2,560 research outputs found
Numerical analysis of the master equation
Applied to the master equation, the usual numerical integration methods, such
as Runge-Kutta, become inefficient when the rates associated with various
transitions differ by several orders of magnitude. We introduce an integration
scheme that remains stable with much larger time increments than can be used in
standard methods. When only the stationary distribution is required, a direct
iteration method is even more rapid; this method may be extended to construct
the quasi-stationary distribution of a process with an absorbing state.
Applications to birth-and-death processes reveal gains in efficiency of two or
more orders of magnitude.Comment: 7 pages 3 figure
Sandpiles with height restrictions
We study stochastic sandpile models with a height restriction in one and two
dimensions. A site can topple if it has a height of two, as in Manna's model,
but, in contrast to previously studied sandpiles, here the height (or number of
particles per site), cannot exceed two. This yields a considerable
simplification over the unrestricted case, in which the number of states per
site is unbounded. Two toppling rules are considered: in one, the particles are
redistributed independently, while the other involves some cooperativity. We
study the fixed-energy system (no input or loss of particles) using cluster
approximations and extensive simulations, and find that it exhibits a
continuous phase transition to an absorbing state at a critical value zeta_c of
the particle density. The critical exponents agree with those of the
unrestricted Manna sandpile.Comment: 10 pages, 14 figure
Path-integral representation for a stochastic sandpile
We introduce an operator description for a stochastic sandpile model with a
conserved particle density, and develop a path-integral representation for its
evolution. The resulting (exact) expression for the effective action highlights
certain interesting features of the model, for example, that it is nominally
massless, and that the dynamics is via cooperative diffusion. Using the
path-integral formalism, we construct a diagrammatic perturbation theory,
yielding a series expansion for the activity density in powers of the time.Comment: 22 pages, 6 figure
Asymptotic behavior of the order parameter in a stochastic sandpile
We derive the first four terms in a series for the order paramater (the
stationary activity density rho) in the supercritical regime of a
one-dimensional stochastic sandpile; in the two-dimensional case the first
three terms are reported. We reorganize the pertubation theory for the model,
recently derived using a path-integral formalism [R. Dickman e R. Vidigal, J.
Phys. A 35, 7269 (2002)], to obtain an expansion for stationary properties.
Since the process has a strictly conserved particle density p, the Fourier mode
N^{-1} psi_{k=0} -> p, when the number of sites N -> infinity, and so is not a
random variable. Isolating this mode, we obtain a new effective action leading
to an expansion for rho in the parameter kappa = 1/(1+4p). This requires
enumeration and numerical evaluation of more than 200 000 diagrams, for which
task we develop a computational algorithm. Predictions derived from this series
are in good accord with simulation results. We also discuss the nature of
correlation functions and one-site reduced densities in the small-kappa
(large-p) limit.Comment: 18 pages, 5 figure
Series expansion for a stochastic sandpile
Using operator algebra, we extend the series for the activity density in a
one-dimensional stochastic sandpile with fixed particle density p, the first
terms of which were obtained via perturbation theory [R. Dickman and R.
Vidigal, J. Phys. A35, 7269 (2002)]. The expansion is in powers of the time;
the coefficients are polynomials in p. We devise an algorithm for evaluating
expectations of operator products and extend the series to O(t^{16}).
Constructing Pade approximants to a suitably transformed series, we obtain
predictions for the activity that compare well against simulations, in the
supercritical regime.Comment: Extended series and improved analysi
Quasi-stationary distributions for the Domany-Kinzel stochastic cellular automaton
We construct the {\it quasi-stationary} (QS) probability distribution for the
Domany-Kinzel stochastic cellular automaton (DKCA), a discrete-time Markov
process with an absorbing state. QS distributions are derived at both the one-
and two-site levels. We characterize the distribuitions by their mean, and
various moment ratios, and analyze the lifetime of the QS state, and the
relaxation time to attain this state. Of particular interest are the scaling
properties of the QS state along the critical line separating the active and
absorbing phases. These exhibit a high degree of similarity to the contact
process and the Malthus-Verhulst process (the closest continuous-time analogs
of the DKCA), which extends to the scaling form of the QS distribution.Comment: 15 pages, 9 figures, submited to PR
N-Site approximations and CAM analysis for a stochastic sandpile
I develop n-site cluster approximations for a stochastic sandpile in one
dimension. A height restriction is imposed to limit the number of states: each
site can harbor at most two particles (height z_i \leq 2). (This yields a
considerable simplification over the unrestricted case, in which the number of
states per site is unbounded.) On the basis of results for n \leq 11 sites, I
estimate the critical particle density as zeta_c = 0.930(1), in good agreement
with simulations. A coherent anomaly analysis yields estimates for the order
parameter exponent [beta = 0.41(1)] and the relaxation time exponent (nu_||
\simeq 2.5).Comment: 12 pages, 7 figure
Paths to Self-Organized Criticality
We present a pedagogical introduction to self-organized criticality (SOC),
unraveling its connections with nonequilibrium phase transitions. There are
several paths from a conventional critical point to SOC. They begin with an
absorbing-state phase transition (directed percolation is a familiar example),
and impose supervision or driving on the system; two commonly used methods are
extremal dynamics, and driving at a rate approaching zero. We illustrate this
in sandpiles, where SOC is a consequence of slow driving in a system exhibiting
an absorbing-state phase transition with a conserved density. Other paths to
SOC, in driven interfaces, the Bak-Sneppen model, and self-organized directed
percolation, are also examined. We review the status of experimental
realizations of SOC in light of these observations.Comment: 23 pages + 2 figure
Wang-Landau sampling in three-dimensional polymers
Monte Carlo simulations using Wang-Landau sampling are performed to study
three-dimensional chains of homopolymers on a lattice. We confirm the accuracy
of the method by calculating the thermodynamic properties of this system. Our
results are in good agreement with those obtained using Metropolis importance
sampling. This algorithm enables one to accurately simulate the usually hardly
accessible low-temperature regions since it determines the density of states in
a single simulation.Comment: 5 pages, 9 figures arch-ive/Brazilian Journal of Physic
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