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Multiscaling in Stochastic Fractals

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Multiscaling in Stochastic Fractals Multiscaling in Stochastic Fractals P. L. Krapivsky1 and E. Ben-Naim2 1Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, USA 2The James Franck Institute, The University of Chicago, Chicago, IL 60637, USA We introduce a simple kinetic model describing the formation of a stochastic Cantor set in arbitrary spatial dimension d. In one dimension, the model exhibits scaling asymptotic behavior. 1=2 For d > 1, the volume distribution is characterized by a single scale t− , while other geometric properties such as the length are characterized by an infinite number of length scales and thus exhibit multiscaling. The notion of a fractal has been widely used to de- with the initial conditions P (x; 0) = δ(x 1). The loss scribe self-similar structures [1]. The simplest way to term on the right-hand side represents the− decrease of construct a fractal is to repeat a given operation over intervals of length x, x-mers, due to the division process. and over again. The classical example of such a repet- Each division event consists of choosing two points at itive consruction is the Cantor's \middle-third erasing" random and thus, the overall breakage rate is quadratic set [1]. Recall the definition of this set: One divides an in the interval length, while the factor 1=2 arises since the interval into three equal intervals and then removes the two points are indistinguishable. The gain term repre- middle interval; on the next step, one repeats the same sents the increase of x-intervals due to breakups of longer procedure with the two remaining intervals; etc.. The intervals. outcome of this process is a counterintuitive uncount- able set having a measure (\length") zero. The Can- tor 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, self- similar structures arising in nature are usually random. Moreover, fractals are usually formed by continuous ki- netic processes while the classical repetitive constructions are discrete in time. In the present letter we introduce a FIG. 1. Fig. 1 Illustration of the process in two dimension. stochastic process which may be considered as a natural kinetic counterpart to the original Cantor construction. The formation of the random Cantor set is equivalent The resulting set turns out to be a random fractal of to random sequential parking on a line with a uniform dimension Df = (p17 3)=2 = 0:56155. We also investi- − ∼ distribution of lengths of parking intervals. While the lat- gate d-dimensional random Cantor sets and find that sev- ter problem has been investigated in Ref. [2], we present eral geometric characteristics such as the average length, an alternative solution method [3] that can be easily gen- surface area, etc., are characterized by different scales. eralized to higher dimensions. This method focuses on In the following, the existence of multipole kinetic ex- the leading asymptotic behavior of the moments of the ponents characterizing the process, will be shortly called length distribution M(s; t), defined by multiscaling. In one dimension, our model can be defined as fol- 1 lows. Starting with the unit interval [0:1], cracks are s 1 M(s; t) = Z P (x; t)x − dx: (2) deposited uniformly on the unit interval with unit rate. When two cracks apear on the initial interval, the mid- 0 dle is removed immediately and two new intervals are The rate equation (1) yield the following kinetic equa- formed. The process continues independently for the sur- tion for the moments, viving intervals such that whenever a surviving interval contains two cracks, the middle interval is removed. In @M(s; t) 1 2 the classical Cantor process, after n stages we are left = + M(s + 2; t): (3) n n @t −2 s(s + 1) with 2 intervals of length 3− . In the stochastic process the number of intervals and their lengths at time t are Asymptotically, the moments exhibit the power-law be- in principle arbitrary. The distribution function P (x; t) havior describing intervals of length x at time t satisfies the fol- lowing linear evolution equation, α(s) M(s; t) A(s)t− : (4) ' 1 @P (x; t) x2 By inserting the anticipated power-law behavior into = P (x; t) + 2Z dy(y x)P (y; t); (1) Eq. (3) and solving the resulting difference equations one @t − 2 − x gets 1 1 s β Γ(s)Γ β + 2 (β s)=2 where s = (s1; : : : ; sd) and 2 = (2;:::; 2). α(s) = − ;A(s) = 2 − : (5) 2 Γ s+β+1 Γ(β) A surprising feature of Eq. (10) is that it implies the 2 existence of an infinite number of conservation laws: on d d d In the above equation we have introduced a shorthand the hypersurface sj(sj + 1) = 2 3 1 , the mo- j=1 − notation β = (p17 1)=2. ments M(s; t) areQ independent of time. Thus the com- − Eq. (4) implies that in the long-time limit P (x; t) ap- petition between creation and destruction of the (hy- proaches the scaling form, per)rectangles gives birth to an infinite number of inte- grals of motion. Similar hidden conserved integrals have P (x; t) tβ=2Φ xpt ; (6) ' been found in recent studies of multidimensional frag- mentation [3,4]. In contrast to the fragmentation prob- with the scaling function Φ(z) being the inverse Mellin lem where at least one integral - the total volume - is transform of A(s). In the limit of small z; Φ(z) ap- an obvious conserved quantity, in the present model we proaches a constant while in the large-z limit, Φ(z) could not physically explain the appearance of any con- z β exp( z2=2). ∼ − served integral in any dimension. Furthermore,− the total number of intervals N(t), (β 1)=2 These integrals play an important role in the dynamics N(t) M(1; t), grows as t − , while the typical ≡ 1=2 of the system, e: g:, they are responsible for the absence interval size x , x = M(2; t)=N(t), decays as t− . This simple scalingh i h relationi follows directly from the rate of scaling solutions to Eq. (8). Indeed, trying a scaling w z equations, since the loss rate is quadratic in the interval solution of the form P (x; t) = t Q(t x), one derives in- finitely many scaling relations, w = z sj, which should length. However, it is interesting that the simple rate P equation (1) leads to non-trivial asymptotic exponents. be valid for all points s on the hypersurface. This in- Knowledge of the asymptotic behavior of the average finite set of scaling relations cannot be satisfied by just length and the average number enables calculation of the two scaling exponents, w and z. (β 1) fractal dimension. Since N x − − , the fractal di- In analogy with one-dimensional case, one can expect mension of the stochastic Cantor∼ h i set is given by a power-law behavior of the moments in the general d- α(s) dimensional situation: M(s; t) t− as t . Sub- Df = β 1 = (p17 3)=2 = 0:56155: (7) ∼ ! 1 − − ∼ stituting this asymptotic form into Eq. (10) we obtain The above dimension is smaller than the fractal dimen- the difference equation for the exponent α(s), sion of the classic Cantor set, Df = ln 2= ln 3 ∼= 0:63093. We turn now to the general d-dimensional version α(s) + 1 = α(s + 2): (11) of the model. In two dimensions, the model de- scribes the formation of the stochastic Cantor gas- In addition, on the hypersurface s (s + 1) = ket. The governing rule of the model is sketched in 1 j d j j d d Q ≤ ≤ Fig. 1 for the two-dimensional situation. Denote by 2 3 1 , one has α(s) = 0. The solution to Eq. (11) with this− boundary condition is given by the formal ex- P (x; t); x = (x1; : : : ; xd), the distribution function for pression (hyper)rectangles of size x1 ::: xd. The rate equation governing P (x; t), is given by× a straightforward× general- ization of Eq. (1), α(s) α(s∗ + k2) = k; (12) ≡ @P (x; t) P (x; t) d = x2 + (8) where the point s lies on the hypersurface. Geomet- @t 2d Y j ∗ j=1 rically, the exponent α(s) gives a (normalized) distance 1 1 d from the point s to the hypersurface in the 1 = (1;:::; 1) d direction. 3 1 Z ::: Z P (y; t) (yj xj)dyj: − Y − j=1 For ordinary scaling distributions the exponent α(s) x1 xd should be linear in the variable sj. This property is Similarly, the moments of the distribution function equivalent to the existence of a singleP length scale in the P (x; t), system. However, in our stochastic process the exponent 1 1 d α(s) is a function of all of its variables. This manifests sj 1 the non-trivial scaling properties of the process. On the M(s; t) = ::: P (x; t) x − dx ; (9) Z Z Y j j other hand, since all the moments still show a power-law j=1 0 0 behavior we conclude that the model exhibits a multi- satisfy the kinetic equation scaling asymptotic behavior. As a manifestation of the existence of multiple length @M(s; t) 1 d 1 = 2 + 3d 1 3 M(s + 2; t); scales in the system let us consider the ratio of the aver- @t −2d − Y s (s + 1) age volume V ; V = M(2; t)=N(t), to the dthpower of j=1 j j 4 5 the averageh lengthi h i l ; l = M(2; 1;:::; 1; t)=N(t).
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