Dissipation in the Abelian Sandpile Model

88 Pages·2016·2.26 MB·English

Technische Universiteit Delft
Faculteit Elektrotechniek, Wiskunde en Informatica
Faculteit Technische Natuurwetenschappen
Delft Institute of Applied Mathematics
Dissipation in the Abelian Sandpile Model
Verslag ten behoeve van het
Delft Institute of Applied Mathematics
als onderdeel ter verkrijging
van de graad van
BACHELOR OF SCIENCE
in
Technische Wiskunde
en
Technische Natuurkunde
door
HENK JONGBLOED
Delft, Nederland
Augustus 2016
Copyright (cid:13)c 2016 door Henk Jongbloed. Alle rechten voorbehouden.

BSc verslag TECHNISCHE WISKUNDE en TECHNISCHE NATUURKUNDE
“Dissipation in the Abelian Sandpile Model”
HENK JONGBLOED
Technische Universiteit Delft
Begeleiders
Prof. dr.ir. F.H.J. Redig
Dr.ir. J.M. Thijssen
Overige commissieleden
Dr. T. Idema Dr. J.L.A. Dubbeldam
Dr. J.A.M. de Groot
Augustus, 2016 Delft

Abstract
The Abelian Sandpile model was originally introduced by Bak, Tang and Wiesenfeld in 1987 as a
paradigm for self-organized criticality. In this thesis, we study a variant of this model, both from the
point of view of mathematics, as well as from the point of view of physics. The effect of dissipation
and creation of mass is investigated. By linking the avalanche dynamics of the infinite-volume sandpile
model to random walks, we derive some criteria on the amount of dissipation and creation of mass
in order for the model to be critical or non-critical. As an example we prove that a finite amount of
conservative sites on a totally dissipative lattice is not critical, and more generally, if the distance to a
dissipativesiteisuniformlyboundedfromabove, thenthemodelisnotcritical. Weapplyalsoapplieda
renormalisationmethodtothemodelinordertodeduceitscriticalexponentsandtodeterminewhether
a constant bulk dissipation destroys critical behaviour. Numerical simulations and a statistical analysis
are performed to estimate critical exponents. Finally, we give a short discussion on self-organized
criticality.

Preface
Sitting here in the pittoresque Zillertal in Austria, the beauty of nature intrigues me. Moreover,
the symphony of light, sound and movement has triggered existential sensations in many people in
history. No wonder that the great Greek philosopher Aristotle has said ‘Wonder is the beginning of all
philosophy’.
The research documented in this report forms my Bachelor Project at the TU Delft for the pro-
grammes Applied Mathematics and Applied Physics. When choosing a project, I wanted to study a
subject contained a robust mathematical aspect as well as an interesting physical interpretation. Also,
numerical simulation was a preference. These properties are all found in the subject of this research,
which is the Abelian Sandpile Model. Proposed for the first time in 1987 by Bak, Tang and Wiesenfeld
in a physical context, it served as an illustration of ‘self-organized criticality’. Since then, the model
andrelatedconceptshavebeenstudiedbothinthemathematicsandphysicsliterature. Althoughmuch
of the model has been understood thus far, there remain open questions as to how complex structure
can arise out of some elementary local interaction rules. This touches on the philosophical concept of
emergence. These three disciplines combined -physics, mathematics and philosophy- make the Abelian
Sandpile model a beautiful subject for me to study.
Some personal notes are in place here. During this project, I had a lot of other activities, like
three courses, extracurricular activities and preparations for next year, which will be a year on the
board of a student association. All of this made it difficult to really focus on the project, especially
since it is on my own. In fact, I have come to better know myself, my interests and my weaknesses in
studying. Projects are different from courses in a lot of ways and require a great deal of independence.
To double-degree bachelor students, the difference can be even greater due to the relative small number
of projects we have had before and necessity of ‘efficient course studying’ in order to graduate in time.
In a project, efficiency is a lot harder to achieve, at least in my case. Delay of thesis defence was one
of the consequences of my behaviour. I do not know exactly how to say it, but the last year of my
bachelor was a lot harder that expected. This project was a real challenge for me; I have learned a lot
but also disappointed myself due to my bad skills in planning. These are all lessons I will take with me
in the future.
I would like to thank my supervisors Frank Redig and Jos Thijssen for the time and effort they
offered to help me in my project. Making appointments always went smoothly and I have learned a
lot from them. The mathematical background of this project I discussed with Frank, and many times
during these meetings we ran into problems that I could not even begin to solve with my relative small
experience in the field of probability. Luckily, Frank always found a way to tackle a specific problem.
WithJos, Imostlydiscussednumericalsimulation, renormalisationtheoryandgeneraltheoryofcritical
phenomena. These meetings were always very informative and relaxed, for which I thank him greatly.
I also thank dr. Johan Dubbeldam, dr. Timon Idema and dr. Joost de Groot for taking place in my
thesis committee.
Finally, I would like to thank my family, roommates and close friends for giving me advice and
helping me.
Henk Jongbloed, August 15th, 2016

Contents
1 Introduction 1
2 The classical Abelian Sandpile Model 3
3 Introduction to mixed dissipative/source systems 7
4 Markov Processes, semigroups and generators 11
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Deriving the generator of Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 The Feynman-Kac formula for countable state space Markov Processes . . . . . . . . . . 16
5 Toppling numbers, Avalanches and Random Walks 18
5.1 Linking avalanche dynamics to random walks . . . . . . . . . . . . . . . . . . . . . . . . 20
5.2 Towards infinite volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.3 Estimating critical behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.4 From a CRW to a DRW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Mathematical results 29
6.1 Conditions on D to obtain non-criticality . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 Adding sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3 Finitely many sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7 A Renormalisation Approach to the BTW model 35
7.1 Introduction: Critical phenomena and renormalisation . . . . . . . . . . . . . . . . . . . 35
7.2 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.3 Renormalisation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7.4 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7.5 Critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7.6 Introducing dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8 Numerical simulation 46
8.1 Simulating the ASM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
9 Critical avalanche data analysis 53
9.1 The BTW approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
9.2 Likelihood estimation of τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.3 Truncated power law MLE estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
9.4 Our recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
10 Self-Organized Criticality 61
11 Conclusions, Discussion, Recommendations and personal notes 64
Appendices 68
A Project Description: Dissipation in the Abelian Sandpile Model 68
B Code 69