54 research outputs found
Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal noise
To understand the sample-to-sample fluctuations in disorder-generated
multifractal patterns we investigate analytically as well as numerically the
statistics of high values of the simplest model - the ideal periodic
Gaussian noise. By employing the thermodynamic formalism we predict the
characteristic scale and the precise scaling form of the distribution of number
of points above a given level. We demonstrate that the powerlaw forward tail of
the probability density, with exponent controlled by the level, results in an
important difference between the mean and the typical values of the counting
function. This can be further used to determine the typical threshold of
extreme values in the pattern which turns out to be given by
with . Such observation provides a
rather compelling explanation of the mechanism behind universality of .
Revealed mechanisms are conjectured to retain their qualitative validity for a
broad class of disorder-generated multifractal fields. In particular, we
predict that the typical value of the maximum of intensity is to be
given by , where is the
corresponding singularity spectrum vanishing at . For the
noise we also derive exact as well as well-controlled approximate
formulas for the mean and the variance of the counting function without
recourse to the thermodynamic formalism.Comment: 28 pages; 7 figures, published version with a few misprints
corrected, editing done and references adde
Explosive crystallization mechanism of ultradisperse amorphous ÿlms
Abstract The explosive crystallization of germanium ultradisperse amorphous ÿlms has been studied experimentally. We show that crystallization may be initiated by local heating at the small ÿlm thickness but is realized spontaneously at large thickness. The fractal pattern of the crystallization phase is discovered to be inherent in the phenomena of di usion-limited aggregation. It is shown that in contrast to the ordinary crystallization mode, the explosive one is connected with the instability which is caused by self-heating. A transition from the ÿrst mechanism to the second one is modelled by the Lorenz system. The process of explosive crystallization is represented on the basis of the self-organized criticality conception. The front movement is described as the e ective di usion in the ultrametric space of hierarchically subordinated avalanches, corresponding to the explosive crystallization of elementary volumes of ultradisperse powder. The expressions for the stationary crystallization heat distribution and the steady-state heat current are obtained. The heat needed for initiation of the explosive crystallization is obtained as a function of the thermometric conductivity. The time dependence of the spontaneous crystallization probability in a thin ÿlm is examined
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