Kramers formulation

The dumbbell is thus called the FENE-P dumbbell ( FENE stands for Finite Extendable Nonlinear Elastic , and P stands for Peterlin, who first suggested this simplification). With this approximation, the Kramers formulation becomes... 

The effects of spin-orbit coupling on geometric phase may be illustrated by imagining the vibronic coupling between the two Kramers doublets arising from a 2E state, spin-orbit coupled to one of symmetry 2A. The formulation given below follows Stone [24]. The four 2E components are denoted by e, a), e a), e+ 3), c p), and those of 2A by coa), cop). The spin-orbit coupling operator has nonzero matrix elements... 

Extensions of the Kramers model are considered necessary [92-94, 97-99] although there are refined versions of the original formulation [100, 101]. Such non-Markovian dynamics has been taken into consideration... 

It has been claimed that reactions in proteins can, as an approximation, be formulated within the Kramers reaction theory of barrier crossing [106]. The highly nonexponential relaxation pattern can now be explained by our model,... 

While the pl does not determine the shape ofthe pH-solubility proLle, it does lx the location of this proLle on the pH coordinate. All other factors being equal, each upward or downward shift in the pl is matched exactly by an upward or downward shift irVrgiN If the solubility ofthe free base is very small relative to that ofthe hydrochloride, the free base limiting curve (curve II) ofthe overall pH-solubility proLle cuts deeply into the acidic pH range. Therefore, the solubility of the free base and th basically determine the maximum pH at which formulation as a solution is possible, assuming the desired concentration exceeds the free base solubility (Kramer and Flynn, 1972). 

The time-dependent formulation of Raman scattering has been introduced by Lee and Heller (1979), Heller, Sundberg, and Tannor (1982), Tannor and Heller (1982), and Myers, Mathies, Tannor, and Heller (1982). Its derivation is strikingly simple. We start from the Kramers-Heisenberg-Dirac formula (14.1) and (14.2) without the nonresonant term and transform it into an integral over time by using the identity... 

The Einstein relation (159) or the expression (157) of the dissipative part ffiep(m) of the mobility constitute another formulation of the first FDT. Indeed they contain the same information as the Kubo formula (156) for the mobility, since p(co) can be deduced from 9ftep(oo) with the help of the usual Kramers-Kronig relations valid for real co [29,30]. Equation (156) on the one hand, and Eq. (157) or Eq. (159) on the other hand, are thus fully equivalent, and they all involve the thermodynamic bath temperature T. Note, however, that while p(oo) as given by Eq. (156) can be extended into an analytic function in the upper complex half-plane, the same property does not hold for D(co). 

Based on the description of craze thickening due to Kramer et al. [31,32], Schirrer [45] proposed a phenomenological viscoplastic formulation for the fibril drawing velocity similar to the Eyring model as... 

For the chemist, the Jahn-Telkr effect is, in a way, the obverse side of the Tanabe-Kamimura theorem . The abstract formulation of the Jahn-Teller theorem is that two or more electronic states having identical energy in a definite symmetry of a non-linear molecule or polyatomic ion (with exception of Kramers doublets in the case of an odd number of electrons) spontaneously separate in energy, such that (at least) the state of... 

Wolf et al. calculated the equilibrium size of droplets formed in a phase-separated system. From a force balance, he derived an expression Indicating that the equilibrium droplet size r is a decreasing function of shear rate and that when r approaches the radius of gyration of the polymer molecules, redissolution will have occurred. Recently Kramer and Wolf have generalized the approach and formulated simple criteria for solution, respectively demixing (16). 

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