The Abelian Sandpile generates complex and beautiful patterns and seems to display allometry. On the plane, beyond patches, patterns periodic in both dimensions, we remark the presence of structures periodic in one dimension, that we call strings. We classify completely their constituents in terms of their principal periodic vector k, that we call momentum. We derive a simple relation between the momentum of a string and its density of particles, E, which is reminiscent of a dispersion relation, E=k^2. Strings interact: they can merge and split and within these processes momentum is conserved. We reveal the role of the modular group SL(2,Z) behind these laws.
Introduction : Since the appearance of the masterpiece by D'Arcy Thompson [1], there have been many attempts to understand the complexity and variety of shapes appearing in Nature at macroscopic scales, in terms of the fundamental laws which govern the dynamics at microscopic level. Because of the second law of thermodynamics, the necessary Self-Organization can be studied only in non-equilibrium statistical mechanics.
In the context of a continuous evolution in a differential manifold, the definition of a shape implies a boundary and thus a discontinuity. This explains why catastrophe theory, the mathematical treatment of continuous action producing a discontinuous result, has been developed in strict connection to the problem of Morphogenesis [2]. More quantitative results have been obtained by the introduction of stochasticity, as for example in the diffusion-limited aggregation [3], where self-similar patterns with fractal scaling dimension [4] emerge which suggest a relation with scaling studies in non-equilibrium.
Cellular automata, that is, dynamical systems with discretized time, space and internal states, were originally introduced by Ulam and von Neumann in the 1940s, and then commonly used as a simplified description of phenomena like crystal growth, Navier-Stokes equations and transport processes [5]. They often exhibit intriguing patterns [6], and, in this regular discrete setting, shapes refer to sharply bounded regions in which periodic patterns appear. Despite very simple local evolution rules, very complex structures can be generated. The well known Conway’s Game of Life can perform computations and can even emulate an universal Turing machine (see [6] also for a historical introduction on cellular automata).
In this Letter, we shall concentrate on a particularly simple cellular automaton, the Abelian Sandpile Model (ASM). Originally introduced as a statistical model of Self-Organized Criticality [7], because it shows scaling laws without any fine-tuning of an external control parameter, it has been shown afterwards to possess remark-able underlying algebraic properties [8,9], and has been studied also in some deterministic approaches, exactly in connection with pattern formation [10]. The ASM has been shown to be able to produce allometry, that is a growth uniform and constant in all the parts of a pattern as to keep the whole shape substantially unchanged, and thus requires some coordination and communication between different parts [11]. This is at variance with diffusion-limited aggregation and other models of growing objects studied in physics literature so far, e.g. the Eden model, KPZ deposition and invasionpercolation [12], which are mainly models of aggregation, where growth occurs by accretion on the surface of the object, and inner parts do not evolve significantly.
In the sandpile, the regions of a configuration periodic in space, called patches, are the ingredients of pattern formation. In [11], a condition on the shape of patch interfaces has been established, and proven at a coarsegrained level.
We discuss how this result is strengthened by avoiding the coarsening, and describe the emerging fine-level structures, including linear interfaces and rigid domain walls with a residual one-dimensional translational invariance. These structures, that we shall call strings, are macroscopically extended in their periodic direction, while showing thickness in a full range of scales between the microscopic lattice spacing and the macroscopic volume size.
The model : While the main structural properties of the ASM can be discussed on arbitrary graphs [8], for the subject at hand here we shall need some extra ingredients (among which a notion of translation), that, for the sake of simplicity, suggest us to concentrate on the original realization on the square lattice [7], within a rectangular region Λ ∈ Z 2 .
We write i ∼ j if i and j are first neighbours. The configurations are vectors z ≡ {z i } i∈Λ ∈ N Λ (z i is the number of sand-grains at vertex i). Let z = 4, the degree of vertices in the bulk, and say that a configuration z is stable if z i < z for all i ∈ Λ. Otherwise, it is unstable on a non-empty set of sites, and undergoes a relaxation process whose elementary steps are called topplings: if i is unstable, we can decrease z i by z, and increase z j by one, for all j ∼ i. The sequence of topplings needed to produce a stable configuration is called an avalanche.
Avalanches always stop after a finite number of steps, which is to say that the diffusion is strictly dissipative. Indeed, the total amount of sand is preserved by topplings at sites far from the boundary of Λ, and strictly decreased by topplings at boundary sites. The stable configuration R(z) obtained from the relaxation of z, is univocally defined, as all valid stabilizing sequences of topplings only differ by permutations.
We call a stable configuration recurrent if it can be obtained through an avalanche invol
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