Power-dependent nonlinear coupling

Figure 11. Power-dependent nonlinear coupling at a nonlinear directional coupler.

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

Based on a perturbation expansion using the KS Hamiltonian [26,27], recently a new systematic scheme for the derivation of orbital-dependent Ec has been proposed [12]. While this representation is exact in principle, an explicit evaluation requires the solution of a highly nonlinear equation, coupling Exc and the corresponding x>xc [19]. For a rigorous treatment of this Exc one thus has to resort to an expansion in powers of e, which allows to establish a recursive procedure for the evaluation of Exc and the accompanying Vxc-... 

Recently, there has been much interest in the development and application of multidimensional coherent nonlinear femtosecond techniques for the study of electronic and vibrational dynamics of molecules [1], In such experiments more than two laser pulses have been used [2-4] and the combination of laser pulses in the sample creates a nonlinear polarization, which in turn radiates an electric field. The multiple laser pulses create wave packets of molecular states and establish a definite phase relationship (or coherence) between the different states. The laser pulses can create, manipulate and probe this coherence, which is strongly dependent on the molecular structure, coupling mechanisms and the molecular environment, making the technique a potentially powerful method for studies of large molecules. 

It is expected that this power law (3 = 2.5 could reflect the increase of the fraction of such coupling resonances forming hubs with the coupling b. The exponent (3 here decreases with K. As K is increased from 0.5 to 0.9, (3 decreases from 3.5 to 2.2. This dependency suggests that the increase of hub coupling resonances is more relevant because the nonlinearity is weaker. 

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