Products of Random Matrices: in Statistical Physics

Overview

This book is devoted to the theory and applications of products of random matrices, which arise naturally in many different fields. It describes analytic and numerical methods for the calculation of the corresponding Lyapunov exponents, which can be used as a tool for the analysis of problems in, for example, statistical mechanics of disordered systems, localization, wave propagation in random media, and chaotic dynamical systems. This book provides an excellent self-contained introduction to the subject for ...
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Paperback (Softcover reprint of the original 1st ed. 1993)
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Overview

This book is devoted to the theory and applications of products of random matrices, which arise naturally in many different fields. It describes analytic and numerical methods for the calculation of the corresponding Lyapunov exponents, which can be used as a tool for the analysis of problems in, for example, statistical mechanics of disordered systems, localization, wave propagation in random media, and chaotic dynamical systems. This book provides an excellent self-contained introduction to the subject for physicists working in condensed-matter and statistical physics.
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Product Details

  • ISBN-13: 9783642849442
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 1/5/2012
  • Series: Springer Series in Solid-State Sciences, #104
  • Edition description: Softcover reprint of the original 1st ed. 1993
  • Edition number: 1
  • Pages: 169
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.40 (d)

Table of Contents

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.
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