Showing papers on "Dissipative system published in 1995"

The relaxation schemes for systems of conservation laws in arbitrary space dimensions

Abstract: We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1-D and 2-D problems are presented. The second-order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.

Dynamical ensembles in stationary states

Abstract: We propose, as a generalization of an idea of Ruelle's to describe turbulent fluid flow, a chaotic hypothesis for reversible dissipative many-particle systems in nonequilibrium stationary states in general. This implies an extension of the zeroth law of thermodynamics to nonequilibrium states and it leads to the identification of a unique distribution μ describing the asymptotic properties of the time evolution of the system for initial data randomly chosen with respect to a uniform distribution on phase space. For conservative systems in thermal equilibrium the chaotic hypothesis implies the ergodic hypothesis. We outline a procedure to obtain the distribution μ: it leads to a new unifying point of view for the phase space behavior of dissipative and conservative systems. The chaotic hypothesis is confirmed in a nontrivial, parameter-free, way by a recent computer experiment on the entropy production fluctuations in a shearing fluid far from equilibrium. Similar applications to other models are proposed, in particular to a model for the Kolmogorov-Obuchov theory for turbulent flow.

Dissipative Structures and Weak Turbulence

Abstract: We present a brief overview of the current understanding of temporal and spatio-temporal chaos, both termed weak turbulence according to the context [1]. The process which allows one to reduce the primitive problem to a low-dimensional dynamical system is discussed. It turns out to be appropriate as long as confinement effects are sufficiently strong to freeze the space dependence of unstable modes, hence temporal chaos only. Otherwise modulated patterns arise, yielding genuine space-time chaos. The corresponding theory rests on envelope equations providing a useful framework for weak turbulence in a globally super-critical setting. spatio-temporal intermittency analyzed next is the relevant scenario in the sub-critical case. Finally, the connection with hydrodynamic turbulence and the more general relevance of some of the ideas developed here are examined.

Nonparabolic dissipative systems modeling the flow of liquid crystals

Abstract: We study a simplified system which retains most of the interesting mathematical properties of the original Ericksen-Leslie equations for the flow of liquid crystals. This is a coupled nonparabolic dissipative dynamic system. We derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions. We also discuss uniqueness and some stability properties of the system. ©1995 John Wiley & Sons, Inc.

Exponential Attractors for Dissipative Evolution Equations

Abstract: Construction of Exponential Attractors for Maps Exponential Attractors for Dissipative Evolution Equations of First Order Approximation of Exponential Attractors Applications Exponential Attractors for Second Order Evolution Equations with Damping and Applications Alternative Construction of Exponential Attractors for Evolution Equations Inertial Manifolds: A Brief Review and Comparison Finite Dimensional Dynamics on Exponential Attractors Mane's Projections and Inertially Equivalent Dynamical Systems.

Changing supply functions in input/state stable systems

Abstract: We consider the problem of characterizing possible supply functions for a given dissipative nonlinear system and provide a result which allows some freedom in the modification of such functions. >

Hydrodynamics from dissipative particle dynamics.

Abstract: Starting from the stochastic differential equations corresponding to the updating algorithm of dissipative particle dynamics we derive, with a standard technique of projection operators, the hydrodynamic equations for the mass and momentum density fields. The connection of the original parameters of the model with the viscosity and speed of sound of the fluid is clarified.

Numerical path integral techniques for long time dynamics of quantum dissipative systems

Abstract: Recent progress in numerical methods for evaluating the real‐time path integral in dissipative harmonic environments is reviewed. Quasi‐adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi‐adiabatic propagator path integral is evaluated by efficient system‐specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunnelingdynamics in the condensed phase are described.

Quantized energy cascade and log-Poisson statistics in fully developed turbulence.

Abstract: It is proposed that the statistics of the inertial range of fully developed turbulence can be described by a quantized random multiplicative process. We then show that (i) the cascade process must be a log-infinitely divisible stochastic process (i.e., stationary independent log-increments); (ii) the inertial-range statistics of turbulent fluctuations, such as the coarse-grained energy dissipation, are log-Poisson; and (iii) a recently proposed scaling model [Z.-S. She and E. Leveque 72, 336 (1994)] of fully developed turbulence can be derived. A general theory using the Levy-Khinchine representation for infinitely divisible cascade processes is presented, which allows for a classification of scaling behaviors of various strongly nonlinear dissipative systems.

Attractors for modulation equations on unbounded domains-existence and comparison

Abstract: We are interested in the long-time behaviour of nonlinear parabolic PDEs defined on unbounded cylindrical domains. For dissipative systems defined on bounded domains, the longtime behaviour can often be described by the dynamics in their finite-dimensional attractors. For systems defined on the infinite line, very little is known at present, since the lack of compactness prevents application of the standard existence theory for attractors. We develop an abstract theorem based on the interaction of a uniform and a localizing (weighted) norm which allows us to define global attractors for some dissipative problems on unbounded domains such as the Swift-Hohenberg and the Ginzburg-Landau equation. The second aim of this paper is the comparison of attractors. The so-called Ginzburg-Landau formalism allows us to approximate solutions of weakly unstable systems which exhibit modulated periodic patterns. Here we show that the attractor of the Swift-Hohenberg equation is upper semicontinuous in a particular limit to the attractor of the associated Ginzburg-Landau equation.

Dissipative MHD solutions for resonant Alfven waves in 1-dimensional magnetic flux tubes

Abstract: The present paper extends the analysis by Sakurai, Goossens, and Hollweg (1991) on resonant Alfven waves in nonuniform magnetic flux tubes. It proves that the fundamental conservation law for resonant Alfven waves found in ideal MHD by Sakurai, Goossens, and Hollweg remains valid in dissipative MHD. This guarantees that the jump conditions of Sakurai, Goossens, and Hollweg, that connect the ideal MHD solutions forξr, andP′ across the dissipative layer, are correct. In addition, the present paper replaces the complicated dissipative MHD solutions obtained by Sakurai, Goossens, and Hollweg forξr, andP′ in terms of double integrals of Hankel functions of complex argument of order\(\frac{1}{3}\) with compact analytical solutions that allow a straightforward mathematical and physical interpretation. Finally, it presents an analytical dissipative MHD solution for the component of the Lagrangian displacement in the magnetic surfaces perpendicular to the magnetic field linesξ⊥ which enables us to determine the dominant dynamics of resonant Alfven waves in dissipative MHD.

On the Structure of Stationary Stable Processes

Abstract: A connection between structural studies of stationary non-Gaussian stable processes and the ergodic theory of nonsingular flows is established and exploited. Using this connection, a unique decomposition of a stationary stable process into three independent stationary parts is obtained. It is shown that the dissipative part of a flow generates a mixed moving average part of a stationary stable process, while the identity part of a flow essentially gives the harmonizable part. The third part of a stationary process is determined by a conservative flow without fixed points and by a related cocycle.

Model of intermittency in magnetohydrodynamic turbulence

Abstract: We extend to the magnetohydrodynamic (MHD) case a recent model of intermittency due to She and L\'ev\^eque [Phys. Rev. Lett. 72, 336 (1994)]. The model that we develop in the framework of the Iroshnikov-Kraichnan theory of MHD turbulence depends on two parameters that are linked to anomalous scaling laws of dissipative structures when their characteristic scale l\ensuremath{\rightarrow}0. A brief comparison with published data stemming from spacecraft observations within the solar wind shows that it is a workable model and that within the framework of the model, dissipative structures in MHD turbulence are sheetlike, as observed in recent numerical simulations in three dimensions at moderate Reynolds numbers.

Constitutive model and wave equations for linear, viscoelastic, anisotropic media

Abstract: Rocks are far from being isotropic and elastic. Such simplifications in modeling the seismic response of real geological structures may lead to misinterpretations, or even worse, to overlooking useful information. It is useless to develop highly accurate modeling algorithms or to naively use amplitude information in inversion processes if the stress-strain relations are based on simplified rheologies. Thus, an accurate description of wave propagation requires a rheology that accounts for the anisotropic and anelastic behavior of rocks. This work presents a new constitutive relation and the corresponding time-domain wave equation to model wave propagation in inhomogeneous anisotropic and dissipative media. The rheological equation includes the generalized Hooke’s law and Boltzmann’s superposition principle to account for anelasticity. The attenuation properties in different directions, associated with the principal axes of the medium, are controlled by four relaxation functions of viscoelastic type. A dissipation model that is consistent with rock properties is the general standard linear solid. This is based on a spectrum of relaxation mechanisms and is suitable for wavefield calculations in the time domain. One relaxation function describes the anelastic properties of the quasi-dilatational mode and the other three model the anelastic properties of the shear modes. The convolutional relations are avoided by introducing memory variables, six for each dissipation mechanism in the 3-D case, two for the generalized SH-wave equation, and three for the qP - qSVwave equation. Two-dimensional wave equations apply to monoclinic and higher symmetries. A plane analysis derives expressions for the phase velocity, slowness, attenuation factor, quality factor and energy velocity (wavefront) for homogeneous viscoelastic waves. The analysis shows that the directional properties of the attenuation strongly depend on the values of the elasticities. In addition, the displacement formulation of the 3-D wave equation is solved in the time domain by a spectral technique based on the Fourier method. The examples show simulations in a transversely-isotropic clayshale and phenolic (orthorhombic symmetry).

A semiclassical self‐consistent‐field approach to dissipative dynamics: The spin–boson problem

Abstract: A semiclassical time‐dependent self‐consistent‐field approach for the description of dissipative quantum phenomena is proposed. The total density operator is approximated by a semiclassical ansatz, which couples the system degrees of freedom to the bath degrees of freedom in a self‐consistent manner, and is thus in the spirit of a classical‐path description. The capability of the approach is demonstrated by comparing semiclassical calculations for a spin–boson model with an Ohmic bath to exact path‐integral calculations. It is shown that the semiclassical model nicely reproduces the complex dissipative behavior of the spin–boson model for a large range of model parameters. The validity and accuracy of the semiclassical approach is discussed in some detail. It is shown that the method is essentially based on the assumption of complete randomization of nuclear phases. In particular, the assumption of phase randomization allows one to perform the trace over the bath variables through quasiclassical sampling of the nuclear initial conditions without invoking any further approximation.

Fluctuation-dissipation relation for semiclassical cosmology.

Abstract: Using the concept of open systems where the classical geometry is treated as the system and the quantum matter field as the environment, we derive a fluctuation-dissipation theorem for semiclassical cosmology. This theorem, which exists under very general conditions for dissipations in the dynamics of the system, and the noise and fluctuations in the environment, can be traced to the formal mathematical relation between the dissipation and noise kernels of the influence functional depicting the open system, and is ultimately a consequence of the unitarity of the closed system. In particular, for semiclassical gravity, it embodies the back reaction effect of matter fields on the dynamics of spacetime. The back reaction equation derivable from the influence action is in the form of an Einstein-Langevin equation. It contains a dissipative term in the equation of motion for the dynamics of spacetime and a noise term related to the fluctuations of particle creation in the matter field. Using the well-studied model of a quantum scalar field in a Bianchi type-I universe we illustrate how this Langevin equation and the noise term are derived and show how the creation of particles and the dissipation of anisotropy during the expansion of the Universe can be understood as a manifestation of this fluctuation-dissipation relation.

Dispersion characteristics of sound waves in a tunnel with an array of Helmholtz resonators

Abstract: This paper examines dispersion characteristics of sound waves propagating in a tunnel with an array of Helmholtz resonators connected axially. Assuming plane waves over the tunnel’s cross section except a thin boundary layer, weakly dissipative effects due to the wall friction and the thermoviscous diffusivity of sound are taken into account. Sound propagation in such a spatially periodic structure may be termed ‘‘acoustic Bloch waves.’’ The dispersion relation derived exhibits peculiar characteristics marked by emergence of ‘‘stopping bands’’ in the frequency domain. The stopping bands inhibit selectively propagation of sound waves even if no dissipative effects are taken into account, and enhance the damping pronouncedly even in a dissipative case. The stopping bands result from the resonance with the resonators as side branches and also from the Bragg reflection by their periodic arrangements. In the ‘‘passing bands’’ outside of the stopping bands, the sound waves exhibit dispersion, though subjected intrinsically not only to the weak damping due to the dissipative effects but also to the weak dispersion due to the wall friction. Taking a plausible example, the dispersion relation and the Bloch wave functions for the pressure are displayed. Finally the validity of the continuum approximation for distribution of the Helmholtz resonators is discussed in terms of the dispersion relations.

Method for solution of Maxwell's equations in non-uniform media

Abstract: SUMMARY It is shown that the L2 norm of electric currents induced in a dissipative medium can never exceed the norm of the external currents. This allows the construction of a simple iteration method to soIve the Maxwell's equations. The method produces a series converging to the solution for an arbitrary conductivity distribution and arbitrary frequency of field variations. The convergence is slow if the lateral contrast of the conductivity distribution is about lo4 or higher. A modification significantly improving the convergence is described in this paper. As an example, electromagnetic fields induced in the model (including the western part of the Northern American continent and the adjacent part of the Pacific Ocean) are calculated.

Exponential Stability of Coupled Beams with Dissipative Joints: A Frequency Domain Approach

Abstract: Two examples of coupled Euler-Bernoulli beams with a dissipative joint are considered. The joint placements that lead to exponential stability for these systems are characterized. The technique used shows input-output stability of a related controlled, observed system, and then shows that in these examples, input-output stability implies exponential stability. In the first example, the energy dissipation arises from a discontinuity in the shear at the joint. In the second example, the energy dissipation arises from a discontinuity in the bending moment at the joint. The analysis of this system involves a complete spectrum analysis of the zero dynamics of the associated controlled, observed system.

N-pulse homoclinic orbits in perturbations of resonant hamiltonian systems

Abstract: In this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable, two-degree-of-freedom resonant Hamiltonian systems. Our energy-phase method applies to both Hamiltonian and dissipative perturbations and reveals families of multi-pulse solutions which are not amenable to Melnikov-type methods. As an example, we study a two-mode approximation of the nonlinear, nonplanar oscillations of a parametrically forced inextensional beam. In this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically.

Foundations of fractional dynamics

Abstract: Time flow in dynamical systems is reconsidered in the ultralong time limit. The ultralong time limit is a limit in which a discretized time flow is iterated infinitely often and the discretization time step is infinite. The new limit is used to study induced flows in ergodic theory, in particular for subsets of measure zero. Induced flows on subsets of measure zero require an infinite renormalization of time in the ultralong time limit. It is found that induced flows are given generically by stable convolution semigroups and not by the conventional translation groups. This could give new insight into the origin of macroscopic irreversibility. Moreover, the induced semigroups are generated by fractional time derivatives of orders less than unity, and not by a first order time derivative. Invariance under the induced semiflows therefore leads to a new form of stationarity, called fractional stationarity. Fractionally stationary states are dissipative. Fractional stationarity also provides the dynamical foundation for a previously proposed generalized equilibrium concept.

Inverse problems for a perturbed dissipative half-space

Abstract: Addresses the scattering of acoustic and electromagnetic waves from a perturbed dissipative half-space. For simplicity, the perturbation is assumed to have compact support. Section 1 discusses the application that motivated this work and explains how the scalar model used here is related to Maxwell's equations. Section 2 introduces three formulations for direct and inverse problems for the half-space geometry. Two of these formulations relate to scattering problems, and the third to a boundary value problem. Section 3 shows how the scattering problems can be related to the boundary value problem. This shows that the three inverse problems are equivalent in a certain sense. In section 4, the boundary value problem is used to outline a simple way to formulate a multi-dimensional layer stripping procedure. This procedure is unstable and does not constitute a practical algorithm for solving the inverse problem. The paper concludes with three appendices, the first two of which carry out a careful construction of solutions of the direct problems and the third of which contains a discussion of some properties of the scattering operator.

Stability constraints in the formulation of viscoelastic constitutive equations

Abstract: This paper discusses the criteria for both global Hadamard and dissipative stabilities of viscoelastic constitutive equations (CEs) with instantaneous elasticity. In order to formulate as generally as possible the conditions of stability analyses, we used some unified forms for CEs of differential Maxwell-like and factorable single integral types. The obtained criteria formulate the constraints which have to be imposed on parameters or functions in CEs to avoid the unphysical instabilities in the whole range of Deborah numbers, independent of the flow situation. The problem of Hadamard stability for the viscoelastic CEs is reduced to that well known in the theory of nonlinear elasticity where the complete solution of the problem was found recently in an algebraic form. The criteria of dissipative stability, which have been established previously, are also discussed. When applied to many popular viscoelastic models, the combined stability criteria impose a set of such severe constraints that no single integral CE with time-strain separability proposed in the literature, could satisfy them, and only few differential models have proved to be stable.

Relativistic theories of dissipative fluids

Abstract: The simple Navier–Stokes equations for dissipative fluids, translated into relativity, form a parabolic—and hence mathematically nonviable—system. There have been formulated numerous alternative theories, consisting instead of hyperbolic equations—theories that necessarily involve more dynamical variables and more free functions than does the simple Navier–Stokes theory. It is argued that, these mathematical differences notwithstanding, the physical content of these hyperbolic theories is in most cases precisely the same as that of Navier–Stokes.