Mode coupling equations approximations

In the remainder of this article, we shall discuss the proper mode-mode coupling,theory. This discussion naturally falls into two parts. The twofold nature of the theory has already been seen in the previous pages. First, it was necessary to make approximations to obtain Eq. (40), the mode-mode coupling equation. Second, it was necessary to make approximations to solve Eq. 

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. 

Although a theoretical approach has been desecrated as to how one can apply the generalized coupled master equations to deal with ultrafast radiationless transitions taking place in molecular systems, there are several problems and limitations to the approach. For example, the number of the vibrational modes is limited to less than six for numerical calculations. This is simply just because of the limitation of the computational resources. If the efficient parallelization can be realized to the generalized coupled master equations, the limitation of the number of the modes can be relaxed. In the present approach, the Markov approximation to the interaction between the molecule and the heat bath mode has been employed. If the time scale of the ultrashort measurements becomes close to the characteristic time of the correlation time of the heat bath mode, the Markov approximation cannot be applicable. In this case, the so-called non-Markov treatment should be used. This, in turn, leads to a more computationally demanding task. Thus, it is desirable to develop a new theoretical approach that allows a more efficient algorithm for the computation of the non-Markov kernels. Another problem is related to the modeling of the interaction between the molecule and the heat bath mode. In our model, the heat bath mode is treated as... 

Unlike ISS, the electro-optic effect (or its inverse) can occur only in noncentrosymmetric media and in general does not lead to any real material excitation. However, if there are low-frequency IR-active modes in the crystal, they may be excited impulsively [36, 59]. Such phonons couple strongly to IR radiation to form mixed modes called polaritons. Impulsive stimulated polariton scattering can be described approximately by coupled equations of motion for the polarization contributions P, and due to ionic motions (i.e., phonons) and electronic motions, respectively [9, 60] ... 

The memory equation contains fluctuating stresses and, like (z,y) in (lid), is calculated in mode coupling approximation using (10a) giving ... 

In the case of a nonreacting fluid, where one is usually interested in macroscopic equations for conserved (in the limit k O) variables, the origin and region of validity of this approximation is clear. In the small k limit the conserved fields do decay much more slowly than other variables in the system, and the limit z- 0, k O has the effect of extracting the decay on this slow time scale. (Mode coupling contributions spoil some of these arguments, but it is now known how to account for these effects. We discuss this aspect of the problem in Section VII.)... 

In this chapter we have described a theory for dynamics of polyatomic fluids based on the memory-function formalism and on the interaction-site representation of molecular liquids. Approximation schemes for memory functions appearing in the generalized Langevin equation have been developed by assuming an exponential form for memory functions and by employing the mode-coupling approach. Numerical results were presented for longitudinal current spectra of a model diatomic liquid and water, and it has been discussed how the results can be interpreted in... 

In contrast to the corresponding coupled equations bijrij = 0 in mass-weighted coordinates, Eq. 6.49 shows that each normal coordinate Q,- oscillates independently with motion which is uncoupled to that in other normal coordinates Qj. This separation of motion into noninteracting normal coordinates is possible only if V contains no cubic or higher-order terms in Eq. 6.4. Anharmonicity will inevitably couple motion between different vibrational modes, and then the concept of normal modes will break down. In the normal mode approximation, no vibrational energy redistribution can take place in an isoFated molecule. 

Tunnel Splitting in the Symmetric Mode Coupling Potential Equation (4.15). The Second (Third) Column Is the Result by Exact Quantum Mechanical (EQM) Calculation [by the Sudden Approximation Equation (4.12)]. Parameters (u)y, a, g) Are Equal to (0.2,0.25,0.04)... 

Finally, we touch upon the adiabatic and sudden approximations in the present model. In the same way as in the case of symmetric mode coupling model, the adiabatic approximation leads to the tunneling splitting independent of Hy. This does not exhibit any characteristic behavior discussed above and can never be reliable. If we apply the sudden approximation, we also encounter a problem since the potential curve in jc direction is not symmetric except when y is zero. Thus we cannot use Equation (4.12) directly anymore. 

To solve Eq. (29-2) we need to know . An expression could be obtained by expanding over the complete set of Pj for the bound and radiation modes of the first fiber, and this would, of course, lead to the infinite set of coupled mode equations derived in Section 33-11. The disadvantage of this description is that each mode of the first fiber by itself is a poor approximation to the field within the second fiber. Consequently large numbers of modes are required for accuracy, and. the set of coupled equations is then intractable. 

In Chapter 19 we introduced local modes to describe the fields of waveguides with large nonuniformities that vary slowly along their length. As an individual local mode is only an approximation to the exact fields, it couples power with other local modes as it propagates. Our purpose here is to derive the set of coupled equations which determines the ampUtude of each mode [10]. First, however, we require the relationships satisfied by the fields of such waveguides. 

In more recent work, Cheng et al. (2002) reported measurements of the low-shear viscosity for dispersions of colloidal hard spheres up to (f) = 0.56. Nonequilibrium theories based on solutions to the two-particle Smoluchowski equation or ideal mode coupling approximations did not capture observed viscosity divergence (Cheng et al. 2002), although the Doolittle and Adam-Gibbs equations still appeared to hold. 

The mode coupling approximation for m (0 yields a set of equations that needs to be solved self-consistently. Hereby the only input to the theory is the static equilibrium structure factor 5, that enters the memory kernel directly and via the direct correlation function that is given by the Ornstein-Zernicke expression = (1 - l/5,)/p, with p being the average density. In MCT, the dynamics of a fluid close to the glass transition is therefore completely determined by equilibrium quantities plus one time scale, here given by the short-time diffusion coefficient. The theory can thus make rather strong predictions as the only input, namely, the equilibrium structure factor, can often be calculated from the particle interactions, or even more directly can be taken from the simulations of the system whose dynamics is studied. 

Finally we shall derive the equation used by Bixon and Jortner. Suppose that an intramolecular vibrational mode, say Qi, plays a very important role in electron transfer. To this mode, we can apply the strong-coupling approximation (or the short-time approximation). From Eq. (3.40), we have... 

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