Phase slowly varying

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

In this brief review of dynamics in condensed phases, we have considered dense systems in various situations. First, we considered systems in equilibrium and gave an overview of how the space-time correlations, arising from the themial fluctuations of slowly varying physical variables like density, can be computed and experimentally probed. We also considered capillary waves in an inliomogeneous system with a planar interface for two cases an equilibrium system and a NESS system under a small temperature gradient. 

Thus the original differential equation (6-90) of the second order has been replaced by the system (6-96) of two first order differential equations in terms of the amplitude a and the phase 9. Moreover, as Eqs. (6-96) contain the small factor (i on the right-hand side, the quantities, a and 9 are small, that is, both a and 9 are slowly varying functions of time and one can assume that during one period T = 2nfca, the trigonometric functions vary but slightly. 

A simple rocking device was tested for routine determination of distribution coefficients [9], Sample cells were constructed for two-phase [9] and three-phase [10] systems. The investigators claim that the rocking action causes the shape of each phase to vary slowly and constantly and that the precision associated with the distribution coefficient is similar to that for shake-out methods. The three-phase cell was tested as an in vitro model to simulate factors involved in the absorption process. Rates of drug transfer and equilibrium drug distribution were evaluated under conditions in which one aqueous phase was maintained at pH 7.4 and the other phase was maintained at another pH. 

The following cases are somewhat more subtle. Figure 2c depicts a slowly varying amplitude Pnw with a very slowly varying phase-... 

Here, the sum is over all possible resonance vectors. Since the leading contribution to 0,(f) is co,t, it follows that near the particular resonance m of interest, other phase factors in Eq. (3.29) will be oscillatory functions of time while due to Eq. (3.27), exp(/m 0) will be slowly varying. Retaining only the one resonance term in Eq. (3.29),... 

It should be noted, however, that the limit 0 is only a formal procedure, which does not necessarily lead to a unique or correct semiclassical limit. In the case of the mapping formulation, this is because of the following reasons (i) For a given molecule, the frequencies f)mi(x) will in general also depend in a nontrivial way on h. (ii) A slowly varying term may as well be included in the stationary phase treatment [147]. (iii) As indicated by the term resulting from the commutator = 8 , the effective action constant ... 

Consider the case when the external field is a monochromatic circularly polarized pulse, E(f) = Aexp(-iojf), where A is a slowly varying envelope function. For this pulse the phase angle of dE/df is rotated by r/2 from the direction of E. From Eqs. (5.17) and (5.18) we then find... 

In the case of quasi-periodic sinusoidal signals, the buzziness can often be linked to the fact that the phase coherence between sinusoidal components is not preserved. Shape invariant modification techniques for quasi-periodic signals are an attempt to tackle this problem. As explained in 9.4.2, quasi-periodic signals such as speech voiced segments or sounds of musical instruments can be thought of as sinusoidal signals whose frequencies are multiples of a common fundamental COo(x), but with additional, slowly varying phases 0 (/) ... 

However, AH, the difference between the molar enthalpy of the gas and the condensed phase, depends in general on both the temperature and the pressure. The enthalpy for an ideal gas is independent of pressure and, fortunately, the enthalpy for the condensed phase is only a slowly varying function of the pressure. It is therefore possible to assume that AH is independent of the pressure and a function of the temperature alone, provided that the limits of integration do not cover too large an interval. With this final assumption, the integration can be carried out. When the molar heat capacities of the two phases are known as functions of the temperature, AR is obtained by integration. If ACP, the difference in the molar heat capacities of the two phases, is expressed as... 

FIGURE 2.15 Comparison among different views of a snapshot of a subsystem satisfying the Swift-Hohenberg equation—a simplified model of convection in the absence of mean flow. Panel (a) shows the detailed flow directions, whereas panels (b) and (c) exhibit the amplitude and phase, respectively. The latter are slowly varying and allow for easier identification of the grain boundaries of the flow. 

Since A is known and D is either given or measured in the experiment, it is possible to extract Aal and 5 from the parameters Q and Q, provided /% and values are also known. Typical values of / , for the uCB liquid crystals are 1.55-1.65 aud, luckily, these are slowly varying fnnctions of T and wavelength in the isotropic phase. For 5CB an r valne of 1.587 can be nsed with an error estimated at less than 0.5%. 

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