245 research outputs found
Linear Stochastic Models of Nonlinear Dynamical Systems
We investigate in this work the validity of linear stochastic models for
nonlinear dynamical systems. We exploit as our basic tool a previously proposed
Rayleigh-Ritz approximation for the effective action of nonlinear dynamical
systems started from random initial conditions. The present paper discusses
only the case where the PDF-Ansatz employed in the variational calculation is
``Markovian'', i.e. is determined completely by the present values of the
moment-averages. In this case we show that the Rayleigh-Ritz effective action
of the complete set of moment-functions that are employed in the closure has a
quadratic part which is always formally an Onsager-Machlup action. Thus,
subject to satisfaction of the requisite realizability conditions on the noise
covariance, a linear Langevin model will exist which reproduces exactly the
joint 2-time correlations of the moment-functions. We compare our method with
the closely related formalism of principal oscillation patterns (POP), which,
in the approach of C. Penland, is a method to derive such a linear Langevin
model empirically from time-series data for the moment-functions. The
predictive capability of the POP analysis, compared with the Rayleigh-Ritz
result, is limited to the regime of small fluctuations around the most probable
future pattern. Finally, we shall discuss a thermodynamics of statistical
moments which should hold for all dynamical systems with stable invariant
probability measures and which follows within the Rayleigh-Ritz formalism.Comment: 36 pages, 5 figures, seceq.sty for sequential numbering of equations
by sectio
Equation-free implementation of statistical moment closures
We present a general numerical scheme for the practical implementation of
statistical moment closures suitable for modeling complex, large-scale,
nonlinear systems. Building on recently developed equation-free methods, this
approach numerically integrates the closure dynamics, the equations of which
may not even be available in closed form. Although closure dynamics introduce
statistical assumptions of unknown validity, they can have significant
computational advantages as they typically have fewer degrees of freedom and
may be much less stiff than the original detailed model. The closure method can
in principle be applied to a wide class of nonlinear problems, including
strongly-coupled systems (either deterministic or stochastic) for which there
may be no scale separation. We demonstrate the equation-free approach for
implementing entropy-based Eyink-Levermore closures on a nonlinear stochastic
partial differential equation.Comment: 7 pages, 2 figure
The supernova-regulated ISM. II. The mean magnetic field
The origin and structure of the magnetic fields in the interstellar medium of
spiral galaxies is investigated with 3D, non-ideal, compressible MHD
simulations, including stratification in the galactic gravity field,
differential rotation and radiative cooling. A rectangular domain, 1x1x2
kpc^{3} in size, spans both sides of the galactic mid-plane. Supernova
explosions drive transonic turbulence. A seed magnetic field grows
exponentially to reach a statistically steady state within 1.6 Gyr. Following
Germano (1992) we use volume averaging with a Gaussian kernel to separate
magnetic field into a mean field and fluctuations. Such averaging does not
satisfy all Reynolds rules, yet allows a formulation of mean-field theory. The
mean field thus obtained varies in both space and time. Growth rates differ for
the mean-field and fluctuating field and there is clear scale separation
between the two elements, whose integral scales are about 0.7 kpc and 0.3 kpc,
respectively.Comment: 5 pages, 10 figures, submitted to Monthly Notices Letter
Fluctuation-Response Relations for Multi-Time Correlations
We show that time-correlation functions of arbitrary order for any random
variable in a statistical dynamical system can be calculated as higher-order
response functions of the mean history of the variable. The response is to a
``control term'' added as a modification to the master equation for statistical
distributions. The proof of the relations is based upon a variational
characterization of the generating functional of the time-correlations. The
same fluctuation-response relations are preserved within moment-closures for
the statistical dynamical system, when these are constructed via the
variational Rayleigh-Ritz procedure. For the 2-time correlations of the
moment-variables themselves, the fluctuation-response relation is equivalent to
an ``Onsager regression hypothesis'' for the small fluctuations. For
correlations of higher-order, there is a new effect in addition to such linear
propagation of fluctuations present instantaneously: the dynamical generation
of correlations by nonlinear interaction of fluctuations. In general, we
discuss some physical and mathematical aspects of the {\it Ans\"{a}tze}
required for an accurate calculation of the time correlations. We also comment
briefly upon the computational use of these relations, which is well-suited for
automatic differentiation tools. An example will be given of a simple closure
for turbulent energy decay, which illustrates the numerical application of the
relations.Comment: 28 pages, 1 figure, submitted to Phys. Rev.
Fluctuations in the Irreversible Decay of Turbulent Energy
A fluctuation law of the energy in freely-decaying, homogeneous and isotropic
turbulence is derived within standard closure hypotheses for 3D incompressible
flow. In particular, a fluctuation-dissipation relation is derived which
relates the strength of a stochastic backscatter term in the energy decay
equation to the mean of the energy dissipation rate. The theory is based on the
so-called ``effective action'' of the energy history and illustrates a
Rayleigh-Ritz method recently developed to evaluate the effective action
approximately within probability density-function (PDF) closures. These
effective actions generalize the Onsager-Machlup action of nonequilibrium
statistical mechanics to turbulent flow. They yield detailed, concrete
predictions for fluctuations, such as multi-time correlation functions of
arbitrary order, which cannot be obtained by direct PDF methods. They also
characterize the mean histories by a variational principle.Comment: 26 pages, Latex Version 2.09, plus seceq.sty, a stylefile for
sequential numbering of equations by section. This version includes new
discussion of the physical interpretation of the formal Rayleigh-Ritz
approximation. The title is also change
Localness of energy cascade in hydrodynamic turbulence, II. Sharp spectral filter
We investigate the scale-locality of subgrid-scale (SGS) energy flux and
inter-band energy transfers defined by the sharp spectral filter. We show by
rigorous bounds, physical arguments and numerical simulations that the spectral
SGS flux is dominated by local triadic interactions in an extended turbulent
inertial-range. Inter-band energy transfers are also shown to be dominated by
local triads if the spectral bands have constant width on a logarithmic scale.
We disprove in particular an alternative picture of ``local transfer by
nonlocal triads,'' with the advecting wavenumber mode at the energy peak.
Although such triads have the largest transfer rates of all {\it individual}
wavenumber triads, we show rigorously that, due to their restricted number,
they make an asymptotically negligible contribution to energy flux and
log-banded energy transfers at high wavenumbers in the inertial-range. We show
that it is only the aggregate effect of a geometrically increasing number of
local wavenumber triads which can sustain an energy cascade to small scales.
Furthermore, non-local triads are argued to contribute even less to the
space-average energy flux than is implied by our rigorous bounds, because of
additional cancellations from scale-decorrelation effects. We can thus recover
the -4/3 scaling of nonlocal contributions to spectral energy flux predicted by
Kraichnan's ALHDIA and TFM closures. We support our results with numerical data
from a pseudospectral simulation of isotropic turbulence with
phase-shift dealiasing. We conclude that the sharp spectral filter has a firm
theoretical basis for use in large-eddy simulation (LES) modeling of turbulent
flows.Comment: 42 pages, 9 figure
Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case
We consider the steady state of an open system in which there is a flux of
matter between two reservoirs at different chemical potentials. For a large
system of size , the probability of any macroscopic density profile
is ; thus generalizes to
nonequilibrium systems the notion of free energy density for equilibrium
systems. Our exact expression for is a nonlocal functional of ,
which yields the macroscopically long range correlations in the nonequilibrium
steady state previously predicted by fluctuating hydrodynamics and observed
experimentally.Comment: 4 pages, RevTeX. Changes: correct minor errors, add reference, minor
rewriting requested by editors and refere
- …