Features arising from randomly multiplicative measures

Under the formalism of annealed averaging of the partition function, a type of random multifractal measures with their multipliers satisfying exponentially distributed is investigated in detail. Branching emerges in the curve of generalized dimensions, and negative values of generalized dimensions arise. Three equivalent methods of classification of the random multifractal measures are proposed, which is based on: (i) the discrepancy between the curves of generalized dimensio...

In this paper we revisit an idea originally proposed by Mandelbrot about the possibility to observe ``negative dimensions'' in random multifractals. For that purpose, we define a new way to study scaling where the observation scale $\tau$ and the total sample length $L$ are respectively going to zero and to infinity. This ``mixed'' asymptotic regime is parametrized by an exponent $\chi$ that corresponds to Mandelbrot ``supersampling exponent''. In order to study the scaling e...

We have found an analytic expression for the multivariate generating function governing all n-point statistics of random multiplicative cascade processes. The variable appropriate for this generating function is the logarithm of the energy density, ln epsilon, rather than epsilon itself. All cumulant statistics become sums over derivatives of ``branching generating functions'' which are Laplace transforms of the splitting functions and completely determine the cascade process...

The multifractal nature of drop breakup in air-blast nozzle atomization process has been studied. We apply the multiplier method to extract the negative and the positive parts of the f(alpha) curve with the data of drop size distribution measured using Dual PDA. A random multifractal model with the multiplier triangularly distributed is proposed to characterize the breakup of drops. The agreement of the left part of the multifractal spectra between the experimental result and...

It is shown phenomenologically that the fractional derivative $\xi=D^\alpha u$ of order $\alpha$ of a multifractal function has a power-law tail $\propto |\xi| ^{-p_\star}$ in its cumulative probability, for a suitable range of $\alpha$'s. The exponent is determined by the condition $\zeta_{p_\star} = \alpha p_\star$, where $\zeta_p$ is the exponent of the structure function of order $p$. A detailed study is made for the case of random multiplicative processes (Benzi {\it et ...

In this paper (Shivamoggi et al.), we explore a variant for the simple model based on a binomial multiplicative process of Meneveau and Sreenivasan that mimics the multi-fractal nature of the energy dissipation field in the inertial range of fully developed turbulence (FDT), and uses the generalized fractal dimension (GFD) prescription of Hentschel and Proccacia, Halsey et al. However, the presence of an even infinitesimal dissipation in the inertial range is shown to lead to...

We present a dynamical log-stable process for the spatio-temporal evolution of the energy-dissipation field in fully developed turbulence. The process is constructed from multifractal scaling relations required for two-point correlators of arbitrary order. n-point correlation functions are calculated analytically and interpreted in terms of generalised fusion rules and in terms of the random multiplicative cascade picture. Multiplier distributions are compared with experime...

Among the statistical models employed to approximate nonlinear interactions in biological and psychological processes, one prominent framework is that of cascades. Despite decades of empirical work using multifractal formalisms, a fundamental question has persisted: Do the observed nonlinear interactions across scales owe their origin to multiplicative interactions, or do they inherently reside within the constituent processes? This study presents the results of rigorous nume...

We propose a new statistical model that can reproduce the hierarchical nature of the ubiquitous filamentary structures of molecular clouds. This model is based on the multiplicative random cascade, which is designed to replicate the multifractal nature of intermittency in developed turbulence. We present a modified version of the multiplicative process where the spatial fluctuations as a function of scales are produced with the wavelet transforms of a fractional Brownian moti...

This contribution addresses the question commonly asked in scientific literature about the sources of multifractality in time series. Two primary sources are typically considered. These are temporal correlations and heavy tails in the distribution of fluctuations. Most often, they are treated as two independent components, while true multifractality cannot occur without temporal correlations. The distributions of fluctuations affect the span of the multifractal spectrum only ...