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Crisanti A., Paladin G., Vulpiani A. — Products of random matrices in statistical physics
Crisanti A., Paladin G., Vulpiani A. — Products of random matrices in statistical physics

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Название: Products of random matrices in statistical physics

Авторы: Crisanti A., Paladin G., Vulpiani A.


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.

Язык: en

Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 184

Добавлена в каталог: 04.12.2013

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Предметный указатель
Adiabatic approximation      87
Affine transformation      128
Affine transformation contractive      128
Anderson model      12 89
Anderson model, two-dimensional      106
Annealed, average      75
Annealed, free energy      see "Free energy"
Arnold — Beltrami — Childress (ABC) map      125
Asymptotic limits for PRM      see "Furstenberg theorem" "Oseledec
Baker transformation      51—52
Barnsley — Elton theorem      129
Bernoullian probability distribution      14
Borland conjecture      91
Cauchy probability distribution      95
Central limit theorem      22
Central limit theorem for i.i.d. matrices in $SL(D,\mathds{R})$      23
Central limit theorem for i.i.d. matrices with positive elements      22
Central limit theorem for i.i.d. symplectic matrices      23
Central limit theorem for random numbers      17
Circular theorem      5
Cohen — Newman theorem      29—30
Collage theorem      129
Conductivity      93 see
Conductivity, maximum entropy approach to fluctuations of      110
Correlation function      75
Correlation length, most probable      78
Density of states integrated      91—92 see
Differentiable dynamical systems, continuous time      43
Differentiable dynamical systems, discrete time      43
Dimension, fractal      132 148
Dimension, generalized fractal      126 149
Dimension, information      149 150
Dimension, topological      148
Directed random polymers      32
Discrete Laplace operator in one dimension      89
Discrete Laplace operator in two dimesions      107
Disordered, harmonic chains      8
Disordered, Ising chain      11
Ecosystems      6
Effective Lyapunov exponent      see "Maximum Lyapunov exponent"
Eigenvalues      see "Statistics of eigenvalues"
Eigenvectors, Lyapunov      see "Lyapunov basis"
Eigenvectors, stability      see "stability basis"
Electrostatics, two-dimensional classi cal      136
Entropy, function for effective Lyapunov exponent      25
Entropy, function for multifractal objects      149
Entropy, information      113
Entropy, Kolmogorov — Sinai or metric      55 146
Entropy, Renyi      148
Entropy, Shannon      145
Entropy, topological      52 146
Extended states      88
Extended states in light transmission      118
Extended states in random 1d potentials      104—106
Fermi — Pasta — Ulam (FPU) model      56
Fluctuations of exponential growth rate      23
Fluctuations of Lyapunov exponent in canonical ensemble      38
Fluctuations of magnetic field in kinematic dynamo      see "Generalized Lyapunov exponents for kinematic dynamo"
Fractal dimension      see "Dimension"
Free energy fluctuations      71—72
Free energy fluctuations of random coupling 1d Ising model      74
Free energy, annealed      71
Free energy, of 1d Ising model      60
Free energy, of random 1d Ising model      12 61
Free energy, of random 2d Ising model      84
Free energy, quenched      70 71
Furstenberg theorem      19
Gaussian, measure in matrix space      135 138
Gaussian, orthonormal ensemble (GOE)      4 135
Gaussian, unitary ensemble (GUE)      4
Generalized correlation function      77 78
Generalized correlation function, Monte Carlo calculation of      80
Generalized correlation lengths      77 78
Generalized correlation lengths for random 1d Ising model      80
Generalized correlation lengths for random d > 1 Ising model      85
Generalized correlation lengths for random field 1d Ising model      79
Generalized localization lengths      100—101
Generalized localization lengths for random potentials with power law distribution      104
Generalized localization lengths, weak disorder expansion for 1d Anderson model      102
Generalized Lyapunov exponents      22—23 101
Generalized Lyapunov exponents for i.i.d. gaussian matrices      31
Generalized Lyapunov exponents for kinematic dynamo      124
Generalized Lyapunov exponents in canonical ensemble      39
Generalized Lyapunov exponents in microcanonical ensemble      38
Generalized semicircular theorem      5
Gram — Schmidt orthonormalization method      27 142
Hamiltonian flow      56
Hausdorff distance      129
Henon map      51
Herbert — Jones — Thouless formula for 1d localization      92
Herbert — Jones — Thouless formula for quasi-1d localization      108
Hilbert space      121
Incidence angle      119
Incidence angle, average critical      120
Incidence angle, critical      120
Independent electrons approximation      87
Independent RM approximation for hamiltonian flows      56—57
Independent RM approximation for kinematic dynamo      126
Independent RM approximation for symplectic maps      46
Induction equation      123
Ising model      59
Ising model, random 1d      11
Ising model, random 2d Ising model      82
Ising model, random coupling 1d      61
Ising model, random field 1d      61
Iterated function system (IFS)      129
Iterated function system, average contractive      129 130
Kinematic dynamo      123
Lagrangian, chaos      124
Lagrangian, motion      124
Landauer formula for 1d localization      93
Landauer formula for quasi-1d localization      108
Large number law for i.i.d. matrices in $SL(D,\mathds{R})$      20
Large number law for i.i.d. matrices with positive elements      20
Large number law for random numbers      17
Legendre tranformation      25 150
Localization length      12 90
Localization length fluctuations      101
Localization length fluctuations, weak disorder expansion for 1d Anderson model      102
Localization length for quasi-1d localization      108
Localization of eigenstates in random harmonic chains      12
Localization of wave functions in correlated random potential      153
Localization of wave functions in random potential      12
Localized states      88
Lognormal probability distribution      18 24
Low temperature expansion for random field 1d Ising model      63—67
Lozi map      52
Lyapunov basis      21
Lyapunov exponents (CLE)      21
Lyapunov exponents for i.i.d. gaussian matrices      28
Lyapunov exponents, numerical calculation of      27 138
Lyapunov spectrum      21
Magnetic diffusivity      123
Magnetic dynamos      see "Kinematic dynamo"
Markov chain      144
Markov chain, ergodic      144
Maximum entropy ansatz      113 see maximum
Maximum Lyapunov exponent      10 19
Maximum Lyapunov exponent for i.i.d. gaussian matrices      31
Maximum Lyapunov exponent for large sparse i.i.d. matrices      31
Maximum Lyapunov exponent, effective      25
Maximum Lyapunov exponent, numerical calculation of      26
Mean return time in Poincare section      45
Metal-insulator transition      109
Microcanonical, ensemble for PRM      36
Microcanonical, method      37
Mott — Twose conjecture      89
Multifractal      126 149
Multifractal measure      125
Nearest-neighbours tight-binding model      89 see
Numerical methods for the computation of Lyapunov exponents      see "Lyapunov exponents"
Optical index      118
Oseledec theorem      20
Oseledec theorem in disordered systems      71
Oseledec theorem in dynamical systems      44
Partition function      11
Partition function for 1d Ising model      60
Partition function for random 2d Ising model      83
Penetration length      119 123
Perturbative expansion for 1d localization      96—97
Perturbative expansion for products of symplectic matrices      48—50
Perturbative expansion for random field 1d Ising model      62—64 67
Pesin relation      55 147
Poincare section      142—143
Poincare section, map      143
Poisson equation      136
Polymers      see "Directed random polymers"
Probability measure      125 133 149
Product of commutative diagonalizable ma trices      13
Product of random numbers      13
Quantum Ohm law      94
Quenched free energy      see "Free energy"
Random energy model      32
Reflection coefficient      94 119
Replica method for 1d localization      98
Replica method for disorderd systems      71
Replica method for product of random matrices      34
Replica method, variational      99
Resonance in optical transmission      see "Extended states in optical localization"
Schroedinger equation      88 118
Schroedinger equation, discretized with random potential      see "Anderson model"
Self-averaging      71 88
Semicircular theorem      4
Semicircular theorem, generalised semicircular theorem      5
Shannon — McMillan theorem      145
Sierpinsky, carpet      131
Sierpinsky, gasket      130
Stability basis      21
Stability matrix for dynamical systems      10 44
Stability matrix for hamiltonian flows      56
Stability matrix for symplectic maps      46
Stability of ecosystems      7—8
Statistics of eigenvalues, complex random matrices      138
Statistics of eigenvalues, real random matrices      5 135
Statistics of eigenvalues, real random symmetric matrices      4
Superlocalization      104
Symbolic dynamics      146
Symplectic, maps      9 45
Symplectic, matrices      46 107
Tangent map      10
Tangent map for 2d symplectic map      10
Tangent map for Baker transformation      52
Tangent map for Lozi map      52
Tangent vector      10 43
Thermal average      75
Thermodynamic limit for Lyapunov exponents      54
Thermodynamic limit for quasi-1d localization      109
Thermodynamic limit, correction to t.l. in disordered systems      71
Tight-binding approximation      see "Anderson model"
Time-evolution operator      43
Transfer matrix for 1d Ising model      60
Transfer matrix for 1d optical media      119 121
Transfer matrix for 2d Anderson model      107
Transfer matrix for Anderson model      90
Transfer matrix for random 1d Ising model      11
Transfer matrix for random 2d Ising model      84
Transfer matrix for random coupling 1d Ising model      61
Transfer matrix for random field 1d Ising model      61
Transfer matrix for random waveguide      122
Transition matrix of Markov process      53 144 152
Transmission coefficient      94 119
Verlet method      56
Virster theorem      22 23
Wave-equation      118
waveguide      see "Transfer matrix for random waveguide"
Weak disorder expansion      33
Weak disorder expansion for 1d localization      97
Wiener, paths      127
Wiener, process      127
Zel’dovic antidynamo theorem      127
Zero temperature entropy of random coupling 1d Ising model      69 70
Zero temperature limit for random coupling 1d Ising model      68 69
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