Landau theory curves

We will consider the cold-gas-convex surface of the flame front as a curved cell of the flame which had been formed after the plane flame lost its stability. The steady state of the convex flame is a result of the nonlinear hydrodynamic interaction with the gas flow field (see Zeldovich, 1966, 1979). In the linear approximation the flame perturbation amplitude grows in time in accordance with Landau theory, but this growth is restricted by nonlinear effects. 

The dependence e vs a derived from Eq. (36) is plotted on Fig. 4 as a dotted line. Solutions of Eq. (36) and of Eq. (27) show good agreement only at small values (e < 0.05) as would be expected, because the assumptions used in Eq. (32) are valid only at sufficiently small amplitudes A /A -C 1. Note that the changes of the gas motion before and after the curved front were considered indirectly through value /(a) taken from the Landau theory for a small disturbance. 

Fig. 5.10.2. Variation of the polarization P with relative temperature in the smectic C phase of DOBAMBC. The curve represents the fit obtained with the generalized Landau theory. (After Dumrongrattana and Huang. )...

The critical field curves and Ginzburg-Landau theory... 

Figure 1.6 Curves of free energy with respect to that of the high-temperature phase, f f, T) - /o(T), as a function oforder parameter plotted according to the Landau theory for second-order phase transitions. The symmetry-breaking phase transition on reducing temperature is signalled by the development of a non-zero order parameter. At T > T, the equilibrium value (i/ ) = 0. T = T defines the phase transition. At T < T, the equilibrium value (i/ ) 0...

The pair potential of colloidal particles, i.e. the potential energy of interaction between a pair of colloidal particles as a function of separation distance, is calculated from the linear superposition of the individual energy curves. When this was done using the attractive potential calculated from London dispersion forces, Fa, and electrostatic repulsion, Ve, the theory was called the DLVO Theory (from Derjaguin, Landau, Verwey and Overbeek). Here we will use the term to include other potentials, such as those arising from depletion interactions, Kd, and steric repulsion, Vs, and so we may write the total potential energy of interaction as... 

Thus a wavepacket initiated in well A passes to well B by a curve crossing. Prof. Fleming showed an interesting case of persistent coherence in such a situation, despite the erratic pattern of the eigenvalue separations. An alternative, possibly more revealing approach, is to employ Stuckelberg-Landau-Zener theory, which relates the interference (i.e., coherence) in the two different wells via the area shown in Fig. 2. A variety of applications to time-independent problems may be found in the literature [1]. 

Deiyagln-Landau-Verwey-Overbeek (DLVO) theory I.1.21ff(intr.), 1.3.59 interaction curve fig. 1.4.2 desalination 1.1.3... 

Nonadiabatic transitions play crucial roles in various fields of physics and chemistry [1. 2. 3. 4. 5, 7, 8 9. 10], and it is quite important to develop basic analytical theories so that we can understand fundamental mechanisms of various dynamics. The most fundamental models among them are the Landau-Zener type curve crossing and the Rosen-Zener-Demkov type non-curve-crossing. Furthermore, there is an interesting intermediate case in which two diabatie exponential potentials arc... 

The basic assumptions of the Landau-Zener theory need to be satisfied. These involve the applicability of classical mechanics (e.g. the neglect of tunneling) for the nuclear dynamics and the locality of the curve crossing event. 

Note that assumptions (2) and (3) are about timescales. Denoting by x, and tlz the characteristic times (inverse rates) of the electron transfer reaction, the solvent relaxation, and the Landau-Zener transition, respectively, (the latter is the duration of a single curve-crossing event) we are assuming that the inequalities Tr A Ts tlz hold. The validity of this assumption has to be addressed, but for now let us consider its consequences. When assumptions (1)—(3) are satisfied we can invoke the extended transition-state theory of Section 14.3.5 that leads to an expression for the electron transfer rate coefficient of the form (cf. Eq. 14.32)... 

The theory of nonadiabatic transition dates back to 1932, when the pioneering works for curve-crossing and noncrossing problems were published by Landau (13), Zener (14), and Stiickelberg (15) and by Rosen and Zener (28), respectively. Since then numerous papers by many authors have been devoted to these subjects, especially to curve-crossing problems (1,2,19,29-33). In this section a brief survey to around 1991 is made, which is informative and gives a good introduction to further developments described in subsequent sections. 

Landau discussed the curve-crossing problem by using the complex contour integral method (13,34). In general, the transition probability p in first-order perturbation theory is given by... 

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