456 research outputs found
Sandpile Prediction on Structured Undirected Graphs
We present algorithms that compute the terminal configurations for sandpile
instances in time on trees and time on paths, where is
the number of vertices. The Abelian Sandpile model is a well-known model used
in exploring self-organized criticality. Despite a large amount of work on
other aspects of sandpiles, there have been limited results in efficiently
computing the terminal state, known as the sandpile prediction problem.
Our algorithm improves the previous best runtime of on trees
[Ramachandran-Schild SODA '17] and on paths [Moore-Nilsson '99].
To do so, we move beyond the simulation of individual events by directly
computing the number of firings for each vertex. The computation is accelerated
using splittable binary search trees. We also generalize our algorithm to adapt
at most three sink vertices, which is the first prediction algorithm faster
than mere simulation on a sandpile model with sinks.
We provide a general reduction that transforms the prediction problem on an
arbitrary graph into problems on its subgraphs separated by any vertex set .
The reduction gives a time complexity of where
denotes the total time for solving on each subgraph. In addition, we give
algorithms in time on cliques and time on pseudotrees.Comment: 66 pages, submitted to SODA2
Smoothing of sandpile surfaces after intermittent and continuous avalanches: three models in search of an experiment
We present and analyse in this paper three models of coupled continuum
equations all united by a common theme: the intuitive notion that sandpile
surfaces are left smoother by the propagation of avalanches across them. Two of
these concern smoothing at the `bare' interface, appropriate to intermittent
avalanche flow, while one of them models smoothing at the effective surface
defined by a cloud of flowing grains across the `bare' interface, which is
appropriate to the regime where avalanches flow continuously across the
sandpile.Comment: 17 pages and 26 figures. Submitted to Physical Review
Threshold phenomena in erosion driven by subsurface flow
We study channelization and slope destabilization driven by subsurface
(groundwater) flow in a laboratory experiment. The pressure of the water
entering the sandpile from below as well as the slope of the sandpile are
varied. We present quantitative understanding of the three modes of sediment
mobilization in this experiment: surface erosion, fluidization, and slumping.
The onset of erosion is controlled not only by shear stresses caused by
surfical flows, but also hydrodynamic stresses deriving from subsurface flows.
These additional forces require modification of the critical Shields criterion.
Whereas surface flows alone can mobilize surface grains only when the water
flux exceeds a threshold, subsurface flows cause this threshold to vanish at
slopes steeper than a critical angle substantially smaller than the maximum
angle of stability. Slopes above this critical angle are unstable to
channelization by any amount of fluid reaching the surface.Comment: 9 pages, 11 figure
On the concept of complexity in random dynamical systems
We introduce a measure of complexity in terms of the average number of bits
per time unit necessary to specify the sequence generated by the system. In
random dynamical system, this indicator coincides with the rate K of divergence
of nearby trajectories evolving under two different noise realizations.
The meaning of K is discussed in the context of the information theory, and
it is shown that it can be determined from real experimental data. In presence
of strong dynamical intermittency, the value of K is very different from the
standard Lyapunov exponent computed considering two nearby trajectories
evolving under the same randomness. However, the former is much more relevant
than the latter from a physical point of view as illustrated by some numerical
computations for noisy maps and sandpile models.Comment: 35 pages, LaTe
25 Years of Self-Organized Criticality: Solar and Astrophysics
Shortly after the seminal paper {\sl "Self-Organized Criticality: An
explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has
been applied to solar physics, in {\sl "Avalanches and the Distribution of
Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring
cross-fertilization from complexity theory to solar and astrophysics took
place, where the SOC concept was initially applied to solar flares, stellar
flares, and magnetospheric substorms, and later extended to the radiation belt,
the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar
glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and
boson clouds. The application of SOC concepts has been performed by numerical
cellular automaton simulations, by analytical calculations of statistical
(powerlaw-like) distributions based on physical scaling laws, and by
observational tests of theoretically predicted size distributions and waiting
time distributions. Attempts have been undertaken to import physical models
into the numerical SOC toy models, such as the discretization of
magneto-hydrodynamics (MHD) processes. The novel applications stimulated also
vigorous debates about the discrimination between SOC models, SOC-like, and
non-SOC processes, such as phase transitions, turbulence, random-walk
diffusion, percolation, branching processes, network theory, chaos theory,
fractality, multi-scale, and other complexity phenomena. We review SOC studies
from the last 25 years and highlight new trends, open questions, and future
challenges, as discussed during two recent ISSI workshops on this theme.Comment: 139 pages, 28 figures, Review based on ISSI workshops "Self-Organized
Criticality and Turbulence" (2012, 2013, Bern, Switzerland
The Lattice structure of Chip Firing Games and Related Models
In this paper, we study a famous discrete dynamical system, the Chip Firing
Game, used as a model in physics, economics and computer science. We use order
theory and show that the set of reachable states (i.e. the configuration space)
of such a system started in any configuration is a lattice, which implies
strong structural properties. The lattice structure of the configuration space
of a dynamical system is of great interest since it implies convergence (and
more) if the configuration space is finite. If it is infinite, this property
implies another kind of convergence: all the configurations reachable from two
given configurations are reachable from their infimum. In other words, there is
a unique first configuration which is reachable from two given configurations.
Moreover, the Chip Firing Game is a very general model, and we show how known
models can be encoded as Chip Firing Games, and how some results about them can
be deduced from this paper. Finally, we define a new model, which is a
generalization of the Chip Firing Game, and about which many interesting
questions arise.Comment: See http://www.liafa.jussieu.fr/~latap
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