Mode coupling theory variables

Even with all of its sophistication, the mode coupling theory is still a perturbation theory where dynamics is described in terms of a subset of dynamical variables chosen from the products of hydrodynamic modes. It fails, for example, to describe rare events, such as the activated processes or stringlike cooperative motions often found to dictate dynamics in glassy liquids [13-15]. 

When the solvent molecules are explicitly included, one needs to treat a ternary system (two ions and the dipolar solvent molecules). The additional slow variables to be included in the mode coupling theory are the products of the ion charge and solvent densities. This will explicitly introduce terms like Fis(k,t), which is the partial dynamic structure factor involving the ion and the solvent molecules. The calculation of the microscocpic terms of the friction, containing the density terms, does not appears to be difficult, but calculation of the current terms now appears to be formidable. 

A simple theory of the concentration dependence of viscosity has recently been developed by using the mode coupling theory expression of viscosity [197]. The slow variables chosen are the center of mass density and the charge density. The final expressions have essentially the same form as discussed in Section X the structure factors now involve the intermolecular correlations among the polyelectrolyte rods. Numerical calculation shows that the theory can explain the plateau in the concentration dependence of the viscosity, if one takes into account the anisotropy in the motion of the rod-like polymers. The problem, however, is far from complete. We are also not aware of any study of the frequency-dependent properties. Work on this problem is under progress [198]. 

Another approach towards a thermodynamics of steady-state systems is presented by Santamaria-Holek et al.193 In this formulation a local thermodynamic equilibrium is assumed to exist. The probability density and associated conjugate chemical potential are interpreted as mesoscopic thermodynamic variables from which the Fokker-Planck equation is derived. Nonequilibrium equations of state are derived for a gas of shearing Brownian particles in both dilute and dense states. It is found that for low shear rates the first normal stress difference is quadratic in strain rate and the viscosity is given as a simple power law in the strain rate, in contrast to standard mode-coupling theory predictions (see Section 6.3). 

As discussed in Chapter 6, molecular modeling and neutron scattering experiments greatly improved understanding of the liquid structures and their variability with T. The mode-coupling theory (MCT) originates from analysis of the flow behavior near a critical point involving nonlinear correlation functions [Kawasaki, 1966]. 

It is well established that the principal results of the generalized kinetic theory, especially the functional form of the slow portion of the memory function, can be obtained also by a direct mode-coupling approach [18, 19, 20]. The basic idea behind the mode-coupling theory is that the fluctuation of a given dynamical variable decays, at intermediate and long times, predominantly into pairs of hydrodynamic modes associated with quasi-conserved dynamical variables. The possible decay channels of a fluctuation are determined by selection rules based, for example, on time-reversal symmetry or on physical considerations. 

A simple explanation of the breakdown of the old ideas about the linear laws, according to the Kawasaki-Kadanoff and Swift mode-mode coupling theory goes as follows. Consider some conserved variable A then (/k(r)A k) = 0. Even if A k is the only slowly varying variable for small k, we may form products Ak+k A k, Ak+k +k A k A k , etc., characterized by wave vector k (the wave vector for a product is additive), which will be slowly varying if k k etc., are also small. There is no reason to expect that, e.g., [Pg.267]

Alternatively the hydrodynamic equations of motion can determine directly the time evolution of the correlation functions instead of the mesoscopic variables, according to the Mori-Zwanzig model [42,52]. These equations are the starting point of the Mode-Coupling Theories (MCT) [60-62]. 

For radical-radical reactions, the full mode coupling and anharmonicity effects for the relative and overall rotational motions must be explicitly accounted for. We have derived a direct variable reaction coordinate transition state theory approach that appears to 3deld accurate rate coefficients for a number of alkyl radical reactions.This approach is analogous to that embodied in Eq. (4.10) for the long-range transition state, but includes variational optimizations of the form of the reaction coordinate and does not make the large orbital moment of inertia assumption. A detailed description of this approach was provided in some of our recent articles. 

Freed and Jortner226 have reworked the formal theory of radiationless transitions described in this paper. They carefully account for the difference between distinguishable and indistinguishable levels, and allow for variable coupling of the sparse system to the dense system of states. Of course, only certain vibrational modes in the dense manifold have the appropriate symmetries to couple to the sparse manifold and thereby contribute to the radiationless transition. Freed and Jortner take this into account in the fashion in which the zero-order manifolds of the molecule are classified. 

Trimerized organic conductors are of special interest, because two electrons per three sites constitute the simplest situation, where both electronic transitions resulting in single- and double-site occupation take place [21]. As one considers larger n-mers, two complications arise. First, the number of equations that should be solved sharply increases. The second complication is the increase in the number of n-meric normal modes, which are coupled to an external electromagnetic field. Recently, Yartsev et al. [22] have proposed using the linear response theory for several variables to describe the optical properties of trimers with arbitrary equilibrium charge density distribution. This approach can be extended to any cluster—the size is limited only by computer facilities. 

Let X = q,p) denote the one-degree-of-freedom reaction coordinate. For M-degrees-of-freedom vibrational modes, 7 e R" and 0 G T" denote their action and angle variables, respectively, where T = [0,27t]. These action and angle variables would be obtained by the Lie transformation, as we have discussed in Section IV. In reaction dynamics, the variables (/, 0) describe the degrees of freedom of the intramolecular and possibly the intermolecular vibrational modes that couple with the reaction coordinate. In the conventional reaction rate theory, these vibrational modes are supposed to play the role of a heat bath for the reaction coordinate x. 

Let us assume that a variable A(t) is coupled to the reaction coordinate and that (A) is its mean value. If a measurement of some property P depends on (A), but not on the particular details of the time dependence of A(t), then we will call it a statistical dependence. If the property P depends on particular details of the dynamics of A(t) we will call it a dynamical dependence. Note that in this definition it is not the mode A(t) alone that causes dynamical effects, but it also depends on the timescale of the measured property P. Promoting vibrations (to be discussed in Sections 2-4) are a dynamic effect in this sense, since their dynamics is coupled to the reaction coordinate and have similar timescales. Conformation fluctuations that enhance tunneling (to be discussed in Section 5) are a statistical effect the reaction rate is the sum of transition state theory (TST) rates for barriers corresponding to some configuration, weighted by the probability that the system reaches that configuration. This distinction between dynamic and statistical phenomena in proteins was first made in the classic paper of Agmon and Hopfield.4 We will discuss three kinds of motions ... 

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