11 research outputs found
Energy avalanches in a rice-pile model
We investigate a one-dimensional rice-pile model. We show that the
distribution of dissipated potential energy decays as a power law with an
exponent . The system thus provides a one-dimensional example of
self-organized criticality. Different driving conditions are examined in order
to allow for comparison with experiments.Comment: 8 pages, elsart sty files (provided
Directed percolation with an absorbing boundary
We consider directed percolation with an absorbing boundary in 1+1 and 2+1
dimensions. The distribution of cluster lifetimes and sizes depend on the
boundary. The new scaling exponents can be related to the exponents
characterizing standard directed percolation in 1+1 dimension. In addition, we
investigate the backbone cluster and red bonds, and calculate the distribution
of living sites along the absorbing boundary.Comment: 10 latex pages, including 4 figure
Self-organized criticality in a rice-pile model
We present a new model for relaxations in piles of granular material. The
relaxations are determined by a stochastic rule which models the effect of
friction between the grains. We find power-law distributions for avalanche
sizes and lifetimes characterized by the exponents and
, respectively. For the discharge events, we find a
characteristic size that scales with the system size as , with . We also find that the frequency of the discharge events
decrease with the system size as with .Comment: 4 pages, RevTex, multicol, epsf, rotate (sty files provided). To
appear Phys. Rev. E Rapid Communication (Nov or Dec 96
Directed Percolation with a Wall or Edge
We examine the effects of introducing a wall or edge into a directed
percolation process. Scaling ansatzes are presented for the density and
survival probability of a cluster in these geometries, and we make the
connection to surface critical phenomena and field theory. The results of
previous numerical work for a wall can thus be interpreted in terms of surface
exponents satisfying scaling relations generalising those for ordinary directed
percolation. New exponents for edge directed percolation are also introduced.
They are calculated in mean-field theory and measured numerically in 2+1
dimensions.Comment: 14 pages, submitted to J. Phys.
Self-Organized Branching Processes: A Mean-Field Theory for Avalanches
We discuss mean-field theories for self-organized criticality and the
connection with the general theory of branching processes. We point out that
the nature of the self-organization is not addressed properly by the previously
proposed mean-field theories. We introduce a new mean-field model that
explicitly takes the boundary conditions into account; in this way, the local
dynamical rules are coupled to a global equation that drives the control
parameter to its critical value. We study the model numerically, and
analytically we compute the avalanche distributions.Comment: 4 pages + 4 ps figure
Noisy Kuramoto-Sivashinsky equation for an erosion model
We derive the continuum equation for a discrete model for ion sputtering. We
follow an approach based on the master equation, and discuss how it can be
truncated to a Fokker-Planck equation and mapped to a discrete Langevin
equation. By taking the continuum limit, we arrive at the Kuramoto-Sivashinsky
equation with a stochastic noise term.Comment: latex (w/ multicol.sty), 4 pages; to appear in Physical Review E (Oct
1996
Universality classes for rice-pile models
We investigate sandpile models where the updating of unstable columns is done
according to a stochastic rule. We examine the effect of introducing nonlocal
relaxation mechanisms. We find that the models self-organize into critical
states that belong to three different universality classes. The models with
local relaxation rules belong to a known universality class that is
characterized by an avalanche exponent , whereas the models
with nonlocal relaxation rules belong to new universality classes characterized
by exponents and . We discuss the values
of the exponents in terms of scaling relations and a mapping of the sandpile
models to interface models.Comment: 4 pages, including 3 figure
Surface Critical Behavior in Systems with Non-Equilibrium Phase Transitions
We study the surface critical behavior of branching-annihilating random walks
with an even number of offspring (BARW) and directed percolation (DP) using a
variety of theoretical techniques. Above the upper critical dimensions d_c,
with d_c=4 (DP) and d_c=2 (BARW), we use mean field theory to analyze the
surface phase diagrams using the standard classification into ordinary,
special, surface, and extraordinary transitions. For the case of BARW, at or
below the upper critical dimension, we use field theoretic methods to study the
effects of fluctuations. As in the bulk, the field theory suffers from
technical difficulties associated with the presence of a second critical
dimension. However, we are still able to analyze the phase diagrams for BARW in
d=1,2, which turn out to be very different from their mean field analog.
Furthermore, for the case of BARW only (and not for DP), we find two
independent surface beta_1 exponents in d=1, arising from two distinct
definitions of the order parameter. Using an exact duality transformation on a
lattice BARW model in d=1, we uncover a relationship between these two surface
beta_1 exponents at the ordinary and special transitions. Many of our
predictions are supported using Monte-Carlo simulations of two different models
belonging to the BARW universality class.Comment: 19 pages, 12 figures, minor additions, 1 reference adde
DIRECTED PERCOLATION AND OTHER SYSTEMS WITH ABSORBING STATES: IMPACT OF BOUNDARIES
We review the critical behavior of nonequilibrium systems, such as directed percolation (DP) and branching-annihilating random walks (BARW), which possess phase transitions into absorbing states. After reviewing the bulk scaling behavior of these models, we devote the main part of this review to analyzing the impact of walls on their critical behavior. We discuss the possible boundary universality classes for the DP and BARW models, which can be described by a general scaling theory which allows for two independent surface exponents in addition to the bulk critical exponents. Above the upper critical dimension dc, we review the use of mean field theories, whereas in the regime d < dc, where fluctuations are important, we examine the application of field theoretic methods. Of particular interest is the situation in d = 1, which has been extensively investigated using numerical simulations and series expansions. Although DP and BARW fit into the same scaling theory, they can still show very different surface behavior: for DP some exponents are degenerate, a property not shared with the BARW model. Moreover, a “hidden ” duality symmetry of BARW in d = 1 is broken by the boundary and this relates exponents and boundary conditions in an intricate way