Ashvin B. Chhabra

Two-point statistics of multifractal measures

The relationship between the ƒ(α) function of a multifractal and the spatial correlations of its singularity strengths is examined. An expression to compute the probability of observing two different singularities α’ and α″ within a distance r is derived for measures arising from isotropic random multiplicative processes. The correlation of αs is shown to decay logarithmically with distance. Possible applications to turbulence models are discussed, and the results are illustrated for a binomial measure, where the scaling of two-point correlation functions is shown to exhibit a phase transition.

Probabilistic Multifractals and Negative Dimensions

We propose that negative dimensions can be best understood using the concept of level-independent multiplier distributions and show that, by utilising them, one can extract the positive and negative parts of the f(α) function with exponentially less work than by using conventional boxcounting methods. When the underlying multiplicative structure is not known, both methods of computing negative dimensions can give spurious results at finite resolution. Applications to fully developed turbulence are discussed briefly.

FRACTAL DIMENSIONALITIES OF BACKBONES AND CLUSTERS IN A KINETIC GELATION MODEL

We present results of a computer simulation study of the fractal dimensionality of the largest cluster, backbone and the elastic backbone of a radical initiated irreversible kinetic gelation model in three dimensions. This work was motivated by earlier observations, that although the bulk exponents of this model are compatible with those of percolation, the cluster size distribution is vastly different (damped oscillatory) and obeys different scaling forms. On contrasting these dimensionalities with those from percolation models we find that while the fractal dimensionalities of the elastic backbone and the largest cluster are similar (to percolation) the dimensionality of the backbone is significantly different.