17 research outputs found
Upper critical dimension of the KPZ equation
Numerical results for the Directed Polymer model in 1+4 dimensions in various
types of disorder are presented. The results are obtained for system size
considerably larger than that considered previously. For the extreme strong
disorder case (Min-Max system), associated with the Directed Percolation model,
the expected value of the meandering exponent, zeta = 0.5 is clearly revealed,
with very week finite size effects. For the week disorder case, associated with
the KPZ equation, finite size effects are stronger, but the value of seta is
clearly seen in the vicinity of 0.57. In systems with "strong disorder" it is
expected that the system will cross over sharply from Min-Max behavior at short
chains to weak disorder behavior at long chains. This is indeed what we find.
These results indicate that 1+4 is not the Upper Critical Dimension (UCD) in
the week disorder case, and thus 4+1 does not seem to be the upper critical
dimension for the KPZ equation
Directed Polymer -- Directed Percolation Transition
We study the relation between the directed polymer and the directed
percolation models, for the case of a disordered energy landscape where the
energies are taken from bimodal distribution. We find that at the critical
concentration of the directed percolation, the directed polymer undergoes a
transition from the directed polymer universality class to the directed
percolation universality class. We also find that directed percolation clusters
affect the characterisrics of the directed polymer below the critical
concentration.Comment: LaTeX 2e; 12 pages, 5 figures; in press, will be published in
Europhys. Let
Method to estimate critical exponents using numerical studies
Computational methods in statistical physics and nonlinear dynamics. PACS. 77.80.Dj -Domain structure; hysteresis. Abstract. -A novel method to estimate the critical point and critical exponents of physical models from numerical studies is presented. The method utilizes linear approximation to compute the values of the characteristic variables in the near vicinity of the critical point from numerical results obtained for only one point in that vicinity. The method is applied to two models: Two-dimensional directed percolation, and one-dimensional reaction-diffusion model. In both cases the critical point and critical exponents are determined with higher accuracy than achieved in former studies. In the reaction-diffusion model, the results strongly suggest simple rational values of 1/3, 1/12, etc., for the characteristic exponents. The estimation of critical exponents of various models from numerical studies is a task which frequently occurs in statistical physics and phase transitions. Different simulation algorithms have been applied to various models, and no method is considered superior to others. This article presents a new method to perform the task, and describes its application to two different models: Two-dimensional directed percolation, and the reaction-diffusion model introduced in [1] and numerically studied in The article starts with the presentation of the method, and follows by the presentation of the numerical results obtained for the two models. The models which are referred to in this study are characterized by the following relation: where V is a dependent variable, q is an independent state variable, q c is the critical point at which the above relation holds, t is the time or space variable, and e is the critical exponent considered. The models are characterized by some type of random process, and numerical simulations are performed in order to estimate the values of q c and e. The common approach to this estimation task is to perform numerical simulations for various values of q, and to find the value of q for which the log-log plot of V (q, t) vs. t is a straight line. However, this search procedure consumes a lot of computer time, and the more precise the value of q c to be determined, the longer the search procedure. Moreover, usually c EDP Science
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
Reaction-controlled diffusion: Monte Carlo simulations
We study the coupled two-species non-equilibrium reaction-controlled
diffusion model introduced by Trimper et al. [Phys. Rev. E 62, 6071 (2000)] by
means of detailed Monte Carlo simulations in one and two dimensions. Particles
of type A may independently hop to an adjacent lattice site provided it is
occupied by at least one B particle. The B particle species undergoes
diffusion-limited reactions. In an active state with nonzero, essentially
homogeneous B particle saturation density, the A species displays normal
diffusion. In an inactive, absorbing phase with exponentially decaying B
density, the A particles become localized. In situations with algebraic decay
rho_B(t) ~ t^{-alpha_B}, as occuring either at a non-equilibrium continuous
phase transition separating active and absorbing states, or in a power-law
inactive phase, the A particles propagate subdiffusively with mean-square
displacement ~ t^{1-alpha_A}. We find that within the accuracy of
our simulation data, \alpha_A = \alpha_B as predicted by a simple mean-field
approach. This remains true even in the presence of strong spatio-temporal
fluctuations of the B density. However, in contrast with the mean-field
results, our data yield a distinctly non-Gaussian A particle displacement
distribution n_A(x,t) that obeys dynamic scaling and looks remarkably similar
for the different processes investigated here. Fluctuations of effective
diffusion rates cause a marked enhancement of n_A(x,t) at low displacements
|x|, indicating a considerable fraction of practically localized A particles,
as well as at large traversed distances.Comment: Revtex, 19 pages, 27 eps figures include
Intermittency of Height Fluctuations and Velocity Increment of The Kardar-Parisi-Zhang and Burgers Equations with infinitesimal surface tension and Viscosity in 1+1 Dimensions
The Kardar-Parisi-Zhang (KPZ) equation with infinitesimal surface tension,
dynamically develops sharply connected valley structures within which the
height derivative is not continuous. We discuss the intermittency issue in the
problem of stationary state forced KPZ equation in 1+1--dimensions. It is
proved that the moments of height increments behave as with for length scales . The length scale is the characteristic length of the
forcing term. We have checked the analytical results by direct numerical
simulation.Comment: 13 pages, 9 figure
Hierarchical model for the scale-dependent velocity of seismic waves
Elastic waves of short wavelength propagating through the upper layer of the
Earth appear to move faster at large separations of source and receiver than at
short separations. This scale dependent velocity is a manifestation of Fermat's
principle of least time in a medium with random velocity fluctuations. Existing
perturbation theories predict a linear increase of the velocity shift with
increasing separation, and cannot describe the saturation of the velocity shift
at large separations that is seen in computer simulations. Here we show that
this long-standing problem in seismology can be solved using a model developed
originally in the context of polymer physics. We find that the saturation
velocity scales with the four-third power of the root-mean-square amplitude of
the velocity fluctuations, in good agreement with the computer simulations.Comment: 7 pages including 3 figure
Statistical Theory for the Kardar-Parisi-Zhang Equation in 1+1 Dimension
The Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension dynamically develops
sharply connected valley structures within which the height derivative {\it is
not} continuous. There are two different regimes before and after creation of
the sharp valleys. We develop a statistical theory for the KPZ equation in 1+1
dimension driven with a random forcing which is white in time and Gaussian
correlated in space. A master equation is derived for the joint probability
density function of height difference and height gradient when the forcing correlation length is much smaller than
the system size and much bigger than the typical sharp valley width. In the
time scales before the creation of the sharp valleys we find the exact
generating function of and . Then we express the time
scale when the sharp valleys develop, in terms of the forcing characteristics.
In the stationary state, when the sharp valleys are fully developed, finite
size corrections to the scaling laws of the structure functions are also obtained.Comment: 50 Pages, 5 figure
Optimal paths as correlated random walks
A numerical study of optimal paths in the directed polymer model shows that the paths
are similar to correlated random walks. It is shown that when a directed optimal path of length
t is divided into 3 segments whose length is , the correlation between the transversal
movements along the first and last path segments is independent of the path length t.
It is also shown that the transversal correlations along optimal paths decrease as the paths
approach their endpoints. The numerical results obtained for optimal paths in 1+4 dimensions are
qualitatively similar to those obtained for optimal paths in lower dimensions,
and the data supplies a strong numerical indication that 1+4 is not the upper critical
dimension of this model, and of the associated KPZ equation