ANOMALOUS FEATURES ARISING FROM RANDOM MULTIFRACTALS

Abstract

Under the formalism of annealed averaging of the partition function, two types of random multifractal measures with their probability of multipliers satisfying power law distribution and triangular distribution are investigated mathematically. In these two illustrations, branching emerges in the curve of generalized dimensions, and more abnormally, negative values of generalized dimensions arise. Therefore, we classify the random multifractal measures into three classes based on the properties of generalized dimensions. Other equivalent classifications are also presented by investigating the location of the zero-point of τ(q) or the relative position either between the f(α) curve and the diagonal f(α) = α or between the f(q) curve and the α(q) curve. We consequently propose phase diagrams to characterize the classification procedure and distinguish the scaling properties between different classes. The branching phenomenon emerging is due to the extreme value condition and the convergency of the generalized dimensions at point q = 1. We conjecture that the branching condition exists and that the classification is universal for any random multifractals. Moreover, the asymptotic behaviors of the scaling properties are studied. We apply the cascade processes studied in this paper to characterizing two stochastic processes, i.e. the energy dissipation field in fully developed turbulence and the droplet breakup in atomization. The agreement between the proposed model and experiments are remarkable.