Effective damping theories

Theories of Effective Damping. If one assumes that a range of temperatures and/or frequencies will be encountered, then the area under the damping curve (E or tan 6) determines the effectiveness of the polymer better than the hight of the transition alone, see Figure 11. For E, the effective area is called the loss area, LA. This is determined after subtracting the background, as in any spectroscopic experiment. There are two theories to determine the quantity LA. 

Stirba and Hurt (S12), 1955 Experimental work on C02 absorption by water films in vertical tubes of length 3 and 6 ft., and dissolution of tubes of solid organic acids by water films. Effective diffusivity exceeds molecular diffusivity, even at 2VRe = 300. Dye streak experiments show that waves cause mixing surfactants damp waves to give continuous dye streak and mass transfer results in agreement with theory. 

Hikita (H12), 1959 Experimental study of effects of rippling on rate of absorption of C02 by water films containing surfactants (0.0005-0.05 wt.%), with film flow inside tubes 1.3 cm. X 15-101 cm. Results approach Emmert and Pigford theory (E4) as rippling is damped by surfactants. 

The theory above has been applied in a variety of realistic situations. The range includes ionic conductance in aqueous solutions and molten alkali chlorides, damped spin-wave behaviour in paramagnetic systems, stimulated emission of radiation in masers, the fractional quantum Hall effect and quantum correlations in high-Tc cuprates and other non-BCS superconductors [4, 5, 7, 8, 14, 30]. In the next section we will also make some comments on the problem of long-range transcorrelations of protons in DNA [31]. 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

Both Hamaker and Lifshitz theories of van der Waals interaction between particles are continuum theories in which the dispersion medium is considered to have uniform properties. At short distances (i.e. up to a few molecular diameters) the discrete molecular nature of the dispersion medium cannot be ignored. In the vicinity of a solid surface, the constraining effect of the solid and the attractive forces between the solid and the molecules of the dispersion medium will cause these molecules to pack, as depicted schematically in Figure 8.5. Moving away from the solid surface, the molecular density will show a damped oscillation about the bulk value. In the presence of a nearby second solid surface, this effect will be even more pronounced. The van der Waals interaction will, consequently, differ from that expected for a continuous dispersion medium. This effect will not be significant at liquid-liquid interfaces where the surface molecules can overlap, and its significance will be difficult to estimate for a rough solid surface. 

Memory effects play an important role for the description of dynamical effects in open quantum systems. As mentioned above, Meier and Tannor [32] developed a time-nonlocal scheme employing the numerical decomposition of the spectral density. The TL approach as discussed above as well as the approaches by Yan and coworkers [33-35] use similar techniques. Few systems exist for which exact solutions are available and can serve as test beds for the various theories. Among them is the damped harmonic oscillator for which a path-integral solution exists [1], In the simple model of an initially excited... 

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