Products of Random Matrices: In Statistical Physics

ISBN-13: 9783642849442 / Angielski / Miękka / 2012 / 169 str.

Andrea Crisanti; Giovanni Paladin; Angelo Vulpiani

Opis

At the present moment, after the success of the renormalization group in providing a conceptual framework for studying second-order phase tran sitions, we have a nearly satisfactory understanding of the statistical me chanics of classical systems with a non-random Hamiltonian. The situation is completely different if we consider the theory of systems with a random Hamiltonian or of chaotic dynamical systems. The two fields are connected; in fact, in the latter the effects of deterministic chaos can be modelled by an appropriate stochastic process. Although many interesting results have been obtained in recent years and much progress has been made, we still lack a satisfactory understanding of the extremely wide variety of phenomena which are present in these fields. The study of disordered or chaotic systems is the new frontier where new ideas and techniques are being developed. More interesting and deep results are expected to come in future years. The properties of random matrices and their products form a basic tool, whose importance cannot be underestimated. They playa role as important as Fourier transforms for differential equations. This book is extremely interesting as far as it presents a unified approach for the main results which have been obtained in the study of random ma trices. It will become a reference book for people working in the subject. The book is written by physicists, uses the language of physics and I am sure that many physicists will read it with great pleasure.

Spis treści

I Background.- 1. Why Study Random Matrices?.- 1.1 Statistics of the Eigenvalues of Random Matrices.- 1.1.1 Nuclear Physics.- 1.1.2 Stability of Large Ecosystems.- 1.1.3 Disordered Harmonic Solids.- 1.2 Products of Random Matrices in Chaotic and Disordered Systems.- 1.2.1 Chaotic Systems.- 1.2.2 Disordered Systems.- 1.3 Some Remarks on the Calculation of the Lyapunov Exponent of PRM.- 2. Lyapunov Exponents for PRM.- 2.1 Asymptotic Limits: the Furstenberg and Oseledec Theorems.- 2.2 Generalized Lyapunov Exponents.- 2.3 Numerical Methods for the Computation of Lyapunov Exponents.- 2.4 Analytic Results.- 2.4.1 Weak Disorder Expansion.- 2.4.2 Replica Trick.- 2.4.3 Microcanonical Method.- II Applications.- 3. Chaotic Dynamical Systems.- 3.1 Random Matrices and Deterministic Chaos.- 3.1.1 The Independent RM Approximation.- 3.1.2 Independent RM Approximation: Perturbative Approach.- 3.1.3 Beyond the Independent RM Approximation.- 3.2 CLE for High Dimensional Dynamical Systems.- 4. Disordered Systems.- 4.1 One-Dimensional Ising Model and Transfer Matrices.- 4.2 Random One-Dimensional Ising Models.- 4.2.1 Ising Chain with Random Field.- 4.2.2 Ising Chain with Random Coupling.- 4.3 Generalized Lyapunov Exponents and Free Energy Fluctuations.- 4.4 Correlation Functions and Random Matrices.- 4.5 Two-and Three-Dimensional Systems.- 5. Localization.- 5.1 Localization in One-Dimensional Systems.- 5.1.1 Exponential Growth and Localization: The Borland Conjecture.- 5.1.2 Density of States in One-Dimensional Systems.- 5.1.3 Conductivity and Lyapunov Exponents: The Landauer Formula.- 5.2 PRMs and One-Dimensional Localization: Some Applications.- 5.2.1 Weak Disorder Expansion.- 5.2.2 Replica Trick and Microcanonical Approximation.- 5.2.3 Generalized Localization Lengths.- 5.2.4 Random Potentials with Extended States.- 5.3 PRMs and Localization in Two and Three Dimensions.- 5.4 Maximum Entropy Approach to the Conductance Fluctuations.- III Miscellany.- 6. Other Applications.- 6.1 Propagation of Light in Random Media.- 6.1.1 Media with Random Optical Index.- 6.1.2 Randomly Deformed Optical Waveguide.- 6.2 Random Magnetic Dynamos.- 6.3 Image Compression.- 6.3.1 Iterated Function System.- 6.3.2 Determination of the IFS Code for Image Compression.- 7. Appendices.- 7.1 Statistics of the Eigenvalues of Real Random Asymmetric Matrices.- 7.2 Program for the Computation of the Lyapunov Spectrum.- 7.3 Poincaré Section.- 7.4 Markov Chain and Shannon Entropy.- 7.5 Kolmogorov-Sinai and Topological Entropies.- 7.6 Generalized Fractal Dimensions and Multifractals.- 7.7 Localization in Correlated Random Potentials.- References.

O autorze

Products of random matrices arise naturally in many areas of condensed-matter and statistical physics. This book provides a self-contained introduction, suitable for graduate students and researchers, to such products and to methods for the calculation of the corresponding Lyapunov exponents.