The concept of multifractality is extended to self-affine fractals in order to provide a more complete description of fractal surfaces. We show that for a class of iteratively constructed self-affine functions there exists an infinite hierarchy of exponents Hq describing the scaling of the qth order height-height correlation function cq(x)~xqHq. Possible applications to random walks and turbulent flows are discussed. It is demonstrated on the example of random walks along a chain that for stochastic lattice models leading to self-affine fractals Hq exhibits phase-transition-like behavior.
©1991 The American Physical Society.
PACS: 05.40.+j 05.45.+b 05.70.Ln
Many recent investigations have demonstrated that a wide class of processes lead to complex objects that can be described in terms of self affine fractals. The examples range from plots of various kinds of random walks [1,2] to interfaces developing in marginally stable, far from equilibrium systems [3ndash6]. Growth processes resulting in self affine interfaces have attracted particular interest during the last few years because of their relevance to a number of phenomena of practical importance, including thin film growth by vapor deposition, two phase viscous flow in porous media, formation of biological patterns, and sedimentation of granular materials (see, e.g., Refs. [3ndash6] and references cited therein).
The properties of a fractal can be best characterized by a set of exponents describing the scaling behavior of the quantities defined for the fractal. Recent studies have shown that in addition to the fractal dimension, there exists an infinite hierarchy of exponents that allows a much more complete representation of the so called fractal measures [7ndash9]. Although this approach has been very successful, up to now the related ideas (multifractality, multiscaling) have only been applied to self similar sets. Here we intend to treat the self affine case and use an analogous analysis for functions which, in general, can be considered as complicated signals of arbitrary origin.
A single valued standard self affine function h ( x ) satisfies the relation h ( x ) apeq lambda^{-H}h ( lambda x ), where lambda is a parameter and H is the Houmllder or roughness exponent. Alternatively, the height height correlation function c ( h ) defined for h ( x ) scales as c ( x ) = langle [ h ( x prime ) - h ( x prime + x ) ]^{2}rangle_{{}x prime } app x^{2H_.}Characterizing h ( x ) with many exponents instead of a single H is expected to result in a more complete description of fractal surfaces and advance our understanding of their relevant features.
In this paper we extend the concept of multifractality to self affine fractals and demonstrate its applicability on selected examples. We show that for a class of iteratively constructed self affine functions there exists an infinite hierarchy of exponents H_{q_}describing the scaling of the qth order correlation function c_{q}( x ) app x^{{}q H_{q}}_, which is defined through the qth moments of the distribution of height differences. Furthermore, it is demonstrated on the example of random walks on a lattice that any discrete model of self affine fractals (including surfaces in growth models) results in a phase transition like behavior of H_{q_.}
The multiscaling properties of the self affine function h ( x ) can be investigated by calculating the qth order height height correlation function, which we define as
c_{q}( x ) = 1 over N ^ sum from i=1 to N ^ | h ( x_{i}) - h ( x_{i}+ x ) |^{q} , (1)
where N >> 1 is the number of points over which the average is taken, and only terms with | h ( x_{i}) - h ( x_{i}+ x ) | > 0_ are considered. This correlation function exhibits a nontrivial multiscaling behavior if
c_{q}( x ) app x^{{}q H_{q}} , (2)
with H_{q_}changing continuously with q at least for some region of the q values. For self affine functions defined on a finite interval the above scaling is expected to hold in the x << 1 limit, while in the discrete case (where there exists a lower cutoff length DELTA x as, e.g., in the lattice growth models) Eq. (2) is satisfied in the x -> inf limit. It can be shown that a continuous spectrum of H_{q_}values is not consistent with the expression h ( x ) apeq lambda^{-H}h ( lambda x ), which is valid for standard self affine functions with a single exponent H. For self affine functions with multifractal properties this expression is modified, since the local scaling of the height differences depends on x in analogy with the local scaling of the measure in a box for fractal measures. A detailed analysis of this type of scaling behavior and its implications will be discussed in Ref. [10].
Let us examine the validity of the assumption (2) on a recursively constructed fractal which is a straightforward generalization of Mandelbrot's deterministic self affine function [11], imitating the main features of a Brownian plot with H = cf12. The iteration procedure is demonstrated in Fig. 1. In each step of the recursion the intervals obtained in the previous step are replaced with the properly rescaled version of the generator, which has the form of an asymmetric letter z made of four intervals. During this procedure every interval is regarded as a diagonal of a rectangle becoming more elongated as the number of iterations k increases. The basis of the rectangle is divided into four parts and the generator replaces the intervals in such a way that its turnovers are always at analogous positions (at the first quarter and the middle of the basis). The function becomes self affine in the k -> inf limit. Depending on the parameter b_{1_}very different structures can be generated.
After the kth step, the number of intervals (boxes) along the x axis is 4^{k}and their length is equal to 4^{-k.}Denoting with N ( DELTA h ) the number of boxes in which | h ( x ) - h ( x + DELTA x ) | = DELTA h, we have N ( b_{1^{n}b}sub 2^{k-n}) = 2^{k}C_{k^{n_,}where}n = 0 , .^.^. , k. Thus
c_{q}( DELTA x ) = ^ sum from n=0 to k ^ 2^{-k}C_{k^{n}b}sub 1^{nq}b_{2^{(k-n)q} }, (3)
with DELTA x = 4^{-k.}Note that here c_{q}( x )_ is given only for discrete values of its argument; however, because of scaling, this is not expected to influence the results obtained for H_{q_.}Since Eq. (3) can be written as c_{q}( DELTA x ) = [ ( b_{1^{q}+}b_{2^{q})}/ 2 ]^{k_}we have
H_{q}= { ln [ ( b_{1^{q}+}b_{2^{q})}/ 2 ] } over { q ^ ln ( case 1 over 4 ) } . (4)
In the present approach the roughness exponent introduced earlier is H = H_{1_.}Figure 2 shows H_{q_}calculated from (4) for b_{1}= 0.8_ and b_{2}= 0.5_ and its comparison with the numerically determined data [using Eqs. (1) and (2)] for a prefractal obtained after k = 9 steps of the construction. There is a good agreement between the results; however, it is possible to carry out the numerical calculations only for q > 0 because of the large uncertainties.
Next we discuss the multifractal aspects of Brownian plots. These self affine functions are obtained by plotting the displacement h ( x ) of a particle randomly walking in one dimension as a function of time (denoted here by x). In this case the probability that h ( x ) changes by DELTA h on an interval of length DELTA x is P ( DELTA h , DELTA x ) = ( 1 / sqrt { 2 pi DELTA x } ) exp ( - DELTA h^{2}/ 2 DELTA x ). Changing the summation in (1) for integration and using the variable u = DELTA h / sqrt { 2 DELTA x } we have
c_{q}( DELTA x ) mark = int_{0^{inf}P}( DELTA h , DELTA x ) DELTA h^{q}d DELTA h
lineup = { ( 2 DELTA x )^{q/2}} over { sqrt pi } int_{0^{inf}u^{q}e^{{}-}u^{2}} d u app I DELTA x^{q/2} . (5)
The integral for u is equal to I = GAMMA [ ( q + 1 ) / 2 ] / 2 if q > -1, while it is divergent for q < - 1. This means that H_{q}= cf12_ for all values of q larger than -1, below which H_{q_}is undefined. The same calculation can be repeated for arbitrary fractional Brownian motion, leading to the same result with H instead of cf12.
The situation is rather different in the discrete case, where we can assume without the loss of generality that the particle makes a jump of unit length at every time step of unit duration. We studied this version by analytical and simulation techniques and found that the results correspond to the following behavior:
c_{q}( x ) apeq exp ( { sgn( q ) } over |q| right ) x^{{}q H_{q}} , (6)
with
H_{q}= { lpile { H_{0} for q > - 1 above - H_{0}/ q for q < - 1 , } (7)
with H_{0}= cf12_. Thus, in this case H_{q_}is defined for all q and it tends to zero as q -> - inf.
It follows from (7) that H_{q_}manifests phase transition like behavior in the vicinity of q = - 1, at which point it is not differentiable. Analogous phase transitions can be observed in some dynamical systems as well. For example, a singular point at q = - 1 followed by a constant behavior for q > - 1 was found in the D_{q_}spectrum of some simple dynamical systems described by a family of one dimensional, piecewise parabolic maps [12]. A similar dependence of the multifractal spectrum was observed in Ref. [13]. The above H_{q_}spectrum is obviously not monotonously decreasing, which is due to the fact that the distribution of the differences in h ( x ) is not a normalized measure. Figure 3 shows the comparison of simulation results with the expression (7).
Equation (7) for q > -1 is just the standard scaling of the distance of a random walker from the origin. For the q < - 1 case the following argument can be used: The terms in the sum for the height correlation function are essentially the probabilities of a given sequence of length x = n = k + l of ``up'' (k) and ``down'' (l) jumps multiplied by the absolute value of the resulting height difference n - k in power of q. Thus, c_{q}( n )_ = (1/2)^{n}tsum_{k=0^{n/2}2}n ! / [ k ! ( n - k ) ! ] | n - k |^{q_.}It is easy to show (by taking the derivative of the logarithm of the kth term of the sum) that for q < - 1 the dominant contribution to c_{q_}comes from the term with k = n / 2 - 1 (for k = n / 2, DELTA h = 0, and these sequences are not considered by definition). From this we get H_{q}= lim_{{}n -> inf } 1 /_ ( q ^ ln n ) ln (2^{q}n ! / lcurl 2^{n}[ ( n / 2 ) ! ]^{2}rcurl ). Using Stirling's formula n ! = ( n / e )^{n}( 2 pi n )^{1/2}leads to the required result H_{q}= - 1 / 2 q_. Thus, the origin of the phase transition in our case is entirely due to the discrete nature of the h ( x ) function for walks on a chain, and it is not analogous to the phase transition like behavior observed for growth models [14].
We expect that the formalism presented in this paper will be useful in the analysis of various experimentally observed physical quantities. As an important example let us mention measurements of the distribution of (i) a passive scalar rho ( x vec ) and (ii) the velocity in turbulent flows. (i) It has recently been shown that the dissipation field in turbulent jets can be described in terms of multifractal spectra [15]. Since the local dissipation rate is proportional to the squared gradient of the passive scalar, on the basis of Eqs. (1) and (2) [in which | h ( x_{i}) - h ( x_{i}+ x ) |_ is related to the local coarse grained derivative of the function] we conclude that the distribution rho ( x vec ) is a self affine function with multifractal properties. (ii) The definition of the so called velocity structure functions in turbulent flows is analogous to Eq. (1). There is experimental and theoretical evidence for the q-dependent scaling of these functions [16]. This means that the velocity field itself has to be a multiscaling self affine fractal and its properties can be investigated by the corresponding methods developed for such geometrical objects.
The examples discussed in this paper demonstrate that the multifractal analysis described above provides relevant additional information about the scaling properties of self affine functions. Furthermore, our approach has potentially interesting implications concerning the relationship of self affine fractals and fractal measures. The elaboration of a complete multifractal type formalism for the self affine case is in progress. Finally, it is straightforward to extend the present multiscaling approach to higher dimensions.
We thank F. Family, J. Kerteaasz, P. Szeaapfalusy, and T. Teaal for useful discussions. The present research was supported by the Hungarian Scientific Research Foundation Grant No. 693.