This book deals with the various thermodynamic concepts used for the analysis of nonlinear dynamical systems. The most important invariants used to characterize chaotic systems are introduced in a way that stresses the interconnections with thermodynamics and statistical mechanics. Among the subjects treated are probabilistic aspects of chaotic dynamics, the symbolic dynamics technique, information measures, the maximum entropy principle, general thermodynamic relations, spin systems, fractals and multifractals, expansion rate and information loss, the topological pressure, transfer operator methods, repellers and escape. The more advanced chapters deal with the thermodynamic formalism for expanding maps, thermodynamic analysis of chaotic systems with several intensive parameters, and phase transitions in nonlinear dynamics.
Author(s): Christian Beck, Friedrich Schögl
Series: Cambridge Nonlinear Science Series
Publisher: Cambridge University Press
Year: 1993
Language: English
Pages: 306
Contents......Page 7
Preface......Page 13
Introduction......Page 15
1.1 Maps and trajectories......Page 21
1.2 Attractors......Page 22
1.3 The logistic map......Page 24
1.4 Maps of Kaplan-Yorke type......Page 31
1.5 The Henon map......Page 34
1.6 The standard map......Page 36
2.1 Relative frequencies and probability......Page 40
2.2 Invariant densities......Page 42
2.3 Topological conjugation......Page 48
3.1 Partitions of the phase space......Page 52
3.2 The binary shift map......Page 53
3.3 Generating partition......Page 55
3.4 Symbolic dynamics of the logistic map......Page 57
3.5 Cylinders......Page 59
3.6 Symbolic stochastic processes......Page 60
4.1 Bit-numbers......Page 64
4.2 The Shannon information measure......Page 66
4.3 The Khinchin axioms......Page 67
5.1 The Rényi information......Page 70
5.2 The information gain......Page 71
5.3 General properties of the Rényi information......Page 73
6.1 The maximum entropy principle......Page 76
6.2 The thermodynamic equilibrium ensembles......Page 79
6.3 The principle of minimum free energy......Page 81
6.4 Arbitrary a priori probabilities......Page 83
7.1 The Legendre transformation......Page 85
7.2 Gibbs' fundamental equation......Page 88
7.3 The Gibbs-Duhem equation......Page 91
7.4 Susceptibilities and fluctuations......Page 93
7.5 Phase transitions......Page 95
8.1 Various types of spin models......Page 98
8.2 Phase transitions of spin systems......Page 102
8.3 Equivalence between one-dimensional spin systems and symbolic stochastic processes......Page 103
8.4 The transfer matrix method......Page 105
9.1 Temperature in chaos theory......Page 108
9.2 Two or more intensities......Page 110
9.3 Escort distributions for general test functions......Page 111
10.1 Simple examples of fractals......Page 114
10.2 The fractal dimension......Page 117
10.3 The Hausdorff dimension......Page 120
10.4 Mandelbrot and Julia sets......Page 122
10.5 Iterated function systems......Page 129
11.1 The grid of boxes of equal size......Page 134
11.2 The Rényi dimensions......Page 135
11.3 Thermodynamic relations in the limit of box size going to zero......Page 137
11.4 Definition of the Rényi dimensions for cells of variable size......Page 143
12.1 Moments and cumulants of bit-numbers......Page 147
12.2 The bit-variance in thermodynamics......Page 149
12.3 The sensitivity to correlations......Page 151
12.4 Heat capacity of a fractal distribution......Page 152
13.1 Boxes of finite size......Page 156
13.2 Thermodynamic potentials for finite volume......Page 159
13.3 An analytically solvable example......Page 163
14.1 The Kolmogorov-Sinai entropy......Page 166
14.2 The Rényi entropies......Page 169
14.3 Equivalence with spin systems......Page 175
14.4 Spectra of dynamical scaling indices......Page 176
15.1 Expansion of one-dimensional systems......Page 178
15.2 The information loss......Page 180
15.3 The variance of the loss......Page 181
15.4 Spectra of local Liapunov exponents......Page 184
15.5 Liapunov exponents for higher-dimensional systems......Page 187
15.6 Stable and unstable manifolds......Page 192
16.1 Definition of the topological pressure......Page 198
16.2 Length scale interpretation......Page 202
16.3 Examples......Page 204
16.4 Extension to higher dimensions......Page 207
16.5 Topological pressure for arbitrary test functions......Page 208
17.1 The Perron-Frobenius operator......Page 210
17.2 Invariant densities as fixed points of the Perron-Frobenius operator......Page 213
17.3 Spectrum of the Perron-Frobenius operator......Page 216
17.4 Generalized Perron-Frobenius operator and topological pressure......Page 217
17.5 Connection with the transfer matrix method of classical statistical mechanics......Page 219
18.1 Repellers......Page 224
18.2 Escape rate......Page 227
18.3 The Feigenbaum attractor as a repeller......Page 228
19.1 A variational principle for the topological pressure......Page 231
19.2 Gibbs measures and SRB measures......Page 234
19.3 Relations between topological pressure and the Rényi entropies......Page 238
19.4 Relations between topological pressure and the generalized Liapunov exponents......Page 241
19.5 Relations between topological pressure and the Rényi dimensions......Page 242
20.1 The pressure ensemble......Page 246
20.2 Equation of state of a chaotic system......Page 249
20.3 The grand canonical ensemble......Page 253
20.4 Zeta functions......Page 257
20.5 Partition functions with conditional probabilities......Page 260
21.1 Static phase transitions......Page 263
21.2 Dynamical phase transitions......Page 268
21.3 Expansion phase transitions......Page 271
21.4 Bivariate phase transitions......Page 274
21.5 Phase transitions with respect to the volume......Page 276
21.6 External phase transitions of second order......Page 280
21.7 Renormalization group approach......Page 282
21.8 External phase transitions of first order......Page 287
References......Page 293
Index......Page 302
