5,403 research outputs found
The probability density function tail of the Kardar-Parisi-Zhang equation in the strongly non-linear regime
An analytical derivation of the probability density function (PDF) tail
describing the strongly correlated interface growth governed by the nonlinear
Kardar-Parisi-Zhang equation is provided. The PDF tail exactly coincides with a
Tracy-Widom distribution i.e. a PDF tail proportional to , where is the the width of the interface. The PDF tail is
computed by the instanton method in the strongly non-linear regime within the
Martin-Siggia-Rose framework using a careful treatment of the non-linear
interactions. In addition, the effect of spatial dimensions on the PDF tail
scaling is discussed. This gives a novel approach to understand the rightmost
PDF tail of the interface width distribution and the analysis suggests that
there is no upper critical dimension.Comment: 17 pages, 2 figure
The instanton method and its numerical implementation in fluid mechanics
A precise characterization of structures occurring in turbulent fluid flows
at high Reynolds numbers is one of the last open problems of classical physics.
In this review we discuss recent developments related to the application of
instanton methods to turbulence. Instantons are saddle point configurations of
the underlying path integrals. They are equivalent to minimizers of the related
Freidlin-Wentzell action and known to be able to characterize rare events in
such systems. While there is an impressive body of work concerning their
analytical description, this review focuses on the question on how to compute
these minimizers numerically. In a short introduction we present the relevant
mathematical and physical background before we discuss the stochastic Burgers
equation in detail. We present algorithms to compute instantons numerically by
an efficient solution of the corresponding Euler-Lagrange equations. A second
focus is the discussion of a recently developed numerical filtering technique
that allows to extract instantons from direct numerical simulations. In the
following we present modifications of the algorithms to make them efficient
when applied to two- or three-dimensional fluid dynamical problems. We
illustrate these ideas using the two-dimensional Burgers equation and the
three-dimensional Navier-Stokes equations
Fronts in randomly advected and heterogeneous media and nonuniversality of Burgers turbulence: Theory and numerics
A recently established mathematical equivalence--between weakly perturbed
Huygens fronts (e.g., flames in weak turbulence or geometrical-optics wave
fronts in slightly nonuniform media) and the inviscid limit of
white-noise-driven Burgers turbulence--motivates theoretical and numerical
estimates of Burgers-turbulence properties for specific types of white-in-time
forcing. Existing mathematical relations between Burgers turbulence and the
statistical mechanics of directed polymers, allowing use of the replica method,
are exploited to obtain systematic upper bounds on the Burgers energy density,
corresponding to the ground-state binding energy of the directed polymer and
the speedup of the Huygens front. The results are complementary to previous
studies of both Burgers turbulence and directed polymers, which have focused on
universal scaling properties instead of forcing-dependent parameters. The
upper-bound formula can be heuristically understood in terms of renormalization
of a different kind from that previously used in combustion models, and also
shows that the burning velocity of an idealized turbulent flame does not
diverge with increasing Reynolds number at fixed turbulence intensity, a
conclusion that applies even to strong turbulence. Numerical simulations of the
one-dimensional inviscid Burgers equation using a Lagrangian finite-element
method confirm that the theoretical upper bounds are sharp within about 15% for
various forcing spectra (corresponding to various two-dimensional random
media). These computations provide a new quantitative test of the replica
method. The inferred nonuniversality (spectrum dependence) of the front speedup
is of direct importance for combustion modeling.Comment: 20 pages, 2 figures, REVTeX 4. Moved some details to appendices,
added figure on numerical metho
Exactly solvable variable parametric Burgers type models
Exactly solvable variable parametric Burgers type equations in one-dimension
are introduced, and two different approaches for solving the corresponding
initial value problems are given. The first one is using the relationship
between the variable parametric models and their standard counterparts. The
second approach is a direct linearization of the variable parametric Burgers
model to a variable parametric parabolic model via a generalized Cole-Hopf
transform. Eventually, the problem of finding analytic and exact solutions of
the variable parametric models reduces to that of solving a corresponding
second order linear ODE with time dependent coefficients. This makes our
results applicable to a wide class of exactly solvable Burgers type equations
related with the classical Sturm-Liouville problems for the orthogonal
polynomials
Entire solutions of hydrodynamical equations with exponential dissipation
We consider a modification of the three-dimensional Navier--Stokes equations
and other hydrodynamical evolution equations with space-periodic initial
conditions in which the usual Laplacian of the dissipation operator is replaced
by an operator whose Fourier symbol grows exponentially as \ue ^{|k|/\kd} at
high wavenumbers . Using estimates in suitable classes of analytic
functions, we show that the solutions with initially finite energy become
immediately entire in the space variables and that the Fourier coefficients
decay faster than \ue ^{-C(k/\kd) \ln (|k|/\kd)} for any . The
same result holds for the one-dimensional Burgers equation with exponential
dissipation but can be improved: heuristic arguments and very precise
simulations, analyzed by the method of asymptotic extrapolation of van der
Hoeven, indicate that the leading-order asymptotics is precisely of the above
form with . The same behavior with a universal constant
is conjectured for the Navier--Stokes equations with exponential
dissipation in any space dimension. This universality prevents the strong
growth of intermittency in the far dissipation range which is obtained for
ordinary Navier--Stokes turbulence. Possible applications to improved spectral
simulations are briefly discussed.Comment: 29 pages, 3 figures, Comm. Math. Phys., in pres
Asymptotic stability of traveling wave solutions for perturbations with algebraic decay
For a class of scalar partial differential equations that incorporate
convection, diffusion, and possibly dispersion in one space and one time
dimension, the stability of traveling wave solutions is investigated. If the
initial perturbation of the traveling wave profile decays at an algebraic rate,
then the solution is shown to converge to a shifted wave profile at a
corresponding temporal algebraic rate, and optimal intermediate results that
combine temporal and spatial decay are obtained. The proofs are based on a
general interpolation principle which says that algebraic decay results of this
form always follow if exponential temporal decay holds for perturbation with
exponential spatial decay and the wave profile is stable for general
perturbations.Comment: 14 page
Statistical properties of the Burgers equation with Brownian initial velocity
We study the one-dimensional Burgers equation in the inviscid limit for
Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian
motion that starts from the origin x=0). We obtain the one-point distribution
of the velocity field in closed analytical form. In the limit where we are far
from the origin, we also obtain the two-point and higher-order distributions.
We show how they factorize and recover the statistical invariance through
translations for the distributions of velocity increments and Lagrangian
increments. We also derive the velocity structure functions and we recover the
bifractality of the inverse Lagrangian map. Then, for the case where the
initial density is uniform, we obtain the distribution of the density field and
its -point correlations. In the same limit, we derive the point
distributions of the Lagrangian displacement field and the properties of
shocks. We note that both the stable-clustering ansatz and the Press-Schechter
mass function, that are widely used in the cosmological context, happen to be
exact for this one-dimensional version of the adhesion model.Comment: 42 pages, published in J. Stat. Phy
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