A New Approach to Characterize Disordered Structures

A New Approach to Characterize Disordered Structures J. Peinke, R. Friedrich, and A. Naert a b c Experimentalphysik II, Universität Bayreuth, D-95440 Bayreuth Institut für Theoretische Physik, Universität Stuttgart, D-70550 Stuttgart R. I. E. C„ Tohoku University, 2-1-1 Katahira, Aobaku, Sendai 980-77, Japan Z. Naturforsch. 52 a, 588-592 (1997); received June 10, 1997 A new application of the theory of Markov processes to the characterization of fractal scaling behavior is presented. We show under which condition distinct stochastic processes of a cascade lead to multifractal scaling behavior. We apply our method to the analysis of the statistical properties of the energy in turbulent fluid flow. 1. I n t r o d u c t i o n T h e r e is an actual interest to characterize disordered structures S exhibiting scaling b e h a v i o r in t e r m s of fractality or affinity, cf. [1]. T h e quantification of fractality is based on the introduction of a m e a s u r a b l e quantity Q(l) d e p e n d i n g on a selected length scale. T h e q-th p o w e r of Q(l), averaged over S, is investigated and it is e x a m i n e d w h e t h e r there exists multiscaling behavior in the limiting case / —• 0: <(Q(0)9)~/C*. (1) U n d e r this scaling assumption, scaling e x p o n e n t s Q are d e d u c e d w h i c h serve as a characterization of the statistical properties of the disordered structure of 5 . If ( q is a nonlinear function of q w e say that 5 is a multifractal or has multiaffine scaling properties [1]. For synthetically constructed c o m p l e x structures like the Cantor-set or the Sierpinski-gasket (just to mention t w o w e l l - k n o w n e x a m p l e s ) , the e x p o n e n t s Q can be calculated explicitly. P r o b l e m s c o m m o n l y arise in the investigation of natural structures with the quality of this characterization m e t h o d . A n o t h e r c o m m o n m e t h o d to characterize the statistical content of the disordered structure 5 is the evaluation of correlations. Let s(x) d e n o t e a f u n c tional value of 5 at the location x (for e x a m ple, x can be thought of as space variable but m a y also be considered as a time variable). T h e n ( • s ^ C r i ) ^ 2 ^ ) - • -sqn(xn)) is the general f o r m of an Reprint requests to Dr. J. Peinke, Fax: +49 921 552621, email: [email protected]. « - p o i n t correlation. T h e k n o w l e d g e of all n - p o i n t correlations can be regarded as a c o m p l e t e statistical characterization of 5 . T h e multiscaling analysis and the correlation analysis is linked. Let us consider the definition Q(l) := <s(rr + I) — s(x). Here Q{1) is d e n o t e d as an increment. In this case the two-point-correlations ( ^ ' ( z O ^ f o ) ) = (sq'{x\)sq2(x\ + / ) ) , where I = X2 — x\ are directly linked to ( ( Q ( l ) ) q ) . In this case it is c o m m o n to call ( ( Q ( l ) ) q ) the q-th order structure function. 2. E v o l u t i o n E q u a t i o n for the Probability Distribution In this paper w e want to present a new m e t h o d to analyze disordered structures in a m o r e general way. We will show that the two stochastic aspects just m e n t i o n e d are incorporated in our approach. Starting f r o m (1), w e k n o w that m m = j m ) ) q p m \ o a qh)- (2) T h u s instead of evaluation the scaling behavior of (1) it is m o r e general to evaluate the / - d e p e n d e n c e of the probability density distribution P(Q(l), /). T h e essential point of our work is that w e present a p r o c e d u r e w h i c h allows us to derive an evolution e q u a t i o n for the probability distribution P(Q(l), I) with respect to the size parameter / directly f r o m e x p e r i m e n t a l data. T h e first step consists in investigating h o w the quantity Q(l) at one fixed point x c h a n g e s with l. (Also other constructions of Q(l) m a y be used, like the midpoint construction: Q(l) := 0932-0784 / 97 / 0600-0561 S 06.00 © - Verlag der Zeitschrift für Naturforschung, D-72027 Tübingen Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Creative Commons Namensnennung-Keine Bearbeitung 3.0 Deutschland Lizenz. This work has been digitalized and published in 2013 by Verlag Zeitschrift für Naturforschung in cooperation with the Max Planck Society for the Advancement of Science under a Creative Commons Attribution-NoDerivs 3.0 Germany License. Zum 01.01.2015 ist eine Anpassung der Lizenzbedingungen (Entfall der Creative Commons Lizenzbedingung „Keine Bearbeitung“) beabsichtigt, um eine Nachnutzung auch im Rahmen zukünftiger wissenschaftlicher Nutzungsformen zu ermöglichen. On 01.01.2015 it is planned to change the License Conditions (the removal of the Creative Commons License condition “no derivative works”). This is to allow reuse in the area of future scientific usage. 589 J. Peinke et al. • A New Approach to Characterize Disordered Structures s(x+l/2) — s(x — 1/2).) F o r m the data w e can evaluate the n - d i m e n s i o n a l probability density distributions pmOJüQ«2),l2;...;Q(ln),ln) 3. M a r k o v - p r o c e s s a n d Multifractality or respective conditional probability distributions p(Q(ln), Qdr), lr\Q(lr-l), lr- 15 • • • i Q(U ), U ), w h e r e w e use the convention ZJ+1 < Next w e ask w h e t h e r these ?i-dimensional distributions can be exp r e s s e d by t w o - d i m e n s i o n a l probability distributions. T h i s is n o t h i n g else but investigating w h e t h e r the statistics of Qil) corresponds to a M a r k o v process. To give evidence that a M a r k o v - p r o c e s s is related to o u r p r o b l e m one should show that P(Q(U, In',...; = p(Q(D, Q(lr), lr\Q(lr-l), /„;...; Q(lr), lr- 1 I • • • i Q(l l), h) lr\Q(lr-i), lr-1). T h i s is evidently an impossible task. However, as is w e l l - k n o w n , a necessary condition for a process to b e M a r k o v i a n is the validity of the C h a p m a n K o l m o g o r o v equation [2] p(Q(h\h\Q(h)Ji)= A n important f e a t u r e of the present approach is based on the fact that these K r a m e r - M o y a l coefficients can actually be evaluated directly f r o m the given data. In the f o l l o w i n g w e w a n t to discuss s o m e further implications b a s e d on w e l l - k n o w n features of M a r k o v - p r o c e s s e s [2] w h i c h , however, have not yet b e e n put into direct c o n n e c t i o n with the p h e n o m e n o n of multifractality. (We w o u l d like to mention that the presented p r o c e d u r e m a y also be seen in analogy to the application of the F r o b e n i u s Peron operator for the d e t e r m i n a t i o n of the natural m e a s u r e of chaotic attractors [1].) 1. If D{4) = 0 P a w u l a ' s t h e o r e m implies that only the drift term D([) and the d i f f u s i o n term D{2) are non-zero. T h e K r a m e r s - M o y a l expansion (4) reduces to the F o k k e r - P l a n c k e q u a t i o n P m i h\Q(hl hMQih), h\Q(h), F o r m the C h a p m a n - K o l m o g o r o v equation an evolution equation for the conditional probabilities can b e d e d u c e d , which is k n o w n as the K r a m e r s - M o y a l expansion (4) —P(Q(l),l) al £ k= 1 D{k)(Q(l), I) P(Q(l)Jy dQ(l) H e r e , the K r a m e r s - M o y a l coefficients D(k\Q(l), I) are d e f i n e d by the f o l l o w i n g conditional m o m e n t s : M ( k ) m \ / , AI) = -L • (Q{1 - Al) - Q(l))k D(k\Q(l),l)= I dQ{1 p(Q(l D(1\Q{1), l)P(Q(l), I). (7) h)dQih). T h e validity of this equations yields strong hints that the stochastic process is Markovian. = 0P(Q(0,0 (3) dQ(l) Jp(Q(h), l) = - Al) - Al), / - Al\Q(l), lim — M l k \ Q ( l ) , l , A l ) Ai-+o n\ (5) I) (6) 2. F r o m (4) and (7) the evolution equation for a single event Q(l) at point x in f o r m of a L a n g e v i n e q u a t i o n can b e derived [2]: - ^Q(i) al =g m ) , o + w ( o , o m . (8) T h e f u n c t i o n g d e s c r i b e s the deterministic evolution of Q{1), w h e r e a s h takes into account the stochastic fluctuations. In the case of the Fokker-Planck equation the f u n c t i o n s g and h are given by D{ 1 > and Di2), w h e r e h oc VD{2\ T h e noise t e r m r has the properties of ^-correlated g a u s s i a n noise. 3. F r o m (4), e q u a t i o n s f o r the m o m e n t s ( ( Q ( l ) ) q ) are obtained by m u l t i p l y i n g (4) with Qq and successively integrating o v e r Q(l) [3]: - > < < » < ) = E T T ^ W - " ) n=l (9 > 4. If the K r a m e r - M o y a l coefficients are D( I (10) 590 J. Peinke et al. • A New Approach to Characterize Disordered Structures the scaling behavior of (1) is g u a r a n t e e d and w e obtain f r o m (9) For the simple case of a F o k k e r - P l a n c k e q u a t i o n w e see n o w that £> (1) = diQ/Und Da) = d2Q2/l'\mplies multiscaling with (q = d\q — d2q{q — 1). T h e quadratic Q - d e p e n d e n c e of D{2) c o r r e s p o n d s to a purely multiplicative noise process, as w e can see f r o m the discussion of point 2 above. 5. For the case of the validity of a F o k k e r - P l a n c k equation the stationary solution <•««>•»= is k n o w n , w h e r e QQ is a constant. Taking the condition for scaling behavior of point four, (10), that the drift term is linear and the d i f f u s i o n term is quadratic in Q, w e find that the probability density distribution P ( Q ( / ) , I) evaluated by (12) is not normalizable. T h e stationary distribution is s i m p l y the ^ - f u n c t i o n 6(Q(l)). Inclusion of a small additive noise term yields a distribution w h i c h for large values of Q(l) has a powerlaw f o r m , which is a characteristic feature of a Levy-distribution. In this case the m o m e n t s ( Q q ) diverge for higher q values and t h u s the derivation of (9) b e c o m e s questionable [4]. A n a r g u m e n t that the above m e n t i o n e d results on ( Q q ) still hold is that the p o w e r law tails of the distribution at infinity are only valid for the stationary solution. For an initial probability distribution at large scales I it w o u l d take an infinite c a s c a d e ( d e v e l o p m e n t with the evolution equation) to create such w i n g s at infinity. T h e r e f o r e , for any natural structure 5 , w h e r e typically scaling behavior is only f o u n d in a finite interval of scales /, w e expect that these divergences d o not affect the results discussed here. 6. We want to point out that h a v i n g s h o w n the Markovian properties any n - p o i n t statistics can be derived f r o m the t w o - p o i n t statistics P(Q(h),h\Q(li),l\)In the case of the applicability of the Fokker-Planck equation the n - p o i n t statistics can be explicitly given by the k n o w l e d g e of the t w o K r a m e r - M o y a l coefficients D ( 1 ) and D{2) [2]. 4. A p p l i c a t i o n to E x p e r i m e n t s Finally, let us mention s o m e verifications and applications of the m e t h o d presented above: F o r the case of fully developed turbulence, a m a j o r challenge consists in explaining the statistics of velocity increments V(l), cf. [5]. Typically the inc r e m e n t s of the velocity c o m p o n e n t in the direction of the distance vector are taken. Investigation of experimental data has recently shown that the statistics of the velocity increments can b e characterized by a F o k k e r - P l a n c k equation with D ( 1 ) = 7 V ( l ) / l and Di2) = a(l) + b(V(l))2/l, [6]. Here, 7 turns out to be a p p r o x i m a t e l y — 1 / 3 , which is clearly related with the scaling behavior of the second m o m e n t s , which scale like ( V ( l ) 2 ) « C2 « 2 / 3 . H e r e w e want to report on further new findings on the disorder in turbulence. It is of central interest to d e t e r m i n e the statistics of the energy dissipation on d i f f e r e n t length scales. Following the suggestion of K o l m o g o r o v and Obukhov, w e evaluate the energy dissipation at scale I according to [7]: 1 5 f i w = i>x+l/2 -L / ^ / 2 \ 2 y ^ (,3> w h e r e u denotes the kinematic viscosity. It has b e e n postulated that nongaussian statistics in the velocity i n c r e m e n t s is d u e to a log-normal distribution of e. We take log(e) as our quantity Q. T h e variance of log(e) w a s a s s u m e d to be logarithmic in the length scale. We have already verified the C h a p m a n - K o l m o g o r o v e q u a t i o n [8]. Next w e have evaluated the K r a m e r s M o y a l coefficients, and w e saw that the fourth c o e f ficient is close to zero. Figure 1 presents the first t w o K r a m e r s - M o y a l coefficients, D{X\Q, I) and D(2\Q, I). Both coefficients have been evaluated for several scales covering the w h o l e inertial range. (The inertial range in turbulence d e n o t e s the range of length scales w h e r e scaling beh a v i o r is e x p e c t e d to occur. This range is b o u n d e d by a large length scale L d o m i n a t e d by the b o u n d a r y conditions, like the size of the flow, and the viscous or K o l m o g o r o v length scale 77, w h e r e dissipation d o m i nates the flow.) For convenience w e have used in o u r evaluation a logarithmic length scale I = In r , w h i c h c o r r e s p o n d s in evaluation rDln) in a linear scale. In the following, r d e n o t e s the linear scale. (Note that these K r a m e r s - M o y a l coefficients presented are o b tained for a finite scale ratio of r\/r2 = 1 0 2 , just 591 J. Peinke et al. • A New Approach to Characterize Disordered Structures Fig. 1. Drift and diffusion coefficients, Z\(1) = D(l) - F{1) and Da) for the values of the scale r/r7=400, 200, 100, 50, r/ being the Kolmogorov viscous scale. These coefficients were evaluated from experimental data of a free jet with a Reynolds number R\ of 328 [9]. b e l o w w h i c h the resolution of the velocity is not sufficient any m o r e . ) To m a k e the drift t e r m s c o i n c i d e an / - d e p e n d e n t additive term F(l) h a s been used. T h e o c c u r e n c e of this additive term can b e justified. In fact it is determined by the constraint of c o n s e r v a t i o n of m e a n energy. In the central region, the drift coefficient is linear in log(er) = Q and the d i f f u s i o n coefficient is constant. F u r t h e r m o r e , the slope of D ( 1 ) and the value of Da) are both scale invariant in the inertial range: Dil\Q) = 7 Da\Q) = D (Q-(Q)) + (14) A s initial condition at the integral scale L w e take a sharp distribution: = HQL ~ log(e)), -(^)-<QM>)2/2/i25 P(Q(r)\Q(L))= (15) (16) A\/2ir where the m e a n and the variance are given by [2] F(l), with 7 « 0 . 2 a n d D « 0 . 0 3 . T h u s , w e can s u m m a r i z e that our e m p i r i c a l results s h o w that the statistics of the quantity l o g ( e r ) is d e t e r m i n e d by a Fokker-Planck equation with a linear drift and a constant d i f f u s i o n coefficient. S u c h a F o k k e r - P l a n c k equation is related to an O r n s t e i n - U h l e n b e c k process. T h e positive slope o f £ > ( 1 ) indicates that there is n o stationary probability distribution [2]. Nevertheless, the f o l l o w i n g conclusions on the evolution of the probability distributions can be d r a w n . P(QL) where (c) d e n o t e s the m e a n dissipation. T h e conditional distribution f u n c t i o n for r < L, P(Q(r)\Q(L)), is then G a u s s i a n : (Q(r)) = ^ = (Q(L))-A2/2, g t e r - ' ) - In the r a n g e of scales r power law: D ( <»> L, A2 can b e written as a r\~2~i Such a d e p e n d e n c e h a s been predicted by C a s t a i n g [10] for a quantity A2 proportional to A2. For large scales r , / l 2 b e c o m e s small and m a y be a p p r o x i m a t e d by a logarithmic law leading to scaling behavior in the velocity increments. 592 J. Peinke et al. • A New Approach to Characterize Disordered Structures 5. S u m m a r y To s u m m a r i z e , w e have presented a new m e t h o d to analyze the statistical content of a disordered structure. We have s h o w n h o w this a p p r o a c h is related to multifractal scaling. F u r t h e r m o r e , is has been s h o w n that this m e t h o d can be applied to e x p e r i m e n t a l data; therefore w e have presented the analysis of the energy statistics in turbulence. F r o m this w e f o u n d that the statistics are very close to a s i m p l e w e l l - k n o w n stochastic process, n a m e l y the O r n s t e i n - U h l e n b e c k process. If o n e verified that the disorder of a s y s t e m can be attributed to a stochastic process, one has the opportunity to use m a n y results explaining details of the disorder. T h u s w e saw for the energy cascade in turbulence that the positive slope of the drift term leads to the nonstationary spreading of the width of [1] R. Badii and A. Politi, Complexity, Cambridge University Press, Cambridge 1997; - B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco 1982; - K.J. Falconer, Fractal Geometry, Mathematical Foundations and Applications, John Wiley, Chichester 1990; - J. Feder, Fractals, Plenum Press, New York 1988); - T. Vicsek, Fractal Growth Phenomena, World Scientific, Singapore 1992. [2] P. Hänggi and H. Thomas, Physics Reports 88. 207 (1982); - N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier, Amsterdam 1990; - H. Risken, The Fokker-Planck Equation, SpringerVerlag Berlin 1984. [3] For the integartion it is assumed that the probability distributions P(Q(l), I) decay at infinity faster than any power of Q. the probability distributions. T h i s b e h a v i o r c a u s e s the often discussed intermittency e f f e c t s in turbulence directly. T h e s e intermittency e f f e c t s are usually discussed in terms of multifractal scaling, but u p to now no satisfying conclusion could be d r a w n of the m u l tifractal scaling ansatz. With the analysis presented here, w e h o p e to have given s o m e new insight into the statistics of the energy c a s c a d e in turbulence. Acknowledgements Helpful discussions with M a r t i n Greiner, Peter Lipa, and the m e m b e r s of E N G A D Y N are a c k n o w l edged. J P a c k n o w l e d g e s financial support by the Deutsche F o r s c h u n g s g e m e i n s c h a f t . T h e p a p e r was presented at 6th A n n u a l M e e t i n g of E N G A D Y N , Tüchersfeld 1996. [4] private communication with Peter Lipa. [5] K. R. Sreenivasan and R. A. Antonia, Annu. Rev. Fluid Mech. 29, 435 (1997). [6] R. Friedrich and J. Peinke, Physica D 102, 147 (1997); - R. Friedrich and J. Peinke, Phys. Rev. Lett. 78. 863 (1997). [7] A. M. Obukhov, J. Fluid Mech. 13, 77 (1962); A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962). [8] A. Naert, R. Friedrich, and J. Peinke, A Stochastic Equation for the Energy Cascade in Turbulence, Phys. Rev. E (in press). [9] B. Chabaud, A. Naert, J. Peinke, F. Chillä, B. Castaing, and B. Hebral, Phys. Rev. Lett., 73,3227 (1994). [10] B. Castaing, Y. Gagne, and E. Hopfinger, Physica D 46. 177, (1990). 

A New Approach to Characterize Disordered Structures J. Peinke, R. Friedrich, and A. Naert a b c Experimentalphysik II, Universität Bayreuth, D-95440 Bayreuth Institut für Theoretische Physik, Universität Stuttgart, D-70550 Stuttgart R. I. E. C„ Tohoku University, 2-1-1 Katahira, Aobaku, Sendai 980-77, Japan Z. Naturforsch. 52 a, 588-592 (1997); received June 10, 1997 A new application of the theory of Markov processes to the characterization of fractal scaling behavior is presented. We show under which condition distinct stochastic processes of a cascade lead to multifractal scaling behavior. We apply our method to the analysis of the statistical properties of the energy in turbulent fluid flow. 1. I n t r o d u c t i o n T h e r e is an actual interest to characterize disordered structures S exhibiting scaling b e h a v i o r in t e r m s of fractality or affinity, cf. [1]. T h e quantification of fractality is based on the introduction of a m e a s u r a b l e quantity Q(l) d e p e n d i n g on a selected length scale. T h e q-th p o w e r of Q(l), averaged over S, is investigated and it is e x a m i n e d w h e t h e r there exists multiscaling behavior in the limiting case / —• 0: <(Q(0)9)~/C*. (1) U n d e r this scaling assumption, scaling e x p o n e n t s Q are d e d u c e d w h i c h serve as a characterization of the statistical properties of the disordered structure of 5 . If ( q is a nonlinear function of q w e say that 5 is a multifractal or has multiaffine scaling properties [1]. For synthetically constructed c o m p l e x structures like the Cantor-set or the Sierpinski-gasket (just to mention t w o w e l l - k n o w n e x a m p l e s ) , the e x p o n e n t s Q can be calculated explicitly. P r o b l e m s c o m m o n l y arise in the investigation of natural structures with the quality of this characterization m e t h o d . A n o t h e r c o m m o n m e t h o d to characterize the statistical content of the disordered structure 5 is the evaluation of correlations. Let s(x) d e n o t e a f u n c tional value of 5 at the location x (for e x a m ple, x can be thought of as space variable but m a y also be considered as a time variable). T h e n ( • s ^ C r i ) ^ 2 ^ ) - • -sqn(xn)) is the general f o r m of an Reprint requests to Dr. J. Peinke, Fax: +49 921 552621, email: [email protected]. « - p o i n t correlation. T h e k n o w l e d g e of all n - p o i n t correlations can be regarded as a c o m p l e t e statistical characterization of 5 . T h e multiscaling analysis and the correlation analysis is linked. Let us consider the definition Q(l) := <s(rr + I) — s(x). Here Q{1) is d e n o t e d as an increment. In this case the two-point-correlations ( ^ ' ( z O ^ f o ) ) = (sq'{x\)sq2(x\ + / ) ) , where I = X2 — x\ are directly linked to ( ( Q ( l ) ) q ) . In this case it is c o m m o n to call ( ( Q ( l ) ) q ) the q-th order structure function. 2. E v o l u t i o n E q u a t i o n for the Probability Distribution In this paper w e want to present a new m e t h o d to analyze disordered structures in a m o r e general way. We will show that the two stochastic aspects just m e n t i o n e d are incorporated in our approach. Starting f r o m (1), w e k n o w that m m = j m ) ) q p m \ o a qh)- (2) T h u s instead of evaluation the scaling behavior of (1) it is m o r e general to evaluate the / - d e p e n d e n c e of the probability density distribution P(Q(l), /). T h e essential point of our work is that w e present a p r o c e d u r e w h i c h allows us to derive an evolution e q u a t i o n for the probability distribution P(Q(l), I) with respect to the size parameter / directly f r o m e x p e r i m e n t a l data. T h e first step consists in investigating h o w the quantity Q(l) at one fixed point x c h a n g e s with l. (Also other constructions of Q(l) m a y be used, like the midpoint construction: Q(l) := 0932-0784 / 97 / 0600-0561 S 06.00 © - Verlag der Zeitschrift für Naturforschung, D-72027 Tübingen 589 J. Peinke et al. • A New Approach to Characterize Disordered Structures s(x+l/2) — s(x — 1/2).) F o r m the data w e can evaluate the n - d i m e n s i o n a l probability density distributions pmOJüQ«2),l2;...;Q(ln),ln) 3. M a r k o v - p r o c e s s a n d Multifractality or respective conditional probability distributions p(Q(ln), Qdr), lr\Q(lr-l), lr- 15 • • • i Q(U ), U ), w h e r e w e use the convention ZJ+1 < Next w e ask w h e t h e r these ?i-dimensional distributions can be exp r e s s e d by t w o - d i m e n s i o n a l probability distributions. T h i s is n o t h i n g else but investigating w h e t h e r the statistics of Qil) corresponds to a M a r k o v process. To give evidence that a M a r k o v - p r o c e s s is related to o u r p r o b l e m one should show that P(Q(U, In',...; = p(Q(D, Q(lr), lr\Q(lr-l), /„;...; Q(lr), lr- 1 I • • • i Q(l l), h) lr\Q(lr-i), lr-1). T h i s is evidently an impossible task. However, as is w e l l - k n o w n , a necessary condition for a process to b e M a r k o v i a n is the validity of the C h a p m a n K o l m o g o r o v equation [2] p(Q(h\h\Q(h)Ji)= A n important f e a t u r e of the present approach is based on the fact that these K r a m e r - M o y a l coefficients can actually be evaluated directly f r o m the given data. In the f o l l o w i n g w e w a n t to discuss s o m e further implications b a s e d on w e l l - k n o w n features of M a r k o v - p r o c e s s e s [2] w h i c h , however, have not yet b e e n put into direct c o n n e c t i o n with the p h e n o m e n o n of multifractality. (We w o u l d like to mention that the presented p r o c e d u r e m a y also be seen in analogy to the application of the F r o b e n i u s Peron operator for the d e t e r m i n a t i o n of the natural m e a s u r e of chaotic attractors [1].) 1. If D{4) = 0 P a w u l a ' s t h e o r e m implies that only the drift term D([) and the d i f f u s i o n term D{2) are non-zero. T h e K r a m e r s - M o y a l expansion (4) reduces to the F o k k e r - P l a n c k e q u a t i o n P m i h\Q(hl hMQih), h\Q(h), F o r m the C h a p m a n - K o l m o g o r o v equation an evolution equation for the conditional probabilities can b e d e d u c e d , which is k n o w n as the K r a m e r s - M o y a l expansion (4) —P(Q(l),l) al £ k= 1 D{k)(Q(l), I) P(Q(l)Jy dQ(l) H e r e , the K r a m e r s - M o y a l coefficients D(k\Q(l), I) are d e f i n e d by the f o l l o w i n g conditional m o m e n t s : M ( k ) m \ / , AI) = -L • (Q{1 - Al) - Q(l))k D(k\Q(l),l)= I dQ{1 p(Q(l D(1\Q{1), l)P(Q(l), I). (7) h)dQih). T h e validity of this equations yields strong hints that the stochastic process is Markovian. = 0P(Q(0,0 (3) dQ(l) Jp(Q(h), l) = - Al) - Al), / - Al\Q(l), lim — M l k \ Q ( l ) , l , A l ) Ai-+o n\ (5) I) (6) 2. F r o m (4) and (7) the evolution equation for a single event Q(l) at point x in f o r m of a L a n g e v i n e q u a t i o n can b e derived [2]: - ^Q(i) al =g m ) , o + w ( o , o m . (8) T h e f u n c t i o n g d e s c r i b e s the deterministic evolution of Q{1), w h e r e a s h takes into account the stochastic fluctuations. In the case of the Fokker-Planck equation the f u n c t i o n s g and h are given by D{ 1 > and Di2), w h e r e h oc VD{2\ T h e noise t e r m r has the properties of ^-correlated g a u s s i a n noise. 3. F r o m (4), e q u a t i o n s f o r the m o m e n t s ( ( Q ( l ) ) q ) are obtained by m u l t i p l y i n g (4) with Qq and successively integrating o v e r Q(l) [3]: - > < < » < ) = E T T ^ W - " ) n=l (9 > 4. If the K r a m e r - M o y a l coefficients are D( I (10) 590 J. Peinke et al. • A New Approach to Characterize Disordered Structures the scaling behavior of (1) is g u a r a n t e e d and w e obtain f r o m (9) For the simple case of a F o k k e r - P l a n c k e q u a t i o n w e see n o w that £> (1) = diQ/Und Da) = d2Q2/l'\mplies multiscaling with (q = d\q — d2q{q — 1). T h e quadratic Q - d e p e n d e n c e of D{2) c o r r e s p o n d s to a purely multiplicative noise process, as w e can see f r o m the discussion of point 2 above. 5. For the case of the validity of a F o k k e r - P l a n c k equation the stationary solution <•««>•»= is k n o w n , w h e r e QQ is a constant. Taking the condition for scaling behavior of point four, (10), that the drift term is linear and the d i f f u s i o n term is quadratic in Q, w e find that the probability density distribution P ( Q ( / ) , I) evaluated by (12) is not normalizable. T h e stationary distribution is s i m p l y the ^ - f u n c t i o n 6(Q(l)). Inclusion of a small additive noise term yields a distribution w h i c h for large values of Q(l) has a powerlaw f o r m , which is a characteristic feature of a Levy-distribution. In this case the m o m e n t s ( Q q ) diverge for higher q values and t h u s the derivation of (9) b e c o m e s questionable [4]. A n a r g u m e n t that the above m e n t i o n e d results on ( Q q ) still hold is that the p o w e r law tails of the distribution at infinity are only valid for the stationary solution. For an initial probability distribution at large scales I it w o u l d take an infinite c a s c a d e ( d e v e l o p m e n t with the evolution equation) to create such w i n g s at infinity. T h e r e f o r e , for any natural structure 5 , w h e r e typically scaling behavior is only f o u n d in a finite interval of scales /, w e expect that these divergences d o not affect the results discussed here. 6. We want to point out that h a v i n g s h o w n the Markovian properties any n - p o i n t statistics can be derived f r o m the t w o - p o i n t statistics P(Q(h),h\Q(li),l\)In the case of the applicability of the Fokker-Planck equation the n - p o i n t statistics can be explicitly given by the k n o w l e d g e of the t w o K r a m e r - M o y a l coefficients D ( 1 ) and D{2) [2]. 4. A p p l i c a t i o n to E x p e r i m e n t s Finally, let us mention s o m e verifications and applications of the m e t h o d presented above: F o r the case of fully developed turbulence, a m a j o r challenge consists in explaining the statistics of velocity increments V(l), cf. [5]. Typically the inc r e m e n t s of the velocity c o m p o n e n t in the direction of the distance vector are taken. Investigation of experimental data has recently shown that the statistics of the velocity increments can b e characterized by a F o k k e r - P l a n c k equation with D ( 1 ) = 7 V ( l ) / l and Di2) = a(l) + b(V(l))2/l, [6]. Here, 7 turns out to be a p p r o x i m a t e l y — 1 / 3 , which is clearly related with the scaling behavior of the second m o m e n t s , which scale like ( V ( l ) 2 ) « C2 « 2 / 3 . H e r e w e want to report on further new findings on the disorder in turbulence. It is of central interest to d e t e r m i n e the statistics of the energy dissipation on d i f f e r e n t length scales. Following the suggestion of K o l m o g o r o v and Obukhov, w e evaluate the energy dissipation at scale I according to [7]: 1 5 f i w = i>x+l/2 -L / ^ / 2 \ 2 y ^ (,3> w h e r e u denotes the kinematic viscosity. It has b e e n postulated that nongaussian statistics in the velocity i n c r e m e n t s is d u e to a log-normal distribution of e. We take log(e) as our quantity Q. T h e variance of log(e) w a s a s s u m e d to be logarithmic in the length scale. We have already verified the C h a p m a n - K o l m o g o r o v e q u a t i o n [8]. Next w e have evaluated the K r a m e r s M o y a l coefficients, and w e saw that the fourth c o e f ficient is close to zero. Figure 1 presents the first t w o K r a m e r s - M o y a l coefficients, D{X\Q, I) and D(2\Q, I). Both coefficients have been evaluated for several scales covering the w h o l e inertial range. (The inertial range in turbulence d e n o t e s the range of length scales w h e r e scaling beh a v i o r is e x p e c t e d to occur. This range is b o u n d e d by a large length scale L d o m i n a t e d by the b o u n d a r y conditions, like the size of the flow, and the viscous or K o l m o g o r o v length scale 77, w h e r e dissipation d o m i nates the flow.) For convenience w e have used in o u r evaluation a logarithmic length scale I = In r , w h i c h c o r r e s p o n d s in evaluation rDln) in a linear scale. In the following, r d e n o t e s the linear scale. (Note that these K r a m e r s - M o y a l coefficients presented are o b tained for a finite scale ratio of r\/r2 = 1 0 2 , just 591 J. Peinke et al. • A New Approach to Characterize Disordered Structures Fig. 1. Drift and diffusion coefficients, Z\(1) = D(l) - F{1) and Da) for the values of the scale r/r7=400, 200, 100, 50, r/ being the Kolmogorov viscous scale. These coefficients were evaluated from experimental data of a free jet with a Reynolds number R\ of 328 [9]. b e l o w w h i c h the resolution of the velocity is not sufficient any m o r e . ) To m a k e the drift t e r m s c o i n c i d e an / - d e p e n d e n t additive term F(l) h a s been used. T h e o c c u r e n c e of this additive term can b e justified. In fact it is determined by the constraint of c o n s e r v a t i o n of m e a n energy. In the central region, the drift coefficient is linear in log(er) = Q and the d i f f u s i o n coefficient is constant. F u r t h e r m o r e , the slope of D ( 1 ) and the value of Da) are both scale invariant in the inertial range: Dil\Q) = 7 Da\Q) = D (Q-(Q)) + (14) A s initial condition at the integral scale L w e take a sharp distribution: = HQL ~ log(e)), -(^)-<QM>)2/2/i25 P(Q(r)\Q(L))= (15) (16) A\/2ir where the m e a n and the variance are given by [2] F(l), with 7 « 0 . 2 a n d D « 0 . 0 3 . T h u s , w e can s u m m a r i z e that our e m p i r i c a l results s h o w that the statistics of the quantity l o g ( e r ) is d e t e r m i n e d by a Fokker-Planck equation with a linear drift and a constant d i f f u s i o n coefficient. S u c h a F o k k e r - P l a n c k equation is related to an O r n s t e i n - U h l e n b e c k process. T h e positive slope o f £ > ( 1 ) indicates that there is n o stationary probability distribution [2]. Nevertheless, the f o l l o w i n g conclusions on the evolution of the probability distributions can be d r a w n . P(QL) where (c) d e n o t e s the m e a n dissipation. T h e conditional distribution f u n c t i o n for r < L, P(Q(r)\Q(L)), is then G a u s s i a n : (Q(r)) = ^ = (Q(L))-A2/2, g t e r - ' ) - In the r a n g e of scales r power law: D ( <»> L, A2 can b e written as a r\~2~i Such a d e p e n d e n c e h a s been predicted by C a s t a i n g [10] for a quantity A2 proportional to A2. For large scales r , / l 2 b e c o m e s small and m a y be a p p r o x i m a t e d by a logarithmic law leading to scaling behavior in the velocity increments. 592 J. Peinke et al. • A New Approach to Characterize Disordered Structures 5. S u m m a r y To s u m m a r i z e , w e have presented a new m e t h o d to analyze the statistical content of a disordered structure. We have s h o w n h o w this a p p r o a c h is related to multifractal scaling. F u r t h e r m o r e , is has been s h o w n that this m e t h o d can be applied to e x p e r i m e n t a l data; therefore w e have presented the analysis of the energy statistics in turbulence. F r o m this w e f o u n d that the statistics are very close to a s i m p l e w e l l - k n o w n stochastic process, n a m e l y the O r n s t e i n - U h l e n b e c k process. If o n e verified that the disorder of a s y s t e m can be attributed to a stochastic process, one has the opportunity to use m a n y results explaining details of the disorder. T h u s w e saw for the energy cascade in turbulence that the positive slope of the drift term leads to the nonstationary spreading of the width of [1] R. Badii and A. Politi, Complexity, Cambridge University Press, Cambridge 1997; - B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco 1982; - K.J. Falconer, Fractal Geometry, Mathematical Foundations and Applications, John Wiley, Chichester 1990; - J. Feder, Fractals, Plenum Press, New York 1988); - T. Vicsek, Fractal Growth Phenomena, World Scientific, Singapore 1992. [2] P. Hänggi and H. Thomas, Physics Reports 88. 207 (1982); - N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier, Amsterdam 1990; - H. Risken, The Fokker-Planck Equation, SpringerVerlag Berlin 1984. [3] For the integartion it is assumed that the probability distributions P(Q(l), I) decay at infinity faster than any power of Q. the probability distributions. T h i s b e h a v i o r c a u s e s the often discussed intermittency e f f e c t s in turbulence directly. T h e s e intermittency e f f e c t s are usually discussed in terms of multifractal scaling, but u p to now no satisfying conclusion could be d r a w n of the m u l tifractal scaling ansatz. With the analysis presented here, w e h o p e to have given s o m e new insight into the statistics of the energy c a s c a d e in turbulence. Acknowledgements Helpful discussions with M a r t i n Greiner, Peter Lipa, and the m e m b e r s of E N G A D Y N are a c k n o w l edged. J P a c k n o w l e d g e s financial support by the Deutsche F o r s c h u n g s g e m e i n s c h a f t . T h e p a p e r was presented at 6th A n n u a l M e e t i n g of E N G A D Y N , Tüchersfeld 1996. [4] private communication with Peter Lipa. [5] K. R. Sreenivasan and R. A. Antonia, Annu. Rev. Fluid Mech. 29, 435 (1997). [6] R. Friedrich and J. Peinke, Physica D 102, 147 (1997); - R. Friedrich and J. Peinke, Phys. Rev. Lett. 78. 863 (1997). [7] A. M. Obukhov, J. Fluid Mech. 13, 77 (1962); A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962). [8] A. Naert, R. Friedrich, and J. Peinke, A Stochastic Equation for the Energy Cascade in Turbulence, Phys. Rev. E (in press). [9] B. Chabaud, A. Naert, J. Peinke, F. Chillä, B. Castaing, and B. Hebral, Phys. Rev. Lett., 73,3227 (1994). [10] B. Castaing, Y. Gagne, and E. Hopfinger, Physica D 46. 177, (1990).