143 research outputs found
Superdiffusivity of the 1D lattice Kardar-Parisi-Zhang equation
The continuum Kardar-Parisi-Zhang equation in one dimension is lattice
discretized in such a way that the drift part is divergence free. This allows
to determine explicitly the stationary measures. We map the lattice KPZ
equation to a bosonic field theory which has a cubic anti-hermitian
nonlinearity. Thereby it is established that the stationary two-point function
spreads superdiffusively.Comment: 21 page
Phase transitions and noise crosscorrelations in a model of directed polymers in a disordered medium
We show that effective interactions mediated by disorder between two directed
polymers can be modelled as the crosscorrelation of noises in the
Kardar-Parisi-Zhang (KPZ) equations satisfied by the respective free energies
of these polymers. When there are two polymers, disorder introduces attractive
interactions between them. We analyze the phase diagram in details and show
that these interactions lead to new phases in the phase diagram. We show that,
even in dimension , the two directed polymers see the attraction only if
the strength of the disorder potential exceeds a threshold value. We extend our
calculations to show that if there are polymers in the system then -body
interactions are generated in the disorder averaged effective free energy.Comment: To appear in Phys. Rev. E(2000
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
A Singular Perturbation Analysis for \\Unstable Systems with Convective Nonlinearity
We use a singular perturbation method to study the interface dynamics of a
non-conserved order parameter (NCOP) system, of the reaction-diffusion type,
for the case where an external bias field or convection is present. We find
that this method, developed by Kawasaki, Yalabik and Gunton for the
time-dependant Ginzburg-Landau equation and used successfully on other NCOP
systems, breaks down for our system when the strength of bias/convection gets
large enough.Comment: 5 pages, PostScript forma
On the solvable multi-species reaction-diffusion processes
A family of one-dimensional multi-species reaction-diffusion processes on a
lattice is introduced. It is shown that these processes are exactly solvable,
provided a nonspectral matrix equation is satisfied. Some general remarks on
the solutions to this equation, and some special solutions are given. The
large-time behavior of the conditional probabilities of such systems are also
investigated.Comment: 13 pages, LaTeX2
Three-dimensional stability of Burgers vortices
Burgers vortices are explicit stationary solutions of the Navier-Stokes
equations which are often used to describe the vortex tubes observed in
numerical simulations of three-dimensional turbulence. In this model, the
velocity field is a two-dimensional perturbation of a linear straining flow
with axial symmetry. The only free parameter is the Reynolds number , where is the total circulation of the vortex and is
the kinematic viscosity. The purpose of this paper is to show that Burgers
vortex is asymptotically stable with respect to general three-dimensional
perturbations, for all values of the Reynolds number. This definitive result
subsumes earlier studies by various authors, which were either restricted to
small Reynolds numbers or to two-dimensional perturbations. Our proof relies on
the crucial observation that the linearized operator at Burgers vortex has a
simple and very specific dependence upon the axial variable. This allows to
reduce the full linearized equations to a vectorial two-dimensional problem,
which can be treated using an extension of the techniques developped in earlier
works. Although Burgers vortices are found to be stable for all Reynolds
numbers, the proof indicates that perturbations may undergo an important
transient amplification if is large, a phenomenon that was indeed observed
in numerical simulations.Comment: 31 pages, no figur
Vortex tubes in velocity fields of laboratory isotropic turbulence: dependence on the Reynolds number
The streamwise and transverse velocities are measured simultaneously in
isotropic grid turbulence at relatively high Reynolds numbers, Re(lambda) =
110-330. Using a conditional averaging technique, we extract typical
intermittency patterns, which are consistent with velocity profiles of a model
for a vortex tube, i.e., Burgers vortex. The radii of the vortex tubes are
several of the Kolmogorov length regardless of the Reynolds number. Using the
distribution of an interval between successive enhancements of a small-scale
velocity increment, we study the spatial distribution of vortex tubes. The
vortex tubes tend to cluster together. This tendency is increasingly
significant with the Reynolds number. Using statistics of velocity increments,
we also study the energetical importance of vortex tubes as a function of the
scale. The vortex tubes are important over the background flow at small scales
especially below the Taylor microscale. At a fixed scale, the importance is
increasingly significant with the Reynolds number.Comment: 8 pages, 3 PS files for 8 figures, to appear in Physical Review
A thermodynamic framework to develop rate-type models for fluids without instantaneous elasticity
In this paper, we apply the thermodynamic framework recently put into place
by Rajagopal and co-workers, to develop rate-type models for viscoelastic
fluids which do not possess instantaneous elasticity. To illustrate the
capabilities of such models we make a specific choice for the specific
Helmholtz potential and the rate of dissipation and consider the creep and
stress relaxation response associated with the model. Given specific forms for
the Helmholtz potential and the rate of dissipation, the rate of dissipation is
maximized with the constraint that the difference between the stress power and
the rate of change of Helmholtz potential is equal to the rate of dissipation
and any other constraint that may be applicable such as incompressibility. We
show that the model that is developed exhibits fluid-like characteristics and
is incapable of instantaneous elastic response. It also includes Maxwell-like
and Kelvin-Voigt-like viscoelastic materials (when certain material moduli take
special values).Comment: 18 pages, 5 figure
Large negative velocity gradients in Burgers turbulence
We consider 1D Burgers equation driven by large-scale white-in-time random
force. The tails of the velocity gradients probability distribution function
(PDF) are analyzed by saddle-point approximation in the path integral
describing the velocity statistics. The structure of the saddle-point
(instanton), that is velocity field configuration realizing the maximum of
probability, is studied numerically in details. The numerical results allow us
to find analytical solution for the long-time part of the instanton. Its
careful analysis confirms the result of [Phys. Rev. Lett. 78 (8) 1452 (1997)
[chao-dyn/9609005]] based on short-time estimations that the left tail of PDF
has the form ln P(u_x) \propto -|u_x|^(3/2).Comment: 10 pages, RevTeX, 10 figure
The solution of Burgers' equation for sinusoidal excitation at the upstream boundary
This paper generates an exact solution to Burgers' nonlinear diffusion equation on a convective stream with sinusoidal excitation applied at the upstream boundary, x =0. The downstream boundary, effectively at x =∞, is assumed to always be far enough ahead of the convective front at x=V s t that no disturbance is felt there. The Hopf-Cole transformation is applied in achieving the analytical solution, but only after integrating the equation and its conditions in x to avoid a nonlinearity in the transformed upstream boundary condition.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42700/1/10665_2006_Article_BF02383570.pd
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