Time correlation function single-particle

Figure 18. (a) <(5V(5V(t)>. The solid curve at the bottom is the total time correlation function and the curves labeled 1-3 are the contributions of individual shells 1-3. The curve marked /i is the single-particle dipole time correlation function (Cld(t) shown for comparison. From Ref. 57 with permission, from J. Chem. Phys. 89, 5044 (1988). Copyright 1988, American Physical Society. 

Das and Bhattacharjee236 derive the frequency and shear dependent viscosity of a simple fluid at the critical point and find good agreement with recent experimental measurements of Berg et al.237 Ernst238 calculates universal power law tails for single and multi-particle time correlation functions and finds that the collisional transfer component of the stress autocorrelation function in a classical dense fluid has the same long-time behaviour as the velocity autocorrelation function for the Lorentz gas, i.e. 

For H in metals the measured QENS intensity, after the necessary ra v data corrections (background subtraction, detector efficiency, etc.) is proportional to the incoherent scattering function S , (Q, co) which can be written as the two-fold Fourier transform (in space and time) of the single-particle, space-time van Hove correlation function, Gg(r,t),... 

Despite extensive investigation of phase behavior of liquid crystals in computer simulation studies [97-99], the literature on computational studies of their dynamics is somewhat limited. The focal point of the latter studies has often been the single-particle and collective orientational correlation functions. The Zth rank single-particle orientational time correlation function (OTCF) is defined by... 

Figure 9. Orientational relaxation in the model liquid crystalline system GB(3, 5, 2, 1) (N = 576) at several densities across the I-N transition along the isotherm at T = 1. (a) Time dependence of the single-particle second-rank orientational time correlation function in a log-log plot. From left to right, the density increases from p = 0.285 to p = 0.315 in steps of 0.005. The continuous line is a fit to the power law regime, (b) Time dependence of the OKE signal, measured by the negative of the time derivative of the collective second-rank orientational time correlation function C t) in a log-log plot at two densities. The dashed line corresponds to p = 0.31 and the continuous line to p = 0.315. (Reproduced from Ref. 112.)...

Figure 12. Orientational dynamics of the discotic system GBDII (N = 500) at several temperatures across the isotropic-nematic transition along the isobar at pressure P — 25. (a) Time evolution of the single-particle second-rank orientational time correlation function in a log-log plot. Temperature decreases from left to right, (b) Time dependence the OKE signal at short-to-intermediate times in a log-log plot. Temperature decreases from top to bottom on the left side of the plot T = 2.991,2.693,2.646, and 2.594. The dashed lines are the simulation data and the continuous lines are the linear fits to the data, showing the power law decay regimes at temperatures. (Reproduced from Ref. 115.)...

As discussed in Appendix 3.A, we employ two kinds of time correlation functions to describe rotational motion. They employ single particle and collective quantities. While they can be quite different in some cases, usually they both measure similar dynamics. Most of the experiments measure the collective response of the liquid. It is, however, important to know the difference. 

If C t) is a time correlation function of a single-particle property, then in addition to time averaging, C t) may be averaged over the N particle labels in the molecular dynamics system. This additional averaging decreases the error by the factor (1/N). Even if C(t) is the time correlation function of a... 

Figure 15 Single-particle time correlation functions of imidazolium cations. Correlation functions include velocity of geometric center of the imidazolium ring, C (t), dihedral angle of the alkyl chain, Cdih(t), reorientation projected along the NN direction, CrNN(f)j NN direction, CrNN( ), and MSD of geometric center of the imidazolium ring. The scale for MSD is on the right of the figure. (From Ref. 100 and used with permission.)...

The time correlation functions for the various thermal transport coefficients can be efficiently computed by MD with low statistical uncertainty. The results for transport coefficients are usually accurate to within 5-10%. One-particle time correlation functions are in general more accurate than collective functions due to the possible averaging over each single particle trajectory (Hansen McDonald 1986 Hoheisel Vogelsang 1988). So self-diffusion coefficients are, for instance, more accurately computable than mutual-diffusion coefficients. 

In the above discussion of the frequency dependent permittivity, the analysis has been based on either the single particle rotational diffusion model of Debye, or empirical extensions of this model. A more general approach can be developed in terms of time correlation functions [6], which in turn have to be interpreted in terms of a suitable molecular model. While using the correlation function approach does not simplify the analysis, it is useful, since experimental correlation functions can be compared with those deduced from approximate theories, and perhaps more usefully with the results of molecular dynamics simulations. Since the use of correlation functions will be mentioned in the context of liquid crystals, they will be briefly introduced here. The dipole-dipole time correlation function C(t) is related to the frequency dependent permittivity through a Laplace transform such that ... 

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

The relation between collective and self-motion in simple monoatomic liquids was theoretically deduced by de Gennes [233] applying the second sum rule to a simple diffusive process. Phenomenological approaches like those proposed by Vineyard [ 194] and Skbld [234] also relate pair and single particle motions and may be applied to non-exponential functions. The first clearly fails to describe the PIB results since it considers the same time dependence for both correlators. Taking into account the stretched exponential forms for Spair(Q.t) (Eq. 4.21) and Sseif(Q>0 (Eq 4.9), the Skold approximation ... 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

The time-dependent density functional theory [38] for electronic systems is usually implemented at adiabatic local density approximation (ALDA) when density and single-particle potential are supposed to vary slowly both in time and space. Last years, the current-dependent Kohn-Sham functionals with a current density as a basic variable were introduced to treat the collective motion beyond ALDA (see e.g. [13]). These functionals are robust for a time-dependent linear response problem where the ordinary density functionals become strongly nonlocal. The theory is reformulated in terms of a vector potential for exchange and correlations, depending on the induced current density. So, T-odd variables appear in electronic functionals as well. 

In multidimensional NMR studies of organic compounds, 2H, 13C and 31P are suitable probe nuclei.3,4,6 For these nuclei, the time evolution of the spin system is simple due to 7 1 and the strengths of the quadrupolar or chemical shift interactions exceed the dipole-dipole couplings so that single-particle correlation functions can be measured. On the other hand, the situation is less favorable for applications on solid-ion conductors. Here, the nuclei associated with the mobile ions often exhibit I> 1 and, hence, a complicated evolution of the spin system requires elaborate pulse sequences.197 199 Further, strong dipolar interactions often hamper straightforward analysis of the data. Nevertheless, it was shown that 6Li, 7Li and 9Be are useful to characterize ion dynamics in crystalline ion conductors by means of 2D NMR in frequency and time domain.200 204 For example, small translational diffusion coefficients D 1 O-20 m2/s became accessible in 7Li NMR stimulated-echo studies.201... 

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