Autocorrelation function conductivity

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

Using linear response theory and noting (according to the results at the end of Section 5.1.3) that the (complex) electrical conductivity a is the Fourier transform of the current density autocorrelation function, we obtain from Eqn. (5.75) (see the equivalent Eqn. (5.21))... 

Let us summarize by modeling the velocity autocorrelation function using Debye-Huckel type interactions between charged point defects in ionic crystals, one can evaluate the frequency-dependent conductivity and give an interpretation of the universal dielectric response. 

In this section it will be outlined how the different molar masses contribute to the TDFRS signal. Of especial interest is the possibility of selective excitation and the preparation of different nonequilibrium states, which allows for a tuning of the relative statistical weights in the way a TDFRS experiment is conducted. Especially when compared to PCS, whose electric field autocorrelation function g t) strongly overestimates high molar mass contributions, a much more uniform contribution of the different molar masses to the heterodyne TDFRS diffraction efficiency t) is found. This will allow for the measurement of small... 

Does this concern ions in solution and electrochemistry It does indeed concern some approaches to diffusion and hence the related properties of conduction and viscous flow. It has been found that the autocorrelation function for the velocity of an ion diffusing in solution decays to zero very quickly, i.e., in about the same time as that of the random force due to collisions between the ion and the solvent. This is awkward because it is not consistent with one of the approximations used to derive analytical expressions for the autocorrelation function. The result of this is that instead of an analytical expression, one has to deal with molecular dynamics simulations. 

Equilibrium molecular dynamics uses the Green-Kubo relationship between the heat current autocorrelation function and the thermal conductivity [66] to obtain the thermal conductivity as ... 

Most research on macroscopic dispersion has focused on the prediction of solute spreading from knowledge of the spatial distribution and covariance of the log-transformed saturated hydraulic conductivity, ln(/csat). Theoretical models have been developed for exponential and fractal autocorrelation functions (Gelhar Ax-ness, 1983 Dagan, 1984, 1994 Koch Brady, 1988 Kemblowski Wen, 1993 Neuman, 1995 Zhan Wheatcraft, 1996 Hassan et al., 1997). While these models may explain the scale-dependency observed in dispersivity, they lack predictive power because multiple spatial measurements o lti(Asll,) ate needed and such data are not easy to collect. Moreover, they are macroscopic avciagcs that depend... 

Quasi-elastic light scattering (QELS) experiments were conducted on both neutralized and unneutralized sols, in a Brookhaven model BI-90 particle sizer. This instrument measured the autocorrelation function, C(t), and fit this function to... 

FIGURE 8.22 Hydrodynamic radius of PPy-DBSA in chloroform containing different amounts of extra DBSA at 25°C, where the concentration of PPy-DBSA is 0.01 g/dL The inset shows autocorrelation functions obtained from the dynamic light scattering analysis. (From Song, K.T., Synthesis of electrically conducting soluble polypyrrole and its characterization. Ph.D. thesis, 2000. With permission.)... 

Transport properties, such as diffusion coefficients, shear viscosity, fhermal or electrical conductivity, can be determined from the time evolution of the autocorrelation function of a particular microscopic flux in a system in equilibrium based on the Green-Kubo formalism [217, 218] or the Einstein equations [219], Autocorrelation functions give an insight into the dynamics of a fluid and their Fourier transforms can be related to experimental spectra. The general Green-Kubo expression for an arbitrary transport coefficient y is given by ... 

One of the predominant sources of fluctuations in addition to diffusion in PCS is from intersystem crossing to the lowest excited triplet state of the fluorophore [38]. Correlations due to triplet crossing tend to be dominant at very short lag times in comparison with the part of the autocorrelation curve dominated by diffusion (see figures 2.9 and 2.12). Numerous triplet state studies have been conducted principally using the organic dyes rhodamine 6G (R6G) and fluorescein [42,54]. A model for the autocorrelation function of a signal whose fluctuations derive from a combination of diffusion and triplet crossing has been developed [54],... 

Fig. 1 Random versus correlated jump diffusion velocity autocorrelation functions and corresponding real parts of the complex conductivity [42]...

The simplest approach to describe the ion dynamics in disordered materials is to assume completely uncorrelated, random ion movements [42]. In this case, the jump of an ion moving in a forward direction is only correlated to itself, thus the velocity autocorrelation function is proportional to a Dirac Delta function at = 0 (see Fig. la). The complex conductivity obtained by Fourier transform is then independent of frequency. This means that the real part of the conductivity shows no dispersion and at all frequencies the ac conductivity < (6 ) can be identified with the dc conductivity. By contrast, conductivity spectra of most ion-conducting materials show that o (o) varies with frequency. This is schematically illustrated... 

Earlier we mentioned that Voth and co-workers conducted equilibrium MD simulations on [C2mim][N03] at 400 K and computed the self-diffusivity and shear viscosity using both a fixed charge and polarizable force field. They computed the viscosity not from integrating the stress-stress autocorrelation function as is normally done, but rather from integrating the so-called transverse current correlation function, details of which are foimd in a work by Hess. ° They used the standard Einstein formula (Eq. [15]) for the self-diffusivity and were careful to ensure that diffusive behavior was achieved when computing the self-diffusivity. Their calculated values of ca. 1 x 10 m /s for the polarizable model and ca. 5 x 10 m /s are reasonable. The finding that the polarizable model yielded faster dynamics than with the nonpolarizable model... 

Before leaving the topic of Green-Kubo integrals for transport properties, we mention briefly the characteristics of the electric current correlation functions that are used to compute the electrical conductivity. Figure 18 shows the electric current and velocity autocorrelation functions for [C2mim][Cl] at 486 K and 1 bar. The current fluctuations decay rapidly and appear to vanish... 

Figure 18 Normalized electric current autocorrelation function of [C2mim][Cl] at 486 K and 1 bat (bold solid line) total velocity autocottelation function (dashed line) and diffetence between them (gray). The inset shows the running integral of the electtical conductivity (gtay line), togethet with the best-fit exponential decay function (black line). (Ftom Ref. 107 and used with petmission.)...

Examples of linear response functions (susceptibilities) include the frequency dependent electrical conductivity (the Fourier transform of an equilibrium current autocorrelation function), dielectric susceptibility, which is the transform of a dipole moment autocorrelation function, along with stress, heat flux, and an assortment of velocity correlation functions. 

Ae in terms of the low-density coefficients (equations (5.3) and (5.4)) accordingly contains additional terms. The first, kinetic contribution is the only important term at low densities and scales in time as for diffusion. The final term is the contribution from the potential part alone and the middle term is the cross contribution of the kinetic and potential part. The presence of these terms, and their functional dependence, can be demonstrated simply from a derivation of these expressions by the fluctuation-dissipation theorem, which gives the transport properties in terms of an autocorrelation function of the appropriate flux (see 5.4.1). For thermal conductivity, for example, the flux involves the sum of kinetic and potential eneigies. The autocorrelation of this flux involves the product of the flux at two different times, producing three different terms which can be shown to have the same dependence on density and g(a) as above. 

Another approach to calculate thermal conductivity is equilibrium molecular dynamics (EMD) [125] that uses the Green-Kubo relation derived from linear response theory to extract thermal conductivity from heat current correlation functions. The thermal conductivity X is calculated by integrating the time autocorrelation function of the heat flux vector and is given by... 

The molecular dynamics calculation of electrical conductivity in ionic fluids has been approached in several different ways. The autocorrelation function of the current may be related to the conductivity by... 

There are two other means of obtaining the trical conductivity from the same computation, amount of heat extracted from the system at each was recorded and divided by the square of the rent. The resulting quantity is also the conductivity. Values obtained in this way are also shown in Fig.l (+ s). Finally the power spectrum of the electrical current in a field free system was computed and the autocorrelation function derived therefrom used to obtain the electrical conductivity via the well-known relationship (U ). The value so obtained is entered on Fig.l as an open circle. All of these methods are, in principle, equivalent. The highest accuracy is attached to the direct measurement of the mean current. 

The electrical conductivity, a, is calculated as the time integral of the electrical current autocorrelation function ... 

This equilibrium MD approach uses current fluctuations to calculate the thermal conductivity k via the fluctuation dissipation theorem. The MD approach is used to compute the autocorrelation function of the heat flux, which is related to the thermal conductivity by the Green-Kubo formula given by ... 

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