately complex systems. Now, someone has come up with a mathematical trick that makes approximating the outcomes that these equations predict possible to do on a practical basis.

It turns out to be much harder to do exact calculations with the strong force that governs how quarks interact by exchanging color charged gluons (QCD) than it is to do exact calculations with the electromagnetic force in which charged particles exchange photons (QED). At first glance, the equations seem very similar. Both involve adding up an infinte series all possible ways that the particles in question would interact, each of which has a Feynmann diagram corresponding to it. When you do it right, this gives you a probability distribution of the next state the system could be in, and you interate. The equations for the probabilities for any given Feynmann diagram are pretty similar, although the QCD equations are a bit more complicated. In practice, you can't get an exact answer for anything but the most stylized situations, but you can do an arbitrarily large number of calculations and get a good approximation. Until a development announced recently at Science Daily, you did that by starting with the biggest terms and working your way down until you were statisfied with the degree of accuracy that you achieved.

In a nutshell, QCD equations are harder because you need to calculate far more terms to get the same level of precision, because the terms in the QCD series get smaller more slowly than the QED series, because there are positive and negative values of three orthogonal color charges instead of one unified color charge that comes in value of zero, +/-1/3 and +/-1 in the same charge dimension, and because gluons interact with each other, while photons don't.

But, a new method has made it possible to do practical calculations in QCD far more efficiently even in systems that aren't grossly oversimplified. In essence, the new method does a statistical sampling of all the possible ways a system could turn out with greater precision each, rather than looking at every single way the system could turn out up to a certain number of terms each that is used as a cutoff.

What they discovered is a trick -- called Diagrammatic Monte Carlo -- of sampling the Feynman series instead of calculating diagrams one by one. Especially powerful is the Bold Diagrammatic Monte Carlo (BDMC) scheme. This deals with a partially summed Feynman series (Dyson's development) in which the diagrams are constructed not from the bare Green's functions of non-interacting system (usually represented by thin lines), but from the genuine Green's functions of the strongly interacting system being looked for (usually represented by bold lines).

"We poll a series of integrals, and the result is fed back to the series to keep improving our knowledge of the Green's function," says Van Houcke, who developed the BDMC code over the past three years.

The BDMC protocol works a bit like sampling to predict the outcome of an election but with the difference that results of polling are being constantly fed back to the "electorate," Prokof'ev and Svistunov add. "We repeat this with several hundred processors over several days until the solution converges. That is, the Green's function doesn't change anymore. And once you know the Green's function, you know all the basic thermodynamic properties of the system. This has never been done before."

Several days with several hundred processors may seem like a lot. But, the old industry standard was to use similar resources to get less accurate answers in several years of non-stop calculations. And, if you made a subtle error in formulating the problem that caused you to tell the computer to calculate the wrong things, then you got the wrong answer and everybody who could understand what you did wrong would hate you. Fortunately, not many people could understood what you did wrong and most of them lived in constant fear that they would screw up so they would snub you in private, rather than on live TV before an audience of millions of people who can understand exactly what you did wrong like a March Madness basketball player. But, those days may now be behind us.

Also, keep in mind that all quarks and all gluons come in a very small number of varieties, many combinations of which are identical to each other due to symmetry. So, once you can solve a given chunk of a problem, you've solved it once and for all, and can fit that piece like a lego into a larger system (well, actually, quantum mechanics isn't quite as kind as a lego set, but the piece about there being interchangable, perfectly understood subcomponents is right). So, for example, once you figure out the behavior of a particular component of a quantum computer at a given temperature, you're good and you never have to do it again. And, if you can pretty accurately model a sophistiated system in the five years that it used to take to figure out a trivally simple one, you may actually get a result that is visibly worth the effort.

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