19 research outputs found
On the random neighbor Olami-Feder-Christensen slip-stick model
We reconsider the treatment of Lise and Jensen (Phys. Rev. Lett. 76, 2326
(1996)) on the random neighbor Olami-Feder-Christensen stik-slip model, and
examine the strong dependence of the results on the approximations used for the
distribution of states p(E).Comment: 6pages, 3 figures. To be published in PRE as a brief repor
Superdiffusivity of the 1D lattice Kardar-Parisi-Zhang equation
The continuum Kardar-Parisi-Zhang equation in one dimension is lattice
discretized in such a way that the drift part is divergence free. This allows
to determine explicitly the stationary measures. We map the lattice KPZ
equation to a bosonic field theory which has a cubic anti-hermitian
nonlinearity. Thereby it is established that the stationary two-point function
spreads superdiffusively.Comment: 21 page
On the robustness of scale invariance in SOC models
A random neighbor extremal stick-slip model is introduced. In the
thermodynamic limit, the distribution of states has a simple analytical form
and the mean avalanche size, as a function of the coupling parameter, is
exactly calculable. The system is critical only at a special point Jc in the
coupling parameter space. However, the critical region around this point, where
approximate scale invariance holds, is very large, suggesting a mechanism for
explaining the ubiquity of scale invariance in Nature.Comment: 6 pages, 4 figures; submitted to Physical Review E;
http://link.aps.org/doi/10.1103/PhysRevE.59.496
Universality classes in Burgers turbulence
We establish necessary and sufficient conditions for the shock statistics to
approach self-similar form in Burgers turbulence with L\'{e}vy process initial
data. The proof relies upon an elegant closure theorem of Bertoin and Carraro
and Duchon that reduces the study of shock statistics to Smoluchowski's
coagulation equation with additive kernel, and upon our previous
characterization of the domains of attraction of self-similar solutions for
this equation
Boundary effects in a random neighbor model of earthquakes
We introduce spatial inhomogeneities (boundaries) in a random neighbor
version of the Olami, Feder and Christensen model [Phys. Rev. Lett. 68, 1244
(1992)] and study the distributions of avalanches starting both from the bulk
and from the boundaries of the system. Because of their clear geophysical
interpretation, two different boundary conditions have been considered (named
free and open, respectively). In both cases the bulk distribution is described
by the exponent . Boundary distributions are instead
characterized by two different exponents and , for free and open boundary conditions, respectively. These
exponents indicate that the mean-field behavior of this model is correctly
described by a recently proposed inhomogeneous form of critical branching
process.Comment: 6 pages, 2 figures ; to appear on PR
Noiseless Collective Motion out of Noisy Chaos
We consider the effect of microscopic external noise on the collective motion
of a globally coupled map in fully desynchronized states. Without the external
noise a macroscopic variable shows high-dimensional chaos distinguishable from
random motion. With the increase of external noise intensity, the collective
motion is successively simplified. The number of effective degrees of freedom
in the collective motion is found to decrease as with the
external noise variance . It is shown how the microscopic noise can
suppress the number of degrees of freedom at a macroscopic level.Comment: 9 pages RevTex file and 4 postscript figure
Critical States in a Dissipative Sandpile Model
A directed dissipative sandpile model is studied in the two-dimension.
Numerical results indicate that the long time steady states of this model are
critical when grains are dropped only at the top or, everywhere. The critical
behaviour is mean-field like. We discuss the role of infinite avalanches of
dissipative models in periodic systems in determining the critical behaviour of
same models in open systems.Comment: 4 pages (Revtex), 5 ps figures (included
Dynamical models for sand ripples beneath surface waves
We introduce order parameter models for describing the dynamics of sand
ripple patterns under oscillatory flow. A crucial ingredient of these models is
the mass transport between adjacent ripples, which we obtain from detailed
numerical simulations for a range of ripple sizes. Using this mass transport
function, our models predict the existence of a stable band of wavenumbers
limited by secondary instabilities. Small ripples coarsen in our models and
this process leads to a sharply selected final wavenumber, in agreement with
experimental observations.Comment: 9 pages. Shortened version of previous submissio
How self-organized criticality works: A unified mean-field picture
We present a unified mean-field theory, based on the single site
approximation to the master-equation, for stochastic self-organized critical
models. In particular, we analyze in detail the properties of sandpile and
forest-fire (FF) models. In analogy with other non-equilibrium critical
phenomena, we identify the order parameter with the density of ``active'' sites
and the control parameters with the driving rates. Depending on the values of
the control parameters, the system is shown to reach a subcritical (absorbing)
or super-critical (active) stationary state. Criticality is analyzed in terms
of the singularities of the zero-field susceptibility. In the limit of
vanishing control parameters, the stationary state displays scaling
characteristic of self-organized criticality (SOC). We show that this limit
corresponds to the breakdown of space-time locality in the dynamical rules of
the models. We define a complete set of critical exponents, describing the
scaling of order parameter, response functions, susceptibility and correlation
length in the subcritical and supercritical states. In the subcritical state,
the response of the system to small perturbations takes place in avalanches. We
analyze their scaling behavior in relation with branching processes. In
sandpile models because of conservation laws, a critical exponents subset
displays mean-field values ( and ) in any dimensions. We
treat bulk and boundary dissipation and introduce a new critical exponent
relating dissipation and finite size effects. We present numerical simulations
that confirm our results. In the case of the forest-fire model, our approach
can distinguish between different regimes (SOC-FF and deterministic FF) studied
in the literature and determine the full spectrum of critical exponents.Comment: 21 RevTex pages, 3 figures, submitted to Phys. Rev.