Peter Carruthers Department of Physics, University of Arizona Tucson, Arizona 85721, USA

Transcription

1 421 FRACTAL ENGINEERING : APPLICATIONS TO HADRONIC MULTIPLICITY DISTRIBUTIONS Peter Carruthers Department of Physics, University of Arizona Tucson, Arizona 85721, USA ABSTRACT We review the use of fractal techniques to quantify the texture of irregular distributions of points, such as occur in rapidity histograms. Even when scaling ( simple or multiple) does not occur, it is possible to formulate the correlation and moment analysis for various resolutions in a form that is useful. For example, the bin averaged factorial moment approach is close to the recently introduced strip moments. The latter are almost the same as the correlation dimensions commonly used in nonlinear dynamics. The strip moments have the advantage of removing fluctuations for high resolution due to the arbitrary placement of bin walls, particularly for spike events and high-order moments.

2 rntroduction Fractals have by now become a household word. "Textbook fractals" are but a special case of more sophisticated methods of analyzing the geometrical texture of highly irregular systems. The simplest fractal structures exhibit scaling structure over a significant range of variables. Many other systems have multiple scales, inspiring analysis by multifractal techniques. The latter have been found useful for many systems, including galaxy distributions, turbulence and multihadron rapidity histograms. We anticipate that such analyses will be useful for designing detectors that can handle large numbers of particles at high event rates. Non-scaling correlations are at least as common as scaling properties. It is of great interest to formulate data analysis in a way that is tuned to reveal scaling when it exists and to exhibit deviations from scaling when true correlation lengths exist. In the case of number counts (e.g., hadrons and galaxies), it is very difficult to measure density correlation functions beyond the third order (even the latter is hard). However, multiplicity moments, which are integrals of the correlation functions over variable volumes of phase space, can be measured to fifth order1 ). In this way, information about scaling, or the lack thereof, can be obtained. The best example is the bin-averaged factorial moment technique of Bialas and Peschanski2). The Tucson group hae proposed3) a modified version of this analysis, which eliminates spurious statistical fluctuations at high resolution and connects directly to the hierarchy of correlation dimensions used in nonlinear dynamics (valid only in scaling regimes). 2. BASICS OF FRACTAL DIMENSIONS By now many formulations of "the" fractal dimension exist. But most mathe matics texts have not kept up with recent research developments, which show that a complete description requires, in general, an infinite number of dimensions, discrete or continuous. "The" fractal dimension has no more content than the average of a function or the average ii of a multiplicity distribution. To review the most useful dimension concepts, consider Fig. 1, where a large set of points generated by a two-dimensional time series Xn+I f(xn, Yn ), Yn+ I g(xn, Yn) is covered by squares of side length e, the latter being variable. (The best example is the Henon map4l. ) If e gets really small, the series has to be run a long time ( > 104 steps) to get good results. (The Henon map is a nice example of self-similarity: choosing tiny boxes inside one of the strips produces the same structure upon magnification.) The primeval questions to ask are: 1. Is the box occupied? Find the total number N( e ) of occupied boxes. :3. How many points are located within a distance e of a chosen point? 2. What is the probability P; that the box is occupied? This gives weight to the number of particles in the box. These questions are closely related, but have different advantages. In the case of simple scaling, the limits given below will exist. We define: 1. Box Counting Dimension -o DH lim ln ( e) ln(l/e) N (1)

3 423 ; p; lnp; Dr I.1m ln E o L::> B( - lx.-x,i) tj N(N-1) V Jim o ln E 2. Information Dimension. 3. Correlation Dimension (2) (3) In (3) N is the number of steps in the time series, taken as N -+ oo, which is > 1 04 typically5). () is of course unity when I x; Xj I < E. Note that if p; 1/N(E) ( 2 ) reduces to (1 ). By Boltzmann's argument, then Dr :::; DH. H stands for Hausdorff, whose definition of dimension is more subtle than box counting. In a large set of examples, numerical experiments on simple time series with scaling show5l that (4 ) Equations (1-3) are connected through the Renyi concept of entropy6) for a con tinuous dimension Dq : 1 1. ln L;; (p; )q Dq. (5) 1m q - 1, o ln E We leave it as an exercise to show that Do DH, D1 Dr and Dz v. Further analysis "leads to a transformation to the f ( o: ) curve, the spectrum of fractal dimensions. The correlation dimension (3) is also the integral of the 2-particle densitiy correlation. Higher correlations lead to (integral) higher correlation dimensions7l. Note that although the original Renyi concept used true probabilities, current works use individual event frequencies, and then average over events. This is also conceptually different from running a long-time series. This analogue of the ergodic "theorem" is somewhat tricky and not well understood. - _ APPLICATIONS TO MULTIHADRON DATA We note the following for data analysis. For hadrons, we have a large number of events, but each has a small number of particles. We cannot superimpose events to reconstruct a fractal because there is an effective noise due to impact parameter and leading particle effects. Hence, Box Counting is not usefuj8). However, the Correlation Dimension cares only about relative distances, and, therefore, can be used to see if the events are samples from a strange attractor. Also, the generalized Renyi ( multifrac tal) method remembers event structure if one uses event frequencies rather than true probabilities. Next, we review the well-known "bin-averaged" factorial moment method2l. There are M bins of equal width oy; the idea i s t o study the dependence of the facto rial moment as a function of resolution oy. With nk denoting the number of charged particles in bin k, and normalizing locally, we have9-io) (6) The domains nk are the squares in Fig. 2. Pk is the average density in bin k. p(y), p2 ( Y1, Y2 ) are the one and two-partice inclusive rapidity distributions. Extensions to higher orders are found in the literature. We remark:

4 Knowledge of (J2 implies knowledge of F2 ; Eq. (6) is an identity. 2. Poissonian count statistics imply F2 1 (this is true to all orders of Fv. The fact that F2 f' 1 implies the existence of correlations. 3. Scaling of p2 implies scaling of F2 (see the later discussion). 4. In most systems, when scaling occurs, it occurs in the cumulant correlation p2 - pf. In that case, F2 has mixed scaling properties. C2 5. Experimental C2 fits to exponential or Gaussian forms do not scale, and, of course, must give (and do) a good description10l of F2 ( oy). 6. For higher order Fp, one expands i n cumulant moments Kp, as in a phase shift analysis10l. For hadrons at collider energy the Kv are decreasing as p increases, and no 1-d scaling is observed. For galaxy counts, Kv increases with p and good scaling is observed11 l. 4. FACTORIAL MOMENTS, STATISTICAL FLUCTUATIONS There are several tricky problems in the statistical analysis of events having a relatively small number of particles. For example, there is the "empty bin effect" : When the resolution is increased sufficiently no bin has more than one particle and all Fv mo ments vanish. An approach to this problem has been proposed1 2l by an Arizona/N A22 collaboration. Another pair of difficulties in the bin approach is: 1. Nearby points separated by a bin wall do not contribute (see Fig. 3). 2. More seriously, the most interesting fluctuations (spikes) give fluctuating contri butions to the Fv in the most interesting region (small oy and large p ). In Fig. 3, we give an example for nine particles in 28y split into 5 and 4 in oy. For F4 we get a weight for 28y and for oy we get This feature accounts mostly for the large errors at small oy. A few (interesting) spike events decrease precision in exactly the most interesting domain. 5. STRIP MOMENTS Now consider integration of p2 over the strip domain of Fig. 2. Earlier9l we used this to get an easy approximation to F2 (and higher moments). Later we realized13l that the strip domain has equal validity and some advantages over the bin method. The ideal event density correlation f>i(y, y2 ; S) with s locating n particles on the rapidity axis, is 1 P2 (Y1, Y2 i S) where Y (y1 + Y2 )/2 and ( (- Y < y < Y; -oy < c < oy) n L 8(y1 - s;) 8(y2 - sj ) i;fjl (7) i;fjl (8) t 8 (Y - s; +2 Sj ) 8(( - (si - s;)) Y2 - Yt Integration over a strip of width 28y gives (9)

5 425 There can be small comer effects13l depending on conventions, but the center of mass condition is automatic. Hence, we get n i2(8y) L 0(8y - I s; - Sj l). ( 10) i;iojl Although averaging ( 10) over events is natural, Eq. ( 10) is well behaved even for single spike events, as shown by Dremin14l. Now we can remark on some features of Eq. (10), which hold in higher orders3l. 1. All particles closer than.sy are treated equally. 2. Fluctuations due to binning are removed. Improvement of an order of magnitude is possible. 3. If scaling occurs, the correlation dimensions are directly accessible. 4. The Tucson group has tested3l this approach for the p-model, Monte Carlo cascade and for unpublished UAl data. 5. No translation invariance was assumed. 6. More computation time is required to use the strip moments. It will be of interest to reassess the existing data for small approach. 6..Sy using the strip moment CONCLUSIONS 1. Hadronic rapidity histograms are not Poissonian noise. 2. Although the individual histograms are suggestive of fractal structure, the 1-d moments saturate as expected from a finite correlation length. Some evidence has been found for scaling in 2-d 8 8iJ> variables15l. y 3. Bin-average factorial moments Fp have error problems for spike events at small.sy. 4. Strip moments are closely related to the FP and have smaller errors. 5. The usual fractal and multifractal analysis does not tell us the relative contribu tions in phase space. With regard to the last point, I should mention the new developments of wavelet transforms16l. These transforms employ a variety of possible basis sets, which act like a microscope. The typical form is - b)/a), where b centers the function and magnifies it. When convoluted with a function it samples that part close to b. Properly chosen wavelets allow enormous data compression possibilities and are especially good at describing contrast. They are often fractal and allow the retention of local information. g((x f(x),

6 426 REFERENCES 1) W. Kittel, 20th International Symposium on Multiparticle Dynamics, Gut Holmecke/Dortmund, 1990, Nijmegen preprint HEN-335/90; B. Buschbeck, In vited talk at the XXVI Rencontre de Moriond, Les Arcs, Savoie, France, March ) A. Bialas and R. Peschanski, Nucl. Phys. B273 ( 1 986) 703; Nucl. Phys. B308 ( 1988) ) P. Lipa, P. Carruthers, H.C. Eggers and B. Buschbeck, University of Arizona preprint AZPH-TH/91-53 (to be published in Phys. Lett. B). 4) M. Henon, Commun. math. Phys. 50 ( 1976) 69. 5) J.D. Farmer, E. Ott and J.A. Yorke, Proc. of the Int. Conference on Order in Chaos, Los Alamos, New Mexico, USA, May 1982, eds. D. Campbell and H. Rose, Physica 7D ( 1983) ) A. Renyi, "Probability Theory", North-Holland, Amsterdam ( 1970). 7) H.G.E. Hentschel and I. Procaccia, Physica SD ( 1983) ) P. Carruthers, Int. J. Mod. Phys. A4 ( 1989) ) P. Carruthers and I. Sarcevic, Phys. Rev. Lett. 63 ( 1989) LO) ll) 12) 13) P. Carruthers, H.C. Eggers and I. Sarcevic, Phys. Lett. B254 (1991) 258. T. Chmaj, W. Doroba and W.Slomir\.ski, Z. Phys. C50 ( 1991 ) 333. H.C. Eggers, P. Carruthers, P. Lipa and I. Sarcevic, Phys. Rev. D44 ( 1991) P. Carruthers, Astrophysical Journal 380 ( 1991) ) I. Dremin, Mod. Phys. Lett. A3 ( 1988) 1333; Mod. Phys. Lett A4 (1989) ) B. Buschbeck, P. Lipa, N. Neumeister and D. Weselka (UAl Collab.), Talk at the Ringberg Workshop, Proc. of the Ringberg Workshop "Fluctuations and Fractal Structure", June 1991, eds. R.C. Hwa, W. Ochs and N. Schmitz, World Scientific, Singapore, ) I. Daubechis, IEEE Transactions on Information Theory 36 ( 1990) 961. FIGURE CAPTIONS Fii.1 The most common ways to check scaling properties of point sets generated by time series are shown in Eqs They involve varying the resolution of covering boxes or considering the number of points separated by a variable distance e. Fig.2 The integration domains n k are shown for Eq. 6. Fig.3 This example shows how strongly clustered particles can generate large fluctua tions in the factorial moments as the resolution increases.

7 , I Fig I.5 +Y Fig. 2 -Y +Y oy - Y Fig. 3 oy