Deriving Relaxation Equations

Until now we have discussed only elementary methods for determining correlation functions, based on ad hoc models. In this chapter a powerful formalism for computing time-correlation functions is presented. As a by-product of this formalism several useful theorems emerge which result from symmetry considerations. Moreover some of the assumptions made in Chapter 10 are shown to be valid. Throughout this chapter we treat classical systems. The methods developed here can also be applied to quantum systems. This is shown in Appendix 11.A. The formalism of this chapter is applied in Chapter 12 to the calculation of the depolarized spectrum. 

---

Recent Trends in Theory of Physical Phenomena in High Magnetic Fields R 89 W.Apel and Yu.A. Bychkov, On Deriving Relaxation Equations for Nuclear Spins , p. 223... 

In the derivation of equations 24—26 (60) it is assumed that the cylinder is made of a material which is isotropic and initially stress-free, the temperature does not vary along the length of the cylinder, and that the effect of temperature on the coefficient of thermal expansion and Young s modulus maybe neglected. Furthermore, it is assumed that the temperatures everywhere in the cylinder are low enough for there to be no relaxation of the stresses as a result of creep. 

Not all oxidants formed biolc cally have the potential to promote lipid peroxidation. The free radicals superoxide and nitric oxide [or endothelium-derived relaxing aor (EDRF)] are known to be formed in ww but are not able to initiate the peroxidation of lipids (Moncada et tU., 1991). The protonated form of the superoxide radical, the hydroperoxy radical, is capable of initiating lipid peroxidation but its low pili of 4.5 effectively precludes a major contribution under most physiological conditions, although this has been suggested (Aikens and Dix, 1991). Interestingly, the reaction product between nitric oxide and superoxide forms the powerful oxidant peroxynitrite (Equation 2.6) at a rate that is essentially difiiision controlled (Beckman eta/., 1990 Huie and Padmaja, 1993). 

The Fukui functions generalize the concept of frontier orbitals by including the relaxation of the orbital upon the net addition or removal of one electron. Because the number of electrons of an isolated system can only change by discrete integer number, the derivative in Equation 24.37 is not properly defined. Only the finite difference approximation of Equation 24.37 allows to define these Fukui functions (noted here by capital letters) F1 (r)... 

The relaxing Gaussian network of Green and Tobolsky (4) is the earliest version of this model. Lodge (12) and Yamamoto (J5) independently derived constitutive equations for similar systems, based on a stress-free state for each newly created strand and a distribution of junction lifetimes which is independent of flow history. For Gaussian strands in an incompressible system ... 

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 ... 

To derive the dynamic structure factor, let us once again write down the generalized relaxation equation in terms of its components and in the frequency plane,... 

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]. 

The relaxation equation derived so far for electrons and nuclei share a common assumption usually called the perturbation regime or Redfield limit [54]. The... 

Fig. 4.3. Relaxation time of absorption of HTO vapour by raindrops. Experimental results of Booker, 1965 (O) and Friedman etal., 1962 ( ). Line is theoretical, derived from equations (4.7) and (4.11).

The dynamics of the macromolecule in the form of a set of differential equations of the first order is convenient for derivation of relaxation equations (Volkov and Vinogradov 1984, 1985 Volkov 1990). As a starting point, we use equations (7.6) and consider m = 0 in this system, so that the second equation allows us to define... 

One has no results for this case derived consequently from the basic equations (7.6) with local anisotropy. Instead, to find conformational relaxation equation, we shall use the Doi-Edwards model, which approximate the large-scale conformational changes of the macromolecule due to reptation. The mechanism of relaxation in the Doi-Edwards model was studied thoroughly (Doi and Edwards 1986 Ottinger and Beris 1999), which allows us to write down the simplest equation for the conformational relaxation for the strongly entangled systems... 

Let us remind that equation (9.24), describing the relaxation of macro-molecular conformation, can be considered only as an assumed results of accurate derivation of the relaxation equation from the macromolecular dynamics. 

To calculate characteristics of linear viscoelasticity, one can consider linear approximation of constitutive relations derived in the previous section. The expression (9.19) for stress tensor has linear form in internal variables x"k and u"k, so that one has to separate linear terms in relaxation equations for the internal variables. This has to be considered separately for weakly and strongly entangled system. 

Thus, one can see that the single-mode approximation allows us to describe linear viscoelastic behaviour, while the characteristic quantities are the same quantities that were derived in Chapter 6. To consider non-linear effects, one must refer to equations (9.52) and (9.53) and retain the dependence of the relaxation equations on the anisotropy tensor. 

In general, it is more convenient to determine the moments from equations which can be derived directly from the diffusion equation (F.18). For example, on multiplying equation (F.18) by pipk and integrating with respect to all the variables, we find the relaxation equation... 

---