Mode coupling equations solution

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 ... 

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... 

When the density of bath modes becomes high, as in the case of a DC, counterparts of the discrete-mode expressions (Equations (3.73)-(3.80)) are readily available, based on the assumption that the solute-solvent coupling can be expressed as a linear functional of solute charge densities (p) [12]. Models for defining or calculating p are discussed in later sections. 

Schweizer and collaborators have elaborated an extensive mode-coupling model of polymer dynamics [52-54]. The model does not make obvious assumptions about the nature of polymer motion or the presence or absence of particular long-lived dynamic structures, e.g., tubes it yields a set of generalized Langevin equations and associated memory functions. Somewhat realistic assumptions are made for the equilibrium structure of the solutions. Extensive calculations were made of the molecular weight dependences for probe diffusion in melts, often leading by calculation rather than assumption to power-law behaviors for various transport coefficients. However, as presented in the papers noted here, the model is applicable to melts rather than solutions Momentum variables have been completely suppressed, so there are no hydrodynamic interactions. Readers should recall that hydrodynamic interactions usually refer to interactions that are solvent-mediated. 

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 ... 

As a first step it is noted that a finite cutoff qo should be retained in the evaluation of both mode-coupling integrals given by equations (6.28) and (6.29), since away from the critical point will no longer be negligible compared to the finite correlation length t. Here only a solution for the transport properties in the macroscopic hydrodynamic limit A 0 is needed. From the information presented in the previous section it... 

FIGURE 3.2 Schematic picture of tunneling in the case of symmetric mode coupling [see Equation (3.24)]. SP is the saddle point and S is the symmetry line. Dashed lines are solutions of the Hamilton-Jacobi equations but not the optimal tunneling path, (a) Case of modest and... 

All the terms in this equation are analytical functions of the parameter z except for A(z), which is the numerical solution of Equation (6.181). Equation (7.27) is a system of coupled nonlinear equations with removable singularity at z = -1 that may cause instability of the numerical solution. Note, however, that the nonlinearity of this equation is totally due to the factor 9 z) [see Equation (6.193)], which is related to the normalization condition for the vector U. Thus, for a given mode y the solution can be written in the form... 

When only a small fraction of the total power of the perturbed fiber is transferred between modes, the coupled mode equations can be solved iteratively. The solution of the first iteration is then identical to the induced-current solution of Chapter 22. To demonstrate this equivalence, we assume that only the /th forward-propagating mode of the unperturbed fiber is excited at z = 0 of the perturbed fiber. To lowest order we ignore coupling to all other modes, whence the solution of Eq. (27-3a) is... 

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. 

We see from both equations 8.32 and 8.33 that the most unstable mode is the mode and that ai t) = 1 - 1/a is stable for 1 < a < 3 and ai t) = 0 is stable for 0 < a < 1. In other words, the diffusive coupling does not introduce any instability into the homogeneous system. The only instabilities present are those already present in the uncoupled local dynamics. A similar conclusion would be reached if we were to carry out the same analysis for period p solutions. The conclusion is that if the uncoupled sites are stable, so are the homogeneous states of the CML. Now what about inhomogeneous states ... 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

The matrix element Gjk (that can be taken as real) measures the coupling strength between the solute and the solvent modes. To get other formulations found in the literature, it is useful to introduce a real coupling parameter Aj for each solute mode via the relationship Gjk = -Aj Wjk With this convention one gets for the solute equations of motion ... 

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