Local modes coupling coefficients

The harmonic contribution to the thermal conductivity is Ki,(r), which we calculated in the previous section. The main purpose of Kjni,(r) is to allow for localized modes, which in a protein such as myoglobin correspond to vibrational modes above 150 cm", to contribute to thermal flow via anharmonic coupling to other vibrational modes. The contribution to the diffusion coefficient due to anharmonic coupling can be thought of as == where l/x is the transfer rate between modes... 

If we substitute the coupling coefficient of Eq. (28-4) into Eq. (28-8a) and assume bi(0) = 1, the resulting expression for b j is identical with that of Eq. (22-35). The latter was derived using induced currents to represent the slight mismatch between the local-mode fields and the exact fields of the waveguide. Thus we deduce the equivalence of the first iteration of the coupled local-mode equation and the induced-current representation. 

In general, the local-mode propagation constants are contained implicitly within the local eigenvalue equation, and their precise values must be obtained numerically. However, analytical expressions can be derived for the coupling coefficients, as we show in the example below. 

The coupling coefficients of Eq. (3 l-65c) are expressible in a more compact form, as we show here. Combining the field components of Eq. (31-58) and substituting the fields for the yth forward-propagating local mode of Eq. (19-2), we have... 

We derived the set of coupled local-mode equations for arbitrary waveguides in Section 31-14. In the weak-guidance approximation, the modal fields in the coupling coefficients of Eq. (31-65c) have only transverse components. If we use Table 13-1, page 288, to relate these components to the corresponding normalized solutions of the scalar wave equation of Eq. (33-45), we find with the help of Eq. (33-48b) that... 

When an atomic system is cooled below its glass temperature, it vitrifies, that is, it forms an amorphous solid [1]. Upon decreasing the temperature, the viscosity of the fluid increases dramatically, as well as the time scale for structural relaxation, until the solid forms concomitantly, the diffusion coefficient vanishes. This process is observed in atomic or molecular systems and is widely used in material processing. Several theories have been developed to rationalize this behavior, in particular, the mode coupling theory (MCT) that describes the fluid-to-glass transition kinetically, as the arrest of the local dynamics of particles. This becomes manifest in (metastable) nondecaying amplitudes in the correlation functions of density fluctuations, which are due to a feedback mechanism that has been called cage effect [2],... 

In Sects. 5.5 and 5.6, we have introduced two 2-DOF models for the lead screw drives. In this section, the equations of motion of these models are transformed into matrix form and linearized with respect to their respective steady-sliding equilibrium point. These equations are then used in the next sections to study the local stability of the equilibrium point and the role of mode coupling instability mechanism. In this chapter, for simplicity, the coefficient of friction, n, is taken as a constant. 

Coupling to these low-frequency modes (at n < 1) results in localization of the particle in one of the wells (symmetry breaking) at T = 0. This case, requiring special care, is of little importance for chemical systems. In the superohmic case at T = 0 the system reveals weakly damped coherent oscillations characterised by the damping coefficient tls (2-42) but with Aq replaced by A ft-If 1 < n < 2, then there is a cross-over from oscillations to exponential decay, in accordance with our weak-coupling predictions. In the subohmic case the system is completely localized in one of the wells at T = 0 and it exhibits exponential relaxation with the rate In k oc - hcoJksTY ". 

If the contribution of the localized eigenfunctions (scars or bouncing ball modes) in (8) is negligible, then all eigenfunctions ipn(x,y) are RGF. The complex coefficients in the superposition, cn, depend on the energy and the coupling between the billiard and leads and are not random as in the Berry function (2). Nonetheless the superposition of RGFs is also a complex RGF (W. Feller, 1971 M.I. Tribelsky, 2002)... 

---