Self-Organized Criticality : Emergent Complex Behavior in Physical and Biological Systems

by Henrik Jeldtoft Jensen

• ISBN13: 9780521483711
• ISBN10: 0521483719
• Format: Paperback
• Copyright: 1998-01-13
• Publisher: Cambridge University Press

Summary

Self-organized criticality (SOC) maintains that complex behavior can develop spontaneously in certain multi-body systems whose dynamics vary abruptly. This is a clear and concise introduction to the field of self-organized criticality, and contains an overview of the main research results. The author begins with an examination of what is meant by SOC, and the systems in which it can occur. He then presents and analyzes computer models to describe a number of systems, explaining the different mathematical formalisms developed to understand SOC. The final chapter assesses the impact of this field of study, and highlights some key areas of new research. The author assumes no previous knowledge of the field, and the book contains several exercises. It will be ideal as a textbook for graduate students taking physics, engineering, or mathematical biology courses in nonlinear science or complexity.

Table of Contents

Preface	xi	(2)
Acknowledgment	xiii
1 Introduction	1	(6)
2 Characterization of the SOC State	7	(5)
2.1 Response Distributions	7	(1)
2.2 Temporal Fluctuations	8	(1)
2.3 Power Spectrum and Distribution of Lifetimes	9	(1)
2.4 Spatial Correlation Functions	10	(2)
3 Systems Exhibiting SOC	12	(17)
3.1 Introduction	12	(2)
3.2 Sandpiles	14	(2)
3.3 Ricepiles	16	(2)
3.4 Superconducting Avalanches	18	(5)
3.5 Droplet Formation	23	(2)
3.6 Earthquakes	25	(2)
3.7 Evolution	27	(2)
4 Computer Models	29	(47)
4.1 Introduction	29	(1)
4.2 Sandpiles: Conservative Model	29	(13)
4.2.1 One-Dimensional Sandpile	30	(3)
4.2.2 Dimensions Larger than I	33	(3)
4.2.3 Critical Response	36	(1)
4.2.4 Numerical Results: Distribution Functions	37	(2)
4.2.5 Power Spectrum	39	(3)
4.3 Earthquake Models: Nonconservative Models	42	(16)
4.3.1 Criticality of the OFC Model	44	(1)
4.3.2 Nearest Neighbor OFC Model	44	(2)
4.3.3 Random Neighbor OFC Model	46	(1)
4.3.4 Distributions and Fluctuations in the Nearest Neighbor OFC Model	47	(4)
4.3.5 The Effect of Disorder on the OFC Model	51	(3)
4.3.6 Physical Relevance of the OFC Model	54	(4)
4.4 Lattice Gas	58	(7)
4.4.1 Definition of the Lattice Gas Model	58	(2)
4.4.2 Properties of Lattice Gas	60	(2)
4.4.3 The Lesson of Lattice Gas	62	(2)
4.4.4 Physical Relevance of the Lattice Gas Model	64	(1)
4.5 Forest Fires	65	(3)
4.5.1 Definition of a Critical Forest Fire Model	66	(1)
4.5.2 Simulation Results for the Forest Fire Model	67	(1)
4.5.3 Physical Relevance of the Forest Fire Model	68	(1)
4.6 Extremum Dynamics	68	(8)
4.6.1 The Model of Interface Growth in a Random Medium	69	(4)
4.6.2 The Evolution Model	73	(3)
5 The Search for a Formalism	76	(49)
5.1 Introduction	76	(1)
5.2 Mean Field Theory	77	(15)
5.2.1 Sandpile Models	78	(3)
5.2.2 Earthquake Models	81	(4)
5.2.3 Diffusive Description of Lattice Gas	85	(1)
5.2.4 Forest Fire Model	86	(3)
5.2.5 Model of Biological Evolution	89	(3)
5.3 Exact Solution of the Abelian Sandpile	92	(8)
5.3.1 The XXX Matrix and the Probability Measure on the Configuration Space	93	(4)
5.3.2 Correlation Functions	97	(3)
5.4 Langevin Equations	100	(7)
5.4.1 Conservative Models	100	(3)
5.4.2 Nonconservative Models	103	(4)
5.5 Dynamically Driven Renormalization Group Calculations	107	(18)
5.5.1 Renormalization Transformation	108	(8)
5.5.2 Exponents	116	(3)
5.5.3 Nonconservative Models	119	(5)
5.5.4 Forest Fire Models	124	(1)
6 Is It SOC or Not?	125	(6)
6.1 Where Is SOC to be Found?	126	(1)
6.2 What Is Tuning?	127	(4)
Appendices	131	(18)
A Code for the BTW Sandpile	131	(3)
B Code for the Lattice Gas	134	(4)
C Code for the Bak-Sneppen Evolution Model	138	(3)
D Power Spectra and the Correlation Function	141	(5)
E Statistical Weights in the DDRG	146	(3)
References	149	(4)
Index	153