Current correlation function

Figure 2. A pictorial representation of the mode coupling theory scheme for the calculation of the time-dependent friction (f) on a tagged molecule at time t. The rest of the notation is as follows Fs(q,t), self-scattering function F(q,t), intermediate scattering function D, self-diffusion coefficient t]s(t), time-dependnet shear viscosity Cu(q,t), longitudinal current correlation function C q,t), longitudinal current correlation functioa...

The transverse current correlation function is assumed to be given by the viscoelastic expression... 

The longitudinal current correlation function, Q(q, t), is related to the dynamic structure factor by the following expression ... 

As the current correlation function in the time integral has sums over all charge velocities z, effects of cross terms between ionic and molecular motions appear which cannot be identified or separated by electromagnetic measurements. In addition to static solvation and saturation effects on permittivity often considered in biological contexts, Hubbard and Onsager have pointed out "kinetic depolarization" effects which need to be considered. In II, we discuss experimental evidence and implications of the theoretical predictions of such effects. 

The heat-current correlation functions in the nematic phase of the prolate ellipsoids are depicted in Fig. 1. The perpendicular component resembles the heat-current correlation function of a Lennard-Jones fluid. The parallel component, which is the largest one, is different. Immediately after the initial decay there is a negative region, the absolute magnitude of which is rather small though, and it does not contribute very much to the time integral of the heat-current correlation function or the thermal conductivity. 

Fig. 1 The heat current correlation functions, Cqqw n(/) = T(yg (r)ygn(0)) (dashed curve) and Cggjj (f) = V

In Fig. 2 we show the heat-current correlation functions of the oblate ellipsoids in the nematic phase. Here the roles of the two components are interchanged. The perpendicular component is the largest one. It has a fairly long... 

Temporary current correlation function and low-frequency noise for adatom diffusion... 

However, the form of time dependence of the temporary current correlation function appears to be similar to the one obtained in [4]. The difference is in the values of coefficients in these expressions. 

Similarly we define the matrix of the site-site longitudinal current correlation functions by... 

Conductivity can be deduced from vibrational spectra in IR spectroscopy, the absorption coefficient a(co) is related to tr(co) a(o)) = 4no(o))/nc, n being the refractive index and c the velocity of light. In Raman spectroscopy, the scattered intensity /(m) is related to conductivity by a(o ) oc o)I (o)/n(a)) + 1, n(co) being the Bose-Einstein population factor . Finally, the inelastic incoherent neutron scattering function P(o)) is proportional to the Fourier transform of the current correlation function of the mobile ions. P co) is homogeneous with a) /(cu) formalism. However, since P(co) reflects mainly single particle motions, its comparison with ff(co) could provide a method for the evaluation of correlation effects. (For further discussion, see also Chapter 9 and p. 333.)... 

In a similar way, quantities like the longitudinal and transverse current correlation functions can be extracted. 

Fig. 3.14. Dispersion curves obtained from the maxima h(o in the longitudinal current correlation function for liquid rubidium at various temperatures along the liquid-vapor coexistence line (Pilgrim et al 1991) A = 42 C V = 800 C = 1100 °C O = 1400 °C. The data for 42 "C are those of Copley and Rowe...

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... 

The electrical conductivity is related to a current-current correlation function. In order to evaluate it we write the interaction hamiltonian in the form... 

The thermal conductivity is related to a heat current - heat current correlation function. It is therefore given by an expression very similar to eq. (17.50) namely... 

---