Time correlation function collective

If the Bath relaxation constant, t, is greater than O.I ps, you should be able to calculate dynamic properties, like time correlation functions and diffusion constants, from data in the SNP and/or CSV files (see Collecting Averages from Simulations on page 85). 

TOWARDS THE HYDRODYNAMIC LIMIT STRUCTURE FACTORS AND SOUND DISPERSION. The collective motions of water molecules give rise to many hydrodynamical phenomena observable in the laboratories. They are most conveniently studied in terms of the spatial Fourier ( ) components of the density, particle currents, stress, and energy fluxes. The time correlation function of those Fourier components detail the decay of density, current, and fluctuation on the length scale of the Ijk. 

Collective Modes and Time-Correlation Functions. Our linear equations (6.15) and (6.16) describe two characteristic kinds of collective modes in gels the longitudinal part of u obeys Tanaka s equation (4.16) and g = V u is governed by the diffusion equation (4.18), while the transverse parts of u and are coupled to form a slow transverse sound at small wave numbers. By assuming the space-time dependence as exp(i[Pg.99]

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

Raman scattering depends on the time correlation function of the many-body polarizability of the liquid, collective dipole moment. In the case of Raman scattering, an external electric field (from a laser) generates an induced collective dipole in the liquid ... 

Keywords Molten salts, ionic solutions, collective dynamics, time correlation function, col-... 

The linearized transport equations (7), the equations for the equilibrium time correlation functions (13), and the equation for collective mode spectrum (14) form a general basis for the study of the dynamic behavior of a multicomponent fluid in the memory function formalism. 

P,-1 ) LyYjf Vl containing the projection operators Vp-. The theory is formulated in matrix form convenient for real calculations and permits within the unique approach to obtain the collective mode spectrum and weight coefficients for different time correlation functions. 

Optic-like collective excitations are not a unique feature of binary mixture of charged particles. Such modes can also be found in binary mixtures of neutral particles. However, the behavior of mode contributions to time correlation functions in small k range in these two cases is quite different. In particular, amplitude of optic-like modes to the mass concentration autocorrelation function tends to zero for the latter case, whereas for the former one these modes produce the finite contribution even in the hydrodynamic limit. 

Gottke et al. [5] offered a theoretical treatment of collective motions of mesogens in the isotropic phase at short to intermediate time scales within the framework of the Mode coupling theory (MCT). The wavenumber-dependent collective orientational time correlation function C/m(, t) is defined as... 

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.)...

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. 

There is an indirect way to detect intermittent local collective motions. In the case of depolarized Raman scattering, the depolarization ratio is sensitive to low-frequency fluctuations in water. Depolarization is the scattering of the polarization of the electric field of light in a direction perpendicular to the original direction of polarization. Each fluctuating state has a distinct depolarization ratio. The intermittent character of the dynamics is known to appear as a so-called 1,/ frequency (f) dependence in a power spectrum. The power spectmm is obtained by Fourier transforming a time correlation function. 

The RMFA is generated by setting the normalized Ith memory function of the time-correlation function of interest equal to the Zth normalized memory function of the time-correlation function of a reference dynamical variable [47, 48]. Specializing this prescription in various ways leads to known approximation schemes, such as the Vineyard and Kerr approximations, and their generalization to molecular liquids. A dielectric form of the RMFA has been applied quite successfully to reproduce the optical mode excitation profiles in liquid water [49, 50]. However, this dielectric approach cannot be applied to the description of another important collective excitation in water, the acoustic mode, and it is desirable to develop a theory that accounts for all the characteristic features of the collective excitations in molecular liquids. 

As a result of the factorization approximation the time dependence of the relaxation functions in Eqs. (5.178) and (5.179) is all in the time-correlation functions of the solute and solvent dynamics. Thus, within the linear response regime and the factorization approximation, the solvent response part due to the solute perturbation is determined by F k,t). As demonstrated in Sec. 5.3, Fxix k,t) in thesmall-A regime comprises contributions from collective excitations of acoustic and optical modes of solvent, and these modes are responsible for the density and dielectric (or polarization) relaxations, respectively. 

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. 

Compared with the collective time correlation functions of atomic fluids the behavior of molecular time correlation functions is significantly different. This is illustrated by plots of the normaUzed correlation function for the shear viscosity of liquid model CF4 in Figure 9.7. In the figure, the time correlation function for the four-center LJ liquid is plotted with that of the common Lennard-Jones liquid modeled to give roughly the same tj value (Hoheisel 1993). All the LJ potential parameters used for the computations are Usted in Table 9.1. 

Luo, H. Hoheisel, C. (1991). Behaviour of collective time correlation functions in liquids composed of polyatomic molecules. J. Chem. Phys., 94,8378-8383. 

The energetics of the recombination process and the concomitant changes in the Li " solvation sheath were followed on a function of f. The main results are collected in Pig. 1 and in Table 1. It is convenient to monitor the state of the electron using the complex time correlation function R (t-t ) where O < t < t = i/ ft.2 The value of this function at t-t = t/2 is expression... 

Dielectric relaxation is sensitive to the time correlation function of the collective variable P(t) that is the total dipole moment, just as quasielastic light scattering is sensitive to the time correlation function of the collective variable YTj= exp(iq r (t)) that is the spatial Fourier component of the concentration. 

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. 

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