Scale Invariance and Beyond

This book is an excellent introduction to the concept of scale invariance, which is a growing field of research with wide applications. It describes where and how symmetry under scale transformation (and its various forms of partial breakdown) can be used to analyze solutions of a problem without the need to explicitly solve it. The first part gives descriptions of tools and concepts; the second is devoted to recent attempts to go beyond the invariance or symmetry breaking, to discuss causes and consequences, and to extract useful information about the system. Examples are carefully worked out in fields as diverse as condensed matter physics, population dynamics, earthquake physics, turbulence, cosmology and finance.

This is a book on scale invariance, a field that has become very fashionable since the work of Mandelbrot on factals and the success of the renormalization group of Kadanoff-Wilson.

Contenu
Lecture 1 Scale Invariance Without Mechanism?.- Lecture 2 Scale Invariance and Beyond: What Can We Learn from Wavelet Analysis?.- Lecture 3 Fractional Derivatives in Static and Dynamic Scaling.- Lecture 4 Multi-Dimensional Self-Similarity, and Self-Gravitating N-Body Systems.- Lecture 5 Scaling in Stock Market Data: Stable Laws and Beyond.- Lecture 6 Hysteresis, Avalanches, and Barkhausen Noise.- Lecture 7 Burgers Turbulence and the Energy Landscape of Randomly Pinned Objects.- Lecture 8 Power Laws and Scale Invariance in Physical Metallurgy.- Lecture 9 Scale Invariance in Fluids with Anticorrelated Entropy-Specific Volume Fluctuations.- Lecture 10 Scale Invariance(s) in the Cosmic Matter Distribution?.- Lecture 11 Scale Invariance of the Avalanche Effect in Phase Transition Modulated Mantle Mixing.- Lecture 12 Scale Invariance of Earthquakes.- Lecture 13 Models for Evolution and Extinction.- Lecture 14 Scale Invariance in Economics and in Finance.- Lecture 15 Models of Artificial Foreign Exchange Markets.- Lecture 16 From Scale-Invariance to Scale-Covariance in Out-of-Equilibrium Systems.- Lecture 17 Turbulence: Statistical Approach.- Lecture 18 Discrete Scale Invariance.- Lecture 19 Scale Relativity.- Lecture 20 Scaling in Turbulent Flows.- Lecture 21 Statistical Scale Symmetry Breaking.