Overdamped regime

Another problem of interest concerns the effect of external radiation fields. In the overdamped regime this will be shown to be reminiscent of the effect of the nonreactive modes. These problems will be the major topics of the present section. 

The model that we are going to consider in this section is given by two spherical rotators, simply called body 1 and body 2. Body 1 is the solute molecule, whereas body 2 is the instantaneous structure of solvent molecules in the immediate surroundings of the solute. The rest of the solvent is described as a homogeneous, isotropic and continuous viscous fluid. In the overdamped regime, the system is described by a Smoluchowski equation in the phase space where ft, and ftj... 

Berezhkovskii A M and Zitserman V Yu 1993 Multi-dimensional Kramers theory of the reaction rate with highly anisotropio friotion. Energy diffusion for the fast coordinate versus overdamped regime for the slow coordinate Chem. Phys. Lett. 212 413-19... 

This theory highlights the importance of the time dependence of the friction and demonstrates how it may be taken into account at the leading order. In the overdamped regime this is done so through the insightful and compact Grote-Hynes (subscript GH) formula for the transmission coefficient [4]. 

Dynamics (31) — (32) are commonly referred to as Rouse dynamics. It can be directly seen that the results for large-scale or low-frequency dynamics of a macromolecule do not depend on the number of particles N. In the overdamped regime m 0), the mean sizes of the macromolecule relax to equilibrium values, whereas relaxation times are the Rouse relaxation times... 

The positive dielectric function in the far-IR indicates that PPy-PF is a dirty metal with WpT 1 the overdamped plasma oscillation prevents the zero-crossing even at Wp. In the critical (sample D) and insulating regimes (sample F), the overdamping of the plasma oscillation is even more clearly evident. 

KkroCT") and Eq. (3.53) for Kst(T ) provide k T) formulas that apply in exactly opposite limiting regimes namely, the regimes of extreme overdamping for Kkrd(T ) and extreme underdamping for Kfs(T ). 

As depicted in Figure 3.7, Eq. (3.58) describes the motion of a harmonic oscillator in a regime of extreme underdamping exactly opposite to the regime of extreme overdamping discussed in Appendix Section F.ll and described by Eq. (A.47b). 

Kramers approach to rate theory in the underdamped and spatial-diffusion-limited regimes spurred extensions which were applicable to the much more complex STGLE. Grote and Hynes (23) used a parabolic barrier approximation to derive the rate expression for the GLE in the spatial diffusion limit. Carmeli and Nitzan derived expressions for the rate of the GLE (24) and the STGLE (25) in the underdamped limit. The overdamped limit for the rate in the presence of delta correlated friction was solved using the mean first passage time expression (26,27). A turnover theory, valid for space- and time-dependent friction, has only been recently presented by Haynes, Voth, and Poliak... 

The overdamped limit gives the golden rule rate, with k proportional to the square of the coupling, J, and inversely proportional to the damping strength, /. Note that the population relaxation rate in the eigenstate representation is simply f M in both regimes. Thus,... 

It predicts, as in the N= 1 case, a coherent exchange of energy between atoms and field at the frequency fiov and one has again to distinguish two regimes depending upon the magnitudes of and u/Q. The overdamped re-... 

The compaction behavior is also indicated by the effective volume fraction of an aggregate (pejf (Fig- 5), calculated from the aspect ratio of an equivalent ellipsoid. The shear-rale dependence on effective volume fraction was small so that a maximum compaction was not observed even at high shear rates. Again, higher shear rates required for higher compaction may violate model assumptions, such as Stokes regime for the hydrodynamics and the conditions for the overdamped motion. 

FIGURE 28.1 The PDF model accurately predicts clinically recorded early rapid- filling transmitral flow velocity contours. Both underdamped and overdamped kinematic regimes are observed clinically. See text for details. In addition to ECM and titin, recent work by Jobsis [27] has demonstrated the striking elastic properties of the visceral pericardium, thus adding a further chapter to the still-developing story of the sources of ventricular elastic recoil. 

FIGURE 28.3 The PDF model reveals how E-wave DT is jointly determined by stiffness and relaxation. The model re-expresses DT in terms of PDF parameters (c, k) for both overdamped and underdamped contours. Taylor series approximation simplifies expressions for DT. The right panel shows lines of constant DT in the c versus k plane. Note that a constant value of DT can be generated by a range of c and k values. The y = 1 curve separates overdamped (top) and underdamped (bottom) regimes. 

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