90 research outputs found
On Growth, Disorder, and Field Theory
This article reviews recent developments in statistical field theory far from
equilibrium. It focuses on the Kardar-Parisi-Zhang equation of stochastic
surface growth and its mathematical relatives, namely the stochastic Burgers
equation in fluid mechanics and directed polymers in a medium with quenched
disorder. At strong stochastic driving -- or at strong disorder, respectively
-- these systems develop nonperturbative scale-invariance. Presumably exact
values of the scaling exponents follow from a self-consistent asymptotic
theory. This theory is based on the concept of an operator product expansion
formed by the local scaling fields. The key difference to standard Lagrangian
field theory is the appearance of a dangerous irrelevant coupling constant
generating dynamical anomalies in the continuum limit.Comment: review article, 50 pages (latex), 10 figures (eps), minor
modification of original versio
Quantized Scaling of Growing Surfaces
The Kardar-Parisi-Zhang universality class of stochastic surface growth is
studied by exact field-theoretic methods. From previous numerical results, a
few qualitative assumptions are inferred. In particular, height correlations
should satisfy an operator product expansion and, unlike the correlations in a
turbulent fluid, exhibit no multiscaling. These properties impose a
quantization condition on the roughness exponent and the dynamic
exponent . Hence the exact values for two-dimensional
and for three-dimensional surfaces are derived.Comment: 4 pages, revtex, no figure
Critical Exponents of the KPZ Equation via Multi-Surface Coding Numerical Simulations
We study the KPZ equation (in D = 2, 3 and 4 spatial dimensions) by using a
RSOS discretization of the surface. We measure the critical exponents very
precisely, and we show that the rational guess is not appropriate, and that 4D
is not the upper critical dimension. We are also able to determine very
precisely the exponent of the sub-leading scaling corrections, that turns out
to be close to 1 in all cases. We introduce and use a {\em multi-surface
coding} technique, that allow a gain of order 30 over usual numerical
simulations.Comment: 10 pages, 8 eps figures (2 figures added). Published versio
Patterns in the Kardar-Parisi-Zhang equation
We review a recent asymptotic weak noise approach to the Kardar-Parisi-Zhang
equation for the kinetic growth of an interface in higher dimensions. The weak
noise approach provides a many body picture of a growing interface in terms of
a network of localized growth modes. Scaling in 1d is associated with a gapless
domain wall mode. The method also provides an independent argument for the
existence of an upper critical dimension.Comment: 8 pages revtex, 4 eps figure
Canonical phase space approach to the noisy Burgers equation
Presenting a general phase approach to stochastic processes we analyze in
particular the Fokker-Planck equation for the noisy Burgers equation and
discuss the time dependent and stationary probability distributions. In one
dimension we derive the long-time skew distribution approaching the symmetric
stationary Gaussian distribution. In the short time regime we discuss
heuristically the nonlinear soliton contributions and derive an expression for
the distribution in accordance with the directed polymer-replica model and
asymmetric exclusion model results.Comment: 4 pages, Revtex file, submitted to Phys. Rev. Lett. a reference has
been added and a few typos correcte
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.
Upper critical dimension, dynamic exponent and scaling functions in the mode-coupling theory for the Kardar-Parisi-Zhang equation
We study the mode-coupling approximation for the KPZ equation in the strong
coupling regime. By constructing an ansatz consistent with the asymptotic forms
of the correlation and response functions we determine the upper critical
dimension d_c=4, and the expansion z=2-(d-4)/4+O((4-d)^2) around d_c. We find
the exact z=3/2 value in d=1, and estimate the values 1.62, 1.78 for z, in
d=2,3. The result d_c=4 and the expansion around d_c are very robust and can be
derived just from a mild assumption on the relative scale on which the response
and correlation functions vary as z approaches 2.Comment: RevTex, 4 page
Fluctuations and correlations in an individual-based model of biological coevolution
We extend our study of a simple model of biological coevolution to its
statistical properties. Staring with a complete description in terms of a
master equation, we provide its relation to the deterministic evolution
equations used in previous investigations. The stationary states of the
mutationless model are generally well approximated by Gaussian distributions,
so that the fluctuations and correlations of the populations can be computed
analytically. Several specific cases are studied by Monte Carlo simulations,
and there is excellent agreement between the data and the theoretical
predictions.Comment: 25 pages, 2 figure
CRIM-TRACK: Sensor system for detection of criminal chemical substances
Detection of illegal compounds requires a reliable, selective and sensitive detection device. The successful device features automated target acquisition, identification and signal processing. It is portable, fast, user friendly, sensitive, specific, and cost efficient. LEAs are in need of such technology. CRIM-TRACK is developing a sensing device based on these requirements. We engage highly skilled specialists from research institutions, industry, SMEs and LEAs and rely on a team of end users to benefit maximally from our prototypes. Currently we can detect minute quantities of drugs, explosives and precursors thereof in laboratory settings. Using colorimetric technology we have developed prototypes that employ disposable sensing chips. Ease of operation and intuitive sensor response are highly prioritized features that we implement as we gather data to feed into machine learning. With machine learning our ability to detect threat compounds amidst harmless substances improves. Different end users prefer their equipment optimized for their specific field. In an explosives-detecting scenario, the end user may prefer false positives over false negatives, while the opposite may be true in a drug-detecting scenario. Such decisions will be programmed to match user preference. Sensor output can be as detailed as the sensor allows. The user can be informed of the statistics behind the detection, identities of all detected substances, and quantities thereof. The response can also be simplified to “yes” vs. “no”. The technology under development in CRIM-TRACK will provide custom officers, police and other authorities with an effective tool to control trafficking of illegal drugs and drug precursors
Scaling of fluctuation for Directed polymers with random interaction
Using a finite size scaling form for reunion probability, we show numerically
the existence of a binding-unbinding transition for Directed polymers with
random interaction. The cases studied are (A1) two chains in 1+1 dimensions,
(A2) two chains in 2+1 dimensions and (B) three chains in 1+1 dimensions. A
similar finite size scaling form for fluctuation establishes a disorder induced
transition with identical exponents for cases A2 and B. The length scale
exponents in all the three cases are in agreement with previous exact
renormalization group results.Comment: Revtex, 4 postscript figures available on request (email:
[email protected]); To appear in J. Phys. A Letter
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