Relaxation equations mode coupling theory

A. Comments on the Relaxation Equation Vm. Structure of Mode Coupling Theory as Applied to Liquid-State Dynamics... 

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. 

It appears, however, that the mode-coupling theory is not able to explain some of the most significant slow-relaxation processes of these more complex glass formers. In particular, it cannot explain the success of the Vogel-Fulcher-Tammann-Hesse (VFTH) equation for the temperature-dependence of the relaxation time near the glass transition. The mode-coupling theory predicts instead a power-law dependence of the longest relaxation... 

One should keep in mind that the rate constant increases monotonically with coupling only if the relaxation time of the fast mode is infinitely small. If it is finite, as it should be, the rate versus couphng dependence starts as the Golden Rule prescribes, then continues according to the Sumi-Marcus theory, but finally saturates at a certain value determined by the relaxation time of the fast mode. This effect is illustrated in Figure 9.18(h) for the case where both fast and slow modes relax exponentially. In this case, the problem is reduced to solution of a system of 2D diffusion equations along parabolic potential surfaces corresponding to two coordinates and... 

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