565 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
Behavior of Coupled Automata
We study the nature of statistical correlations that develop between systems of interacting self-organized critical automata (sandpiles). Numerical and analytical findings are presented describing the emergence of synchronization between sandpiles and the dependency of this synchronization on factors such as variations in coupling strength, toppling rule probabilities, symmetric versus asymmetric coupling rules, and numbers of sandpiles
The Behavior of Coupled Automata
We study the nature of statistical correlations that develop between systems of interacting self-organized critical automata (sandpiles). Numerical and analytical findings are presented describing the emergence of “synchronization” between sandpiles and the dependency of this synchronization on factors such as variations in coupling strength, toppling rule probabilities, symmetric versus asymmetric coupling rules, and numbers of sandpiles
Sandpiles and Dominos
We consider the subgroup of the abelian sandpile group of the grid graph
consisting of configurations of sand that are symmetric with respect to central
vertical and horizontal axes. We show that the size of this group is (i) the
number of domino tilings of a corresponding weighted rectangular checkerboard;
(ii) a product of special values of Chebyshev polynomials; and (iii) a
double-product whose factors are sums of squares of values of trigonometric
functions. We provide a new derivation of the formula due to Kasteleyn and to
Temperley and Fisher for counting the number of domino tilings of a 2m x 2n
rectangular checkerboard and a new way of counting the number of domino tilings
of a 2m x 2n checkerboard on a M\"obius strip.Comment: 35 pages, 24 figure
Activated Random Walkers: Facts, Conjectures and Challenges
We study a particle system with hopping (random walk) dynamics on the integer
lattice . The particles can exist in two states, active or
inactive (sleeping); only the former can hop. The dynamics conserves the number
of particles; there is no limit on the number of particles at a given site.
Isolated active particles fall asleep at rate , and then remain
asleep until joined by another particle at the same site. The state in which
all particles are inactive is absorbing. Whether activity continues at long
times depends on the relation between the particle density and the
sleeping rate . We discuss the general case, and then, for the
one-dimensional totally asymmetric case, study the phase transition between an
active phase (for sufficiently large particle densities and/or small )
and an absorbing one. We also present arguments regarding the asymptotic mean
hopping velocity in the active phase, the rate of fixation in the absorbing
phase, and survival of the infinite system at criticality. Using mean-field
theory and Monte Carlo simulation, we locate the phase boundary. The phase
transition appears to be continuous in both the symmetric and asymmetric
versions of the process, but the critical behavior is very different. The
former case is characterized by simple integer or rational values for critical
exponents (, for example), and the phase diagram is in accord with
the prediction of mean-field theory. We present evidence that the symmetric
version belongs to the universality class of conserved stochastic sandpiles,
also known as conserved directed percolation. Simulations also reveal an
interesting transient phenomenon of damped oscillations in the activity
density
Exact solutions for a mean-field Abelian sandpile
We introduce a model for a sandpile, with N sites, critical height N and each
site connected to every other site. It is thus a mean-field model in the
spin-glass sense. We find an exact solution for the steady state probability
distribution of avalanche sizes, and discuss its asymptotics for large N.Comment: 10 pages, LaTe
Avalanches at rough surfaces
We describe the surface properties of a simple lattice model of a sandpile
that includes evolving structural disorder. We present a dynamical scaling
hypothesis for generic sandpile automata, and additionally explore the kinetic
roughening of the sandpile surface, indicating its relationship with the
sandpile evolution. Finally, we comment on the surprisingly good agreement
found between this model, and a previous continuum model of sandpile dynamics,
from the viewpoint of critical phenomena.Comment: 8 Pages, 7 Figures (in 15 parts); accepted for publication in
Physical Review
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