25 research outputs found
Short-time scaling behavior of growing interfaces
The short-time evolution of a growing interface is studied within the
framework of the dynamic renormalization group approach for the
Kadar-Parisi-Zhang (KPZ) equation and for an idealized continuum model of
molecular beam epitaxy (MBE). The scaling behavior of response and correlation
functions is reminiscent of the ``initial slip'' behavior found in purely
dissipative critical relaxation (model A) and critical relaxation with
conserved order parameter (model B), respectively. Unlike model A the initial
slip exponent for the KPZ equation can be expressed by the dynamical exponent
z. In 1+1 dimensions, for which z is known exactly, the analytical theory for
the KPZ equation is confirmed by a Monte-Carlo simulation of a simple ballistic
deposition model. In 2+1 dimensions z is estimated from the short-time
evolution of the correlation function.Comment: 27 pages LaTeX with epsf style, 4 figures in eps format, submitted to
Phys. Rev.
A pseudo-spectral method for the Kardar-Parisi-Zhang equation
We discuss a numerical scheme to solve the continuum Kardar-Parisi-Zhang
equation in generic spatial dimensions. It is based on a momentum-space
discretization of the continuum equation and on a pseudo-spectral approximation
of the non-linear term. The method is tested in (1+1)- and (2+1)- dimensions,
where it is shown to reproduce the current most reliable estimates of the
critical exponents based on Restricted Solid-on-Solid simulations. In
particular it allows the computations of various correlation and structure
functions with high degree of numerical accuracy. Some deficiencies which are
common to all previously used finite-difference schemes are pointed out and the
usefulness of the present approach in this respect is discussed.Comment: 12 pages, 13 .eps figures, revetx4. A few equations have been
corrected. Erratum sent to Phys. Rev.
Nonequilibrium critical dynamics of the relaxational models C and D
We investigate the critical dynamics of the -component relaxational models
C and D which incorporate the coupling of a nonconserved and conserved order
parameter S, respectively, to the conserved energy density rho, under
nonequilibrium conditions by means of the dynamical renormalization group.
Detailed balance violations can be implemented isotropically by allowing for
different effective temperatures for the heat baths coupling to the slow modes.
In the case of model D with conserved order parameter, the energy density
fluctuations can be integrated out. For model C with scalar order parameter, in
equilibrium governed by strong dynamic scaling (z_S = z_rho), we find no
genuine nonequilibrium fixed point. The nonequilibrium critical dynamics of
model C with n = 1 thus follows the behavior of other systems with nonconserved
order parameter wherein detailed balance becomes effectively restored at the
phase transition. For n >= 4, the energy density decouples from the order
parameter. However, for n = 2 and n = 3, in the weak dynamic scaling regime
(z_S <= z_rho) entire lines of genuine nonequilibrium model C fixed points
emerge to one-loop order, which are characterized by continuously varying
critical exponents. Similarly, the nonequilibrium model C with spatially
anisotropic noise and n < 4 allows for continuously varying exponents, yet with
strong dynamic scaling. Subjecting model D to anisotropic nonequilibrium
perturbations leads to genuinely different critical behavior with softening
only in subsectors of momentum space and correspondingly anisotropic scaling
exponents. Similar to the two-temperature model B the effective theory at
criticality can be cast into an equilibrium model D dynamics, albeit
incorporating long-range interactions of the uniaxial dipolar type.Comment: Revtex, 23 pages, 5 eps figures included (minor additions), to appear
in Phys. Rev.
Interface Scaling in the Contact Process
Scaling properties of an interface representation of the critical contact
process are studied in dimensions 1 - 3. Simulations confirm the scaling
relation beta_W = 1 - theta between the interface-width growth exponent beta_W
and the exponent theta governing the decay of the order parameter. A scaling
property of the height distribution, which serves as the basis for this
relation, is also verified. The height-height correlation function shows clear
signs of anomalous scaling, in accord with Lopez' analysis [Phys. Rev. Lett.
83, 4594 (1999)], but no evidence of multiscaling.Comment: 10 pages, 9 figure
Vortex wandering in a forest of splayed columnar defects
We investigate the scaling properties of single flux lines in a random
pinning landscape consisting of splayed columnar defects. Such correlated
defects can be injected into Type II superconductors by inducing nuclear
fission or via direct heavy ion irradiation. The result is often very efficient
pinning of the vortices which gives, e.g., a strongly enhanced critical
current. The wandering exponent \zeta and the free energy exponent \omega of a
single flux line in such a disordered environment are obtained analytically
from scaling arguments combined with extreme-value statistics. In contrast to
the case of point disorder, where these exponents are universal, we find a
dependence of the exponents on details in the probability distribution of the
low lying energies of the columnar defects. The analytical results show
excellent agreement with numerical transfer matrix calculations in two and
three dimensions.Comment: 11 pages, 9 figure
Critical Behavior of O(n)-symmetric Systems With Reversible Mode-coupling Terms: Stability Against Detailed-balance Violation
We investigate nonequilibrium critical properties of -symmetric models
with reversible mode-coupling terms. Specifically, a variant of the model of
Sasv\'ari, Schwabl, and Sz\'epfalusy is studied, where violation of detailed
balance is incorporated by allowing the order parameter and the dynamically
coupled conserved quantities to be governed by heat baths of different
temperatures and , respectively. Dynamic perturbation theory and the
field-theoretic renormalization group are applied to one-loop order, and yield
two new fixed points in addition to the equilibrium ones. The first one
corresponds to and leads to model A critical
behavior for the order parameter and to anomalous noise correlations for the
generalized angular momenta; the second one is at and is
characterized by mean-field behavior of the conserved quantities, by a dynamic
exponent equal to that of the equilibrium SSS model, and by
modified static critical exponents. However, both these new fixed points are
unstable, and upon approaching the critical point detailed balance is restored,
and the equilibrium static and dynamic critical properties are recovered.Comment: 18 pages, RevTeX, 1 figure included as eps-file; submitted to Phys.
Rev.
Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model
The Langevin equation for a particle (`random walker') moving in
d-dimensional space under an attractive central force, and driven by a Gaussian
white noise, is considered for the case of a power-law force, F(r) = -
Ar^{-sigma}. The `persistence probability', P_0(t), that the particle has not
visited the origin up to time t, is calculated. For sigma > 1, the force is
asymptotically irrelevant (with respect to the noise), and the asymptotics of
P_0(t) are those of a free random walker. For sigma < 1, the noise is
(dangerously) irrelevant and the asymptotics of P_0(t) can be extracted from a
weak noise limit within a path-integral formalism. For the case sigma=1,
corresponding to a logarithmic potential, the noise is exactly marginal. In
this case, P_0(t) decays as a power-law, P_0(t) \sim t^{-theta}, with an
exponent theta that depends continuously on the ratio of the strength of the
potential to the strength of the noise. This case, with d=2, is relevant to the
annihilation dynamics of a vortex-antivortex pair in the two-dimensional XY
model. Although the noise is multiplicative in the latter case, the relevant
Langevin equation can be transformed to the standard form discussed in the
first part of the paper. The mean annihilation time for a pair initially
separated by r is given by t(r) \sim r^2 ln(r/a) where a is a microscopic
cut-off (the vortex core size). Implications for the nonequilibrium critical
dynamics of the system are discussed and compared to numerical simulation
results.Comment: 10 pages, 1 figur
Comparison of Different Parallel Implementations of the 2+1-Dimensional KPZ Model and the 3-Dimensional KMC Model
We show that efficient simulations of the Kardar-Parisi-Zhang interface
growth in 2 + 1 dimensions and of the 3-dimensional Kinetic Monte Carlo of
thermally activated diffusion can be realized both on GPUs and modern CPUs. In
this article we present results of different implementations on GPUs using CUDA
and OpenCL and also on CPUs using OpenCL and MPI. We investigate the runtime
and scaling behavior on different architectures to find optimal solutions for
solving current simulation problems in the field of statistical physics and
materials science.Comment: 14 pages, 8 figures, to be published in a forthcoming EPJST special
issue on "Computer simulations on GPU
Reaction Diffusion Models in One Dimension with Disorder
We study a large class of 1D reaction diffusion models with quenched disorder
using a real space renormalization group method (RSRG) which yields exact
results at large time. Particles (e.g. of several species) undergo diffusion
with random local bias (Sinai model) and react upon meeting. We obtain the
large time decay of the density of each specie, their associated universal
amplitudes, and the spatial distribution of particles. We also derive the
spectrum of exponents which characterize the convergence towards the asymptotic
states. For reactions with several asymptotic states, we analyze the dynamical
phase diagram and obtain the critical exponents at the transitions. We also
study persistence properties for single particles and for patterns. We compute
the decay exponents for the probability of no crossing of a given point by,
respectively, the single particle trajectories () or the thermally
averaged packets (). The generalized persistence exponents
associated to n crossings are also obtained. Specifying to the process or A with probabilities , we compute exactly the exponents
and characterizing the survival up to time t of a domain
without any merging or with mergings respectively, and and
characterizing the survival up to time t of a particle A without
any coalescence or with coalescences respectively.
obey hypergeometric equations and are numerically surprisingly close to pure
system exponents (though associated to a completely different diffusion
length). Additional disorder in the reaction rates, as well as some open
questions, are also discussed.Comment: 54 pages, Late
Renormalization group and nonequilibrium action in stochastic field theory
We investigate the renormalization group approach to nonequilibrium field
theory. We show that it is possible to derive nontrivial renormalization group
flow from iterative coarse graining of a closed-time-path action. This
renormalization group is different from the usual in quantum field theory
textbooks, in that it describes nontrivial noise and dissipation. We work out a
specific example where the variation of the closed-time-path action leads to
the so-called Kardar-Parisi-Zhang equation, and show that the renormalization
group obtained by coarse graining this action, agrees with the dynamical
renormalization group derived by directly coarse graining the equations of
motion.Comment: 33 pages, 3 figures included in the text. Revised; one reference
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