631 research outputs found
Improved linear response for stochastically driven systems
The recently developed short-time linear response algorithm, which predicts
the average response of a nonlinear chaotic system with forcing and dissipation
to small external perturbation, generally yields high precision of the response
prediction, although suffers from numerical instability for long response times
due to positive Lyapunov exponents. However, in the case of stochastically
driven dynamics, one typically resorts to the classical fluctuation-dissipation
formula, which has the drawback of explicitly requiring the probability density
of the statistical state together with its derivative for computation, which
might not be available with sufficient precision in the case of complex
dynamics (usually a Gaussian approximation is used). Here we adapt the
short-time linear response formula for stochastically driven dynamics, and
observe that, for short and moderate response times before numerical
instability develops, it is generally superior to the classical formula with
Gaussian approximation for both the additive and multiplicative stochastic
forcing. Additionally, a suitable blending with classical formula for longer
response times eliminates numerical instability and provides an improved
response prediction even for long response times
A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears
Variational principle for Kolmogorov-Petrovsky-Piskunov (KPP) minimal front
speeds provides an efficient tool for statistical speed analysis, as well as a
fast and accurate method for speed computation. A variational principle based
analysis is carried out on the ensemble of KPP speeds through spatially
stationary random shear flows inside infinite channel domains. In the regime of
small root mean square (rms) shear amplitude, the enhancement of the ensemble
averaged KPP front speeds is proved to obey the quadratic law under certain
shear moment conditions. Similarly, in the large rms amplitude regime, the
enhancement follows the linear law. In particular, both laws hold for the
Ornstein-Uhlenbeck process in case of two dimensional channels. An asymptotic
ensemble averaged speed formula is derived in the small rms regime and is
explicit in case of the Ornstein-Uhlenbeck process of the shear. Variational
principle based computation agrees with these analytical findings, and allows
further study on the speed enhancement distributions as well as the dependence
of enhancement on the shear covariance. Direct simulations in the small rms
regime suggest quadratic speed enhancement law for non-KPP nonlinearities.Comment: 28 pages, 14 figures update: fixed typos, refined estimates in
section
Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves
The influence of an underlying current on 3-wave interactions of capillary
water waves is studied. The fact that in irrotational flow resonant 3-wave
interactions are not possible can be invalidated by the presence of an
underlying current of constant non-zero vorticity. We show that: 1) wave trains
in flows with constant non-zero vorticity are possible only for two-dimensional
flows; 2) only positive constant vorticities can trigger the appearance of
three-wave resonances; 3) the number of positive constant vorticities which do
trigger a resonance is countable; 4) the magnitude of a positive constant
vorticity triggering a resonance can not be too small.Comment: 6 pages, submitte
The Vortex-Wave equation with a single vortex as the limit of the Euler equation
In this article we consider the physical justification of the Vortex-Wave
equation introduced by Marchioro and Pulvirenti in the case of a single point
vortex moving in an ambient vorticity. We consider a sequence of solutions for
the Euler equation in the plane corresponding to initial data consisting of an
ambient vorticity in and a sequence of concentrated blobs
which approach the Dirac distribution. We introduce a notion of a weak solution
of the Vortex-Wave equation in terms of velocity (or primitive variables) and
then show, for a subsequence of the blobs, the solutions of the Euler equation
converge in velocity to a weak solution of the Vortex-Wave equation.Comment: 24 pages, to appea
Classification of conservation laws of compressible isentropic fluid flow in n>1 spatial dimensions
For the Euler equations governing compressible isentropic fluid flow with a
barotropic equation of state (where pressure is a function only of the
density), local conservation laws in spatial dimensions are fully
classified in two primary cases of physical and analytical interest: (1)
kinematic conserved densities that depend only on the fluid density and
velocity, in addition to the time and space coordinates; (2) vorticity
conserved densities that have an essential dependence on the curl of the fluid
velocity. A main result of the classification in the kinematic case is that the
only equation of state found to be distinguished by admitting extra
-dimensional conserved integrals, apart from mass, momentum, energy, angular
momentum and Galilean momentum (which are admitted for all equations of state),
is the well-known polytropic equation of state with dimension-dependent
exponent . In the vorticity case, no distinguished equations of
state are found to arise, and here the main result of the classification is
that, in all even dimensions , a generalized version of Kelvin's
two-dimensional circulation theorem is obtained for a general equation of
state.Comment: 24 pages; published version with misprints correcte
Universality of Probability Distributions Among Two-Dimensional Turbulent Flows
We study statistical properties of two-dimensional turbulent flows. Three
systems are considered: the Navier-Stokes equation, surface quasi-geostrophic
flow, and a model equation for thermal convection in the Earth's mantle. Direct
numerical simulations are used to determine 1-point fluctuation properties.
Comparative study shows universality of probability density functions (PDFs)
across different types of flow. Especially for the derivatives of the
``advected'' quantity, the shapes of the PDFs are the same for the three flows,
once normalized by the average size of fluctuations. Theoretical models for the
shape of PDFs are briefly discussed.Comment: 5 pages, 7 figure
A relative entropy rate method for path space sensitivity analysis of stationary complex stochastic dynamics
We propose a new sensitivity analysis methodology for complex stochastic
dynamics based on the Relative Entropy Rate. The method becomes computationally
feasible at the stationary regime of the process and involves the calculation
of suitable observables in path space for the Relative Entropy Rate and the
corresponding Fisher Information Matrix. The stationary regime is crucial for
stochastic dynamics and here allows us to address the sensitivity analysis of
complex systems, including examples of processes with complex landscapes that
exhibit metastability, non-reversible systems from a statistical mechanics
perspective, and high-dimensional, spatially distributed models. All these
systems exhibit, typically non-gaussian stationary probability distributions,
while in the case of high-dimensionality, histograms are impossible to
construct directly. Our proposed methods bypass these challenges relying on the
direct Monte Carlo simulation of rigorously derived observables for the
Relative Entropy Rate and Fisher Information in path space rather than on the
stationary probability distribution itself. We demonstrate the capabilities of
the proposed methodology by focusing here on two classes of problems: (a)
Langevin particle systems with either reversible (gradient) or non-reversible
(non-gradient) forcing, highlighting the ability of the method to carry out
sensitivity analysis in non-equilibrium systems; and, (b) spatially extended
Kinetic Monte Carlo models, showing that the method can handle high-dimensional
problems
The Cosmic No-Hair Theorem and the Nonlinear Stability of Homogeneous Newtonian Cosmological Models
The validity of the cosmic no-hair theorem is investigated in the context of
Newtonian cosmology with a perfect fluid matter model and a positive
cosmological constant. It is shown that if the initial data for an expanding
cosmological model of this type is subjected to a small perturbation then the
corresponding solution exists globally in the future and the perturbation
decays in a way which can be described precisely. It is emphasized that no
linearization of the equations or special symmetry assumptions are needed. The
result can also be interpreted as a proof of the nonlinear stability of the
homogeneous models. In order to prove the theorem we write the general solution
as the sum of a homogeneous background and a perturbation. As a by-product of
the analysis it is found that there is an invariant sense in which an
inhomogeneous model can be regarded as a perturbation of a unique homogeneous
model. A method is given for associating uniquely to each Newtonian
cosmological model with compact spatial sections a spatially homogeneous model
which incorporates its large-scale dynamics. This procedure appears very
natural in the Newton-Cartan theory which we take as the starting point for
Newtonian cosmology.Comment: 16 pages, MPA-AR-94-
- …