Mode-coupling equation

There are certain subtleties associated with the order to which the expansion of eqn (5.67) is carried and for a discussion of which we refer the perspicacious reader to chap. 25 of Ashcroft and Mermin (1976). If we maintain the use of normal coordinates, but now including the anharmonic terms which are represented via these coordinates, it is seen that the various modes are no longer independent. For the purposes of the present discussion, we see that these anharmonic terms have the effect of insuring that the equation of motion for the n mode is altered by coupling to the rest of the mode variables. This result bears formal resemblance to the Navier-Stokes equations which when written in terms of the Fourier variables yield a series of mode coupled equations. We have already noted that the physics of both thermal expansion and thermal conductivity demand the inclusion of these higher-order terms. 

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. 

In principle the memory kernel in the mode-coupling equation contains contributions from three particle correlations, however, for all systems studied so far in computer simulations, these only slightly modified the predictions of the theory and helped improve agreement between simulation and theory [31]. Also, there has been an extension of the theory taking chain connectivity into account [32] which improved agreement with the simulations of the bead-spring model, but it remains to be seen whether an application of this theory to the two models presented here can account for their strongly different dynamic behavior. 

There are many different solutions for X1 and X2 to this pair of coupled equations, but it proves possible to find two particularly simple ones called normal modes of vibration. These have the property that both particles execute simple harmonic motion at the same angular frequency. Not only that, every possible vibrational motion of the two particles can be described in terms of the normal modes, so they are obviously very important. 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

Figure 1. Schematic illustration of waveguiding structures, a ray-picture b modal intensity distribution obtained with the aid of Maxwell s equation c mode coupling c whispering gallery mode.

The coefficients of this mode-coupling functional are the basic control parameters of this idealized version of MCT. One sees that Eqs. [46] and [47] amount to a set of nonlinear equations for the correlators S(q,t) that must be solved self-consistently. 

Numerics has been used intensively in the field of integrated optics (10) since its early days, simply due to the fact that even the basic example of the slab waveguide requires the solution of a transcendental equation in order to calculate the propagation constants of the slab- guided modes. Of course, the focus was directed to analytical methods, primarily, as long as the power of a desktop computer did allow for a few coupled equations and special functions only e.g. to describe the nonlinear directional coupler in a coupled mode theory (CMT) picture. During the years, lots of analytic and semi-analytic approaches to solve the wave equation have been developed in order... 

Of more concern are the comments by De Schepper et al. [528] and Resibois and De Leener [490]. They have discussed whether such a fourth-order derivative can have meaning. A mode-coupling theory and a kinetic theory of hard spheres both indicate that the Burnett coefficient diverges at tin. There seems little or no reason for the continued use of the Burnett equation in discussing chemical reaction rates in solution. Other effects are clearly more important and far more reasonable from a theoretical point of view. 

These equations are solved by separating out the time dependence through the substitutions x(r, t) = x (r)eX1, y(r, t) = y (r)eKl, and diagonalizing the resulting pair of spatially dependent coupled equations. These two separated equations are Helmholtz-type equations whose solutions can be straightforwardly obtained in different coordinate systems.28,49 The complete space-time-dependent solutions are sums of spatial modes or patterns, each with a characteristic temporal behavior. For example, the complete solution on a circle can be written... 

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. 

In a similar way the contribution for all the different modes to the three transport coefficients can be calculated. Equations (58) and (61) are the classic mode coupling theory expressions that provide general expressions for the shear viscosity and thermal conductivity, respectively. Using these general expressions and the ideas of static scaling laws, Kadanoff and Swift have calculated the transport coefficients near the critical point. 

Note that the above study is performed for a simple system. There exists a large body of literature on the study of diffusion in complex quasi-two-dimensional systems—for example, a collodial suspension. In these systems the diffusion can have a finite value even at long time. Schofield, Marcus, and Rice [17] have recently carried out a mode coupling theory analysis of a quasi-two-dimensional colloids. In this work, equations for the dynamics of the memory functions were derived and solved self-consistently. An important aspect of this work is a detailed calculation of wavenumber- and frequency-dependent viscosity. It was found that the functional form of the dynamics of the suspension is determined principally by the binary collisions, although the mode coupling part has significant effect on the longtime diffusion. 

Mode coupling theory of binary mixtures where the constituents are of rather different sizes is a challenging task, as we have already discussed while addressing the mass depenence of diffusion. In addition to the problem with proper formulation of mode coupling terms, there is an additional difficulty of the nonavailability of the equilibrium two-particle correlation functions The existing integral equation theories become unstable when the size ratio exceeds a certain (low) value, like 1.5 or so [195],... 

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