Fluctuation-relaxation model

The data for the Kerr-effect and dielectric relaxations for the solutes of Table 1 and for TnBP suggest that the cooperative a-process occurring just above Tg involves many molecules, and their motions are Just like those of a system of enmeshing cogs. The fluctuation-relaxation model, where c(t) is described by a defect-diffusion model, rationalizes our observations. However, other models may also give the result Ki(t) = K2(t) e.g. the diffusional motions of a bond on a tetrahedral lattice (94). 

The diffusion equations involved in our theoretical analysis, which are derived via the AEP from the fluctuating reduced model of Section IV, can be regarded as a five-state version of the Anderson two-state model supplemented by a quantitative description of the secondary process. We would especially stress that, in accordance with the point of view of other authors, the characteristic time of the principal dielectric relaxation band roughly coincides with the residence time in the structured part of the liquid. ... 

Gaussian axial fluctuation (GAP) model for peptide plane reorientation about the C —axis, which was initially proposed to interpret spin relaxation derived order parameters [110], is useful to describe a common anisotropic component of protein backbone dynamics [111]. A complete three-dimensional GAP (3D GAP) analysis of local motion was conducted using an extensive set of RDCs from the third immunoglobin binding domain of streptococcal protein G (GB3) [112]. The averaged coupling is calculated by using Eq. (1.39), as a function of Oy and <7, and the amph-... 

The equations for the tube motion (eqn [60]) remain the same as in the pure reptation case the tube segments are aeated and destroyed at the ends. However, the number of tube segments Z becomes a random variable Z(t), as stressed by the model name contour length fluctuations (CLFs). These fluctuations relax stress and orientation faster than the reptation mode alone. To solve eqns [63] and [65], we should first subtract the equilibrium stretch from Xi coordinates y,=X( - i b /a) and then use eqn [23] to transform yi to the Rouse modes. These modes will again satisfy the same Omstein-Uhlenbeck equation [25]. 

Wiesenfeld, et. al. [wiesen89] compare the simplicity of the independent relaxation time interpretation of l//-noise fluctuations in the saudpile CA model to other recent models yielding 1// noise ... 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

Ammonium alums undergo phase transitions at Tc 80 K. The phase transitions result in critical lattice fluctuations which are very slow close to Tc. The contribution to the relaxation frequency, shown by the dotted line in Fig. 6.7, was calculated using a model for direct spin-lattice relaxation processes due to interaction between the low-energy critical phonon modes and electronic spins. 

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. 

Outer sphere relaxation arises from the dipolar intermolecular interaction between the water proton nuclear spins and the gadolinium electron spin whose fluctuations are governed by random translational motion of the molecules (106). The outer sphere relaxation rate depends on several parameters, such as the closest approach of the solvent water protons and the Gdm complex, their relative diffusion coefficient, and the electron spin relaxation rate (107-109). Freed and others (110-112) developed an analytical expression for the outer sphere longitudinal relaxation rate, (l/Ti)os, for the simplest case of a force-free model. The force-free model is only a rough approximation for the interaction of outer sphere water molecules with Gdm complexes. 

Even if we consider a single solvent, e g., water, at a single temperature, say 298K, depends on the solute and in fact on the coordinate of the solute which is under consideration, and we cannot take xF as a constant. Nevertheless, in the absence of a molecular dynamics simulation for the solute motion of interest, XF for polar solvents like water is often approximated by the Debye model. In this model, the dielectric polarization of the solvent relaxes as a single exponential with a relaxation time equal to the rotational (i.e., reorientational) relaxation time of a single molecule, which is called Tp) or the Debye time [32, 347], The Debye time may be associated with the relaxation of the transverse component of the polarization field. However the solvent fluctuations and frictional relaxation occur on a faster scale given by [348,349]... 

---