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E-raamat: Universal Fluctuations: The Phenomenology Of Hadronic Matter

Marek Ploszajczak (Grand Accelerateur Natl D'ions Lourds, France), Robert Botet (Univ Paris-sud, France)

Formaat: 392 pages
Sari: World Scientific Lecture Notes In Physics 65
Ilmumisaeg: 02-Aug-2002
Kirjastus: World Scientific Publishing Co Pte Ltd
Keel: eng
ISBN-13: 9789814488952

Teised raamatud teemal:

Nuclear physics

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Formaat: 392 pages
Sari: World Scientific Lecture Notes In Physics 65
Ilmumisaeg: 02-Aug-2002
Kirjastus: World Scientific Publishing Co Pte Ltd
Keel: eng
ISBN-13: 9789814488952

Teised raamatud teemal:

Nuclear physics

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The main purpose of this book is to present, in a comprehensive and progressive way, the appearance of universal limit probability laws in physics, and their connection with the recently developed scaling theory of fluctuations. Arising from the probability theory and renormalization group methods, this novel approach has been proved recently to provide efficient investigative tools for the collective features that occur in any finite system.The mathematical background is self-contained and is formulated in terms which are easy to apply to the physical context. After illustrating the problem of anomalous diffusion, the book reviews recent advances in nuclear and high energy physics, where the limit laws are now recognized as being able to classify different phases of a system undergoing the pseudo-critical behaviour. A new description of the hadronic matter in terms of the fluctuation scaling is appearing as a consequence of this approach.

Preface

vii

Introduction

(6)

Central Limit Theorem and Stable Laws

(38)

Central limit theorem for broad distributions

(5)

Central limit theorem for the sum of uncorrelated variables

(5)

Stable laws for sum of uncorrelated variables

(17)

The stability problem

(4)

Complete solution of the stability problem for uncorrelated variables

(1)

The ensemble of one-dimensional stable distributions

(1)

Alternative formulas for the stable distributions

(1)

Range of values for μ

(1)

Range of values for β

(1)

Gaussian distribution as a stable law

(1)

Moments of the stable distributions

(2)

Explicit examples of stable distributions

(1)

Symmetric stable distributions (β = 0)

(2)

Asymmetric stable distributions (β = 1)

(1)

The reciprocity relation for stable distributions

(1)

The tail of stable distributions

(1)

Moments of stable distributions

(1)

Asymptotically stable laws - domains of attraction

(2)

The concept of the Δ-scaling

(1)

Limit theorems for more complicated combinations of uncorrelated variables

(9)

Product of uncorrelated variables

(4)

The Kesten variable

(1)

The Gumbel distribution

(3)

The arc-sine law

(1)

Two examples of physical applications

(6)

The Holtsmark problem

(2)

The stretched-exponential relaxation

(4)

Stable Laws for Correlated Variables

(30)

Weakly and strongly correlated random variables

(5)

Correlated random Gaussian processes

(2)

Taqqu's reduction theorem

(1)

Rosenblatt's model

(1)

Dyson's hierarchical model

(3)

The renormalization group

(7)

The renormalization group and the stability problem

(1)

Scaling features

(1)

ε-expansion

(2)

Multiplicative structure of the renormalization group

(2)

Self-similar probability distributions

(5)

Self-similar processes

(1)

Euler theorem

(1)

Self-similarity of fractals in the renormalization group approach

(1)

The power spectral density function

(1)

Δ-scaling framework

(1)

Critical systems

(9)

Anomalous dimension

(1)

First scaling

(2)

Second scaling

(1)

Δ-scaling

(2)

Studies of criticality in finite systems

(2)

Diffusion Problems

(38)

Brownian motion

(6)

Fick's representation

(2)

Ornstein-Uhlenbeck representation

(2)

Fokker-Planck representation

(2)

Random walks

(16)

Gaussian random walks and Gaussian Levy flights

(3)

St. Petersburg paradox

(2)

Non-Gaussian Levy flights

(1)

Anomalous diffusion

(3)

Continuous Levy flights

(1)

Return to the origin of the random walk

(2)

Random walk in a random environment

(4)

Sinai billiard

(1)

Random walks with memory

(9)

Random walks with Gaussian memory

(2)

Fractional Brownian motion

(3)

Flory's approach for linear polymers

102

(4)

Random walk as a critical phenomenon

106

(3)

Criticality of the Brownian motion

106

(1)

Criticality of the Levy flight

107

(1)

Criticality of the self-avoiding walk

108

(1)

Random walk as a self-similar process

109

(4)

Self-similarity of the Brownian motion

109

(1)

Anomalous diffusion in the fractal space

110

(3)

Poisson-Transform Distributions

113

(26)

The class of poisson transforms

114

(6)

General functional relations for the Poisson transforms

116

(1)

Examples of Poisson transforms

117

(2)

Generating function for the Poisson transforms

119

(1)

Pascal distribution

120

(7)

Definition and moments of the Pascal distribution

121

(1)

Recurrence relations for the Pascal distribution

121

(2)

Limit cases of the Poisson distribution

123

(1)

Stability of the Pascal distribution

123

(2)

Origins of the Pascal distribution

125

(1)

Stochastic differential equation leading to the Pascal distribution

126

(1)

Stacy distribution

127

(8)

The generalized Gamma distribution and its moments

128

(1)

Langevin and Fokker-Planck equations leading to the generalized Gamma function

129

(1)

One-dimensional Langevin equation with the multiplicative noise

130

(1)

Explicit physical processes leading to the one-dimensional Langevin equation with the multiplicative noise

131

(1)

Solution of the one-dimensional Langevin equation with the multiplicative noise

132

(1)

The limit case with vanishing random force

133

(2)

Other examples of integral transforms

135

(1)

KNO scaling limit

135

(4)

Extension of the KNO scaling rule

136

(3)

Featuring the Correlations

139

(24)

Moments and their generating function

139

(7)

Moments

140

(1)

Cumulant moments

140

(1)

Factorial moments

141

(1)

Cumulant factorial moments

142

(1)

Normalized moments

142

(2)

Bunching parameters

144

(1)

Combinants

144

(1)

Existence of the generating functions

145

(1)

Some tools specific to the moment generating functions

146

(2)

Singularities of the moment generating function

146

(1)

The Stieltjes series

147

(1)

One example: the poisson distribution

148

(3)

Infinitely divisible distribution functions

151

(2)

Truncating the multiplicity distribution

153

(1)

Composite distributions

153

(5)

Conditional and joint probabilities

154

(1)

Clan structures

155

(3)

More about the pascal distribution

158

(5)

The limiting forms

159

(2)

High-energy phenomenology

161

(2)

Exclusive and Inclusive Densities

163

(36)

Generalities and variables

163

(3)

Cumulant correlation functions

166

(2)

Scaled factorial moments

168

(8)

Intermittency with the scaled factorial moments

169

(2)

Correcting for the shape of the one-particle distribution and the lack of the translational invariance

171

(1)

Unphysical correlations due to the mixing of events of different multiplicities

172

(1)

Dimensional projection

173

(3)

Scaled factorial correlators and bin-split moments

176

(2)

Scaled factorial cumulants

178

(5)

Correlation integral

180

(3)

Linked structure of the correlations

183

(6)

Linked pair approximation

183

(1)

Linked approximation in the conformal theory

184

(2)

Linked approximation for the Δ-scaling

186

(1)

Counts and their fluctuations

187

(2)

Erraticity concept

189

(10)

Wavelet representation

192

(3)

Simple examples of wavelets

195

(4)

Bose-Einstein Correlations in Nuclear and Particle Physics

199

(22)

Basic features of bose-einstein quantum statistical correlations

200

(2)

Parametrization of the HBT data

202

(6)

The space-time structure of the multiparticle system

204

(2)

HBT measurements in condensed matter and atomic physics

206

(2)

Bose-Einstein interference in models

208

(1)

Idealized picture of independent particle production

209

(5)

Monte-Carlo simulations

212

(2)

Bose-Einstein correlations in high-energy collisions

214

(7)

Higher order cumulants in pp collisions

214

(3)

Small-scale Bose-Einstein correlations

217

(2)

Density dependence of the correlations

219

(2)

Random Multiplicative Cascades

221

(30)

Multiplicative cascade models

222

(5)

Weak intermittency regime

223

(2)

Strong intermittency regime

225

(1)

Regularization of the scaled factorial moments in the strong intermittency limit

226

(1)

Multifractals and intermittency

227

(2)

Correlations in random cascading

229

(10)

Some examples of the branching generating functions

235

(1)

Link to the multifractal formalism

236

(2)

Relation between branching generating function and multifractal mass exponents

238

(1)

Non-ideal random cascades: the cut-off effect

239

(6)

Multiscaling dependence on the cut-off parameters

240

(3)

α-model with the cut-off at small scales

243

(2)

QCD cascade

245

(6)

Random Cascades with Short-Scale Dissipation

251

(34)

Basic features of the fragmentation-inactivation binary model

254

(3)

Shattering transition

255

(1)

Scale-independent dissipation effects: the phase diagramme

256

(1)

Various approaches to the fragmentation-inactivation binary model

257

(4)

Fragmentation-inactivation binary model as a random multiplicative cascade

257

(1)

Fragmentation-inactivation binary model as a mean-field branching process

258

(1)

Cascade equation for the multiplicity evolution

259

(1)

Master equation

260

(1)

Moment analysis of the fragmentation-inactivation binary equations

261

(10)

General equations for the factorial moments and cumulant moments

261

(1)

Moments of the multiplicity distribution at the transition line

262

(1)

Brand - Schenzle fragmentation domain (pF > 1/2, α > --1)

263

(2)

Marginal case : pF = 1/2, α > --1

265

(1)

Cayley fragmentation domain: pF < 1/2, α > --1

266

(2)

Evaporative fragmentation domain: pF > 0, α < --1

268

(2)

Structure of higher-order cumulant correlations at the transitional line

270

(1)

Binary cascading with scale-dependent inactivation mechanism

271

(6)

First example : binary cascading with α = --1 and the Gaussian inactivation

272

(3)

Second example: binary fragmentation with α = +1 and the Gaussian inactivation

275

(1)

Δ-scaling vs value of exponent τ

275

(1)

Multiplicity fluctuations in different physical systems and in the binary fragmentation

276

(1)

Perturbative quantum chromodynamics including inactivation mechanism

277

(4)

Multiplicity distributions in the dissipative gluodynamics

280

(1)

Phenomenology of the multiplicity distributions in e+e- reactions

281

(4)

Fluctuations of the Order Parameter

285

(36)

Order parameter fluctuations in self-similar systems

286

(4)

The anomalous dimension

286

(2)

Critical cluster-size

288

(1)

Note about the correct order parameter

289

(1)

Example of the non-critical model

290

(3)

The weight functions

290

(1)

Check of the linked pair approximation

290

(1)

Second scaling law

291

(1)

Note about the average size-distribution

292

(1)

Mean-field critical model: the Landau-Ginzburg model

293

(4)

Landau-Ginzburg free energy

293

(1)

Distribution of the extensive order parameter

293

(1)

First scaling at the pseudo-critical point

294

(1)

Gaussian first scaling in the disordered phase

295

(1)

Second scaling in the ordered phase

295

(1)

Correlation pattern in the Landau-Ginzburg theory

296

(1)

Example of the critical model: the potts model

297

(3)

Scaling laws for the order-parameter distribution

298

(2)

Reversible aggregation: example of the percolation model

300

(8)

Order parameter in the percolation on the Bethe lattice

301

(3)

The three-dimensional percolation model

304

(1)

Multiplicity distributions

304

(1)

Order-parameter distribution

304

(1)

Shifted order parameter

304

(1)

Outside of the critical point

305

(2)

Close to the critical point

307

(1)

Irreversible aggregation: example of the smoluchowski kinetic model

308

(9)

Basic behaviour of the order parameter

308

(2)

Scalings of the order-parameter distributions

310

(1)

Tails of the scaling functions

311

(1)

Scaling for the shifted order parameter

312

(1)

Origin of fluctuations in non-equilibrium aggregation

313

(1)

Argument of Van Kampen

313

(2)

Gelling systems

315

(1)

Scaling of the second moments for gelling systems

315

(2)

Non-gelling systems

317

(1)

Off-equilibrium fragmentation

317

(4)

Universal Fluctuations in Nuclear and Particle Physics

321

(24)

Phenomenology of high energy collisions in the scaled factorial moments analysis

322

(8)

Nonsingular parts in the correlations

322

(1)

Choice of the variables

323

(1)

General phenomenology and experimental results

324

(4)

Self-similarity or self-affinity in multiparticle production?

328

(1)

Self-affine analysis of π+ / K+p data

329

(1)

Δ-scaling in pp collisions?

330

(5)

Aggregation scenario for pp and AA collisions?

333

(2)

Universal fluctuations in excited nuclear matter

335

(10)

Δ-scaling in nucleus-nucleus collisions in the Fermi energy domain

337

(8)

Final Remarks

345

(4)

Bibliography

349

(14)

Index

363

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