Stokes phenomenon

One is the Stokes phenomenon [31-35]. The Stokes phenomenon is a very important subject of semiclassical method, but we won t describe it in detail in this chapter (for details see Ref. [22,35]). 

It should be noted that the parts of branches drawn by broken lines indicate noncontributing parts that make unphysical contributions, and they can be removed by the proper treatment of the Stokes phenomenon. After such a procedure, we can sum up all the contributions from the physically legal parts of the branches in the -set. We find the probabilistic weights of the branches together with the total probability obtained by the sum formula [Eq. (6)] in Fig. 5c. In Fig. 5d, the tunneling probability obtained by the sum formula is... 

G. G. Stokes, Trans. Cambridge Philos. Soc. 10, 106 (1864) R. B. Dingle, Asymptotic Expansions Their Derivation and Interpretation, Academic Press, London, 1973 for a very recent development on the study of Stokes phenomenon, see, for example, B. Y. Sternin and V. E. Shatalov, Borel-Laplace Transform and Asymptotic Theory, CRC Press, Boca Raton, EL, 1996. 

Stuckelberg did the most elaborate analysis (15). He applied the approximate complex WKB analysis to the fourth-order differential equation obtained from the original second-order coupled Schrodinger equations. In the complex / -plane he took into account the Stokes phenomenon associated with the asymptotic solutions in an approximate way, and finally derived not only the Landau-Zener transition probability p but also the total inelastic transition probability Pn as... 

Thus the key point is to obtain the connections of the asymptotic solutions of Eq. (60) at - oo. The underlying mathematics to carry this out analytically is the Stokes phenomenon of asymptotic solutions of differential equations and is explained briefly in Sec. V. This is very important mathematics for the general semiclassical theory and various physical phenomena. It is interesting that the apparent small differences in q(%) of Eqs. (61) between the LZ and NT cases make a big difference mathematically and physically. The mathematical differences are not detailed here, but the physical differences are obvious, as pointed out in the Introduction. 

Recently, Zhu and Nakamura carefully analyzed the Stokes phenomenon of Eq. (60) and succeeded in deriving not only quantum mechanically exact, but also new semiclassical expressions of SR. The quantum mechanically exact solutions in the LZ case (20-23) are... 

The very basic mathematics, i.e., Stokes phenomenon, which underlies semiclassical theory, is briefly explained in this section by taking the Airy function as an example. The Stokes constant and connection matrix in the case of the Weber function are provided, since the Weber function is useful in many applications. Finally, the Stokes phenomenon of the linear curve-crossing model discussed in Sec. IV is explained briefly. 

C. Zhu, H. Nakamura, N. Re, and V. Aquilanti, The two-state linear curve crossing problems revisited. I Analysis of Stokes phenomenon and expressions for scattering matrices, J. Chem. Phys. 97 1892 (1992). 

Stokes phenomenon plays a key role in the theory of Zhu and Naka-mma [510 512, 514], which is the only complete solution of the linear curve... 

This point was taken up by Reynolds in a letter addressed to G. G. Stokes, in the latter s capacity as Secretary of the Royal Society [83]. Reynolds pointed out that Maxwell s theory evaluated the effects of thermal transpiration only in circumstances where they were too small to be measured, and complained that Maxwell had misrepresented his own theoretical treat ment of the phenomenon. However, this incipient controversy never developed... 

Coherent anti-Stokes Raman scatttering, or CARS as it is usually known, depends on the general phenomenon of wave mixing, as occurs, for example, in a frequency doubling crystal (see Section 9.1.6). In that case three-wave mixing occurs involving two incident waves of wavenumber v and the outgoing wave of wavenumber 2v. 

Further support for this approach is provided by modern computer studies of molecular dynamics, which show that much smaller translations than the average inter-nuclear distance play an important role in liquid state atom movement. These observations have conhrmed Swalin s approach to liquid state diffusion as being very similar to the calculation of the Brownian motion of suspended particles in a liquid. The classical analysis for this phenomenon was based on the assumption that the resistance to movement of suspended particles in a liquid could be calculated by using the viscosity as the frictional force in the Stokes equation... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

In Spite of the existence of numerous experimental and theoretical investigations, a number of principal problems related to micro-fluid hydrodynamics are not well-studied. There are contradictory data on the drag in micro-channels, transition from laminar to turbulent flow, etc. That leads to difficulties in understanding the essence of this phenomenon and is a basis for questionable discoveries of special microeffects (Duncan and Peterson 1994 Ho and Tai 1998 Plam 2000 Herwig 2000 Herwig and Hausner 2003 Gad-el-Hak 2003). The latter were revealed by comparison of experimental data with predictions of a conventional theory based on the Navier-Stokes equations. The discrepancy between these data was interpreted as a display of new effects of flow in micro-channels. It should be noted that actual conditions of several experiments were often not identical to conditions that were used in the theoretical models. For this reason, the analysis of sources of disparity between the theory and experiment is of significance. 

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

Sedimentation of particles follows the principle outlined above [Eq. (1)] in which particles in the Stokes regime of flow have attained terminal settling velocity. In the airways this phenomenon occurs under the influence of gravity. The angle of inclination, t /, of the tube of radius R, on which particles might impact, must be considered in any theoretical assessment of sedimentation [14,19]. Landahl s expression for the probability, S, of deposition by sedimentation took the form ... 

If the electric field E is applied to a system of colloidal particles in a closed cuvette where no streaming of the liquid can occur, the particles will move with velocity v. This phenomenon is termed electrophoresis. The force acting on a spherical colloidal particle with radius r in the electric field E is 4jrerE02 (for simplicity, the potential in the diffuse electric layer is identified with the electrokinetic potential). The resistance of the medium is given by the Stokes equation (2.6.2) and equals 6jtr]r. At a steady state of motion these two forces are equal and, to a first approximation, the electrophoretic mobility v/E is... 

The phenomena were reinvestigated by Stokes, who published a famous paper entitled On the refrangibility of light in 18523 . He demonstrated that the phenomenon was an emission of light following absorption of light. It is worth describing one of Stokes experiments, which is spectacular and remarkable for its... 

In his first paper3 , Stokes called the observed phenomenon dispersive reflexion, but in a footnote, he wrote I confess I do not like this term. I am almost inclined to coin a word, and call the appearance fluorescence, from fluorspar, as the analogous term opalescence is derived from the name of a mineral. Most of the varieties of fluorspar or fluorspath (minerals containing calcium fluoride (fluorite)) indeed exhibit the property described above. In his second paper7, Stokes definitely resolved to use the word fluorescence (Scheme 1.2). 

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