^{1,a)}, Sathish K. Sukumaran

^{2}, Bart Vorselaars

^{3}and Alexei E. Likhtman

^{3}

### Abstract

Time correlation functions yield profound information about the dynamics of a physical system and hence are frequently calculated in computer simulations. For systems whose dynamics span a wide range of time, currently used methods require significant computer time and memory. In this paper, we discuss the multiple-tau correlator method for the efficient calculation of accurate time correlation functions on the fly during computer simulations. The multiple-tau correlator is efficacious in terms of computational requirements and can be tuned to the desired level of accuracy. Further, we derive estimates for the error arising from the use of the multiple-tau correlator and extend it for use in the calculation of mean-square particle displacements and dynamic structure factors. The method described here, in hardware implementation, is routinely used in light scattering experiments but has not yet found widespread use in computer simulations.

We gratefully acknowledge financial support from EPSRC, through the project (J.R. and B.V.), Grant No.GR/S94711/01 (S.K.S.), Grant No. EP/H016686/1 (B.V.), and an Advanced Research Fellowship (A.E.L.), and B. Loppinet for drawing our attention to the multiple-tau correlators.

I. INTRODUCTION

II. METHODS

A. Standard calculation of time correlation functions

B. Time correlation of a time-averaged function

C. Mean-square displacement

D. Algorithm proposed by Frenkel

E. Multiple-tau correlator

F. Nonaveraged algorithm

III. CASE STUDIES

A. Stress from molecular dynamics

B. Brownian dynamics simulations of Rouse chains

IV. CONCLUSION

### Key Topics

- Correlation functions
- 39.0
- Computer simulation
- 13.0
- Molecular dynamics
- 13.0
- Brownian dynamics
- 7.0
- Stress relaxation
- 6.0

## Figures

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.

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 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 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.

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. 3 but for the orientation tensor autocorrelation function.

Similar to Fig. 4 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.

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.

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 .

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.

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.

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.

## Tables

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 .

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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