Singular diffusion in a confined sandpile

Abstract

We investigate the behavior of a two-state sandpile model subjected to a confining potential in one and two dimensions. From the microdynamical description of this simple model with its intrinsic exclusion mechanism, it is possible to derive a continuum nonlinear diffusion equation that displays singularities in both the diffusion and drift terms. The stationary-state solutions of this equation, which maximizes the Fermi-Dirac entropy, are in perfect agreement with the spatial profiles of time-averaged occupancy obtained from model numerical simulations in one as well as in two dimensions. Surprisingly, our results also show that, regardless of dimensionality, the presence of a confining potential can lead to the emergence of typical attributes of critical behavior in the two-state sandpile model, namely, a power-law tail in the distribution of avalanche sizes.Physical processes involving anomalous diffusion are typically associated with systems in which the mean square displacement of their elementary units follows a nonlinear power-law relationship with time, σ 2 ∝ t α , with an exponent α = 1, in contrast with linear standard diffusion (α = 1). Instead of being a rare phenomenon, as suggested by its own denomination, anomalous diffusion, however, appears rather ubiquitously in Nature, playing an important role in a variety of scientific and technological applications, such as fluid flow through disordered porous media [1], surface growth [2], diffusion in fractal-like substrates [3][4][5][6][7], turbulent diffusion in the atmosphere [8,9], spatial spreading of cells [10] and biological populations [11], cellular transport [12], and cytoplasmic crowding in cells [13]. Anomalous diffusion can also manifest its non-Gaussian behavior in terms of nonlinear Fokker-Plank equations [14][15][16][17][18], which is the case, for example, of the dynamics of interacting vortices in disordered superconductors [19][20][21][22], diffusion in dusty plasma [23,24], and pedestrian motion [24].The extreme case of nonlinear behavior in diffusive systems certainly corresponds to singular diffusion. For instance, in some physical conditions, the diffusion of adsorbates on a surface can be strongly nonlinear [25][26][27], with a surface diffusion coefficient that depends on the local coverage θ as, D ∝ |θ − θ c | −α . The study of surfacediffusion mechanisms is crucial for the understanding of technologically important processes related with physical adsorption [28] and catalytic surface reactions [29][30][31].In particular, a singularity in the coverage dependence of the diffusion coefficient is frequently associated to continuous phase transitions [27].A direct connection between singular diffusion and selforganized criticality [32] has been disclosed by Carlson et al. [33,34] in terms of a two-state one-dimensional sandpile model with a driving mechanism, where grains are added at one end of the pile and fall off at the other end. Besides exhibiting a self-organized state, the continuum limit of this simple model leads to a nonli...

Confined sandpile in two dimensions: Percolation and singular diffusion

“…Some characteristics of this model change drastically when a confining potential is applied [38]. The jump-size distribution, for instance, starts to exhibit a power-law behavior which suggests a scale-invariant behavior of the system [38]. Scale-invariant behavior in diffusive systems were also observed in gradient percolation diffusion fronts in 2D [39], that have been shown to display fractal diffusion fronts with characteristic dimension similar to the boundary of critical percolation clusters [39].…”

“…The continuous limit of Eq. ( 2) can be obtained similarly to what was done for 1D [38], resulting in the following non-linear diffusion equation…”

“…They suggest that some open driven systems present selforganized criticality [37] because in their continuum limit singularities appear in the diffusion constant at a critical point [27]. Some characteristics of this model change drastically when a confining potential is applied [38]. The jump-size distribution, for instance, starts to exhibit a power-law behavior which suggests a scale-invariant behavior of the system [38].…”

“…In this paper we investigate a 2D confined sandpile model. Our model is the extension of the model introduced in [38] to the case of two dimensions. We are able to deduce the continuous limit for the model, which culminate in a diffusion equation with a singular diffusion coefficient.…”

Stochastic partial differential equations arising in self-organized criticality

“…Despite their seeming simplicity, the analysis of the dynamics of these sandpile models is challenging and the microscopic models contain several artificial degrees of freedom, such as the structure of the underlying grid and its size. Consequently, already shortly after the introduction of the concept of SOC, continuum models mimicking the discrete systems have been introduced in the physics literature, see e. g. [37,33,22,72,73,53,36,24,67,55,19,21,56,74,47]. Informally, these are thought to correspond to continuum limits of the discrete sytems (1.2), (1.5), leading to singular-degenerate (stochastic) PDE of the type dX(t) = ∆ φ(X(t))dt + B(X(t))dW (t) on (0, T ] × (0, 1).…”