50 research outputs found
Solitons in the noisy Burgers equation
We investigate numerically the coupled diffusion-advective type field
equations originating from the canonical phase space approach to the noisy
Burgers equation or the equivalent Kardar-Parisi-Zhang equation in one spatial
dimension. The equations support stable right hand and left hand solitons and
in the low viscosity limit a long-lived soliton pair excitation. We find that
two identical pair excitations scatter transparently subject to a size
dependent phase shift and that identical solitons scatter on a static soliton
transparently without a phase shift. The soliton pair excitation and the
scattering configurations are interpreted in terms of growing step and
nucleation events in the interface growth profile. In the asymmetrical case the
soliton scattering modes are unstable presumably toward multi soliton
production and extended diffusive modes, signalling the general
non-integrability of the coupled field equations. Finally, we have shown that
growing steps perform anomalous random walk with dynamic exponent z=3/2 and
that the nucleation of a tip is stochastically suppressed with respect to
plateau formation.Comment: 11 pages Revtex file, including 15 postscript-figure
Aspects of the Noisy Burgers Equation
The noisy Burgers equation describing for example the growth of an interface
subject to noise is one of the simplest model governing an intrinsically
nonequilibrium problem. In one dimension this equation is analyzed by means of
the Martin-Siggia-Rose technique. In a canonical formulation the morphology and
scaling behavior are accessed by a principle of least action in the weak noise
limit. The growth morphology is characterized by a dilute gas of nonlinear
soliton modes with gapless dispersion law with exponent z=3/2 and a superposed
gas of diffusive modes with a gap. The scaling exponents and a heuristic
expression for the scaling function follow from a spectral representation.Comment: 23 pages,LAMUPHYS LaTeX-file (Springer), 13 figures, and 1 table, to
appear in the Proceedings of the XI Max Born Symposium on "Anomalous
Diffusion: From Basics to Applications", May 20-24, 1998, Ladek Zdroj, Polan
Correlations, soliton modes, and non-Hermitian linear mode transmutation in the 1D noisy Burgers equation
Using the previously developed canonical phase space approach applied to the
noisy Burgers equation in one dimension, we discuss in detail the growth
morphology in terms of nonlinear soliton modes and superimposed linear modes.
We moreover analyze the non-Hermitian character of the linear mode spectrum and
the associated dynamical pinning and mode transmutation from diffusive to
propagating behavior induced by the solitons. We discuss the anomalous
diffusion of growth modes, switching and pathways, correlations in the
multi-soliton sector, and in detail the correlations and scaling properties in
the two-soliton sector.Comment: 50 pages, 15 figures, revtex4 fil
Damped finite-time-singularity driven by noise
We consider the combined influence of linear damping and noise on a dynamical
finite-time-singularity model for a single degree of freedom. We find that the
noise effectively resolves the finite-time-singularity and replaces it by a
first-passage-time or absorbing state distribution with a peak at the
singularity and a long time tail. The damping introduces a characteristic
cross-over time. In the early time regime the probability distribution and
first-passage-time distribution show a power law behavior with scaling exponent
depending on the ratio of the non linear coupling strength to the noise
strength. In the late time regime the behavior is controlled by the damping.
The study might be of relevance in the context of hydrodynamics on a nanometer
scale, in material physics, and in biophysics.Comment: 9 pages, 4 eps-figures, revtex4 fil
Power laws and stretched exponentials in a noisy finite-time-singularity model
We discuss the influence of white noise on a generic dynamical
finite-time-singularity model for a single degree of freedom. We find that the
noise effectively resolves the finite-time-singularity and replaces it by a
first-passage-time or absorbing state distribution with a peak at the
singularity and a long time tail exhibiting power law or stretched exponential
behavior. The study might be of relevance in the context of hydrodynamics on a
nanometer scale, in material physics, and in biophysics.Comment: 10 pages revtex file, including 4 postscript-figures. References
added and a few typos correcte
Does strange kinetics imply unusual thermodynamics?
We introduce a fractional Fokker-Planck equation (FFPE) for Levy flights in
the presence of an external field. The equation is derived within the framework
of the subordination of random processes which leads to Levy flights. It is
shown that the coexistence of anomalous transport and a potential displays a
regular exponential relaxation towards the Boltzmann equilibrium distribution.
The properties of the Levy-flight FFPE derived here are compared with earlier
findings for subdiffusive FFPE. The latter is characterized by a
non-exponential Mittag-Leffler relaxation to the Boltzmann distribution. In
both cases, which describe strange kinetics, the Boltzmann equilibrium is
reached and modifications of the Boltzmann thermodynamics are not required
Subordination Pathways to Fractional Diffusion
The uncoupled Continuous Time Random Walk (CTRW) in one space-dimension and
under power law regime is splitted into three distinct random walks: (rw_1), a
random walk along the line of natural time, happening in operational time;
(rw_2), a random walk along the line of space, happening in operational
time;(rw_3), the inversion of (rw_1), namely a random walk along the line of
operational time, happening in natural time. Via the general integral equation
of CTRW and appropriate rescaling, the transition to the diffusion limit is
carried out for each of these three random walks. Combining the limits of
(rw_1) and (rw_2) we get the method of parametric subordination for generating
particle paths, whereas combination of (rw_2) and (rw_3) yields the
subordination integral for the sojourn probability density in space-time
fractional diffusion.Comment: 20 pages, 4 figure
Reaction, Levy Flights, and Quenched Disorder
We consider the A + A --> emptyset reaction, where the transport of the
particles is given by Levy flights in a quenched random potential. With a
common literature model of the disorder, the random potential can only increase
the rate of reaction. With a model of the disorder that obeys detailed balance,
however, the rate of reaction initially increases and then decreases as a
function of the disorder strength. The physical behavior obtained with this
second model is in accord with that for reactive turbulent flow, indicating
that Levy flight statistics can model aspects of turbulent fluid transport.Comment: 6 pages, 5 pages. Phys. Rev. E. 65 (2002) 011109--1-
Vacancy-assisted domain-growth in asymmetric binary alloys: a Monte Carlo study
A Monte Carlo simulation study of the vacancy-assisted domain-growth in
asymmetric binary alloys is presented. The system is modeled using a
three-state ABV Hamiltonian which includes an asymmetry term, not considered in
previous works. Our simulated system is a stoichiometric two-dimensional binary
alloy with a single vacancy which evolves according to the vacancy-atom
exchange mechanism. We obtain that, compared to the symmetric case, the
ordering process slows down dramatically. Concerning the asymptotic behavior it
is algebraic and characterized by the Allen-Cahn growth exponent x=1/2. The
late stages of the evolution are preceded by a transient regime strongly
affected by both the temperature and the degree of asymmetry of the alloy. The
results are discussed and compared to those obtained for the symmetric case.Comment: 21 pages, 9 figures, accepted for publication in Phys. Rev.
Diffusion over a saddle with a Langevin equation
The diffusion problem over a saddle is studied using a multi-dimensional
Langevin equation. An analytical solution is derived for a quadratic potential
and the probability to pass over the barrier deduced. A very simple solution is
given for the one dimension problem and a general scheme is shown for higher
dimensions.Comment: 13 pages, use revTeX, to appear in Phys. Rev. E6