Mode coupling equations general form

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] ... 

Other structural effect, such as a soft optic mode, which provides the driving mechanism. Lattice distortions then occur by coupling of the spontaneous strain to the order parameter. Under these conditions, the general form of Equation (4) may be given as... 

Li et al. subsequently developed a schematic mode coupling theory description of the short-time as well as long-time dynamics of mesogens in the isotropic phase near the I-N transition [91]. Their treatment started with a very general form [92] for the kinetic equation of the autocorrelation function (j)2(t) of the anisotropy of polarizability... 

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... 

The mode-coupling theory (MCT) " approaches glass-forming systems by a liquid of hard spheres. Its molecular dynamics is described as the solution of a generalized nonlinear oscillator equation ... 

Equation 31b obviously is a system of -coupled equations, since the row vector C = C4>, (1 X ), has in general nonzero elements that introduce in Eq. 31b apart from also terms cij (f) withi 7 i. Thus, contrary to the principle and target of modal analysis, it is not possible to deal with -independent generalized SDOE modal systems. It actually seems that there is no benefit in transforming the initial coupled equation of motion to its modal form using the undamped modes. There are a number of approaches to tackle this problem ... 

Our findings lead us in a number of useful directions. One of these directions is a generalization of our basic instantaneous approach to dynamics. Our original linear INM formalism assumed the potential energy was instantaneously harmonic, but that the coupling was instantaneously linear [Equation (15)]. We still need to retain the harmonic character of the potential to justify the existence of independent normal modes (at least inside the band), but we are free to represent the coupling by any instantaneously nonlinear function we wish. A rather accurate choice for vibrational relaxation, for example, is the instantaneous exponential form ... 

This paper describes some new developments and applications based on the reaction path model. In section II the original form of the reaction path Hamiltonian [of. equation (1) below] is transformed to a new representation that has a more desirable structure for some applications. Section III shows how the reaction path model makes application of the unified statistical model for chemical reactions especially simple, and a generalized version of the unified statistical model is also developed there. Finally, in section IV the fact that the reaction path model consists of one special degree of freedom — i.e., the reaction coordinate — coupled to a number of harmonic oscillators is exploited to derive a generalized Langevin equation (GLE) for motion along the reaction path. This is a reduced equation of motion for only the reaction coordinate, but it experiences friction" and a "random force" because of coupling to the transverse vibrational modes. 

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