Deterministic Abelian Sandpile Models and Patterns

The model investigated in this work, a particular cellular automaton with stochastic evolution, was introduced as the simplest case of self-organized-criticality, that is, a dynamical system which shows algebraic long-range correlations without any tuning of parameters. The author derives exact results which are potentially also interesting outside the area of critical phenomena. Exact means also site-by-site and not only ensemble average or coarse graining. Very complex and amazingly beautiful periodic patterns are often generated by the dynamics involved, especially in deterministic protocols in which the sand is added at chosen sites. For example, the author studies the appearance of allometric structures, that is, patterns which grow in the same way in their whole body, and not only near their boundaries, as commonly occurs. The local conservation laws which govern the evolution of these patterns are also presented. This work has already attracted interest, not only in non-equilibrium statistical mechanics, but also in mathematics, both in probability and in combinatorics. There are also interesting connections with number theory. Lastly, it also poses new questions about an old subject. As such, it will be of interest to computer practitioners, demonstrating the simplicity with which charming patterns can be obtained, as well as to researchers working in many other areas.

Nominated as an outstanding PhD thesis by the University of Pisa Includes many beautiful and highly detailed color images explaining the topic Presents numerical protocols in detail, allowing readers to independently test them Illustrates the recently discovered connection between Sierpinski-like patterns and the Abelian Sandpile Model Includes supplementary material: sn.pub/extras

Autorentext
Guglielmo Paoletti is a post-doc researcher at the Laboratoire d'Informatique de Paris 6 in Paris since January 2013. Born in 1982, he graduated in Physics in 2007 at University of Milan and got his doctorate in Physics at University of Pisa in 2012. He has been a post-doc at the Laboratory of Theoretical Physics and Statistical Models in Orsay-Paris in 2012. He is interested in Statistical Mechanics and Critical Phenomena in relation with combinatorics and graph theory. He published several peer-reviewed papers on international journals. His PhD thesis has been awarded and published by Springer.

Klappentext
The model investigated in this work, a particular cellular automaton with stochastic evolution, was introduced as the simplest case of self-organized-criticality, that is, a dynamical system which shows algebraic long-range correlations without any tuning of parameters.
The author derives exact results which are potentially also interesting outside the area of critical phenomena. Exact means also site-by-site and not only ensemble average or coarse graining. Very complex and amazingly beautiful periodic patterns are often generated by the dynamics involved, especially in deterministic protocols in which the sand is added at chosen sites.
For example, the author studies the appearance of allometric structures, that is, patterns which grow in the same way in their whole body, and not only near their boundaries, as commonly occurs. The local conservation laws which govern the evolution of these patterns are also presented.
This work has already attracted interest, not only in non-equilibrium statistical mechanics, but also in mathematics, both in probability and in combinatorics. There are also interesting connections with number theory. Lastly, it also poses new questions about an old subject. As such, it will be of interest to computer practitioners, demonstrating the simplicity with which charming patterns can be obtained, as well as to researchers working in many other areas.

Inhalt

Introduction.- The Abelian Sandpile Model.- Algebraic structure.- Identity characterization.- Pattern formation.- Conclusions.- SL(2, Z).- Complex notation for vectors in R2.- Generalized quadratic B´ezier curve.- Tessellation.