Schematic view of the data structure the correlator proposed by Frenkel (left) and the multiple-tau correlator reviewed in this paper (right). Data arrays at different levels are represented, and the arrow symbolically represents the averaging and transfer of data between different levels of the correlator.
Stress autocorrelation function averaged over ten runs from the Kremer–Grest model, , calculating the exact time correlation function (see the text) and using the multiple-tau correlator with and . The bottom curves show the statistical error (the absolute error-bars) for the two methods. The gaps at early time are due to the use of a logarithmic scale and the negative value of G(t).
Stress autocorrelation function averaged over 100 runs obtained using multiple-tau correlators with , statistical error (lines at the bottom) and systematic error for and 4. Systematic error for higher is below statistical error (not shown).
Stress autocorrelation function averaged over 100 runs obtained with fixed preaveraging over points. Statistical errors are shown by horizontal curves, whereas systematic errors for and are shown by symbols.
Similar to Fig. 3 but for the orientation tensor autocorrelation function.
Similar to Fig. 4 but for the orientation tensor autocorrelation function.
Relaxation modulus, calculated from the stress-stress autocorrelation function of Rouse chains of length with the chain ends fixed at the origin. The bold line shows the analytical solution obtained from normal mode analysis; the thin lines at the bottom show the error bars of the multiple-tau correlators with and calculated from Brownian dynamics simulations; the symbols show the systematic error from the simulations, in good agreement with the predictions of Eq. (5) shown in lines.
Same as Fig. 7 with and . The error bars (dashed line) are the same for all correlators. Again, symbols and lines show the systematic error from the simulations and the predictions of Eq. (5), respectively.
Mean-squared displacement of the middle monomer of the Rouse simulations shown in previous figures. The bold line shows the analytical solution; the thin lines at the bottom show the error bars of the multiple-tau correlators with and ; the symbols show the systematic error from the simulations, in good agreement with the predictions of Eq. (11) shown with lines. The dashed lines at the bottom show the statistical error when no averaging is applied (in which case the systematic error equals zero). Arrows point toward increasing .
Same as Fig. 9 but adding the second term of the correction of Eq. (11). The error bars (dashed line) are the same for all correlators. The systematic errors for after correction with Eq. (17) are shown with symbols; the lines show the systematic errors using a simpler correction detailed in the text.
Error bar of the relaxation modulus (black) and the mean-squared displacement of the middle monomer , as percentage of the value of the function at the terminal time, plotted as a function of the number of experiments for BD and for MD. The numbers show the slope of the least-squares linear fitting to the BD data.
Summary of relevant features that describe Frenkel’s and multiple-tau correlators: number of correlation points per level , minimum and maximum lag time, and , and averaging time , in number of time-steps, at each block average level .
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