普适起伏
2002-12
World Scientific Pub Co Inc
Botet, Robert/ Poszajczak, Marek/ Ploszajczak, M.
369
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 Chapter 1 IntroductionChapter 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 # 2.2.2.4 Range of values for fl 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 (B = 0) 2.2.3.2 Asymmetric stable distributions (B = 1) 2.2.4 The reciprocity relation for stable distributions 2.2.5 The tail of stable distributions 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 relaxationChapter 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……Chapter 4 Diffusion ProblemsChapter 5 Poisson-Transform DistributionsChapter 6 Feauring the CorrelationsChapter 7 Exclusive and Inclusive DensitiesChapter 8 Bose-Einstein Correlations in Nuclear and Particle PhysicsChapter 9 Random Multiplicative CascadesChapter 10 Random Cascades with Short-Scale DissipationChapter 11 Fluctuations of the Order ParameterChapter 12 Universal fluctuations in Nuclear and Particle PhysicsChapter 13 Final RemarksBibliographyIndex
