Chaos, Nonlinearity, Complexity (eBook)

Preface 8
Contents 14
List of Contributors 16
1 Chaos, Periodicity and Complexity on Dynamical Systems 18
1.1 Introduction 18
1.2 On the periodic structure of some dynamical systems 23
1.3 Metric entropy or Kolmogorov-Sinai entropy (KS) 37
1.4 Properties of the metric entropy of measurable sequences 46
1.5 Topological entropy 63
References 65
2 Critical Attractors and the Physical Realm of q-statistics 70
2.1 Outline 70
2.2 Part I. Anomalous dynamics at onset of chaos and other critical attractors in unimodal maps 73
2.3 Part II. Critical attractor dynamics in thermal systems at phase transitions and glass formation 87
2.4 Clarifying remarks 105
References 109
3 Foundations of Nonextensive Statistical Mechanics 112
3.1 Introduction 112
3.2 Maximum Tsallis entropy principle 114
3.3 Method of steepest descents 117
3.4 Counting algorithm 120
3.5 Evaluation of the density of states 123
3.6 Generalized Boltzmann equation 127
3.7 Concluding remarks 129
Acknowledgment 130
References 130
4 Non-Boltzmannian Entropies for Complex Classical Systems, Quantum Coherent States and Black Holes 131
4.1 Introduction 131
4.2 Helmholtz free energy and Renyi entropy 134
4.3 On stability of Renyi and Tsallis entropies 138
4.4 Axiomatics of an information entropy 144
4.5 MEP for Renyi entropy 148
4.6 MEP for Tsallis entropy 151
4.7 Small subsystem with .uctuating temperature 153
4.8 Power-law Hamiltonian 159
4.9 Transfer to Renyi thermostatistics as a phase transition 161
4.10 The most probable value of the Renyi parameter 162
4.11 Conclusive Notes on the Renyi entropy and Self- Organization 164
4.12 Thermodynamics of a Quantum Mechanical system in Coherent State 166
References 175
5 Modelling and Analysis of LRD in Packet Networks: Tsallis Entropy Framework 179
5.1 Introduction 179
5.2 Tsallis Entropy and Some Other Parametric Entropies 182
5.3 Axiomatic Foundations of Parametric Entropies 185
5.4 Tsallis Entropy and Network Tra.c 188
5.5 Internet Traffic and Tsallis Entropy 191
5.6 Broadband Networks and Non-Extensive Thermodynamics 192
5.7 Conclusion 193
References 193
6 The Role of Chaos and Resonances in Brownian Motion 197
6.1 Introduction 197
6.2 The Prigogine school 198
6.3 Mechanical model for Brownian motion 201
6.4 Entropy and information 205
6.5 Results 211
6.6 Thermalization of the lattice 220
6.7 Conclusion 222
References 224
7 Models of Finite Bath and Generalised Thermodynamics 225
7.1 Introduction 225
7.2 Derivation of canonical ensemble from microcanonical ensemble 226
7.3 Gaussian ensemble 227
7.4 q-exponential distributions and model of .nite heat bath 230
7.5 A new model for .nite bath 231
7.6 Discussion 232
Acknowledgements 234
References 234
8 Quantum Black Hole Thermodynamics 236
8.1 Introduction 236
8.2 Black holes in general relativity 239
8.3 Black hole thermodynamics 242
8.4 Microcanonical entropy 245
8.5 Canonical entropy 256
8.6 Conclusions 262
References 263
9 Complexity in Organizations: A Paradigm Shift 265
9.1 Complexity in organizations: a paradigm shift 265
9.2 A paradigm shift 266
9.3 Core paradigmatic focus 267
9.4 The functions of organization 270
9.5 Organizational control 271
9.6 Emergence 277
9.7 Motivation 278
9.8 De.nition of leadership 280
9.9 Complexity versus top-down? 281
9.10 Conclusions 282
References 282
10 Chaos, Nonlinearity, Complexity: A Uni.ed Perspective 288
10.1 Introduction 289
10.2 ChaNoXity: Chaos, Nonlinearity, Complexity 294
10.4 Conclusions: The Mechanics of Thermodynamics 365
References 368
Index 372

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