494 research outputs found
Broken scaling in the Forest Fire Model
We investigate the scaling behavior of the cluster size distribution in the
Drossel-Schwabl Forest Fire model (DS-FFM) by means of large scale numerical
simulations, partly on (massively) parallel machines. It turns out that simple
scaling is clearly violated, as already pointed out by Grassberger [P.
Grassberger, J. Phys. A: Math. Gen. 26, 2081 (1993)], but largely ignored in
the literature. Most surprisingly the statistics not seems to be described by a
universal scaling function, and the scale of the physically relevant region
seems to be a constant. Our results strongly suggest that the DS-FFM is not
critical in the sense of being free of characteristic scales.Comment: 9 pages in RevTEX4 format (9 figures), submitted to PR
The self-organized critical forest-fire model on large scales
We discuss the scaling behavior of the self-organized critical forest-fire
model on large length scales. As indicated in earlier publications, the
forest-fire model does not show conventional critical scaling, but has two
qualitatively different types of fires that superimpose to give the effective
exponents typically measured in simulations. We show that this explains not
only why the exponent characterizing the fire-size distribution changes with
increasing correlation length, but allows also to predict its asymptotic value.
We support our arguments by computer simulations of a coarse-grained model, by
scaling arguments and by analyzing states that are created artificially by
superimposing the two types of fires.Comment: 26 pages, 7 figure
Phase Transitions in a Forest-Fire Model
We investigate a forest-fire model with the density of empty sites as control
parameter. The model exhibits three phases, separated by one first-order phase
transition and one 'mixed' phase transition which shows critical behavior on
only one side and hysteresis. The critical behavior is found to be that of the
self-organized critical forest-fire model [B. Drossel and F. Schwabl, Phys.
Rev. Lett. 69, 1629 (1992)], whereas in the adjacent phase one finds the spiral
waves of the Bak et al. forest-fire model [P. Bak, K. Chen and C. Tang, Phys.
Lett. A 147, 297 (1990)]. In the third phase one observes clustering of trees
with the fire burning at the edges of the clusters. The relation between the
density distribution in the spiral state and the percolation threshold is
explained and the implications for stationary states with spiral waves in
arbitrary excitable systems are discussed. Furthermore, we comment on the
possibility of mapping self-organized critical systems onto 'ordinary' critical
systems.Comment: 30 pages RevTeX, 9 PostScript figures (Figs. 1,2,4 are of reduced
quality), to appear in Phys. Rev.
Synchronization and Coarsening (without SOC) in a Forest-Fire Model
We study the long-time dynamics of a forest-fire model with deterministic
tree growth and instantaneous burning of entire forests by stochastic lightning
strikes. Asymptotically the system organizes into a coarsening self-similar
mosaic of synchronized patches within which trees regrow and burn
simultaneously. We show that the average patch length grows linearly with
time as t-->oo. The number density of patches of length L, N(L,t), scales as
^{-2}M(L/), and within a mean-field rate equation description we find
that this scaling function decays as e^{-1/x} for x-->0, and as e^{-x} for
x-->oo. In one dimension, we develop an event-driven cluster algorithm to study
the asymptotic behavior of large systems. Our numerical results are consistent
with mean-field predictions for patch coarsening.Comment: 5 pages, 4 figures, 2-column revtex format. To be submitted to PR
A solvable non-conservative model of Self-Organized Criticality
We present the first solvable non-conservative sandpile-like critical model
of Self-Organized Criticality (SOC), and thereby substantiate the suggestion by
Vespignani and Zapperi [A. Vespignani and S. Zapperi, Phys. Rev. E 57, 6345
(1998)] that a lack of conservation in the microscopic dynamics of an SOC-model
can be compensated by introducing an external drive and thereby re-establishing
criticality. The model shown is critical for all values of the conservation
parameter. The analytical derivation follows the lines of Broeker and
Grassberger [H.-M. Broeker and P. Grassberger, Phys. Rev. E 56, 3944 (1997)]
and is supported by numerical simulation. In the limit of vanishing
conservation the Random Neighbor Forest Fire Model (R-FFM) is recovered.Comment: 4 pages in RevTeX format (2 Figures) submitted to PR
Corrections to scaling in the forest-fire model
We present a systematic study of corrections to scaling in the self-organized
critical forest-fire model. The analysis of the steady-state condition for the
density of trees allows us to pinpoint the presence of these corrections, which
take the form of subdominant exponents modifying the standard finite-size
scaling form. Applying an extended version of the moment analysis technique, we
find the scaling region of the model and compute the first non-trivial
corrections to scaling.Comment: RevTeX, 7 pages, 7 eps figure
Forest fires and other examples of self-organized criticality
We review the properties of the self-organized critical (SOC) forest-fire
model. The paradigm of self-organized criticality refers to the tendency of
certain large dissipative systems to drive themselves into a critical state
independent of the initial conditions and without fine-tuning of the
parameters. After an introduction, we define the rules of the model and discuss
various large-scale structures which may appear in this system. The origin of
the critical behavior is explained, critical exponents are introduced, and
scaling relations between the exponents are derived. Results of computer
simulations and analytical calculations are summarized. The existence of an
upper critical dimension and the universality of the critical behavior under
changes of lattice symmetry or the introduction of immunity are discussed. A
survey of interesting modifications of the forest-fire model is given. Finally,
several other important SOC models are briefly described.Comment: 37 pages RevTeX, 13 PostScript figures (Figs 1, 4, 13 are of reduced
quality to keep download times small
Phase Transition in a Stochastic Forest Fire Model and Effects of the Definition of Neighbourhood
We present results on a stochastic forest fire model, where the influence of
the neighbour trees is treated in a more realistic way than usual and the
definition of neighbourhood can be tuned by an additional parameter.
This model exhibits a surprisingly sharp phase transition which can be
shifted by redefinition of neighbourhood. The results can also be interpreted
in terms of disease-spreading and are quite unsettling from the epidemologist's
point of view, since variation of one crucial parameter only by a few percent
can result in the change from endemic to epidemic behaviour.Comment: 23 pages, 13 figure
Universal Behavior of the Coefficients of the Continuous Equation in Competitive Growth Models
The competitive growth models involving only one kind of particles (CGM), are
a mixture of two processes one with probability and the other with
probability . The dependance produce crossovers between two different
regimes. We demonstrate that the coefficients of the continuous equation,
describing their universality classes, are quadratic in (or ). We show
that the origin of such dependance is the existence of two different average
time rates. Thus, the quadratic dependance is an universal behavior of all
the CGM. We derive analytically the continuous equations for two CGM, in 1+1
dimensions, from the microscopic rules using a regularization procedure. We
propose generalized scalings that reproduce the scaling behavior in each
regime. In order to verify the analytic results and the scalings, we perform
numerical integrations of the derived analytical equations. The results are in
excellent agreement with those of the microscopic CGM presented here and with
the proposed scalings.Comment: 9 pages, 3 figure
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