Universal Fluctuations

The following sections are included:

• Preface

• Contents

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…

The following sections are included:

• Central limit theorem for broad distributions
  • Central limit theorem for the sum of uncorrelated variables

• Stable laws for sum of uncorrelated variables
  • The stability problem
  • Complete solution of the stability problem for uncorrelated variables
    • The ensemble of one-dimensional stable distributions
    • Alternative formulas for the stable distributions
    • Range of values for μ
    • Range of values for β
    • Gaussian distribution as a stable law
    • Moments of the stable distributions
  • Explicit examples of stable distributions
    • Symmetric stable distributions (β = 0)
    • Asymmetric stable distributions (β = 1)
  • The reciprocity relation for stable distributions
  • The tail of stable distributions
  • Moments of stable distributions
  • Asymptotically stable laws - domains of attraction
  • The concept of the Δ-scaling

• Limit theorems for more complicated combinations of uncorrelated variables
  • Product of uncorrelated variables
  • The Kesten variable
  • The Gumbel distribution
  • The arc-sine law

• Two examples of physical applications
  • The Holtsmark problem
  • The stretched-exponential relaxation

The following sections are included:

• Weakly and strongly correlated random variables
  • Correlated random Gaussian processes
  • Taqqu's reduction theorem
  • Rosenblatt's model

• Dyson's hierarchical model

• The renormalization group
  • The renormalization group and the stability problem
  • Scaling features
  • ε-expansion
  • Multiplicative structure of the renormalization group

• Self-similar probability distributions
  • Self-similar processes
  • Euler theorem
  • Self-similarity of fractals in the renormalization group approach
  • The power spectral density function
  • Δ-scaling framework

• Critical systems
  • Anomalous dimension
  • First scaling
  • Second scaling
  • Δ-scaling
  • Studies of criticality in finite systems

The following sections are included:

• Brownian motion
  • Fick's representation
  • Ornstein-Uhlenbeck representation
  • Fokker-Planck representation

• Random walks
  • Gaussian random walks and Gaussian Lévy flights
  • St. Petersburg paradox
  • Non-Gaussian Lévy flights
    • Anomalous diffusion
    • Continuous Lévy flights
    • Return to the origin of the random walk
  • Random walk in a random environment
  • Sinai billiard

• Random walks with memory
  • Random walks with Gaussian memory
  • Fractional Brownian motion
  • Flory's approach for linear polymers

• Random walk as a critical phenomenon
  • Criticality of the Brownian motion
  • Criticality of the Lévy flight
  • Criticality of the self-avoiding walk

• Random walk as a self-similar process
  • Self-similarity of the Brownian motion
  • Anomalous diffusion in the fractal space

The following sections are included:

• The class of Poisson transforms
  • General functional relations for the Poisson transforms
  • Examples of Poisson transforms
  • Generating function for the Poisson transforms

• Pascal distribution
  • Definition and moments of the Pascal distribution
  • Recurrence relations for the Pascal distribution
  • Limit cases of the Poisson distribution
  • Stability of the Pascal distribution
  • Origins of the Pascal distribution
  • Stochastic differential equation leading to the Pascal distribution

• Stacy distribution
  • The generalized Gamma distribution and its moments
  • Langevin and Fokker-Planck equations leading to the generalized Gamma function
    • One-dimensional Langevin equation with the multiplicative noise
    • Explicit physical processes leading to the one-dimensional Langevin equation with the multiplicative noise
    • Solution of the one-dimensional Langevin equation with the multiplicative noise
    • The limit case with vanishing random force

• Other examples of integral transforms

• KNO scaling limit
  • Extension of the KNO scaling rule

The following sections are included:

• Moments and their generating function
  • Moments
  • Cumulant moments
  • Factorial moments
  • Cumulant factorial moments
  • Normalized moments
  • Bunching parameters
  • Combinants
  • Existence of the generating functions

• Some tools specific to the moment generating functions
  • Singularities of the moment generating function
  • The Stieltjes series

• One example: the Poisson distribution

• Infinitely divisible distribution functions
  • Truncating the multiplicity distribution

• Composite distributions
  • Conditional and joint probabilities
  • Clan structures

• More about the Pascal distribution
  • The limiting forms
  • High-energy phenomenology

The following sections are included:

• Generalities and variables

• Cumulant correlation functions

• Scaled factorial moments
  • Intermittency with the scaled factorial moments
  • Correcting for the shape of the one-particle distribution and the lack of the translational invariance
  • Unphysical correlations due to the mixing of events of different multiplicities
  • Dimensional projection

• Scaled factorial correlators and bin-split moments

• Scaled factorial cumulants
  • Correlation integral

• Linked structure of the correlations
  • Linked pair approximation
  • Linked approximation in the conformal theory
  • Linked approximation for the Δ-scaling
  • Counts and their fluctuations

• Erraticity concept
  • Wavelet representation
    • Simple examples of wavelets

The following sections are included:

• Basic features of Bose-Einstein quantum statistical correlations

• Parametrization of the HBT data
  • The space-time structure of the multiparticle system
  • HBT measurements in condensed matter and atomic physics

• Bose-Einstein interference in models

• Idealized picture of independent particle production
  • Monte-Carlo simulations

• Bose-Einstein correlations in high-energy collisions
  • Higher order cumulants in collisions
  • Small-scale Bose-Einstein correlations
  • Density dependence of the correlations

The following sections are included:

• Multiplicative cascade models
  • Weak intermittency regime
  • Strong intermittency regime
  • Regularization of the scaled factorial moments in the strong intermittency limit

• Multifractals and intermittency

• Correlations in random cascading
  • Some examples of the branching generating functions
  • Link to the multifractal formalism
  • Relation between branching generating function and multifractal mass exponents

• Non-ideal random cascades: the cut-off effect
  • Multiscaling dependence on the cut-off parameters
  • α-model with the cut-off at small scales

• QCD cascade

The following sections are included:

• Basic features of the fragmentation-inactivation binary model
  • Shattering transition
  • Scale-independent dissipation effects : the phase diagramme

• Various approaches to the fragmentation-inactivation binary model
  • Fragmentation-inactivation binary model as a random multiplicative cascade
  • Fragmentation-inactivation binary model as a mean-field branching process
  • Cascade equation for the multiplicity evolution
  • Master equation

• Moment analysis of the fragmentation-inactivation binary equations
  • General equations for the factorial moments and cumulant moments
  • Moments of the multiplicity distribution at the transition line
    • Brand - Schenzle fragmentation domain (p_F > 1/2,α > −1)
    • Marginal case : p_F = 1/2, α > −1
    • Cayley fragmentation domain : p_F < 1/2, α > −1
    • Evaporative fragmentation domain : p_F > 0, α < −1
  • Structure of higher-order cumulant correlations at the transitional line

• Binary cascading with scale-dependent inactivation mechanism
  • First example : binary cascading with α = −1 and the Gaussian inactivation
  • Second example : binary fragmentation with α = +1 and the Gaussian inactivation
  • Δ-scaling vs value of exponent τ
  • Multiplicity fluctuations in different physical systems and in the binary fragmentation

• Perturbative quantum chromodynamics including inactivation mechanism
  • Multiplicity distributions in the dissipative gluodynamics

• Phenomenology of the multiplicity distributions in e⁺e⁻ reactions

The following sections are included:

• Order parameter fluctuations in self-similar systems
  • The anomalous dimension
  • Critical cluster-size
  • Note about the correct order parameter

• Example of the non-critical model
  • The weight functions
  • Check of the linked pair approximation
  • Second scaling law
  • Note about the average size-distribution

• Mean-field critical model: the Landau-Ginzburg model
  • Landau-Ginzburg free energy
  • Distribution of the extensive order parameter
  • First scaling at the pseudo-critical point
  • Gaussian first scaling in the disordered phase
  • Second scaling in the ordered phase
  • Correlation pattern in the Landau-Ginzburg theory

• Example of the critical model: the Potts model
  • Scaling laws for the order-parameter distribution

• Reversible aggregation: example of the percolation model
  • Order parameter in the percolation on the Bethe lattice
  • The three-dimensional percolation model
    • Multiplicity distributions
    • Order-parameter distribution
    • Shifted order parameter
    • Outside of the critical point
    • Close to the critical point

• Irreversible aggregation: example of the Smoluchowski kinetic model
  • Basic behaviour of the order parameter
  • Scalings of the order-parameter distributions
  • Tails of the scaling functions
  • Scaling for the shifted order parameter
  • Origin of fluctuations in non-equilibrium aggregation
    • Argument of Van Kampen
    • Gelling systems
    • Scaling of the second moments for gelling systems
    • Non-gelling systems

• Off-equilibrium fragmentation

The following sections are included:

• Phenomenology of high energy collisions in the scaled factorial moments analysis
  • Nonsingular parts in the correlations
  • Choice of the variables
  • General phenomenology and experimental results
  • Self-similarity or self-affinity in multiparticle production?
    • Self-affine analysis of π⁺/K⁺p data

• Δ-scaling in collisions?
  • Aggregation scenario for and AA collisions?

• Universal fluctuations in excited nuclear matter
  • Δ-scaling in nucleus-nucleus collisions in the Fermi energy domain

There is still a long way to understand the multihadron-production processes and, in particular, those aspects of these processes which are related to the strong-coupling long-distance regime of QCD. Experimental studies of correlations in small domains of the phase-space, posed the problem of the information contained in the multiparticle correlations of produced hadrons and, more generally, in the fluctuation pattern of observables. In this book we have stressed at several places how limited is the information contained in the multiparticle total cross section. The crucial phase information is hidden in the correlation data, particularly in the Bose-Einstein correlations. We have shown that fluctuations (correlations) of observables in complex multiparticle systems have universal properties even in small finite systems, independently of whether the studied process is classical or quantal, equilibrium or non-equilibrium, dissipative or conservative, whether the underlying microscopic variables are strongly correlated as in critical phenomena or, on the contrary, are mutually independent, etc. We dispose a complete characterization of the convergence (scaling) of probability distributions in finite systems of different sizes. In this sense, these general results and associated mathematical tools can be applied in any physical, economical, social, etc. systems which can appear at different sizes…

The following sections are included:

• Bibliography

• Index