Emergence and complexity in theoretical models of self-organized criticality

28.5 MB·

by  Tridib Sadhu

Emergence and Complexity in
Theoretical Models of Self-Organized
Criticality
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2 A Thesis

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Submitted to the
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Tata Institute of Fundamental Research, Mumbai

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c in Physics
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m Tridib Sadhu
- Department of theoretical physics
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n Tata Institute of Fundamental Research
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July 2011
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Statutory Declarations
Name of the Candidate : Tridib Sadhu
Title of the Thesis : Emergence and Complexity in
Theoretical Models of Self-
Organized Criticality.
Degree : Doctor of Philosophy (Ph.D.)
Subject : Physics
Name of the advisor : Prof. Deepak Dhar
Registration number : PHYS–154
Place of Research : Tata Institute of Fundamental
Research, Mumbai 400005
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Declaration Of Authorship
This thesis is a presentation of my original research work. Wherever con-
tributions of others are involved, every effort is made to indicate this
clearly, with due reference to the literature, and acknowledgement of col-
laborative research and discussions.
The work was done under the guidance of Professor Deepak Dhar, at
the Tata Institute of Fundamental Research, Mumbai.
Signed: ........................ Date:........................
Name: Tridib Sadhu
In my capacity as supervisor of the candidate’s thesis, I certify that the
above statements are true to the best of my knowledge.
Signed: ........................ Date:........................
Name: Prof. Deepak Dhar
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Acknowledgements
I would like to express my gratitude to all those who helped me in the
completion of this work. Most importantly, I gratefully acknowledge the
guidance and support of my thesis advisor Prof. Deepak Dhar. His em-
phasis on perfection and strong sense of work ethics have had a deep
influence on me.
I also acknowledge the financial support I received from TIFR, Mum-
bai.
My special thanks to Shaista for proofreading, and for helping me in
many technical aspects of preparing the thesis. Finally, I would like to
thank my family for their unconditional love and support.
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Contents
Statutory Declarations 1
Declaration of Authorship 3
Acknowledgements 5
1 Synopsis 19
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 Spatial patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.1 The characterization of the pattern . . . . . . . . . . . 23
1.2.2 The effect of multiple sources or sinks . . . . . . . . . 24
1.2.3 The compact and non-compact growth . . . . . . . . 26
1.3 Zhang model . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4 Stochastic models . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 List of publications 33
3 Introduction 35
3.1 Self-organized criticality (SOC) . . . . . . . . . . . . . . . . . 35
3.2 Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Deterministic abelian Sandpile Model (DASM) . . . 38
3.2.2 Zhang model . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.3 Manna model . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Universality in the sandpile models. . . . . . . . . . . . . . . 43
3.4 Experimental models of SOC . . . . . . . . . . . . . . . . . . 45
3.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 Overview of the later chapters . . . . . . . . . . . . . . . . . 48
4 Pattern formation on growing sandpiles 51
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Definition of the model . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Characterizing asymptotic pattern: A general theory . . . . 58
4.4 Determination of the potential function . . . . . . . . . . . . 62
4.5 Other patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 Effectofmultiplesourcesandsinksonthegrowingsandpilepat-
tern 75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Rate of growth of the patterns. . . . . . . . . . . . . . . . . . 77
5.3 A single sink site . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 85
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CONTENTS
5.5 Characterization of the pattern with a line sink . . . . . . . . 87
5.6 Patterns with two sources . . . . . . . . . . . . . . . . . . . . 90
5.7 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 95
5.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6 Pattern Formation in Fast-Growing Sandpiles 101
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Compact and non-compact growth . . . . . . . . . . . . . . . 107
6.3 Examples of non-compact growth . . . . . . . . . . . . . . . 110
6.4 Piece-wise linearity of the toppling function . . . . . . . . . 113
6.5 Characterizing the class I asymptotic patterns . . . . . . . . 116
6.6 Non-compact patterns with exponent α < 1. . . . . . . . . . 125
6.7 Summary and concluding remarks . . . . . . . . . . . . . . . 126
7 A continuous height sandpile model 135
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Definition and preliminaries . . . . . . . . . . . . . . . . . . . 137
7.3 The propagator, and its relation to the discrete abelian model 138
7.4 Calculation of Σ2(x) in large-L limit . . . . . . . . . . . . . . 140
7.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 146
8 Stochastic sandpile models 147
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.3 Determination of the steady state for a DASM . . . . . . . . 150
8.4 Algebra of the addition operators for SASM . . . . . . . . . 152
8.5 Jordan Block structure of the addition operators . . . . . . . 154
8.6 Matrix representation in the configuration basis . . . . . . . 157
8.7 Determination of the steady state vector . . . . . . . . . . . . 159
8.8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 159
8.9 Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . 163
A Solution of Laplace’s equation on hexagonal lattice 165
B Relation to the theory of discrete analytic functions 169
C Solution of the Eq. (5.18) 173
D Jordon Block for L=3 SASM 175
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