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Pressure-energy correlations in liquids. I. Results from computer simulations
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6.If this seems unphysical, the argument could be given in terms of arbitrary deviations from equilibrium, . See, however, Paper II.
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28.Choosing this measure of the slope is equivalent to diagonalizing the correlation matrix (the covariance matrix where the variables have been scaled to have unit variance) to identify the independently fluctuating variable. This is often done in multivariate analysis (see, e.g., Ref. 34), rather than diagonalizing the covariance matrix, when different variables have widely differing variances.
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Image of FIG. 1.

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FIG. 1.

Equilibrium fluctuations of (a) pressure and energy and (b) virial and potential energy , in a single-component Lennard-Jones system simulated in the ensemble at and (argon units). The time-averaged pressure was close to zero . The correlation coefficient between and is 0.94, whereas the correlation coefficient is only 0.70 between and . Correlation coefficients were calculated over the total simulation time .

Image of FIG. 2.

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FIG. 2.

(a) Scatter plot of instantaneous virial and potential energy from the simulation of Fig. 1. The dashed line is a guide to the eyes, with a slope determined by the ratio of standard deviations of and [Eq. (7)]. (b) Example of a system with almost no correlation between and : TIP5P water at and density of (NVT). This system has Coulomb, in addition to Lennard-Jones, interactions.

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FIG. 3.

Illustration of the “effective inverse power law” chosen in this case to match the Lennard-Jones potential and its first two derivatives at the point . The vertical line marks the division into the repulsive and attractive parts of the Lennard-Jones potential.

Image of FIG. 4.

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FIG. 4.

Scatter plots of the configurational parts of pressure and energy—virial vs potential energy—for several state points of the SCLJ liquid (NVT). Each oval represents simulations at one particular temperature and density where each data point marks instantaneous values of virial and potential energy. The dashed lines mark constant density paths with the highest density to the upper left (densities: 39.8, 37.4, 36.0, 34.6, and ). State points on the dotted line have zero average pressure. The plot includes three crystallized samples (lower left corner), discussed at the end of Sec. III A and, in more detail, in Paper II (reproduced from Ref. 10).

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FIG. 5.

Upper plot, correlation coefficient for the SCLJ system as a function of temperature for several densities (NVT). This figure makes clear the different effects of density and temperature on . Lower plot, effective slope as a function of . Simulations at temperatures higher than those shown here indicate that the slope slowly approaches the value 4 as increases. This is to be expected because as collisions become harder, involving shorter distances, the effective inverse power-law exponent approaches the 12 from the repulsive term of the Lennard-Jones potential.

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FIG. 6.

Scatter plot of the -correlations for a perfect face-centered-cubic (fcc) crystal of Lennard-Jones atoms at temperatures 1, 2, 3, 5, 10, 20, 30, 40, 50, 60, 70, and , as well as for defective crystals (i.e., crystallized from the liquid) at temperatures 50, 70, and (NVT). The dashed line gives the best fit to the (barely visible) lowest-temperature data . The inset shows the temperature dependence of at very low temperatures. The crystalline case is examined in detail in Paper II, where we find that does not converge to unity at , but rather to a value very close to unity. All state points refer to the highest density of Fig. 4, 39.8 mol/l.

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FIG. 7.

A plot of the Dzugutov pair potential, with the Lennard-Jones potential (shifted by a constant) shown for comparison.

Image of FIG. 8.

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FIG. 8.

Scatter plot of -correlations for the Dzugutov system at density 0.88 and temperatures 0.65 and 0.70 (NVE). The dashed lines indicate the best-fit line using linear regression. These are consistent with the temperature dependence of the mean values of and , as they should be (see Appendix B), but they clearly do not represent the direction of greatest variance. The full lines have slopes equal to the ratio of standard deviations of the two quantities [Eq. (7)]. The correlation coefficient is 0.585 and 0.604 for and , respectively.

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FIG. 9.

Plot of for TIP5P water in simulations with densities chosen to give an average pressure of . Not only is the magnitude of low (less than 0.2) in the temperature range shown, but it changes sign around the density maximum. The vertical arrow indicates the state point used for Fig. 2(b).

Image of FIG. 10.

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FIG. 10.

-correlations for all simulated liquids; plotted vs . Both quantities correspond to a pressure, which is given in units of GPa; for model systems not specifically corresponding to real systems, such as SCLJ, KABLJ, and SQW, argon units were used to set the energy and length scales. If the correlation is perfect the data fall on the diagonal marked by a dashed line. For the TIP5P model of water only temperatures with are included; volumes were chosen to give a pressure close to zero.

Image of FIG. 11.

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FIG. 11.

Illustration of the square-well potential, indicating the four microscopic processes, which contribute to the virial.

Image of FIG. 12.

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FIG. 12.

Energy-energy, virial-virial, and energy-virial correlation functions for SQW at packing fraction and temperature (normalized to unity at ). The cross correlation has been multiplied by . The arrow marks the time , roughly of the relaxation time (determined from the long-time part of the energy-virial cross-correlation function). This time was used for averaging.

Image of FIG. 13.

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FIG. 13.

Illustration of replacement of discontinuous step by a finite slope for the square-well potential for the purpose of calculating the virial. The limit is taken at the end.


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Table I.

Correlation coefficients and effective slopes for the SCLJ system for the state points in Fig. 4. is the thermally averaged pressure. The last five states were chosen to approximately follow the isobar .

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Table II.

Correlation coefficients and slopes at selected state points for all investigated systems besides SCLJ. Argon units were used for DZ, EXP, KABLJ, and SQW by choosing the length parameter (of the larger particle when there were two types) to be and the energy parameter to be . The phase is indicated as liquid or glass. SQW data involve time averaging over periods 3.0, 3.0, 8.0, and 9.0, respectively, for the four listed state points. A minus sign has been included with the slope when ; note that the values only really make sense as slopes when is close to unity. The ensemble was NVT except for CU, DZ, MGCU, and SQW, where it was NVE.


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We show that a number of model liquids at fixed volume exhibit strong correlations between equilibrium fluctuations of the configurational parts of (instantaneous) pressure and energy. We present detailed results for 13 systems, showing in which systems these correlations are significant. These include Lennard-Jones liquids (both single- and two-component) and several other simple liquids, neither hydrogen-bonding liquids such as methanol and water, nor the Dzugutov liquid, which has significant contributions to pressure at the second nearest neighbor distance. The pressure-energy correlations, which for the Lennard-Jones case are shown to also be present in the crystal and glass phases, reflect an effective inverse power-law potential dominating fluctuations, even at zero and slightly negative pressure. An exception to the inverse power-law explanation is a liquid with hard-sphere repulsion and a square-well attractive part, where a strong correlation is observed, but only after time averaging. The companion paper [N. P. Bailey et al., J. Chem. Phys.129, 184508 (2008)] gives a thorough analysis of the correlations, with a focus on the Lennard-Jones liquid, and a discussion of some experimental and theoretical consequences.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Pressure-energy correlations in liquids. I. Results from computer simulations