Power-law decay

Power-law scattering features will be discussed in relation to mass-fractal scaling laws. Fractal scaling concepts used to interpret the power-law decay are well published in the literature. 

The power, [P], in the fractal power-law regime gives as the fractal dimension, d(. P = —df for each level of the fit, the parameters obtained using the unified model are G, Rg, B, and P. P is the exponent of the power-law decay. When more than one level is fitted, numbered subscripts are used to indicate the level—i.e., G —level 1 Guinier pre-factor. The scattering analysis in the studies summarized here uses two-level fits, as they apply to scattering from the primary particles (level 1) and the aggregates (level 2). 

This power law decay is captured in MPC dynamics simulations of the reacting system. The rate coefficient kf t) can be computed from — dnA t)/dt)/nA t), which can be determined directly from the simulation. Figure 18 plots kf t) versus t and confirms the power law decay arising from diffusive dynamics [17]. Comparison with the theoretical estimate shows that the diffusion equation approach with the radiation boundary condition provides a good approximation to the simulation results. 

In the model of the previous section, we have encountered the situation that outside the regime bounded by the disorder line the correlation function, Eq. (53), is characterized by an angle which is incommensurate with the lattice. Now it is well known that in d 2 dimensions, incommensurate long-range order (which would be described by <5,5 )x = cos[ r -Tjl(/ -F A< ] instead of Eq. (53)) is unstabie against thermal fluctuations (this is a two-component ordering - amplitude tj/, phase A0 - similar to the XY model). Nevertheless, it is possible to have a phase transition where in Eq. (53) diverges, but for T [Pg.126]

We see that the transit signal does not change appreciably with temperature. Also note that at the final stage (t > 1 ), the TOF signal exhibits a power-law decay and appears to be dispersive. According to the Scher-Montroll theory [33], in the case of the dispersive transport process, I(t) should exhibit power-law dependences rd-a) and 7 d+a) fpj. j respectively, where a is the disorder parameter. The... 

One can show that the block length distribution function /b( ) is characterized for asymptotically large l by the power-law decay of its density /bCO a ta with the exponent a = 3/2 [88]. The exponent a estimated from the simulation data (a % 1.6) is, up to the experimental uncertainty, quite close to that predicted by the exact analytical model. As has been noted above, such a power-law decay is a characteristic of the Levy probabilistic processes [36]. 

Beyond the schematic picture sketched above, the two-step density correlation functions emerging from the idealized theory exhibit characteristic power-law decays toward and from the plateau /. Their experimental identification and analysis allows one to determine the crossover temperature... 

The asymptotic scaling laws of MCT describe the crossover from the fast relaxation to the onset of the slow relaxation (a-process)—that is, a power-law decay of 4>(f) toward the plateau with an exponent a, and another power-law decay away from the plateau with an exponent b. For the purpose of the present review, we again ignore the -dependence. 

As discussed at length by Shimizu et al. [7], the on time distributions show temperature and laser power dependencies—for example, exponential cutoffs of power-law behavior. Although no direct observations of cutoffs in the off time distribution have been reported, ensemble measurements by Chung and Bawendi [3] demonstrate that such a cutoff should also exist, but at times of the order of tens of minutes to hours. Our analysis here, employing power-law decaying distributions, is of course applicable in time windows where power-law statistics holds. 

The standard deviation figures for a here and for OpSCi in Fig. 13 represent the standard deviations of the distributions of the corresponding exponents, and not the errors in determination of their mean value. We also note that the on time distributions are less close to the power-law decays than the off times, partly due to the exponential cutoffs and partly due to varying intensities in the on state (cf. 

Hillermeier, C.F., Bliimel, R. and Smilansky, U. (1992). Ionization of H Rydberg atoms Fractals and power-law decay, Phys. Rev. A45, 3486-3502. 

As the fracture propagates, the elastic energy released due to the micro-fractures occurring within the sample can be measured. This ultrasonic emission due to micro-fracture aftershock relaxation has recently been measured for various laboratory samples. Petri et al (1994) measured the ultrasonic emission amplitude distribution in a large number of stressed solid samples under different experimental conditions. A power law decay for the cumulative energy release distribution n Er) with the released energy amplitude Er was observed in all cases n Er) E (see Fig. 3.21). This is indeed very similar to the Guttenberg-Richter law for the frequency distribution of earthquakes, as discussed briefly in Chapter 1, and will be discussed in detail in the next chapter. 

One-particle spectral properties of a Luttinger liquid. The absence of quasiparticles in a Luttinger liquid is also manifest in the one-particle spectral properties. As we have seen in the framework of the ID renormalization group method in the gapless case, there is a power-law decay of the one-particle spectral weight at the Fermi level. This power law behavior is also confirmed in the framework of the bosonization technique [146, 147], namely... 

The mass-fractal dimension of the ramified agglomerates is determined from the slope of the weak power law decay in between the power law regimes that follow Porod s law (Equation 10.9) ... 

The generation parameter defining the generation of ionizing trajectories in the self-similar structure in Fig. 10 is related to the number w of encounters of the two electrons at ri = T2 rather than to the ionization time. This interpretation is confirmed in Fig. 11 which shows the density n of trajectories starting with initial conditions uniformly distributed in the middle panel of Fig. 10 as function of the number w of encounters of the two electrons and of the ionization time T. The density n is proportional to minus the derivative of the survival probability with respect to the relevant variable (w or T). The logarithmic plot in Fig. 11a reveals an exponential decay of the density, n(w)ocexp(—0.27w), and hence also of the survival probability, as a function of the number of encounters of the two electrons, just as expected for a self-similar fractal set of trapped trajectories. The doubly logarithmic plot of the density of trajectories in Fig. 11b reveals a power-law decay of the density, (T) oc and hence... 

In any case, an interpretation based on our working hypothesis is consistent with the fact that the power-law decay is particularly observed in limited observables such as the energy transfer between two subsystems. Slow relaxation is observed if one monitors quantities associated with such a pathway. 

Second, we investigate the dependence of residence time distributions on nonlinearity of elements, as shown in Fig. 10. We get power-law decay of the distribution again. The exponent a is 3/2 for weak nonlinearity, and it approaches 2 for stronger nonlinearity. 

In Fig. 21 we show the asymptotic behavior of the stationary PDF Pst(V) for three different sets of parameters. Clearly, in all three cases the predicted power-law decay is obtained, with exponents that, within the estimated error bars agree well with the predicted relation for p according to Eq. (128).8... 

This power law character of the form factor is related to the power law decay of the pair correlation function of an ideal chain [Eq. (2.121)]. Quite generally, the form factor is related to the Fourier transform of the intramolecular pair correlation function g(r) ... 

Fig. 3.13. For adsorption of real chains, the actual concentration inside each adsorption blob decays as a power law in distance from the surface. This power law decay modifies the exponent in Eqs (3.62) and (3.70) (see Problem 3.22).

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