213 research outputs found
Directed polymers and interfaces in random media : free-energy optimization via confinement in a wandering tube
We analyze, via Imry-Ma scaling arguments, the strong disorder phases that
exist in low dimensions at all temperatures for directed polymers and
interfaces in random media. For the uncorrelated Gaussian disorder, we obtain
that the optimal strategy for the polymer in dimension with
involves at the same time (i) a confinement in a favorable tube of radius with (ii) a superdiffusive behavior with for the wandering of the best favorable
tube available. The corresponding free-energy then scales as with and the left tail of the probability
distribution involves a stretched exponential of exponent .
These results generalize the well known exact exponents ,
and in , where the subleading transverse length is known as the typical distance between two replicas in the Bethe
Ansatz wave function. We then extend our approach to correlated disorder in
transverse directions with exponent and/or to manifolds in dimension
with . The strategy of being both confined and
superdiffusive is still optimal for decaying correlations (), whereas
it is not for growing correlations (). In particular, for an
interface of dimension in a space of total dimension with
random-bond disorder, our approach yields the confinement exponent . Finally, we study the exponents in the presence of an
algebraic tail in the disorder distribution, and obtain various
regimes in the plane.Comment: 19 page
Statistical Physics of Fracture Surfaces Morphology
Experiments on fracture surface morphologies offer increasing amounts of data
that can be analyzed using methods of statistical physics. One finds scaling
exponents associated with correlation and structure functions, indicating a
rich phenomenology of anomalous scaling. We argue that traditional models of
fracture fail to reproduce this rich phenomenology and new ideas and concepts
are called for. We present some recent models that introduce the effects of
deviations from homogeneous linear elasticity theory on the morphology of
fracture surfaces, succeeding to reproduce the multiscaling phenomenology at
least in 1+1 dimensions. For surfaces in 2+1 dimensions we introduce novel
methods of analysis based on projecting the data on the irreducible
representations of the SO(2) symmetry group. It appears that this approach
organizes effectively the rich scaling properties. We end up with the
proposition of new experiments in which the rotational symmetry is not broken,
such that the scaling properties should be particularly simple.Comment: A review paper submitted to J. Stat. Phy
Hysteretic dynamics of domain walls at finite temperatures
Theory of domain wall motion in a random medium is extended to the case when
the driving field is below the zero-temperature depinning threshold and the
creep of the domain wall is induced by thermal fluctuations. Subject to an ac
drive, the domain wall starts to move when the driving force exceeds an
effective threshold which is temperature and frequency-dependent. Similarly to
the case of zero-temperature, the hysteresis loop displays three dynamical
phase transitions at increasing ac field amplitude . The phase diagram in
the 3-d space of temperature, driving force amplitude and frequency is
investigated.Comment: 4 pages, 2 figure
Non-Linear Stochastic Equations with Calculable Steady States
We consider generalizations of the Kardar--Parisi--Zhang equation that
accomodate spatial anisotropies and the coupled evolution of several fields,
and focus on their symmetries and non-perturbative properties. In particular,
we derive generalized fluctuation--dissipation conditions on the form of the
(non-linear) equations for the realization of a Gaussian probability density of
the fields in the steady state. For the amorphous growth of a single height
field in one dimension we give a general class of equations with exactly
calculable (Gaussian and more complicated) steady states. In two dimensions, we
show that any anisotropic system evolves on long time and length scales either
to the usual isotropic strong coupling regime or to a linear-like fixed point
associated with a hidden symmetry. Similar results are derived for textural
growth equations that couple the height field with additional order parameters
which fluctuate on the growing surface. In this context, we propose
phenomenological equations for the growth of a crystalline material, where the
height field interacts with lattice distortions, and identify two special cases
that obtain Gaussian steady states. In the first case compression modes
influence growth and are advected by height fluctuations, while in the second
case it is the density of dislocations that couples with the height.Comment: 9 pages, revtex
Dynamics of driven interfaces near isotropic percolation transition
We consider the dynamics and kinetic roughening of interfaces embedded in
uniformly random media near percolation treshold. In particular, we study
simple discrete ``forest fire'' lattice models through Monte Carlo simulations
in two and three spatial dimensions. An interface generated in the models is
found to display complex behavior. Away from the percolation transition, the
interface is self-affine with asymptotic dynamics consistent with the
Kardar-Parisi-Zhang universality class. However, in the vicinity of the
percolation transition, there is a different behavior at earlier times. By
scaling arguments we show that the global scaling exponents associated with the
kinetic roughening of the interface can be obtained from the properties of the
underlying percolation cluster. Our numerical results are in good agreement
with theory. However, we demonstrate that at the depinning transition, the
interface as defined in the models is no longer self-affine. Finally, we
compare these results to those obtained from a more realistic
reaction-diffusion model of slow combustion.Comment: 7 pages, 9 figures, to appear in Phys. Rev. E (1998
Jamming transition in a homogeneous one-dimensional system: the Bus Route Model
We present a driven diffusive model which we call the Bus Route Model. The
model is defined on a one-dimensional lattice, with each lattice site having
two binary variables, one of which is conserved (``buses'') and one of which is
non-conserved (``passengers''). The buses are driven in a preferred direction
and are slowed down by the presence of passengers who arrive with rate lambda.
We study the model by simulation, heuristic argument and a mean-field theory.
All these approaches provide strong evidence of a transition between an
inhomogeneous ``jammed'' phase (where the buses bunch together) and a
homogeneous phase as the bus density is increased. However, we argue that a
strict phase transition is present only in the limit lambda -> 0. For small
lambda, we argue that the transition is replaced by an abrupt crossover which
is exponentially sharp in 1/lambda. We also study the coarsening of gaps
between buses in the jammed regime. An alternative interpretation of the model
is given in which the spaces between ``buses'' and the buses themselves are
interchanged. This describes a system of particles whose mobility decreases the
longer they have been stationary and could provide a model for, say, the flow
of a gelling or sticky material along a pipe.Comment: 17 pages Revtex, 20 figures, submitted to Phys. Rev.
From dynamical scaling to local scale-invariance: a tutorial
Dynamical scaling arises naturally in various many-body systems far from
equilibrium. After a short historical overview, the elements of possible
extensions of dynamical scaling to a local scale-invariance will be introduced.
Schr\"odinger-invariance, the most simple example of local scale-invariance,
will be introduced as a dynamical symmetry in the Edwards-Wilkinson
universality class of interface growth. The Lie algebra construction, its
representations and the Bargman superselection rules will be combined with
non-equilibrium Janssen-de Dominicis field-theory to produce explicit
predictions for responses and correlators, which can be compared to the results
of explicit model studies.
At the next level, the study of non-stationary states requires to go over,
from Schr\"odinger-invariance, to ageing-invariance. The ageing algebra admits
new representations, which acts as dynamical symmetries on more general
equations, and imply that each non-equilibrium scaling operator is
characterised by two distinct, independent scaling dimensions. Tests of
ageing-invariance are described, in the Glauber-Ising and spherical models of a
phase-ordering ferromagnet and the Arcetri model of interface growth.Comment: 1+ 23 pages, 2 figures, final for
Novel non-equilibrium critical behavior in unidirectionally coupled stochastic processes
Phase transitions from an active into an absorbing, inactive state are
generically described by the critical exponents of directed percolation (DP),
with upper critical dimension d_c = 4. In the framework of single-species
reaction-diffusion systems, this universality class is realized by the combined
processes A -> A + A, A + A -> A, and A -> \emptyset. We study a hierarchy of
such DP processes for particle species A, B,..., unidirectionally coupled via
the reactions A -> B, ... (with rates \mu_{AB}, ...). When the DP critical
points at all levels coincide, multicritical behavior emerges, with density
exponents \beta_i which are markedly reduced at each hierarchy level i >= 2.
This scenario can be understood on the basis of the mean-field rate equations,
which yield \beta_i = 1/2^{i-1} at the multicritical point. We then include
fluctuations by using field-theoretic renormalization group techniques in d =
4-\epsilon dimensions. In the active phase, we calculate the fluctuation
correction to the density exponent for the second hierarchy level, \beta_2 =
1/2 - \epsilon/8 + O(\epsilon^2). Monte Carlo simulations are then employed to
determine the values for the new scaling exponents in dimensions d<= 3,
including the critical initial slip exponent. Our theory is connected to
certain classes of growth processes and to certain cellular automata, as well
as to unidirectionally coupled pair annihilation processes. We also discuss
some technical and conceptual problems of the loop expansion and their possible
interpretation.Comment: 29 pages, 19 figures, revtex, 2 columns, revised Jan 1995: minor
changes and additions; accepted for publication in 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.
Disorder-Induced Depinning Transition
The competition in the pinning of a directed polymer by a columnar pin and a
background of random point impurities is investigated systematically using the
renormalization group method. With the aid of the mapping to the noisy-Burgers'
equation and the use of the mode-coupling method, the directed polymer is shown
to be marginally localized to an arbitrary weak columnar pin in 1+1 dimensions.
This weak localization effect is attributed to the existence of large scale,
nearly degenerate optimal paths of the randomly pinned directed polymer. The
critical behavior of the depinning transition above 1+1 dimensions is obtained
via an -expansion.Comment: 47 pages in revtex; postscript files of 6 figures include
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