1,412 research outputs found
Diffusive epidemic process: theory and simulation
We study the continuous absorbing-state phase transition in the
one-dimensional diffusive epidemic process via mean-field theory and Monte
Carlo simulation. In this model, particles of two species (A and B) hop on a
lattice and undergo reactions B -> A and A + B -> 2B; the total particle number
is conserved. We formulate the model as a continuous-time Markov process
described by a master equation. A phase transition between the (absorbing)
B-free state and an active state is observed as the parameters (reaction and
diffusion rates, and total particle density) are varied. Mean-field theory
reveals a surprising, nonmonotonic dependence of the critical recovery rate on
the diffusion rate of B particles. A computational realization of the process
that is faithful to the transition rates defining the model is devised,
allowing for direct comparison with theory. Using the quasi-stationary
simulation method we determine the order parameter and the survival time in
systems of up to 4000 sites. Due to strong finite-size effects, the results
converge only for large system sizes. We find no evidence for a discontinuous
transition. Our results are consistent with the existence of three distinct
universality classes, depending on whether A particles diffusive more rapidly,
less rapidly, or at the same rate as B particles.Comment: 19 pages, 5 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
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
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
Nonuniversality in the pair contact process with diffusion
We study the static and dynamic behavior of the one dimensional pair contact
process with diffusion. Several critical exponents are found to vary with the
diffusion rate, while the order-parameter moment ratio m=\bar{rho^2}
/\bar{rho}^2 grows logarithmically with the system size. The anomalous behavior
of m is traced to a violation of scaling in the order parameter probability
density, which in turn reflects the presence of two distinct sectors, one
purely diffusive, the other reactive, within the active phase. Studies
restricted to the reactive sector yield precise estimates for exponents beta
and nu_perp, and confirm finite size scaling of the order parameter. In the
course of our study we determine, for the first time, the universal value m_c =
1.334 associated with the parity-conserving universality class in one
dimension.Comment: 9 pages, 5 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
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
Phase diagrams in the lattice RPM model: from order-disorder to gas-liquid phase transition
The phase behavior of the lattice restricted primitive model (RPM) for ionic
systems with additional short-range nearest neighbor (nn) repulsive
interactions has been studied by grand canonical Monte Carlo simulations. We
obtain a rich phase behavior as the nn strength is varied. In particular, the
phase diagram is very similar to the continuum RPM model for high nn strength.
Specifically, we have found both gas-liquid phase separation, with associated
Ising critical point, and first-order liquid-solid transition. We discuss how
the line of continuous order-disorder transitions present for the low nn
strength changes into the continuum-space behavior as one increases the nn
strength and compare our findings with recent theoretical results by Ciach and
Stell [Phys. Rev. Lett. {\bf 91}, 060601 (2003)].Comment: 7 pages, 10 figure
Kinetic description of avalanching systems
Avalanching systems are treated analytically using the renormalization group
(in the self-organized-criticality regime) or mean-field approximation,
respectively. The latter describes the state in terms of the mean number of
active and passive sites, without addressing the inhomogeneity in their
distribution. This paper goes one step further by proposing a kinetic
description of avalanching systems making use of the distribution function for
clusters of active sites. We illustrate application of the kinetic formalism to
a model proposed for the description of the avalanching processes in the
reconnecting current sheet of the Earth magnetosphere.Comment: 9 page
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