A Monte Carlo simulation of the two-dimensional Ising model records the energy per spin,

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Scientific Computing for Physicists, Ch The uncertainty of a simulated number: inputs, sensitivity and correlated output

Problem

A Monte Carlo simulation of the two-dimensional Ising model records the energy per spin, ei=Ei/(JL2)e_i = E_i/(JL^2), once per sweep after equilibration, for N=216=65536N = 2^{16} = 65536 sweeps. The sample mean is eˉ=−1.9600 J\bar e = -1.9600\,J and the sample standard deviation of the individual readings is σe=0.128 J\sigma_e = 0.128\,J. Pairwise blocking of the series gives the standard error of the mean, σ^b\hat\sigma_b, as a function of block size bb (consecutive raw samples averaged into one block before the block means' standard error is computed): bb: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 σ^b\hat\sigma_b (JJ): 0.00050, 0.00075, 0.00108, 0.00148, 0.00188, 0.00220, 0.00241, 0.00248, 0.00250, 0.00249 An ordinary bootstrap that resamples the 6553665536 raw readings independently with replacement gives a standard error for the sample-variance estimator σ^e2\hat\sigma_e^2 of 9.1×10−5 J29.1\times10^{-5}\,J^2. A moving-block bootstrap, which resamples overlapping blocks of 512512 raw readings with replacement and recomputes σ^e2\hat\sigma_e^2 on each resampled series, gives 3.2×10−4 J23.2\times10^{-4}\,J^2. Two further runs, identical except for warm-up length, give eˉ=−1.9600 J\bar e = -1.9600\,J after 50005000 sweeps of warm-up and eˉ=−1.9603 J\bar e = -1.9603\,J after 20 00020\,000 sweeps of warm-up. The exchange coupling JJ itself, fixed by a fit to the material's Curie temperature, is known only to 0.2%0.2\%; propagating this uncertainty linearly through the model's known dependence of ⟨e⟩\langle e\rangle on JJ at this state point gives an input uncertainty of uinput=0.0015 Ju_{\rm input} = 0.0015\,J on eˉ\bar e. ((a)) Using the chapter's plateau criterion, identify the plateau of the blocking curve and state the statistical inefficiency ss and the integrated autocorrelation time τint\tau_{\rm int} (in sweeps) it implies. ((b)) State the effective sample size NeffN_{\rm eff} of the run. ((c)) Find the ratio of the moving-block to the ordinary bootstrap standard error of σ^e2\hat\sigma_e^2 and compare it with s\sqrt{s}. Explain in one sentence why the ordinary bootstrap is too small, and in another why the two factors need not agree. ((d)) Assemble the uncertainty budget of eˉ\bar e from a statistical entry (the blocking plateau), a numerical entry (the warm-up-length comparison) and the input entry given above, and quote the final result.

Browse the problem library