A student repeats the chapter's principal-component estimate of the transition on…
PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Thermal, Statistical & Complex Systems · High-performance and data-driven computational physics
Scientific Computing for Physicists, Ch Learning from physical data: regression, classification and principal components
Problem
A student repeats the chapter's principal-component estimate of the transition on lattices of L=24 and L=48. At each temperature there are four independent Metropolis runs, each saving 1250 configurations ten sweeps apart. The size of the first principal component, proportional to ∣m∣, is averaged over each run and then over the four runs. At T=1.3J/kB both sizes give a run-averaged ∣m∣ of 0.995. Near the transition the run-averaged ∣m∣ is as follows. For L=24: 0.80, 0.62, 0.45, 0.33 at T=2.2,2.3,2.4,2.5J/kB respectively. For L=48: 0.72, 0.49, 0.28, 0.17 at the same four temperatures.
((a)) Find, by linear interpolation between neighbouring temperatures, the half-way temperature T1/2(L) at which the run-averaged ∣m∣ has fallen to half its value at 1.3J/kB, for each size.
((b)) Leaving out one run at a time at L=48 and repeating (a) gives 2.296, 2.299, 2.294 and 2.301J/kB. Compute the jackknife uncertainty of T1/2(48), the root-mean-square scatter of these n=4 values about their mean multiplied by n−1.
((c)) A colleague instead bootstraps the 5000 saved configurations at each temperature individually and reports T1/2(48)=2.297±0.0015J/kB. Near the transition τint(∣m∣) is about 50 sweeps. Explain which assumption of the bootstrap fails, and estimate from τint the factor by which it understates the uncertainty.
((d)) Onsager's exact value is Tc=2.269J/kB. Decide whether the gap between T1/2(48) and Tc can be statistical, and explain its sign and why it is smaller at L=48 than at L=24.