Theoretical studies of self-organized criticality

doi:10.1016/j.physa.2006.04.004

Physica A: Statistical Mechanics and its Applications

Volume 369, Issue 1, 1 September 2006, Pages 29-70

https://doi.org/10.1016/j.physa.2006.04.004 Get rights and content

Abstract

These notes are intended to provide a pedagogical introduction to the abelian sandpile model of self-organized criticality, and its related models. The abelian group, the algebra of particle addition operators, the burning test for recurrent states, equivalence to the spanning trees problem are described. The exact solution of the directed version of the model in any dimension is explained. The model's equivalence to Scheidegger's model of river basins, Takayasu's aggregation model and the voter model is discussed. For the undirected case, the solution for one-dimensional lattices and the Bethe lattice is briefly described. Known results about the two dimensional case are summarized. Generalization to the abelian distributed processors model is discussed. Time-dependent properties and the universality of critical behavior in sandpiles are briefly discussed. I conclude by listing some still-unsolved problems.

Introduction

These notes started as a written version of the lectures given at the Ecole Polytechnique Federale, Lausanne in 1998 [1]. I have updated and reorganized the material somewhat, and added a discussion of some more recent developments once before. The aim is to provide a pedagogical introduction to the abelian sandpile model and other related models of self-organized criticality. In the last nearly two decades, there has been a good deal of work in this area, and some selection of topics, and choice of level of detail has to be made to keep the size of notes manageable. I shall try to keep the discussion self-contained, but algebraic details will often be omitted in favor of citation to original papers. It is hoped that these notes will be useful to students wanting to learn about the subject in detail, and also to others only seeking an overview of the subject.

In these notes, the main concern is the study of the so-called abelian sandpile model (ASM), and its related models: the loop-erased random walks, the $q \to 0$ limit of the q-state Potts model, Scheidegger's model of river networks, the Eulerian walkers model, the abelian distributed processors model, etc. The main appeal of these models is that they are analytically tractable. One can explicitly calculate many quantities of interest, such as properties of the steady state, and some critical exponents, without too much effort. As such, they are very useful for developing our understanding of the basic principles and mechanisms underlying the general theory. A student could think of the ASM as a ‘base camp’ for the explorations into the uncharted areas of non-equilibrium statistical mechanics. The exact results in this case can also serve as proving grounds for developing approximate treatments for more realistic problems.

While these models are rather simple to define, and not too complicated to solve (at least partly), they are non-trivial, and cannot be said to be well understood yet. For example, it has not been possible so far to determine the critical exponents for avalanche distributions for the oldest and best known member of this class: the undirected sandpile model in two dimensions. There are many things we do not understand. Some will be discussed later in the lectures.

Our focus here will be on the mathematical development of these models. However, it is useful to start with a brief discussion of their origin as simplified models of physical phenomena.

Section snippets

Self-organized criticality in nature

It was the great insight of Mandelbrot that fractals are not mathematical curiosities, but that many naturally occurring objects are best described as fractals. Examples include mountain ranges, river networks, coastlines, etc. His book [2] remains the best introduction to fractals, for its clear exposition, evocative pictures, and literary style. The word ‘fractal structure’ here means that some correlation functions show non-trivial power law behavior. For example, in the case of mountain

Models of self-organized criticality

The first step in the study of SOC would be the precise mathematical formulation of some simple model which exhibits it. Such a system should have the necessary features of SOC systems: it should show non-equilibrium steady state of an extended system with a steady drive, but irregular burst-like relaxations, and long-ranged spatio-temporal correlations.

Criticality of the BTW model

It is easy to see that conservation of sand grains in the BTW model implies that on adding a single grain to a stable configuration in the steady state, events involving many topplings must occur with significant probability. Since every particle added anywhere in the system finally leaves the system from the boundary, and on the average it must take at least order L steps to reach the boundary, we see that the average number of topplings per added particle is at least of order L. In other

Definition

The BTW model has an important abelian property that simplifies its analysis considerably. To bring out this property in its full generality, it is preferable to work in more general setting. To emphasize the abelian property, which lies at the heart of theoretical analysis, I shall use the descriptive name ASM, rather than the historical, more conventional, nomenclature ‘the BTW model’ in the following [24]. The ASM is defined as follows [25]:

We consider the model defined on a graph with N

Recurrent and transient configurations

Given a stable configuration of the pile, how can we tell if it is recurrent or transient? A first observation is that there are some forbidden subconfigurations which can never be created by addition of sand and relaxation, if not already present in the initial state. The simplest example on the square lattice case is a subconfiguration of two adjacent sites of height 1, . Since $z_{i} > 0$ , a site of height 1 may only be created as a result of a toppling at one of the two sites (toppling anywhere

Potts model, spanning trees, resistor networks, and loop-erased random walks

In this section, we shall use the burning test to relate the undirected ASM to a well-known model of classical statistical mechanics: the Potts model. The generating function for the number of recurrent configurations with a given number of grains can be expressed in terms of the partition function of the Potts model in a particular limit. This relationship also allows one to relate the ASM problem to other well-studied problems in lattice statistics and graph theory: the spanning trees

The directed abelian sandpile model

The abelian group structure of the original BTW model still does not allow us to deal effectively with the avalanches in the model. The reason is that the avalanche statistics is very ‘coordinate-dependent’. For an abelian group generated by two operators ${a_{1}, a_{2}}$ , an equally good choice of generators is ${a_{1} a_{2}, a_{2}}$ . But this set will clearly have a different avalanche statistics!

There is a variant of the BTW model, the directed ASM (DASM) that is much more tractable analytically. The model was

One dimensional chains and ladders

The simplest undirected model is defined on a one dimensional chain of length L. The matrix $Δ$ is tridiagonal $Δ = (\begin{matrix} 2 & - 1 \\ - 1 & 2 & - 1 \\ - 1 & 2 & ⋱ \\ ⋱ & ⋱ \end{matrix})$ and its determinant is easily seen to be $\det Δ = L + 1 .$

It is thus a degenerate case where the number of recurrent configurations grows only polynomially and not exponentially with L. It is not hard to identify the $L + 1$ recurrent configurations. They are either a configuration with all ones, or a configuration with a single zero somewhere along the chain.

The result of an addition

Models related to the directed sandpile model

In Section 7, we studied the directed ASM (DASM) as an example of a self-organized system in which the invariant state is easily characterized, and all the avalanche exponents can be determined exactly, in all dimensions. The model is also equivalent to other statistical models which were proposed in other contexts. Here I will discuss three such models.

The abelian distributed processors model

It is useful to see how far ASM can be generalized, still retaining its abelian group structure. In this section, I shall discuss an interesting generalization of the ASM. The model consist of a network of nodes and links (a graph), on whose nodes sit processors, which are finite state automata. Each of these has an input stack of messages. When the input stack of a processor is not empty, it pops one of the messages according to a predetermined order, and processes it. As a result of the

Time-dependent properties

Though an explanation of $1 / f$ noise was one of the main motivations for the initial proposal of SOC, time-dependent properties of self-organized critical systems have not been studied much theoretically so far. Two types of correlations should be distinguished: within a single avalanche, and over time scales greater than the interval between successive particle additions, which measures correlations between avalanches. For the latter case, while a power spectrum of mass-fluctuations of $1 / f$ type

The generic behavior of sandpiles

The behavior of sandpile-like models seems very confusing, given the large number of different models already defined (and many more could have been), each with its own set of critical exponents. We now try to address the question of universality of critical behavior of sandpile models [85].

A very general paradigm in non-equilibrium statistical physics is directed percolation (DP), which describes the inactive–active state phase transition in a wide class of reaction–diffusion systems [86]. The

Open problems

Let me conclude by discussing some of the interesting open problems in this field. The list is necessarily incomplete, and influenced by personal taste and prejudices.

We can start with the problem of directed abelian sandpile models. In this case, we can calculate all the avalanche exponents in all dimensions. However, time-dependent correlation functions in the model have not been studied analytically so far. In particular, the power spectrum of the stochastic outflux of particles subjected to

References (100)

A. Dhar et al.
Phys. Rev. E
(1997)
H. Agrawal et al.
Phys. Rev. E
(2001)
G.F. Lawler
Intersections of Random Walks
(1996)
H. Takayasu
Phys. Rev. Lett.
(1989)
D. Dhar et al.
J. Phys. A
(1995)
N. Jacobson
Basic Algebra
(1974)
D. Dhar
W. Feller
An Introduction to Probability Theory and Its Applications
(1970)
S. Lubeck
Phys. Rev. E
(1998)
C. Tang et al.
J. Stat. Phys.
(1988)
B. Gaveau et al.
J. Phys. A
(1991)
S.A. Janowski et al.
J. Phys. A
(1993)
H. Flybjerg
Phys. Rev. Lett.
(1996)
A.A. Ali
Phys. Rev. E
(1995)
S. Maslov et al.
Phys. Rev. Lett.
(1999)

D. Dhar

Physica A

(1999)

Available at...

B.B. Mandelbrot

Fractals: Form, Chance and Dimension

(1977)

A good introduction to various scaling laws in river-networks and references to earlier literature may be found in P.S....P.S. Dodds, D.H. Rothman, Phys. Rev. E 63 (2001)...P.S. Dodds, D.H. Rothman, Phys. Rev. E 016116 (2001); P.S. Dodds, D.H. Rothman, Phys. Rev. E 016117 (2001). See also F....J.R. Banavar, F. Colaiori, A. Flammini, A. Rinaldo, J. Stat. Phys. (2001)...

B. Gutenberg et al.

Ann. Geophys.

(1956)

P. Bak et al.

Phys. Rev. Lett.

(2002)

O. Peters et al.

Phys. Rev. Lett.

(2002)

K.R. Sreenivasan

Ann. Rev. Fluid Mech.

(1991)

P. Bak et al.

Phys. Rev. Lett.

(1987)

The actual behavior of real sand is more complicated than this rather idealized description. Our concern here is to...

P. Bak

How Nature Works

(1997)

R. Dickman et al.

Brazilian J. Phys.

(2000)

E.V. Ivashkevich et al.

Physica A

(1998)

D. Dhar

Physica A

(1999)

D.L. Turcotte

Rep. Prog. Phys.

(1999)

F. Redig, Les Houches Lectures, preprint,...

Models of this type were studied earlier by R. Burridge, L. Knopoff, Bull. Seismol. Soc. Am. 57 (1967)...J.M. Carlson et al.

Phys. Rev. A

(1989)

M. de Sousa Vieira

Phys. Rev. A

(1992)

M. Paczuski et al.

Phys. Rev. Lett.

(1996)

H. Flybjerg

Physica A

(2004)

K. Sneppen

Phys. Rev. Lett.

(1992)

P. Grassberger et al.

Physica A

(1996)

M. Paczuski et al.

Physica A

(2004)

In fact, we may assume that at each toppling, the assignment of which grain goes to which neighbor is done randomly,...

V. Frette et al.

Nature

(1996)

G. Grinstein

C. Maes et al.

Commun. Math. Phys.

(2004)

A. Corral et al.

Phys. Rev. Lett.

(1999)

“My name means the shape I am- and a good handsome shape it is too. With a name like yours, you might be any shape,...

D. Dhar

Phys. Rev. Lett.

(1990)

R. Meester et al.

Markov Proc. Rel. Fields

(2001)

B. Drossel et al.

Phys. Rev. Lett.

(1992)

Y.C. Zhang

Phys. Rev. Lett.

(1989)

This procedure is equivalent to the script test first proposed in Speer E. R., J. Phys. A 71 (1993)...

S.N. Majumdar et al.

Physica A

(1992)

C.M. Fortuin et al.

Physica

(1972)

L.C. Merino

Ann. Comb.

(1997)

R. Cori et al.

Adv. Appl. Math.

(2003)

For a recent treatment of this classical result, see F.Y. Wu, Rev. Mod. Phys. 54 (1982)...

H. Saleur et al.

Phys. Rev. Lett.

(1987)

A. Coniglio

Phys. Rev. Lett.

(1989)

F. Harary

Graph Theory

(1990)

E.V. Ivashkevich et al.

Physica A

(1994)

D. Dhar et al.

Phys. Rev. E

(1994)

F. Spitzer

Principles of Random Walk

(1964)

A.Z. Broder

S.N. Majumdar

Phys. Rev. Lett.

(1992)

D. Dhar et al.

Phys. Rev. Lett.

(1989)

Cited by (278)

Combinatorial aspects of sandpile models on wheel and fan graphs
2023, European Journal of Combinatorics
We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations of the model’s recurrent configurations on these families. For wheel graphs, we exhibit a bijection between these recurrent configurations and the set of subgraphs of the cycle graph which maps the level of the configuration to the number of edges of the subgraph. This bijection relies on two key ingredients. The first consists in considering a stochastic variant of the standard Abelian sandpile model (ASM), rather than the ASM itself. The second ingredient is a mapping from a given recurrent state to a canonical minimal recurrent state, exploiting similar ideas to previous studies of the ASM on complete bipartite graphs and Ferrers graphs. We also show that on the wheel graph with $2 n$ vertices, the number of recurrent states with level $n$ is given by the first differences of the central Delannoy numbers. Finally, using similar tools, we exhibit a bijection between the set of recurrent configurations of the ASM on fan graphs and the set of subgraphs of the path graph containing the right-most vertex of the path. We show that these sets are also equinumerous with certain lattice paths, which we name Kimberling paths after the author of the corresponding entry in the Online Encyclopedia of Integer Sequences.
A physical study of the LLL algorithm
2023, Journal of Number Theory
This paper presents a study of the LLL algorithm from the perspective of statistical physics. Based on our experimental and theoretical results, we suggest that interpreting LLL as a sandpile model may help understand much of its mysterious behavior. In the language of physics, our work presents evidence that LLL and certain 1-d sandpile models with simpler toppling rules belong to the same universality class.
This paper consists of three parts. First, we introduce sandpile models whose statistics imitate those of LLL with compelling accuracy, which leads to the idea that there must exist a meaningful connection between the two. Indeed, on those sandpile models, we are able to prove the analogues of some of the most desired statements for LLL, such as the existence of the gap between the theoretical and the experimental RHF bounds. Furthermore, we test the formulas from finite-size scaling theory (FSS) against the LLL algorithm itself, and find that they are in excellent agreement. This in particular explains and refines the geometric series assumption (GSA), and allows one to extrapolate various quantities of interest to the dimension limit. In particular, we obtain the estimate that the empirical average RHF converges to ≈1.02265 as the dimension goes to infinity.
A method to calculate the number of spanning connected unicyclic(bicyclic) subgraphs in 2-separable networks
2022, Theoretical Computer Science
The numbers of spanning connected unicyclic subgraphs and spanning connected bicyclic subgraphs are essential parameters in studying network reliability. However, these two parameters have hardly been studied. Only in 1983, Liu studied the enumeration of spanning connected unicyclic(bicyclic) subgraphs, which is mainly calculated by the determinant of the cycle adjacency matrix. But for large networks, the calculation of the determinant of cycle adjacency matrix is extremely complex. In this paper, we present a reduction method to compute the number of spanning connected unicyclic(bicyclic) subgraphs of 2-separable networks. As an application, we enumerate the number of spanning connected unicyclic(bicyclic) subgraphs in fractal scale-free networks by linear algorithm. The exact expression of the number of spanning connected unicyclic(bicyclic) subgraphs is obtained. On this basis, the entropy of the number of spanning connected unicyclic(bicyclic) subgraphs is determined. This research provides helpful guidance for computing the number of spanning connected unicyclic(bicyclic) subgraphs of general 2-connected networks.
Interlacement limit of a stopped random walk trace on a torus
2024, Advances in Applied Probability
The sandpile model on the complete split graph: q,t-Schröder polynomials, sawtooth polyominoes, and a cyclic lemma
2024, arXiv
NEW COMBINATORIAL PERSPECTIVES ON MVP PARKING FUNCTIONS AND THEIR OUTCOME MAP
2023, arXiv

View all citing articles on Scopus

View full text