Universal fluctuations: the phenomenology of hadronic matter

World Scientific Lecture Notes in Physics

Vol.

65

Fluctuations

UNIVERSAL FLUCTUATIONS The Phenomenology of Hadronic Matter

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World Scientific Lecture Notes in Physics

Vol. 65

UNIVERSAL FLUCTUATIONS The Phenomenology of Hadronic Matter

Robert Botet CNRS/Universite Paris-Sud, France

Marek Pioszajczak Grand AccSlerateur National d'lons Lourds, France

\fe « •

World Scientific • Singapore • Hong Kong New Jersey • London London'Singapore*

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A

CATHERINE & MARTINE

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Preface

"Los pasos que da un hombre, desde el dia de su nacimiento hasta el de su muerte, dibujan en el tiempo una inconcebible figura. La Inteligencia Divina intuye esa figura inmediatamente, como la de los hombres un tridngulo. Esa figura (acaso) tiene su determinada funcion en la economia del Universo" * [J. L. Borges (1952)]. The objective of this book is to outline a similar quest for scientific comprehension. Searching for figures just a bit more complicated than triangles. This is the quest for structures, hidden into random signals. Intention is not new - whole modern theory of Probabilities comes from this remarkable hunt -, but consequences appear slowly. At its flush of youth, some three centuries ago, the usual purpose of the probability calculations was gambling. Then little by little, maturity came, and we are now trying to comprehend Nature, even if gambling survives through econophysics. Understandable and reasonable evolution. For once in the history of Sciences, it is worth noting here that the way used by Probabilities to introduce themselves in Physics can be resumed in a few words : as long as classical Physics was rational and deterministic, great effort for mathematical analysis was needed but certainly not for the probabilities. And consequently, the classical iV-body physics was soon achieved for the small values of N, even if full solutions could be, or were expected ""The steps taken by a man, from his birth to his death, draw in the time an inconceivable figure. Divine knowledge understands this figure immediately, as we see a triangle. This figure (maybe) has its definite function in arrangement of the Universe" vii

Vlll

Preface

to be, mathematically tedious. Then next step in complexity consisted in playing with larger values of N, as it should for macroscopic objects built with microscopic entities. At this stage, randomness appears spontaneously, as Nature wriggles off the dull solutions of the linear equations, as interactions between individuals open new horizons of freedom, as duality engenders hesitation, as bifurcations lead to labyrinth. To write short, borders of classical physics are full of chaos. And since everything which surrounds us is built with gigantic number of quantic particles, we know why chaos should be present everywhere. Amazingly, when looking around us, we are well forced to accept that Nature is only gently random, and chaoticity is (fortunately) not the leading behaviour. McKeen discussing this paradox, wrote with his punchy and concise style [K. McKeen (1981)] : "In a world as crazy as this one, it ought to be easy to find something that happens solely by chance. It isn't". Without going too far into anthropic principles, one can assert that if chaos is the rule for individuals, some sort of regularization hold advisedly in the collective systems. In Probability theory, the hidden structures to achieve such regularization are the limit distribution laws. This was recognized clearly by Gauss, that fluctuations of almost any physical quantity around its mean value, were distributed according to the normal law. Beyond randomness was hidden a magic formula : exp(—z2). With his sense of the beautiful turn of phrase, he named this general phenomenon : the 'law of error', and was the first one to have intuition of the deep and practical importance of this discovery. Such conceptual jump in the knowledge of universality of random processes did not occur for a long time. The 'law of error' was renamed as : the 'Central Limit Theorem', and its conditions were so weakened that scientists were then convinced to see the Gaussian function everywhere. At the beginning of the XX century Poincare wrote with some humour : "Tout le monde croit (la loi des erreurs) parce que les mathematiciens s 'imaginent que c'est unfait d'observation, et les observateurs que c'est un theoreme de mathematiques." * [H. Poincare (1892)]. But, by the years 30s, extension of the Central Limit Theorem was performed when its conditions are going to fail. This achievement resulted from the collective effort of the prolific •"Everybody believe (the law of error) because mathematicians imagine that this is experimental fact, and experimentalists that this is a mathematical theorem."

Preface

IX

Russian school of Probability as well as from the deep works of Levy justifying and making rigorous the powerful method of moments introduced by Markov. This spectacular progress laid to the complete and explicit characterization of the stable limit distributions, giving reliable access to a precise image of the random process. Comprehension through simple visual representation. But above all, a new complementary and more subtle problem arise spontaneously : what is the correct normalization of a random variable in order to get the limit law ? These thoughts laid to the fundamental concept of scaling. It is difficult to trace the very beginning of this notion, since this idea of scaling seems to have emerged independently in various scientific areas. Anyway, all these results gave profusion of new concepts in the modern characterization of random processes, and made complicated things easier. The present book is about scaling of the fluctuations. ' Imagination ', in its first sense, is to replace abstract entity by an image, expected to be more directly understandable to our human intelligence. Modern Probabilities supply us exactly such imagination, to transform unreasonable tossing of a coin into a universal curve, and unconceivable Brownian motion into Hydrodynamics. "Apprivoiser I'absurdite du monde" * said the singer [M. Farmer (1999)].

Robert Botet & Marek Ploszajczak Orsay, Caen, October 2001

*" Taming absurdity of the world."

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Contents

Preface Chapter 1

vii Introduction

1

Chapter 2 Central Limit Theorem and Stable Laws 2.1 Central limit theorem for broad distributions 2.1.1 Central limit theorem for the sum of uncorrelated variables 2.2 Stable laws for sum of uncorrelated variables 2.2.1 The stability problem 2.2.2 Complete solution of the stability problem for uncorrelated variables 2.2.2.1 The ensemble of one-dimensional stable distributions 2.2.2.2 Alternative formulas for the stable distributions 2.2.2.3 Range of values for p, 2.2.2.4 Range of values for /? 2.2.2.5 Gaussian distribution as a stable law 2.2.2.6 Moments of the stable distributions 2.2.3 Explicit examples of stable distributions 2.2.3.1 Symmetric stable distributions (/3 = 0) . . . . 2.2.3.2 Asymmetric stable distributions [fi = 1) . . . 2.2.4 The reciprocity relation for stable distributions 2.2.5 The tail of stable distributions xi

7 8 8 13 13 17 17 17 18 19 20 20 22 22 24 25 26

xii

Contents

2.2.6 Moments of stable distributions 2.2.7 Asymptotically stable laws - domains of attraction . . . 2.2.8 The concept of the A-scaling 2.3 Limit theorems for more complicated combinations of uncorrelated variables 2.3.1 Product of uncorrelated variables 2.3.2 The Kesten variable 2.3.3 The Gumbel distribution 2.3.4 The arc-sine law 2.4 Two examples of physical applications 2.4.1 The Holtsmark problem 2.4.2 The stretched-exponential relaxation Chapter 3 Stable Laws for Correlated Variables 3.1 Weakly and strongly correlated random variables 3.1.1 Correlated random Gaussian processes 3.1.2 Taqqu's reduction theorem 3.1.3 Rosenblatt's model 3.2 Dyson's hierarchical model 3.3 The renormalization group 3.3.1 The renormalization group and the stability problem . . 3.3.2 Scaling features 3.3.3 e-expansion 3.3.4 Multiplicative structure of the renormalization group . . 3.4 Self-similar probability distributions 3.4.1 Self-similar processes 3.4.2 Euler theorem 3.4.3 Self-similarity of fractals in the renormalization group approach 3.4.4 The power spectral density function 3.4.5 A-scaling framework 3.5 Critical systems 3.5.1 Anomalous dimension 3.5.2 First scaling 3.5.3 Second scaling 3.5.4 A-scaling 3.5.5 Studies of criticality in finite systems

26 27 29 30 30 34 35 38 39 39 41 45 46 47 49 50 51 54 55 56 57 59 61 61 62 63 64 65 66 67 68 70 71 73

Contents

Chapter 4 Diffusion Problems 4.1 Brownian motion 4.1.1 Fick's representation 4.1.2 Ornstein-Uhlenbeck representation 4.1.3 Fokker-Planck representation 4.2 Random walks 4.2.1 Gaussian random walks and Gaussian Levy flights . . . 4.2.2 St. Petersburg paradox 4.2.3 Non-Gaussian Levy flights 4.2.3.1 Anomalous diffusion 4.2.3.2 Continuous Levy flights 4.2.3.3 Return to the origin of the random walk . . . 4.2.4 Random walk in a random environment 4.2.5 Sinai billiard 4.3 Random walks with memory 4.3.1 Random walks with Gaussian memory 4.3.2 Fractional Brownian motion 4.3.3 Flory's approach for linear polymers 4.4 Random walk as a critical phenomenon 4.4.1 Criticality of the Brownian motion 4.4.2 Criticality of the Levy flight 4.4.3 Criticality of the self-avoiding walk 4.5 Random walk as a self-similar process 4.5.1 Self-similarity of the Brownian motion 4.5.2 Anomalous diffusion in the fractal space

xiii

75 75 75 77 79 81 81 84 86 86 89 90 92 96 97 97 99 102 106 106 107 108 109 109 110

Chapter 5 Poisson-Transform Distributions 113 5.1 The class of poisson transforms 114 5.1.1 General functional relations for the Poisson transforms 116 5.1.2 Examples of Poisson transforms 117 5.1.3 Generating function for the Poisson transforms 119 5.2 Pascal distribution 120 5.2.1 Definition and moments of the Pascal distribution . . . 121 5.2.2 Recurrence relations for the Pascal distribution 121 5.2.3 Limit cases of the Poisson distribution 123 5.2.4 Stability of the Pascal distribution 123 5.2.5 Origins of the Pascal distribution 125

xiv

Contents

5.2.6

5.3

5.4 5.5

Stochastic differential equation leading to the Pascal distribution 126 Stacy distribution 127 5.3.1 The generalized Gamma distribution and its moments 128 5.3.2 Langevin and Fokker-Planck equations leading to the generalized Gamma function 129 5.3.2.1 One-dimensional Langevin equation with the multiplicative noise 130 5.3.2.2 Explicit physical processes leading to the one-dimensional Langevin equation with the multiplicative noise 131 5.3.2.3 Solution of the one-dimensional Langevin equation with the multiplicative noise 132 5.3.2.4 The limit case with vanishing random force . 133 Other examples of integral transforms 135 KNO scaling limit 135 5.5.1 Extension of the KNO scaling rule 136

Chapter 6 Featuring the Correlations 6.1 Moments and their generating function 6.1.1 Moments 6.1.2 Cumulant moments 6.1.3 Factorial moments 6.1.4 Cumulant factorial moments 6.1.5 Normalized moments 6.1.6 Bunching parameters 6.1.7 Combinants 6.1.8 Existence of the generating functions 6.2 Some tools specific to the moment generating functions 6.2.1 Singularities of the moment generating function 6.2.2 The Stieltjes series 6.3 One example: the poisson distribution 6.4 Infinitely divisible distribution functions 6.4.1 Truncating the multiplicity distribution 6.5 Composite distributions 6.5.1 Conditional and joint probabilities 6.5.2 Clan structures

....

139 139 140 140 141 142 142 144 144 145 146 146 147 148 151 153 153 154 155

Contents

6.6

More about the pascal distribution 6.6.1 The limiting forms 6.6.2 High-energy phenomenology

Chapter 7 Exclusive and Inclusive Densities 7.1 Generalities and variables 7.2 Cumulant correlation functions 7.3 Scaled factorial moments 7.3.1 Intermittency with the scaled factorial moments . . . . 7.3.2 Correcting for the shape of the one-particle distribution and the lack of the translational invariance 7.3.3 Unphysical correlations due to the mixing of events of different multiplicities 7.3.4 Dimensional projection 7.4 Scaled factorial correlators and bin-split moments 7.5 Scaled factorial cumulants 7.5.1 Correlation integral 7.6 Linked structure of the correlations 7.6.1 Linked pair approximation 7.6.2 Linked approximation in the conformal theory 7.6.3 Linked approximation for the A-scaling 7.6.4 Counts and their fluctuations 7.7 Erraticity concept 7.7.1 Wavelet representation 7.7.1.1 Simple examples of wavelets Chapter 8 8.1 8.2

8.3 8.4

xv

158 159 161 163 163 166 168 169

171 172 173 176 178 180 183 183 184 186 187 189 192 195

Bose-Einstein Correlations in Nuclear and Particle Physics 199 Basic features of bose-einstein quantum statistical correlations 200 Parametrization of the HBT data 202 8.2.1 The space-time structure of the multiparticle system . . 204 8.2.2 HBT measurements in condensed matter and atomic physics 206 Bose-Einstein interference in models 208 Idealized picture of independent particle production 209 8.4.1 Monte-Carlo simulations 212

xvi

8.5

Contents

Bose-Einstein correlations in high-energy collisions 8.5.1 Higher order cumulants in pp collisions 8.5.2 Small-scale Bose-Einstein correlations 8.5.3 Density dependence of the correlations

214 214 217 219

Chapter 9 Random Multiplicative Cascades 221 9.1 Multiplicative cascade models 222 9.1.1 Weak intermittency regime 223 9.1.2 Strong intermittency regime 225 9.1.3 Regularization of the scaled factorial moments in the strong intermittency limit 226 9.2 Multifractals and intermittency 227 9.3 Correlations in random cascading 229 9.3.1 Some examples of the branching generating functions . 235 9.3.2 Link to the multifractal formalism 236 9.3.3 Relation between branching generating function and multifractal mass exponents 238 9.4 Non-ideal random cascades: the cut-off effect 239 9.4.1 Multiscaling dependence on the cut-off parameters . . . 240 9.4.2 a-model with the cut-off at small scales 243 9.5 QCD cascade 245 Chapter 10

Random Cascades with Short-Scale Dissipation 10.1 Basic features of the fragmentation-inactivation binary model 10.1.1 Shattering transition 10.1.2 Scale-independent dissipation effects: the phase diagramme 10.2 Various approaches to the fragmentation-inactivation binary model 10.2.1 Fragmentation-inactivation binary model as a random multiplicative cascade 10.2.2 Fragmentation-inactivation binary model as a mean-field branching process 10.2.3 Cascade equation for the multiplicity evolution 10.2.4 Master equation

251 254 255 256 257 257 258 259 260

Contents

10.3 Moment analysis of the fragmentation-inactivation binary equations 10.3.1 General equations for the factorial moments and cumulant moments 10.3.2 Moments of the multiplicity distribution at the transition line 10.3.2.1 Brand - Schenzle fragmentation domain (pF > l / 2 , a > - l ) 10.3.2.2 Marginal case : pF = 1/2 , a > - 1 10.3.2.3 Cayley fragmentation domain : pF < 1/2, a > - 1 10.3.2.4 Evaporative fragmentation domain : pF > 0 , a < - 1 10.3.3 Structure of higher-order cumulant correlations at the transitional line 10.4 Binary cascading with scale-dependent inactivation mechanism 10.4.1 First example : binary cascading with a = — 1 and the Gaussian inactivation 10.4.2 Second example : binary fragmentation with a = +1 and the Gaussian inactivation 10.4.3 A-scaling vs value of exponent r 10.4.4 Multiplicity fluctuations in different physical systems and in the binary fragmentation 10.5 Perturbative quantum chromodynamics including inactivation mechanism 10.5.1 Multiplicity distributions in the dissipative gluodynamics 10.6 Phenomenology of the multiplicity distributions in e+e~ reactions Chapter 11 Fluctuations of the Order Parameter 11.1 Order parameter fluctuations in self-similar systems 11.1.1 The anomalous dimension 11.1.2 Critical cluster-size 11.1.3 Note about the correct order parameter 11.2 Example of the non-critical model 11.2.1 The weight functions

xvii

261 261 262 263 265 266 268 270 271 272 275 275 . 276 277 280 281 285 286 286 288 289 290 290

xviii

11.3

11.4 11.5

11.6

11.7

Contents

11.2.2 Check of the linked pair approximation 11.2.3 Second scaling law 11.2.4 Note about the average size-distribution Mean-field critical model: the Landau-Ginzburg model 11.3.1 Landau-Ginzburg free energy 11.3.2 Distribution of the extensive order parameter 11.3.3 First scaling at the pseudo-critical point 11.3.4 Gaussian first scaling in the disordered phase 11.3.5 Second scaling in the ordered phase 11.3.6 Correlation pattern in the Landau-Ginzburg theory . . Example of the critical model: the potts model 11.4.1 Scaling laws for the order-parameter distribution . . . . Reversible aggregation: example of the percolation model . . . 11.5.1 Order parameter in the percolation on the Bethe lattice 11.5.2 The three-dimensional percolation model 11.5.2.1 Multiplicity distributions 11.5.2.2 Order-parameter distribution 11.5.2.3 Shifted order parameter 11.5.2.4 Outside of the critical point 11.5.2.5 Close to the critical point Irreversible aggregation: example of the smoluchowski kinetic model 11.6.1 Basic behaviour of the order parameter 11.6.2 Scalings of the order-parameter distributions 11.6.3 Tails of the scaling functions 11.6.4 Scaling for the shifted order parameter 11.6.5 Origin of fluctuations in non-equilibrium aggregation . . 11.6.5.1 Argument of Van Kampen 11.6.5.2 Gelling systems 1T.6.5.3 Scaling of the second moments for gelling systems 11.6.5.4 Non-gelling systems Off-equilibrium fragmentation

290 291 292 293 293 293 294 295 295 296 297 298 300 301 304 304 304 304 305 307 308 308 310 311 312 313 313 315 315 317 317

Contents

Universal Fluctuations in Nuclear and Particle Physics 12.1 Phenomenology of high energy collisions in the scaled factorial moments analysis 12.1.1 Nonsingular parts in the correlations 12.1.2 Choice of the variables 12.1.3 General phenomenology and experimental results . . . . 12.1.4 Self-similarity or self-affinity in multiparticle production? 12.1.4.1 Self-affine analysis of n+/K+p data 12.2 A-scaling in pp collisions? 12.2.0.2 Aggregation scenario for pp and AA collisions? 12.3 Universal fluctuations in excited nuclear matter 12.3.1 A-scaling in nucleus-nucleus collisions in the Fermi energy domain

xix

Chapter 12

Chapter 13

Final Remarks

321 322 322 323 324 328 329 330 333 335 337 345

Bibliography

349

Index

363

Chapter 1

Introduction

Ubiquitous Gaussian law is so spreaded over the Sciences that it seems unreasonable to seek for too many informations in it. Gauss derived its distribution in order to help astronomers to compute accurate asteroid trajectories. Quetelet, appreciating the universality of the normal function, used extensively this law to fit a large collection of data from all the areas of sciences (especially in biology and sociology) stressing the variance as an important parameter independent of the mean value [J. M. Keynes (1921)]. This law arises deeply in the number theory too. To stay in Aristotelian logic : Gaussian form is the quality, and mean and variance are the quantities. But amount of information ends here for this law. So, much more interesting and recent applications should result in the cases where fluctuations of the investigated variable are not Gaussian. Important part of this book will then be devoted to the strongly correlated systems with many degrees of freedom, such as those at the critical point of the second-order phase transition. In the statistical physics, this domain is closely associated with development of the renormalization group ideas and scaling theory of critical phenomena, as they are strongly linked to limit theorems. This approach allows for the mathematically rigorous treatment of renormalization group theory and, hence, the phase transitions, in the language of the probability theory. This particular point will be emphasized. It stems from the works of Bleher and Sinai (1973) (see also Baker Jr. (1972)) and further developed by Jona-Lasinio (1975). The basic idea is splitting the whole system into correlated blocks. Then the probability distribution of the investigated physical quantity is calculated for the blocks, and the renormalization procedure is expressing the distri1

2

Introduction

bution for large blocks in terms of the distributions for the smaller ones. These probability distributions dealing with larger and larger number of subunits, are assumed to be given by probabilistic limit laws, while the parameters of these distributions are properly renormalized at each step. When this scheme is realized, the renormalization procedures gives the universal behaviour of the infinite system. Our presentation aims at developing such concepts and techniques, useful for attacking new problems in statistical physics of systems either atequilibrium or out-of-equilibrium. These insights are exemplified by recent advances in nuclear and high energy physics, where scaling theory provided effective methods for analyzing suspected criticality. But similar reasoning and tools could be employed in any situation where a correlated system close to criticality is at stake. It is worth specifying here that this approach can afford new aids in the exciting novel field of dynamic critical phenomena. We shall now proceed to outline the general plan : The monograph divides itself into three distinct (though connected) parts : the mathematical tools (regrouped in the Chapters 2, 3, 5, 6 and 7), and their physical applications, either fundamental (Chapters 4, 9, 10 and 11) or applied (Chapters 8 and 12). The applied examples and data have been chosen in the areas of particle and nuclear physics, because they correspond to domains of research both presently very active and perfectly at the center of our subject. In particular, they deal with out-of-equilibrium finite systems expected to be close to some critical behaviour. And even if criticality is not proven in many cases, they are known to be strongly correlated at the microscopic level. Note however that in principle, no specialized background in these parts of Physics is needed to read and understand the corresponding sections. This is the same for the mathematical parts which do not require more than the undergraduate level in the mathematics of Probabilities. Even if the precise examples are rather specific to gain homogeneity, the approach and results developed in the present book are by no means attached to some definite areas of Physics. Applications exist in all other fields, such as turbulence, polymers, atomic physics, to cite just a few among the most evident. Let us now review the chapters in greater details : Chapter 2.- The mathematical framework of the stable laws as limit distributions of uncorrelated random variables and their fundamental scaling properties, is presented following the probability theory. There are no

Introduction

3

rigorous demonstrations here, but instead an attempt to give a synthetic and precise panorama of the main general results according to the point of view of the physicist. Nevertheless, one writes explicitly the precise assumptions required for the main theorems to hold. In addition, a large number of particular stable laws are given explicitly. Chapter 3.- When the random variables are correlated, much less is known about the nature of the possible limit probability distributions. Some fundamental cases have been studied in mathematical physics and are discussed here. Most of them are connected to the critical phenomena. Emphasis is deliberately put on important physical implications, such as the renormalization group or the critical scaling theory. Chapter 4.- This section is about immediate applications of the mathematical results presented in Chapters 2 and 3, to a general and important problem of Physics : diffusion, as modelled by random walks. Some more precise applied problems are treated in details within the framework of the probabilistic limit laws, as the linear polymers or diffusion in a random potential. This chapter is a collection of such standard examples of limit laws in the actual physics, as they have been deeply studied by mathematical methods. Its plan follows roughly the same logical structure as for Chapters 2 and 3. Chapter 5.- Often, the experimental probability distributions are discrete. One studies in this chapter a general mathematical transform - the Poisson-transform -, allowing to pass naturally from a continuous distribution to a discrete one, while introducing minimum additional correlations. Some of these distributions are known to fit remarkably well some experimental data, even if no precise argument exists at present to explain such an agreement. Chapters 6 and 7.- These chapters introduce important tools to extract correlations from experimental data. Essentially, they appear as combination of particular moments of probability distributions. Some of them are specific and focus on some particular characteristics of the distribution. Other are more general and serve to reveal global feature, such as intermittency. Real examples are given to illustrate how these tools should be used. With the Chapters 2, 3 and 5, these two chapters compose the mathematical core of the theory developed in this book, and the starting-point of the whole discussion for physical applications. Chapter 8.- Currently, the Bose-Einstein interferometry is the centre of a very active domain of research in several areas of Physics. Here one

4

Introduction

discusses the successes and the shadows of this method in various examples of high-energy collisions. This requires use of the results detailed in the two preceding chapters about correlations. Some general trends in the current investigations on this problem are outlined. Chapter 9.- The random multiplicative cascade model is presented here. It corresponds to a very general approach of strongly-fluctuating processes. The most important application is certainly turbulence, but this model stems too from various areas of physics dealing with multiplicative processes. This should be read as a precise introduction to the following Chapter 10. Note however that a complete discussion of the turbulence problem is beyond the scope of this section. Chapter 10.- This Chapter concerns with a detailed contribution of the foregoing chapter to the problem of random successive fragmentation of an object. One deals here with dynamical, far-from-equilibrium systems, which are known to be difficult to broach. A particularly simple, though rich, model is presented with its critical and scaling properties, as well as more specific consequences in quantum chromodynamics theory. Chapter 11.- The mean-field theory of the second order phase transition provides an elegant application of the ideas developed throughout this book. Even if this is a simple analytical model, it acts as a guide to discuss the other models of critical systems presented here. Some of them are standard at-equilibrium models (percolation, Potts, Ising models), but they are systematically treated with the tools developed in the chapters above. Some others are non-standard and far-from-equilibrium (Smoluchowski solgel transition, shattering transition), in order to exhibit the robust features of the theory of universal fluctuations. Chapter 12.- This chapter is more specifically centered on analysis of recent experimental data from high-energy or nuclear Physics, within the framework of the universal fluctuations and the A-scaling. Emphasis is on conclusions which can be done, as well as ambiguities which cannot be solved, by use of these tools at the present time.

Acknowledgements It is impossible to acknowledge the many influences on our present

Introduction

5

understanding and the ideas we have exposed from other theorists who contributed to our work via encouragement, discussion, opposition or alternative points of view. This book would never be completed without a stimulating interest and encouragement of the multiparticle production community. Here we want to acknowledge a particular role played by N.G. Antoniou, A. Bialas, B. Buschbeck, P. Carruthers, S. Hegyi, R. Hwa, A. Giovannini, W. Kittel, R. Peschanski. Much of the discussion contained in this monograph stems from our collaboration with P. Bozek, whom we thank for many years of fruitful exchanges. Finally, we would like to thank warmly X. Campi, A. Chbihi, K. Gudima, H. Krivine, J. Lang, R. Parades V. and A. Tucholski for many friendly exchanges.

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Chapter 2

Central Limit Theorem and Stable Laws

Macroscopy is our natural human scale. But to gain in flexibility and freedom, Nature likes to play with a lot of tiny objects coupled together at the microscopic scale. This is why most of the interesting physical macroscopic variables are defined as sum of many microscopic random variables, and also why our only hope to understand such macroscopic behaviour is to seek for some universal limit behaviour. Without the universality, no investigation would be possible because of the quasi-infinite number of the parameters which should be taken into account. This is the land of statistical physics, but most of the other areas of modern physics are dealing with the same micro/macro problematic. To be more explicit, let us consider a large number of real random variables Xi depending on the integer index i. This set is regrouped into different domains of N variables, and we are asking for conditions under which the global variable M^ = X\ + ... + XN defined for each of these domains, has a non-trivial limit probability distribution. In the following, Xi will be sometimes called the microscopic random variable, while the global variable MN will be called the macroscopic random variable. Let us be still more precise. The indices i could be related to space location, and the space is divided into non-overlapping domains V; of size N. The random variable M/v attached to the macroscopic volume Vi will then correspond to the extensive physical quantity observed globally in this domain V;. Properly normalized, this variable will be considered as the averaged value of Xi over V;. We have then to ask the following question : how should we choose the shift and normalization factors, AN and BN in

7

8

Central Limit Theorem and Stable Laws

order that the random variable (the arithmetic mean) :

A

MN=^'f-

"

(2.j,

has a smooth positive probability distribution when N goes to oo ? Answering this question will allow to define unambiguously the only way to define properly the averaging protocol (2.1) of Xj. This problem is connected to the relative magnitude of the fluctuations of random variables. 2.1

Central limit theorem for broad distributions

Under certain general assumptions, one can prove that the random variable Mjv, as defined in (2.1) approaches the Gaussian shape for large values of N. This important result is generically known as the Central Limit Theorem. It demonstrates in particular that many features of the Xj—distributions are removed by this averaging procedure, since they are no more present in the final shape of the limit distribution. We detail below the usual conditions of applicability of this theorem, and also what can happen when these conditions go to fail. 2.1.1

Central limit variables

theorem

for the sum of

uncorrelated

Let us consider the random variable MJV as the sum of N statistically independent random continuous real variables Xj. Each variable Xj is assumed to be distributed according to the same probability density fx{x), defined as : fx{x)dx

= Prob[a; < Xj < x + dx]

(2.2)

within the infinitely small range dx. The normalization of probability implies that : / fx(x)dx

=1

for the positive distribution fx- Integration is performed over the whole definition range of the random variable X. To consider only microscopic variables of the same probability distributions is a very general practical case indeed, but this case is also needed if

Central Limit Theorem for Broad

Distributions

9

one wants to find the general limit distribution. Otherwise, there is often one particular variable Xj with a distribution which dominates the other ones and hence the averaging is meaningless. Note however that, if all the Xj—distributions are close enough, even though not exactly identical, most of the Central Limit Theorem may hold true, as shown by Lindeberg's theorems [W. Feller (1971)]. These extended Central Limit Theorems allow, to a certain extent, for fluctuations of the parameters of the Xj—distributions. Even if conceptually important, these theorems will not be considered in the following, in order to keep formulation of both ideas and results as simple as possible. The Central Limit Theorem states : If the common probability density of the Xj is such that : f\M x lim x / , * ' , = 0 x ~*°° J\x'\>xx '2fx{x')dx'

(2.3)

then the limit probability density of the variable MJV, properly shifted and normalized according to (2.1), approaches the normal distribution, that is to say : lim fMN(rn)

= fnorm,ai{m)

w-»°°

1 f m2\ = —== exp —— ^/2-K

\

2

.

(2.4)

/

This is the Gaussian function with the second moment equal to 1 (see figure 2.1): {m2)normal = / m2fnormai{m)dm

:

When a mean and a variance of Xj are both finite, the term 'properly shifted and normalized' means that the normal distribution occurs for the reduced variable :

The moments of rank q are defined generally as : {xq)=

fxqfx{x)da

Central Limit Theorem and Stable Laws

10 J nnrmnl'

^

Fig. 2.1 Plot of the normal distribution (2.4). For the Gaussian function with variance (m 2 ), the two inflexion points where the derivative takes its extremal value ± l / ^ / 2 e 7 r ( m 2 ) 2 are at the abscissa m = ± - y / ( m 2 ) , and the corresponding ordinates are ± l / y ^ 2 e 7 r ( m 2 ) . These values can be used to calculate the variance of an experimental Gaussian distribution.

when these quantities are mathematically correctly defined. On the other hand, when the mean and/or the variance is infinite, the shift and the normalization are less obvious to characterize but they exist in some sense as long as (2.3) takes place. The complete shift and normalization will be given quite generally in Section 2.2.7. The reciprocal of this theorem is also true, so all the distributions of random variables such as (2.1) and leading to the normal law (2.4) are characterized by the condition (2.3). The ensemble of all distribution functions verifying the condition (2.3) is called the domain of attraction of the normal law. This asymptotic condition is indeed very general, and explains a posteriori why the normal law is so widely spread in the Nature. In particular, it is the case for any bounded random variables Xj. One comment about condition (2.3) is important to be noticed here : only the tail of the distribution fx(x), i.e. only the large-z behaviour, is needed to conclude about the appearance of the normal law. In this sense, the normal law depends only on the frequency of appearance of large - and

Central Limit Theorem for Broad

Distributions

11

therefore rare -, values of the microscopic variables. Figure 2.2 shows for some values of N, the probability distributions of the variables MN, defined as in (2.1) for all the Xj uniformly distributed over the interval [—\/3, v^3]. Hence, the mean value of the Xj equals 0 and

fy(m)

0.4

0.2

0.0

Fig. 2.2 Plots of the probability distribution functions of the MN random variables (2.1). The normal distribution is shown by the light continuous curve for comparison, a) for the Mi and M2 variables; b) for the M4 and Ms variables. The M4— and Ms—distributions have been computed using 10 8 times the pseudo-random generator described in the footnote of the page 12.

the variance is 1 i.e. (Xj) = 0 and (Xf) = 1. The uniform distribution corresponding to the Mi—variable, is shown in figure 2.2.a alongside the M2—distribution : fM2{™) = (V6 — m)/6 fM2(jn) = (V6 + m)/6

when when

m >0 m

The following remarks help to understand the essential elements of the convergence to the normal law. First, note that the moments of the common Xj—distribution are :

(Xf+1)

=0