166 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
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.
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
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
Intermittency in the Joint Cascade of Energy and Helicity
The statistics of the energy and helicity fluxes in isotropic turbulence are
studied using high resolution direct numerical simulation. The scaling
exponents of the energy flux agree with those of the transverse velocity
structure functions through refined similarity hypothesis, consistent with
Kraichnan's prediction \cite{Kr74}. The helicity flux is even more intermittent
than the energy flux and its scaling exponents are closer to those of the
passive scalar. Using Waleffe's helical decomposition, we demonstrate that the
existence of positive mean helicity flux inhibits the energy transfer in the
negative helical modes, a non-passive effect
Turbulence Fluctuations and New Universal Realizability Conditions in Modelling
General turbulent mean statistics are shown to be characterized by a
variational principle. The variational functionals, or ``effective actions'',
have experimental consequences for turbulence fluctuations and are subject to
realizability conditions of positivity and convexity. An efficient
Rayleigh-Ritz algorithm is available to calculate approximate effective actions
within PDF closures. Examples are given for Navier-Stokes and for a 3-mode
system of Lorenz. The new realizability conditions succeed at detecting {\em a
priori} the poor predictions of PDF closures even when the classical 2nd-order
moment realizability conditions are satisfied.Comment: 4 pages, LaTeX (Version 2.09), 3 figures, Postscript, Submitted to
Phys. Rev. Let
Rayleigh-Ritz Calculation of Effective Potential Far From Equilibrium
We demonstrate the utility of a Rayleigh-Ritz scheme recently proposed to
compute the nonequilibrium effective potential nonperturbatively in a strong
noise regime far from equilibrium. A simple Kramers model of an ionic conductor
is used to illustrate the efficiency of the method.Comment: 4 pages, Latex (Version 2.09), 2 figures (Postscript),
tar+gzip+uuencoded. Submitted to Phys. Rev. Let
Fluctuations relations for semiclassical single-mode laser
Over last decades, the study of laser fluctuations has shown that laser
theory may be regarded as a prototypical example of a nonlinear nonequilibrium
problem. The present paper discusses the fluctuation relations, recently
derived in nonequilibrium statistical mechanics, in the context of the
semiclassical laser theory.Comment: 11 pages, 3 figure
Hydrodynamic limit for a boundary driven stochastic lattice gas model with many conserved quantities
We prove the hydrodynamic limit for a particle system in which particles may
have different velocities. We assume that we have two infinite reservoirs of
particles at the boundary: this is the so-called boundary driven process. The
dynamics we considered consists of a weakly asymmetric simple exclusion process
with collision among particles having different velocities
Matrix exponential-based closures for the turbulent subgrid-scale stress tensor
Two approaches for closing the turbulence subgrid-scale stress tensor in terms of matrix exponentials are introduced and compared. The first approach is based on a formal solution of the stress transport equation in which the production terms can be integrated exactly in terms of matrix exponentials. This formal solution of the subgrid-scale stress transport equation is shown to be useful to explore special cases, such as the response to constant velocity gradient, but neglecting pressure-strain correlations and diffusion effects. The second approach is based on an Eulerian-Lagrangian change of variables, combined with the assumption of isotropy for the conditionally averaged Lagrangian velocity gradient tensor and with the recent fluid deformation approximation. It is shown that both approaches lead to the same basic closure in which the stress tensor is expressed as the matrix exponential of the resolved velocity gradient tensor multiplied by its transpose. Short-time expansions of the matrix exponentials are shown to provide an eddy-viscosity term and particular quadratic terms, and thus allow a reinterpretation of traditional eddy-viscosity and nonlinear stress closures. The basic feasibility of the matrix-exponential closure is illustrated by implementing it successfully in large eddy simulation of forced isotropic turbulence. The matrix-exponential closure employs the drastic approximation of entirely omitting the pressure-strain correlation and other nonlinear scrambling terms. But unlike eddy-viscosity closures, the matrix exponential approach provides a simple and local closure that can be derived directly from the stress transport equation with the production term, and using physically motivated assumptions about Lagrangian decorrelation and upstream isotropy
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