Sengupta Chaos, Nonlinearity, Complexity

The Dynamical Paradigm of Nature
1. Auflage 2006
ISBN: 978-3-540-31757-9
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 206, 371 Seiten

Reihe: Studies in Fuzziness and Soft Computing

This carefully edited book presents a focused debate on the mathematics and physics of chaos, nonlinearity and complexity in nature. It explores the role of non-extensive statistical mechanics in non-equilibrium thermodynamics, and presents an overview of the strong nonlinearity of chaos and complexity in natural systems that draws on the relevant mathematics from topology, measure-theory, inverse and ill-posed problems, set-valued analysis, and nonlinear functional analysis. It presents a self-contained scientific theory of complexity and complex systems as the steady state of non-equilibrium systems, denoting a homeostatic dynamic equilibrium between stabilizing order and destabilizing disorder.

Written for: Researchers, engineers, graduate students in Soft Computing, Fuzziness, Complexity
Keywords: Chaos, Complexity, Nonlinear Functional Analysis, Nonlinearity.

Autoren/Hrsg.

Sengupta, Ashok

Inhaltsverzeichnis

1;Preface;8
2;Contents;14
3;List of Contributors;16
4;1 Chaos, Periodicity and Complexity on Dynamical Systems;18
4.1;1.1 Introduction;18
4.2;1.2 On the periodic structure of some dynamical systems;23
4.3;1.3 Metric entropy or Kolmogorov-Sinai entropy (KS);37
4.4;1.4 Properties of the metric entropy of measurable sequences;46
4.5;1.5 Topological entropy;63
4.6;References;65
5;2 Critical Attractors and the Physical Realm of q-statistics;70
5.1;2.1 Outline;70
5.2;2.2 Part I. Anomalous dynamics at onset of chaos and other critical attractors in unimodal maps;73
5.3;2.3 Part II. Critical attractor dynamics in thermal systems at phase transitions and glass formation;87
5.4;2.4 Clarifying remarks;105
5.5;References;109
6;3 Foundations of Nonextensive Statistical Mechanics;112
6.1;3.1 Introduction;112
6.2;3.2 Maximum Tsallis entropy principle;114
6.3;3.3 Method of steepest descents;117
6.4;3.4 Counting algorithm;120
6.5;3.5 Evaluation of the density of states;123
6.6;3.6 Generalized Boltzmann equation;127
6.7;3.7 Concluding remarks;129
6.8;Acknowledgment;130
6.9;References;130
7;4 Non-Boltzmannian Entropies for Complex Classical Systems, Quantum Coherent States and Black Holes;131
7.1;4.1 Introduction;131
7.2;4.2 Helmholtz free energy and Renyi entropy;134
7.3;4.3 On stability of Renyi and Tsallis entropies;138
7.4;4.4 Axiomatics of an information entropy;144
7.5;4.5 MEP for Renyi entropy;148
7.6;4.6 MEP for Tsallis entropy;151
7.7;4.7 Small subsystem with .uctuating temperature;153
7.8;4.8 Power-law Hamiltonian;159
7.9;4.9 Transfer to Renyi thermostatistics as a phase transition;161
7.10;4.10 The most probable value of the Renyi parameter;162
7.11;4.11 Conclusive Notes on the Renyi entropy and Self- Organization;164
7.12;4.12 Thermodynamics of a Quantum Mechanical system in Coherent State;166
7.13;References;175
8;5 Modelling and Analysis of LRD in Packet Networks: Tsallis Entropy Framework;179
8.1;5.1 Introduction;179
8.2;5.2 Tsallis Entropy and Some Other Parametric Entropies;182
8.3;5.3 Axiomatic Foundations of Parametric Entropies;185
8.4;5.4 Tsallis Entropy and Network Tra.c;188
8.5;5.5 Internet Traffic and Tsallis Entropy;191
8.6;5.6 Broadband Networks and Non-Extensive Thermodynamics;192
8.7;5.7 Conclusion;193
8.8;References;193
9;6 The Role of Chaos and Resonances in Brownian Motion;197
9.1;6.1 Introduction;197
9.2;6.2 The Prigogine school;198
9.3;6.3 Mechanical model for Brownian motion;201
9.4;6.4 Entropy and information;205
9.5;6.5 Results;211
9.6;6.6 Thermalization of the lattice;220
9.7;6.7 Conclusion;222
9.8;References;224
10;7 Models of Finite Bath and Generalised Thermodynamics;225
10.1;7.1 Introduction;225
10.2;7.2 Derivation of canonical ensemble from microcanonical ensemble;226
10.3;7.3 Gaussian ensemble;227
10.4;7.4 q-exponential distributions and model of .nite heat bath;230
10.5;7.5 A new model for .nite bath;231
10.6;7.6 Discussion;232
10.7;Acknowledgements;234
10.8;References;234
11;8 Quantum Black Hole Thermodynamics;236
11.1;8.1 Introduction;236
11.2;8.2 Black holes in general relativity;239
11.3;8.3 Black hole thermodynamics;242
11.4;8.4 Microcanonical entropy;245
11.5;8.5 Canonical entropy;256
11.6;8.6 Conclusions;262
11.7;References;263
12;9 Complexity in Organizations: A Paradigm Shift;265
12.1;9.1 Complexity in organizations: a paradigm shift;265
12.2;9.2 A paradigm shift;266
12.3;9.3 Core paradigmatic focus;267
12.4;9.4 The functions of organization;270
12.5;9.5 Organizational control;271
12.6;9.6 Emergence;277
12.7;9.7 Motivation;278
12.8;9.8 De.nition of leadership;280
12.9;9.9 Complexity versus top-down?;281
12.10;9.10 Conclusions;282
12.11;References;282
13;10 Chaos, Nonlinearity, Complexity: A Uni.ed Perspective;288
13.1;10.1 Introduction;289
13.2;10.2 ChaNoXity: Chaos, Nonlinearity, Complexity;294
13.3;10.4 Conclusions: The Mechanics of Thermodynamics;365
13.4;References;368
14;Index;372