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 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 dissipative site is uniformly bounded from above, then the model is not critical. We apply also applied a renormalisation method to the model in order to deduce its critical exponents and to determine whether 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 and related concepts have been studied both in the mathematics and physics literature. Although much 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. With Jos, I mostly discussed numerical simulation, renormalisation theory and general theory of critical 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 1 Introduction It has been almost 30 years since Per Bak, Chao Tang and Kurt Wiesenfeld (BTW) proposed the sandpile model as a paradigm of self-organized criticality (SOC) [1]. It serves as the simplest and best- studied example of a non-equilibrium system, driven at a slow steady rate by adding particles, with local threshold relaxation rules, which in its critical state shows power-law behaviour obtained without fine-tuning of any control parameters. BTW claimed that the concept of SOC is an explanation of many different physical phenomena: from the formation of the Earth's crust to the dynamics of solar flares to the distribution of skyscrapers in the world's biggest cities [2]. Figure 1: Per Bak, Chao Tang and Kurt Wiesenfeld.1 This immediately explains the relevance of studying `toy models' such as the sandpile model. Through detailed understanding of these relatively easy models, we may be able to make predictions about certain phenomena in the real world. The model was originally defined as follows. In one dimension (d = 1), consider a connected subset of Z, which we will call Vn. Without loss of generality we can take Vn = [−n; n] \ Z, and thus 2n + 1 is the size of the system. We denote by x 2 Vn a site, and we define a height function η on Vn. Choosing a site x at random, we consequently keep adding particles to the system, increasing local heights by one unit. η(x) ! η(x) + 1 Given the fixed threshold value 2, when a local height becomes greater or equal to 1, one particle tumbles, which is referred to as a toppling event. η(x) ! η(x) − 2 η(x ± 1) ! η(n ± 1) + 1 if η(x) ≥ 2 Boundaries can be either absorbing or periodic. With absorbing boundaries, the one-dimensional BTW model naturally evolves to a minimally stable state, where all but one site have a height of 1. The toppling rules in higher dimensions are essentially the same as in one dimension. In most papers, a d sandpile on a simply connected subset of Z is defined as a height function η with a critical height of 2d determines the toppling condition. Upon toppling, a site distributes its height to its 2d nearest- neighbours. On boundary sites, particles are lost. Thereafter, if another site becomes unstable from a previous toppling, it topples, until the time that no unstable sites exist in the system anymore. At this point, a particle is again added to a random location in Vn. A series of connected toppling events is called an avalanche. However, the dynamics of higher-dimensional BTW models is fundamentally 1Sources: http://www.eoht.info/page/Per+Bak, https://en.wikipedia.org/wiki/Chao_Tang, https://www. physics.gatech.edu/user/kurt-wiesenfeld 1 INTRODUCTION different from the one-dimensional model. Rather than evolving to its minimally stable configuration, it naturally evolves to a critical state in which avalanches of all sizes, up to the size of the system itself, occur.