4 ms·
The simulation in question was created using Lattice QCD (which many people in the HPC world mostly know as a benchmark to optimize their compilers/systems on,
by KenoFischer 10y ago
The simulation in question was created using Lattice QCD (which many people in the HPC world mostly know as a benchmark to optimize their compilers/systems on, but it's also a useful physics technique).
Basically what you end up doing is discretizing spacetime on some grid, maybe e.g. 100x100x100x100 (not much larger than that last I checked or the problem becomes intractable even on very large computers) and then sample field configurations monte-carlo style (the way to do it is actually really clever - it turns out that there is a mathematical trick that can take you from the description of your theory right to a distribution from which you can draw your field configurations). The number of fields is decently large (some discretization introduce extra auxiliary fields for every physical one) and in general you need to consider the entire field configuration to compute an observable. Then do that a few billion times (to bring down statistical error) at a couple different lattice spacings (to be able to extrapolate to the continuum limit) and you quickly end up with a very large computations.
There is a number of clever ways to reduce the amount of computation required, but in general it's a very different problem.
Hope that makes some sense. I had looked into this in detail about two years ago, but looking back, I feel like I only know every other word in my write-up ;).
- evanb 10y agoI do lattice QCD, and this is a pretty decent explanation. A few minor things: People typically use lattice sizes that have lots of factors of 2, because they fit on the supercomputers in a better way. So rather than 100 people do 96 or 128. Only some calculations require lattices this large. The field content is usually a few fermions, and the gluons. On each site there is (usually, depends on the discretization as you say) a Dirac spinor (2 spins * 2 particle/antiparticle * 3 spins = 12 numbers) for each fermion and on each link (ie. the edges) there is an SU(3) matrix (3x3 complex doubles) for the glue. The field configuration is simply the glue. But this already is a large amount of data. volume * (9 complex doubles per link) * (1 link per spacetime direction) * (4D spacetime) ~= 9 gigs for a (32^3 * 64 spacetime volume). This is effectively a giant (sparse) matrix, which we want to invert. The mathematical trick is called importance sampling, https://en.wikipedia.org/wiki/Importance_sampling https://en.wikipedia.org/wiki/Importance_sampling which people accomplish with the metropolis algorithm https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_al... . If only we could do a few billion samples! People usually make a few thousand to a few hundred thousand samples.
- KenoFischer 10y agoThanks for the correction! I never did any actual Lattice QCD calculations, just looked at the theory, so I was taking a bit of an educated guess on the feasible problem sizes.
- semi-extrinsic 10y agoHaving done some QFT, now doing hydrodynamics, I never knew that lattice people use a staggered grid for the fermions and gluons. Do you know if it's for the same reason as staggering velocity and pressure in CFD; to avoid checkerboarding of the spinors (pressure)?
- KenoFischer 10y agoIf I remember correctly, there is no way to obtain a gauge invariant discretization if you try to put the gluons at the lattice sites. I can try to find a proper reference.
- evanb 10y agoKenoFischer is right---it comes down to gauge invariance. Gauge symmetry is a local symmetry that dictates a lot of the structure of QCD. If your discretization breaks it (and say, only recovers it in the continuum limit) you will have very bad renormalization and fine-tuning problems. Putting the gauge on the links automatically guarantees that gauge symmetry is respected, even at the discretized level, and jibes very well with the fact that the gauge field / gauge connection describes, in some way, how to perform parallel transport. So, it's very natural for the gauge to connect the sites together in that you need to know the value of the link in order to compare quantities on two adjacent spacetime sites. https://en.wikipedia.org/wiki/Gauge_theory https://en.wikipedia.org/wiki/Gauge_theory https://en.wikipedia.org/wiki/Connection_form https://en.wikipedia.org/wiki/Connection_form https://en.wikipedia.org/wiki/Parallel_transport https://en.wikipedia.org/wiki/Parallel_transport
- evanb 10y agoMinor erratum: 's/3 spins/3 colors/'.