935 research outputs found
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
Activated Random Walkers: Facts, Conjectures and Challenges
We study a particle system with hopping (random walk) dynamics on the integer
lattice . The particles can exist in two states, active or
inactive (sleeping); only the former can hop. The dynamics conserves the number
of particles; there is no limit on the number of particles at a given site.
Isolated active particles fall asleep at rate , and then remain
asleep until joined by another particle at the same site. The state in which
all particles are inactive is absorbing. Whether activity continues at long
times depends on the relation between the particle density and the
sleeping rate . We discuss the general case, and then, for the
one-dimensional totally asymmetric case, study the phase transition between an
active phase (for sufficiently large particle densities and/or small )
and an absorbing one. We also present arguments regarding the asymptotic mean
hopping velocity in the active phase, the rate of fixation in the absorbing
phase, and survival of the infinite system at criticality. Using mean-field
theory and Monte Carlo simulation, we locate the phase boundary. The phase
transition appears to be continuous in both the symmetric and asymmetric
versions of the process, but the critical behavior is very different. The
former case is characterized by simple integer or rational values for critical
exponents (, for example), and the phase diagram is in accord with
the prediction of mean-field theory. We present evidence that the symmetric
version belongs to the universality class of conserved stochastic sandpiles,
also known as conserved directed percolation. Simulations also reveal an
interesting transient phenomenon of damped oscillations in the activity
density
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
Self-organized Criticality and Absorbing States: Lessons from the Ising Model
We investigate a suggested path to self-organized criticality. Originally,
this path was devised to "generate criticality" in systems displaying an
absorbing-state phase transition, but closer examination of the mechanism
reveals that it can be used for any continuous phase transition. We used the
Ising model as well as the Manna model to demonstrate how the finite-size
scaling exponents depend on the tuning of driving and dissipation rates with
system size.Our findings limit the explanatory power of the mechanism to
non-universal critical behavior.Comment: 5 pages, 2 figures, REVTeX
Entropy of chains placed on the square lattice
We obtain the entropy of flexible linear chains composed of M monomers placed
on the square lattice using a transfer matrix approach. An excluded volume
interaction is included by considering the chains to be self-and mutually
avoiding, and a fraction rho of the sites are occupied by monomers. We solve
the problem exactly on stripes of increasing width m and then extrapolate our
results to the two-dimensional limit to infinity using finite-size scaling. The
extrapolated results for several finite values of M and in the polymer limit M
to infinity for the cases where all lattice sites are occupied (rho=1) and for
the partially filled case rho<1 are compared with earlier results. These
results are exact for dimers (M=2) and full occupation (\rho=1) and derived
from series expansions, mean-field like approximations, and transfer matrix
calculations for some other cases. For small values of M, as well as for the
polymer limit M to infinity, rather precise estimates of the entropy are
obtained.Comment: 6 pages, 7 figure
Generalized mean-field study of a driven lattice gas
Generalized mean-field analysis has been performed to study the ordering
process in a half-filled square lattice-gas model with repulsive nearest
neighbor interaction under the influence of a uniform electric field. We have
determined the configuration probabilities on 2-, 4-, 5-, and 6-point clusters
excluding the possibility of sublattice ordering. The agreement between the
results of 6-point approximations and Monte Carlo simulations confirms the
absence of phase transition for sufficiently strong fields.Comment: 4 pages (REVTEX) with 4 PS figures (uuencoded
Theory of the NO+CO surface reaction model
We derive a pair approximation (PA) for the NO+CO model with instantaneous
reactions. For both the triangular and square lattices, the PA, derived here
using a simpler approach, yields a phase diagram with an active state for
CO-fractions y in the interval y_1 < y < y_2, with a continuous (discontinuous)
phase transition to a poisoned state at y_1 (y_2). This is in qualitative
agreement with simulation for the triangular lattice, where our theory gives a
rather accurate prediction for y_2. To obtain the correct phase diagram for the
square lattice, i.e., no active state, we reformulate the PA using sublattices.
The (formerly) active regime is then replaced by a poisoned state with broken
symmetry (unequal sub- lattice coverages), as observed recently by Kortluke et
al. [Chem. Phys. Lett. 275, 85 (1997)]. In contrast with their approach, in
which the active state persists, although reduced in extent, we report here the
first qualitatively correct theory of the NO+CO model on the square lattice.
Surface diffusion of nitrogen can lead to an active state in this case. In one
dimension, the PA predicts that diffusion is required for the existence of an
active state.Comment: 15 pages, 9 figure
On the absorbing-state phase transition in the one-dimensional triplet creation model
We study the lattice reaction diffusion model 3A -> 4A, A -> 0 (``triplet
creation") using numerical simulations and n-site approximations. The
simulation results provide evidence of a discontinuous phase transition at high
diffusion rates. In this regime the order parameter appears to be a
discontinuous function of the creation rate; no evidence of a stable interface
between active and absorbing phases is found. Based on an effective mapping to
a modified compact directed percolation process, shall nevertheless argue that
the transition is continuous, despite the seemingly discontinuous phase
transition suggested by studies of finite systems.Comment: 23 pages, 11 figure
Scaling in self-organized criticality from interface depinning?
The avalanche properties of models that exhibit 'self-organized criticality'
(SOC) are still mostly awaiting theoretical explanations. A recent mapping
(Europhys. Lett.~53, 569) of many sandpile models to interface depinning is
presented first, to understand how to reach the SOC ensemble and the
differences of this ensemble with the usual depinning scenario. In order to
derive the SOC avalanche exponents from those of the depinning critical point,
a geometric description is discussed, of the quenched landscape in which the
'interface' measuring the integrated activity moves. It turns out that there
are two main alternatives concerning the scaling properties of the SOC
ensemble. These are outlined in one dimension in the light of scaling arguments
and numerical simulations of a sandpile model which is in the quenched
Edwards-Wilkinson universality class.Comment: 7 pages, 3 figures, Statphys satellite meeting in Merida, July 200
Critical behavior and Griffiths effects in the disordered contact process
We study the nonequilibrium phase transition in the one-dimensional contact
process with quenched spatial disorder by means of large-scale Monte-Carlo
simulations for times up to and system sizes up to sites. In
agreement with recent predictions of an infinite-randomness fixed point, our
simulations demonstrate activated (exponential) dynamical scaling at the
critical point. The critical behavior turns out to be universal, even for weak
disorder. However, the approach to this asymptotic behavior is extremely slow,
with crossover times of the order of or larger. In the Griffiths region
between the clean and the dirty critical points, we find power-law dynamical
behavior with continuously varying exponents. We discuss the generality of our
findings and relate them to a broader theory of rare region effects at phase
transitions with quenched disorder.Comment: 10 pages, 8 eps figures, final version as publishe
- …