Relaxation equations

The hyperbolic relaxation equation (A-5-2.4.1 a) contains charge carrier mobility as a variable, which should be sensitive to oil viscosity. This is found to be the case for some viscous nonconductive liquids. These have much slower rates of charge dissipation equivalent to an Ohmic liquid whose conductivity is 0.02 pS/m (5-2.5.4). 

The NMRD profiles of water solution of Ti(H20)g" have been shown in Section I.C.7 and have been already discussed. We only add here that the best fit procedures provide a constant of contact interaction of 4.5 MHz (61), and a distance of the twelve water protons from the metal ion of 2.62 A. If a 10% outer-sphere contribution is subtracted from the data, the distance increases to 2.67 A, which is a reasonably good value. The increase at high fields in the i 2 values cannot in this case be ascribed to the non-dispersive term present in the contact relaxation equation, as in other cases, because longitudinal measurements do not indicate field dependence in the electron relaxation time. Therefore they were related to chemical exchange contributions (see Eq. (3) of Chapter 2) and indicate values for tm equal to 4.2 X 10 s and 1.2 X 10 s at 293 and 308 K, respectively. 

In the more general case where there may be surface interactions, i.e., a chemical exchange with surface sites where more efficient relaxation may occur, a term is added to the relaxation equation that is proportional to Ijd. The relaxation may generally be written as the sum of contributions ... 

If the paramagnetic center is part of a solid matrix, the nature of the fluctuations in the electron nuclear dipolar coupling change, and the relaxation dispersion profile depends on the nature of the paramagnetic center and the trajectory of the nuclear spin in the vicinity of the paramagnetic center that is permitted by the spatial constraints of the matrix. The paramagnetic contribution to the relaxation equation rate constant may be generally written as... 

In order to compute absolute rate constant values from photostationary parameters such as yp, yg, and [M]i/2, it is necessary to obtain independent measurements of the over-all decay constants l/rf and 1/rp of the emitting species (cf. Eqs. 4 and 12). In the absence of photoassociation ([M] [M] ) the appropriate relaxation equation for the 2-state system... 

We reemphasize that the foregoing relaxation equations containing the general shift-variant response-function element denoted by [s] m are equally valid for the special case of convolution, whether discrete or continuous. Cast in the continuous notation for convolution, the relaxation methods are epitomized by the repeated application of... 

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized... 

Frequency-Dependent Specific Heat. We mention measurements of volume relaxation through the frequency-dependent specific heat Cn(co) as in fluids near the glass transition [52]. This is feasible when the experimental frequency co is of the order fi0 in small gels. The deviations of the entropy, temperature, and volume are related by SS = CV5T + (dH/dT)v8Vand the relaxation equation reads... 

A. Comments on the Relaxation Equation Vm. Structure of Mode Coupling Theory as Applied to Liquid-State Dynamics... 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

Finally, note that the method used by Kadanoff and Swift is a very general scheme. For example, the expression of ris[1H is similar to the expression of viscosity derived later by Geszti [39]. In addition, the projection operator technique used in their study is the same used to derive the relaxation equation [20], and the expression of Ly and Uy are equivalent to the elements of the frequency and memory kernel matrices, respectively. 

Zwanzig showed how a powerful but simple technique, known as the projection operator technique, can be used to derive the relaxation equations [12]. Let us consider a vector A(t), which represents an arbitrary property of the system. Since the time evolution of the system is given by the Liouville operator, the time evolution of the vector can be written as /4(q, t) = e,uA q), where A is the initial value. A projection operator P is defined such that it projects an arbitrary vector on A(q). P can be written as... 

An important step in the derivation of relaxation equation is to apply the identity satisfied by the propagator eiU ... 

The generalized relaxation equation in terms of its components and in the frequency plane can be written as... 

The frequency matrix Qy and the memory function matrix Ty, in the relaxation equation are equivalent to the Liouville operator matrix Ly and the Uy matrix, respectively. The later two matrices were introduced by Kadanoff and Swift [37] (see Section V). Thus the frequency matrix can be identified with the static variables (the wavenumber-dependent thermodynamic quantities) associated with the nondissipative part, and the memory kernel matrix can be identified with the transport coefficients associated with the dissipative part. 

The equations of motion in the extended hydrodynamic theory (Section IV) are obtained from the relaxation equation, where the correlation function is normalized. As mentioned before, in the extended hydrodynamic theory, the memory kernel matrix is considered to be independent of frequency thus the transport coefficients are replaced by their corresponding Enskog values. 

Finally, note that the relaxation equation [Eq. (76)] is usually written in terms of the hydrodynamic modes. In many problems of chemical interest, nonhydrodynamic modes such as intramolecular vibration, play an important role [50]. Presence of such coupling creates an extra channel for dissipation. Thus, the memory kernel, T, gets renormalized and acquires an additional frequency-dependent term [16, 43]. 

In the previous section we find that the time correlation function can be obtained from the relaxation equation. When A is not a single variable but represents a column matrix containing a set of coupled variables, then the... 

The velocity autocorrelation function can be obtained from the relaxation equation [Eq. (76)], where Cv(z) = Cjt(q = 0z). Here the suffix s stands for single-particle property. For zero wavenumber, there is no contribution from the frequency matrix [that is, D v(q = 0) = 0] and the memory function matrix becomes diagonal. If we write (z) = Tfj (q = 0z), then the VACF in the frequency plane can be written as... 

With this consideration die relaxation equation will give rise to a set of coupled equations involving the time autocorrelation function of the density and the longitudinal current fluctuation, and also there will be cross terms that involve the correlation between the density fluctuation and the longitudinal current fluctuation. This set of coupled equations can be written in matrix notation, which becomes identical to that derived by Gotze from the Liouvillian resolvent matrix [3]. 

We note in passing that the Mittag-Leffler function is the solution of the fractional relaxation equation [84]... 

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