Multiscaling in Stochastic Fractals

The notion of a fractal has been widely used to describe self-similar structures [1]. The simplest way to construct a fractal is to repeat a given operation over and over again. The classical example of such a repetitive consruction is the Cantor’s “middle-third erasing” set [1]. Recall the definition of this set: One divides an interval into three equal intervals and then removes the middle interval; on the next step, one repeats the same procedure with the two remaining intervals; etc.. The outcome of this process is a counterintuitive uncountable set having a measure (“length”) zero. The Cantor set turns out to be a perfect fractal of dimension Df = ln(2)/ ln(3) ∼= 0.63093. The Cantor set is a regular fractal. In contrast, selfsimilar structures arising in nature are usually random. Moreover, fractals are usually formed by continuous kinetic processes while the classical repetitive constructions are discrete in time. In the present letter we introduce a stochastic process which may be considered as a natural kinetic counterpart to the original Cantor construction. The resulting set turns out to be a random fractal of dimension Df = ( √ 17−3)/2 ∼= 0.56155. We also investigate d-dimensional random Cantor sets and find that several geometric characteristics such as the average length, surface area, etc., are characterized by different scales. In the following, the existence of multipole kinetic exponents characterizing the process, will be shortly called multiscaling. In one dimension, our model can be defined as follows. Starting with the unit interval [0:1], cracks are deposited uniformly on the unit interval with unit rate. When two cracks apear on the initial interval, the middle is removed immediately and two new intervals are formed. The process continues independently for the surviving intervals such that whenever a surviving interval contains two cracks, the middle interval is removed. In the classical Cantor process, after n stages we are left with 2 intervals of length 3−n. In the stochastic process the number of intervals and their lengths at time t are in principle arbitrary. The distribution function P (x, t) describing intervals of length x at time t satisfies the following linear evolution equation,