Frequency-Dependent Phenomenological Coefficients

If exp(— lost) time dependence is assumed for all fields, and the constitutive relations (2.7)-(2.9) are substituted into (2.1)-(2.4), we obtain 

If e = = 0, the electric field is divergence free this is the general condition for a transverse field. Except possibly at frequencies where e = 0, therefore, the medium cannot support longitudinal fields. 

Equations (2.12)—(2.16) will usually be our point of departure in scattering problems. However, to avoid a cluttered notation, we shall often omit the subscript c from the complex fields. In those instances where confusion might result, the subscript will be retained, although it should usually be clear from the context if we are dealing with real fields or their complex representations. 

Consider a real function of time F(t). The Fourier transform (to) of F(t) is defined as 

According to (2.18), an arbitrary time-dependent function can be expressed as a superposition of time-harmonic functions exp( — o/), where the complex amplitude %(u ) depends on the frequency to. The condition that F(t) be real is that 3r (w) = 3r( — w) therefore, F(t) can be expressed as a superposition of time-harmonic functions with positive frequency  

---

In Ref. 76, self-consistent expressions were derived for the frequency-dependent electrolyte friction and the conductivity. Unlike our approach, the effect of the surrounding solvent (water) was described using the phenomenological coefficients. However, no oscillatory component of ct co) or cr"(co) were discovered in Ref. 157, while our Figs. 48 and 49 show typical damped oscillations. A reservation should be made that no damped oscillations are seen in the case of a solution (see Fig. 50), if we employ the additivity approximation Eq. (387). 

Linear nonequilibrium thermodynamics has some fundamental limitations (i) it does not incorporate mechanisms into its formulation, nor does it provide values for the phenomenological coefficients, and (ii) it is based on the local equilibrium hypothesis, and therefore it is confined to systems in the vicinity of equilibrium. Also, properties not needed or defined in equilibrium may influence the thermodynamic relations in nonequilibrium situations. For example, the density may depend on the shearing rate in addition to temperature and pressure. The local equilibrium hypothesis holds only for linear phenomenological relations, low frequencies, and long wavelengths, which makes the application of the linear nonequilibrium thermodynamics theory limited for chemical reactions. In the following sections, some of the attempts that have been made to overcome these limitations are summarized. 

Another important result deals with the temperature dependence of the correlation times of the elementary motions, which agrees fairly well with the prediction of the William, Landel, Ferry equation, using the phenomenological coefficients obtained from low frequency viscoelastic measurements. Tlf s means that the elementary motions which are observed by FAD and... 

The only difference compared to the Taylor series of Eq. [4] is that the numerical factors have been moved into the coefficients of the electric field. Properties based on this convention (with a tilde) are related to those of Eq. [4] (without a tilde) by p = 2 ft and y = 3 y. A third common phenomenological convention is based on expanding a frequency-dependent field, EqCOs(io), as in Eq. [3] and absorbing all constants into the coefficient of E at each order. This convention is unsatisfactory because the different frequency-dependent quantities in each order do not converge to the same limit as cu - 0. A review of these and other conventions has been given by Willetts et al. ... 

---