206 research outputs found
On symmetric sandpiles
A symmetric version of the well-known SPM model for sandpiles is introduced.
We prove that the new model has fixed point dynamics. Although there might be
several fixed points, a precise description of the fixed points is given.
Moreover, we provide a simple closed formula for counting the number of fixed
points originated by initial conditions made of a single column of grains.Comment: Will be presented at ACRI2006 conferenc
Multifractal properties of power-law time sequences; application to ricepiles
We study the properties of time sequences extracted from a self-organized
critical system, within the framework of the mathematical multifractal
analysis. To this end, we propose a fixed-mass algorithm, well suited to deal
with highly inhomogeneous one dimensional multifractal measures. We find that
the fixed mass (dual) spectrum of generalized dimensions depends on both the
system size L and the length N of the sequence considered, being however stable
when these two parameters are kept fixed. A finite-size scaling relation is
proposed, allowing us to define a renormalized spectrum, independent of size
effects.We interpret our results as an evidence of extremely long-range
correlations induced in the sequence by the criticality of the systemComment: 12 pages, RevTex, includes 9 PS figures, Phys. Rev. E (in press
Absorbing-state phase transitions in fixed-energy sandpiles
We study sandpile models as closed systems, with conserved energy density
playing the role of an external parameter. The critical energy density,
, marks a nonequilibrium phase transition between active and absorbing
states. Several fixed-energy sandpiles are studied in extensive simulations of
stationary and transient properties, as well as the dynamics of roughening in
an interface-height representation. Our primary goal is to identify the
universality classes of such models, in hopes of assessing the validity of two
recently proposed approaches to sandpiles: a phenomenological continuum
Langevin description with absorbing states, and a mapping to driven interface
dynamics in random media. Our results strongly suggest that there are at least
three distinct universality classes for sandpiles.Comment: 41 pages, 23 figure
Approximation properties of the -sine bases
For the eigenfunctions of the non-linear eigenvalue problem
associated to the one-dimensional -Laplacian are known to form a Riesz basis
of . We examine in this paper the approximation properties of this
family of functions and its dual, in order to establish non-orthogonal spectral
methods for the -Poisson boundary value problem and its corresponding
parabolic time evolution initial value problem. The principal objective of our
analysis is the determination of optimal values of for which the best
approximation is achieved for a given problem.Comment: 20 pages, 11 figures and 2 tables. We have fixed a number of typos
and added references. Changed the title to better reflect the conten
Abelian sandpiles: an overview and results on certain transitive graphs
We review the Majumdar-Dhar bijection between recurrent states of the Abelian
sandpile model and spanning trees. We generalize earlier results of Athreya and
Jarai on the infinite volume limit of the stationary distribution of the
sandpile model on Z^d, d >= 2, to a large class of graphs. This includes: (i)
graphs on which the wired spanning forest is connected and has one end; (ii)
transitive graphs with volume growth at least c n^5 on which all bounded
harmonic functions are constant. We also extend a result of Maes, Redig and
Saada on the stationary distribution of sandpiles on infinite regular trees, to
arbitrary exhaustions.Comment: 44 pages. Version 2 incorporates some smaller changes. To appear in
Markov Processes and Related Fields in the proceedings of the meeting:
Inhomogeneous Random Systems, Stochastic Geometry and Statistical Mechanics,
Institut Henri Poincare, Paris, 27 January 201
Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks
Adding sand grains at a single site in Abelian sandpile models produces
beautiful but complex patterns. We study the effect of sink sites on such
patterns. Sinks change the scaling of the diameter of the pattern with the
number of sand grains added. For example, in two dimensions, in presence of
a sink site, the diameter of the pattern grows as for large
, whereas it grows as if there are no sink sites. In presence of
a line of sink sites, this rate reduces to . We determine the growth
rates for these sink geometries along with the case when there are two lines of
sink sites forming a wedge, and its generalization to higher dimensions. We
characterize one such asymptotic patterns on the two-dimensional F-lattice with
a single source adjacent to a line of sink sites, in terms of position of
different spatial features in the pattern. For this lattice, we also provide an
exact characterization of the pattern with two sources, when the line joining
them is along one of the axes.Comment: 27 pages, 17 figures. Figures with better resolution is available at
http://www.theory.tifr.res.in/~tridib/pss.htm
Apollonian structure in the Abelian sandpile
The Abelian sandpile process evolves configurations of chips on the integer
lattice by toppling any vertex with at least 4 chips, distributing one of its
chips to each of its 4 neighbors. When begun from a large stack of chips, the
terminal state of the sandpile has a curious fractal structure which has
remained unexplained. Using a characterization of the quadratic growths
attainable by integer-superharmonic functions, we prove that the sandpile PDE
recently shown to characterize the scaling limit of the sandpile admits certain
fractal solutions, giving a precise mathematical perspective on the fractal
nature of the sandpile.Comment: 27 Pages, 7 Figure
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