3,752 research outputs found
Chaos in Sandpile Models
We have investigated the "weak chaos" exponent to see if it can be considered
as a classification parameter of different sandpile models. Simulation results
show that "weak chaos" exponent may be one of the characteristic exponents of
the attractor of \textit{deterministic} models. We have shown that the
(abelian) BTW sandpile model and the (non abelian) Zhang model posses different
"weak chaos" exponents, so they may belong to different universality classes.
We have also shown that \textit{stochasticity} destroys "weak chaos" exponents'
effectiveness so it slows down the divergence of nearby configurations. Finally
we show that getting off the critical point destroys this behavior of
deterministic models.Comment: 5 pages, 6 figure
Order Parameter and Scaling Fields in Self-Organized Criticality
We present a unified dynamical mean-field theory for stochastic
self-organized critical models. We use a single site approximation and we
include the details of different models by using effective parameters and
constraints. We identify the order parameter and the relevant scaling fields in
order to describe the critical behavior in terms of usual concepts of non
equilibrium lattice models with steady-states. We point out the inconsistencies
of previous mean-field approaches, which lead to different predictions.
Numerical simulations confirm the validity of our results beyond mean-field
theory.Comment: 4 RevTex pages and 2 postscript figure
Sandpile Model with Activity Inhibition
A new sandpile model is studied in which bonds of the system are inhibited
for activity after a certain number of transmission of grains. This condition
impels an unstable sand column to distribute grains only to those neighbours
which have toppled less than m times. In this non-Abelian model grains
effectively move faster than the ordinary diffusion (super-diffusion). A novel
system size dependent cross-over from Abelian sandpile behaviour to a new
critical behaviour is observed for all values of the parameter m.Comment: 11 pages, RevTex, 5 Postscript figure
Dynamics of bootstrap percolation
Bootstrap percolation transition may be first order or second order, or it
may have a mixed character where a first order drop in the order parameter is
preceded by critical fluctuations. Recent studies have indicated that the mixed
transition is characterized by power law avalanches, while the continuous
transition is characterized by truncated avalanches in a related sequential
bootstrap process. We explain this behavior on the basis of a through
analytical and numerical study of the avalanche distributions on a Bethe
lattice.Comment: Proceedings of the International Workshop and Conference on
Statistical Physics Approaches to Multidisciplinary Problems, IIT Guwahati,
India, 7-13 January 200
Proof by analogy in mural
One of the most important advantages of using a formal method of developing software is that one can prove that development steps are correct with respect to their specification.
Conducting proofs by hand, however,can be time consuming to the extent that designers have to judge whether a proof of a particular obligation is worth conducting.
Even if hand proofs are worth conducting, how do we know that they are correct?
One approach to overcoming this problem is to use an automatic theorem proving system to develop and check our proofs. However, in order to enable present day
theorem provers to check proofs, one has to conduct
them in much more detail than hand proofs. Carrying out more detailed proofs is of course more time consuming.
This paper describes the use of proof by analogy in an attempt to reduce the time spent on proofs.
We develop and implement a proof follower based on analogy and present two examples to illustrate its
characteristics. One example illustrates the successful use of the proof follower. The other example illustrates that the follower's failure can provide a hint that enables the user to complete a proof
A nonlinear Schr\"odinger equation for water waves on finite depth with constant vorticity
A nonlinear Schr\"odinger equation for the envelope of two dimensional
surface water waves on finite depth with non zero constant vorticity is
derived, and the influence of this constant vorticity on the well known
stability properties of weakly nonlinear wave packets is studied. It is
demonstrated that vorticity modifies significantly the modulational instability
properties of weakly nonlinear plane waves, namely the growth rate and
bandwidth. At third order we have shown the importance of the coupling between
the mean flow induced by the modulation and the vorticity. Furthermore, it is
shown that these plane wave solutions may be linearly stable to modulational
instability for an opposite shear current independently of the dimensionless
parameter kh, where k and h are the carrier wavenumber and depth respectively
Universality Classes in Isotropic, Abelian and non-Abelian, Sandpile Models
Universality in isotropic, abelian and non-abelian, sandpile models is
examined using extensive numerical simulations. To characterize the critical
behavior we employ an extended set of critical exponents, geometric features of
the avalanches, as well as scaling functions describing the time evolution of
average quantities such as the area and size during the avalanche. Comparing
between the abelian Bak-Tang-Wiesenfeld model [P. Bak, C. Tang and K.
Wiensenfeld, Phys. Rev. Lett. 59, 381 (1987)], and the non-abelian models
introduced by Manna [S. S. Manna, J. Phys. A. 24, L363 (1991)] and Zhang [Y. C.
Zhang, Phys. Rev. Lett. 63, 470 (1989)] we find strong indications that each
one of these models belongs to a distinct universality class.Comment: 18 pages of text, RevTeX, additional 8 figures in 12 PS file
Mean-field behavior of the sandpile model below the upper critical dimension
We present results of large scale numerical simulations of the Bak, Tang and
Wiesenfeld sandpile model. We analyze the critical behavior of the model in
Euclidean dimensions . We consider a dissipative generalization
of the model and study the avalanche size and duration distributions for
different values of the lattice size and dissipation. We find that the scaling
exponents in significantly differ from mean-field predictions, thus
suggesting an upper critical dimension . Using the relations among
the dissipation rate and the finite lattice size , we find that a
subset of the exponents displays mean-field values below the upper critical
dimensions. This behavior is explained in terms of conservation laws.Comment: 4 RevTex pages, 2 eps figures embedde
Renormalization group approach to an Abelian sandpile model on planar lattices
One important step in the renormalization group (RG) approach to a lattice
sandpile model is the exact enumeration of all possible toppling processes of
sandpile dynamics inside a cell for RG transformations. Here we propose a
computer algorithm to carry out such exact enumeration for cells of planar
lattices in RG approach to Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett.
{\bf 59}, 381 (1987)] and consider both the reduced-high RG equations proposed
by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. {\bf 72}, 1690
(1994)] and the real-height RG equations proposed by Ivashkevich [Phys. Rev.
Lett. {\bf 76}, 3368 (1996)]. Using this algorithm we are able to carry out RG
transformations more quickly with large cell size, e.g. cell for
the square (sq) lattice in PVZ RG equations, which is the largest cell size at
the present, and find some mistakes in a previous paper [Phys. Rev. E {\bf 51},
1711 (1995)]. For sq and plane triangular (pt) lattices, we obtain the only
attractive fixed point for each lattice and calculate the avalanche exponent
and the dynamical exponent . Our results suggest that the increase of
the cell size in the PVZ RG transformation does not lead to more accurate
results. The implication of such result is discussed.Comment: 29 pages, 6 figure
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