Dynamical scaling hypothesis

Figure 34. At the early stages of the phase separation, the scaling of the curvature distributions is in accordance with the dynamic scaling hypothesis [Eqs. (4)—(7)] (the order parameter is nonconserved).

For the symmetric system (< )0 = 0.5) the scaling exponent for the Euler characteristic has been found in accordance with the dynamic scaling hypothesis x L(t) 3 (see Section I.G). The homogeneity index, HI, of the interface defined as [222]... 

The scaled structure factor F(x) or F(x) characterizes morphological aspects of the growing pattern such as those shown in Fig. 2, while the size of the pattern at t is characterized by qm(t) or Am(t) (eq.2). If the pattern grows with the dynamical self-similarity at a given T and (t)> is invariant with t, F(x) is universal with t as shown schematically in Fig.3(b). Obtaining this universal scaling function F(x) confirms the validity of the dynamical scaling hypothesis. The... 

It is obvious that the dynamical scaling hypothesis is not valid through the dynamical percolation-to-cluster transition. This is because the pattern changes from the bicontinuous periodic structure to the cluster of spheres. In fact the scaled structure factor was found to substantially broaden before and after the transition. 

To get a first crude estimate of the frequency dependence of the modulus we put forward a simple dynamic scaling argument. It uses d in the same spirit as the exponent z in the dynamic scaling hypothesis in phase transitions according to the theory of Halperin and Hohenberg. Consider the modulus of an entropy-dominated or Brownian system which can be written from dimensional analysis as G kT/V, where V is the volume and kT the thermal energy unit. Here K is a fractal... 

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