Relaxation equations derivation

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

In the first case, the following equation, derived from (5), is important for a good adjustment of the individual relaxations (see (.5)). 

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 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... 

In this case, the K s are the intersystem crossing rate constants. A detailed description of the different experimental techniques used at the present time, together with equations derived for molecules whose emission results from two zf levels, is given in a recent review of the subject (87). In this section we will give only a brief description of these PMDR methods. In all but the pulsed-excitation method, these assumptions are made (a) the spin-lattice relaxation is absent and (b) the microwave radiation either saturates or inverts the population of... 

In the various formulations of the mathematical theory of linear viscoelasticity, one should differentiate clearly the measurable and non-measurable fimctions, especially when it comes to modelling apart from the material functions quoted above, one may also define non measurable viscoelastic functions which Eu-e pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and tiie memory function. These mathematical tools may prove to be useful in some situations for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the difierential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 

Under certain assumptions in equation (A3), the time dependence in the lattice interaction tensors is sufficient to describe the relaxation and derive expressions for the relaxation times. This is the basis of Redfield theory, in which first the Master equation is expanded... 

The adsorption behavior of intermediates is usually related to the difference ofTafel (dV/d logi) and potential relaxation ( — dV/d log t) slopes. In the simple case of potential relaxation of a process that does not involve appreciable coverage by intermediates, namely, 9 < 0.02, say, as for the HER at Hg, the kinetics of potential relaxation are derived from the following differential equation ... 

The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208-215). Apparently, Blumen et al. [189] were the first to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200-203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like profile of dielectric permittivity, etc.), and the fractal dimension of a medium. 

The parallel relaxation time may be found by setting = 0, whence using the equation derived in Section III, Eq. (3.84),... 

In this chapter results of the picosecond laser photolysis and transient spectral studies on the photoinduced electron transfer between tryptophan or tyrosine and flavins and the relaxation of the produced ion pair state in some flavoproteins are discussed. Moreover, the dynamics of quenching of tryptophan fluorescence in proteins is discussed on the basis of the equations derived by the present authors talcing into account the internal rotation of excited tryptophan which is undergoing the charge transfer interaction with a nearby quencher or energy transfer to an acceptor in proteins. The results of such studies could also help to understand primary processes of the biological photosynthetic reactions and photoreceptors, since both the photoinduced electron transfer and energy transfer phenomena between chromophores of proteins play essential roles in these systems. 

The procedure above has not in any sense derived the macroscopic relaxation equations only some formal conditions have been stated under which the structures of the microscopic and macroscopic equations become the same. One crucial point, which certainly deserves further comment, is the physical basis of the Markov approximation. This approximation removes the memory effects from (5.5) so that the structures of the microscopic and macroscopic equations become similar. For this approximation to be useful, the memory kernel must decay much more rapidly than the density fields. The projected time evolution will guarantee that this is the case, provided these fields decay much more slowly than other variables in the system. 

The equations derived for creep are even more complicated to handle than those of relaxation since they involve the solution of a system of linked, iterating differential equations. However, for a given (and small) number of saltating chains, a solution can be found, although it is still extremely unwieldy, and it appears at this juncture that no practical benefit will result from attempting to obtain approximate solutions. 

In Section 10.2 we saw that the macroscopic relaxation equations can be used to determine correlation functions. In this section we summarize the traditional methods for deducing the macroscopic relaxation equations of fluid mechanics. In subsequent sections these equations are used to determine the Rayleigh-Brillouin spectrum. The first step in the derivation of the relaxation equation involves a discussion of conservation laws. 

In previous chapters phenomenological relaxation equations were used together with the Onsager regression hypothesis to compute time correlation functions. In this section we present a microscopic derivation of generalized relaxation equations (Zwan-zig, 1961 Berne, Mori, 1965 and 1971). These equations can be used to compute time-correlation functions under circumstances where the usual phenomenological equations do not apply. 

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