Showing papers on "Dissipative system published in 1991"

Resonant behaviour of MHD waves on magnetic flux tubes I. Connection formulae at the resonant surfaces

Abstract: A basic procedure is presented for dealing with the resonance problems that appear in MHD of which resonant absorption of waves at the Alfven resonance point is the best known example in solar physics. The procedure avoids solving the full fourth-order differential equation of dissipative MHD by using connection formulae across the dissipation layer

New Inequality and Functional for Contact with Friction: The Implicit Standard Material Approach∗

Abstract: This paper is devoted to the analysis of the two- or three-dimensional elastic contact problem with Coulomb friction, quasi-static equilibrium, and small displacements. The classical approach is based on two minimum principles, or variational inequalities: the first for unilateral contact and the second for friction. In practical applications, this leads to an algorithm of alternately solving the two problems until convergence is achieved. A coupled approach using one principle or one inequality only is presented. This new approach, based on a model of material called implicit standard, allows for extension of the notion of a normality law to dissipative behavior with a nonassociated flow rule, such as surface friction. For numerical time integration of the laws, Moreau's implicit method is considered. Nondifferentiable potentials are regularized by means of the augmented Lagrangian technique. A discretized formulation using the finite element method and numerical applications are reported in a s...

Exponential decay of energy of evolution equations with locally distributed damping

Abstract: Consider an evolution equation with energy dissipation, \[ \frac{\partial }{{\partial t}}y( {x,t} ) = Ay( {x,t} ) + By( {x,t} ) \] on a bounded x-domain $\Omega $, where $By$ signifies a dissipative perturbation to an otherwise energy-conserving system. This dissipation may be due to medium impurities, viscous effects, or artificially imposed dampers and stabilizers. It is distributed over only part of the domain $\Omega $. The question of when the dissipation is effective enough to cause uniform exponential decay of energy is examined.Because of the locally distributed nature of energy dissipation, the problem lacks coercivity and is not directly solvable by energy identities. Thus, to get conditions sufficient for uniform exponential decay, a different approach needs to be taken. Provided here is a set of tight sufficient conditions in terms of the influence of the dissipative operatorBon the separated eigenmodes or clustered eigenmodes ofA. The main theorems are general enough to treat the wave and bea...

Stability Diagram of the Phase-Locked Solitons in the Parametrically Driven, Damped Nonlinear Schrödinger Equation

Abstract: The parametrically driven, damped NLS equation is shown to describe the dynamics of small-amplitude breathers of the easy-plane ferromagnet and the long Josephson junction under the influence of the parametric pumping and dissipation. The soliton solutions are found exactly and stability problem for the dissipative case is reduced rigorously to the one for the undamped soliton.

Near-minimum-time control of distributed parameter-systems - analytical and experimental results

Abstract: We present a method for generating globally stable feedback control laws for maneuvers of distributed parameter structural systems. The method can accommodate system nonlinearity, and our proof of Lyapunov stability does not rely upon spatially discretizing distributed parameter systems. The approach applies directly to controllable distributed parameter systems that are open-loop conservative or dissipative. The most fundamental version of the formulation leads to controls that drive the system to a fixed point in the state space, but more generally, we develop tracking-type control laws to null the departure of the system state from a smooth target trajectory. Both analytical developments and experimental results are presented. The analytical results provide a theoretical foundation for the approach, whereas the experimental results provide conclusive evidence that the approach can be efficiently realized in actual hardware.

Dissipative Boltzmann-Robertson-Walker cosmologies.

Abstract: The equations governing a flat Robertson-Walker cosmological model containing a dissipative Boltzmann gas are integrated numerically The bulk viscous stress is modeled using the Eckart and Israel-Stewart theories of dissipative relativistic fluids; the resulting cosmologies are compared and contrasted The Eckart models are shown to always differ in a significant quantitative way from the Israel-Stewart models It thus appears inappropriate to use the pathological (nonhyperbolic) Eckart theory for cosmological applications For large bulk viscosities, both cosmological models approach asymptotic nonequilibrium states; in the Eckart model the total pressure is negative, while in the Israel-Stewart model the total pressure is asymptotically zero The Eckart model also expands more rapidly than the Israel-Stewart models These results suggest that bulk-viscous'' inflation may be an artifact of using a pathological fluid theory such as the Eckart theory

Turbulent motion and the structure of chaos : a new approach to the statistical theory of open systems

Abstract: I.1. The Criteria of the Relative Degree of Order of the States of Open Systems.- I.2. Connection Between the Statistical and the Dynamic Descriptions of Motions in Macroscopic Open Systems. The Constructive Role of Dynamic Instability of Motion.- I.3. The Transition from Reversible to Irreversible Equations. The Gibbs Ensemble in the Statistical Theory of Nonequilibrium Processes.- I.4. The Role of Fluctuations at Different Levels of Description. Fluctuation Dissipation Relations.- I.5. Brownian Motion in Open Systems. Molecular and Turbulent Sources of Fluctuations.- I.6. Laminar and Turbulent Motion.- 1. Evolution of Entropy and Entropy Production in Open Systems.- 1.1. Chaos and Order. The Controlling Parameters. Physical Chaos. Evolution and Self-Organization in Open systems.- 1.2. Boltzmann-Shannon-Gibbs Entropy.- 1.3. Entropy Distribution Function.- 1.4. The Gibbs Ensemble. Smoothing over the Physically Infinitesimal Volume. Entropy Including Fluctuations. Space (Time) and Phase Averages. Local Ergodicity Condition.- 1.5. The Kinetic Boltzmann Equation for Statistical and Smoothed Distribuions. Physically Infinitesimal Scales. The Constructive Role of the Dynamic Instability of Motion of Atoms in a Gas.- 1.6. The Role of Nonequilibrium Fluctuations in a Boltzmann Gas. Molecular and Turbulent Sources of Fluctuations.- 1.7. The Kinetic Equations for the N-Particle Distribution Functions. The Leontovich Equation.- 1.8. Boltzmann's H-Theorem for Smoothed (Pulsating) and Deterministic Distributions.- 1.9. Entropy and Entropy Production for Smoothed and Deterministic Distributions.- 1.10. The Gibbs' Theorem.- 1.11. H-Theorem for Open Systems. Kulback Entropy.- 1.12. Evolution in the Space of Controlling Parameters. The S-Theorem.- 1.13. The S-Theorem. Local Formulation.- 1.14. The Comparison of the Relative Degree of Order of States on the Basis of the S-Theorem Using Experimental Data.- 1.15. Dynamic and Statistical Descriptions of Complex Motions. K-Entropy, Lyapunov Indices. Nonlinear Characteristics of the Trajectory Divergence.- 1.16. Criteria of Dynamic Instability of Motion in Statistical Theory.- 1.17. Entropy as Measure of Diversity in Biological Evolution.- 1.18. The Principle of Minimum Entropy Production in Self-Organization Processes.- 2. Transition From the Reversible Equations of Mechanics to the Irreversible Equations of the Statistical Theory.- 2.1. Two Types of Reversible Processes. Symmetry Properties of Distribution Functions.- 2.2. Liouville Equation and Vlasov Equation. The First Moments and the "Collisionless" Approximations.- 2.3. Reversible Equations in Quantum Statistical Theory.- 2.4. Two Types of Dissipative Kinetic Equations for N-Particle Distributions.- 2.5. Measure of Deficiency (Incompleteness) of the Statistical Description.- 2.6. The Hierarchy of Equations of Fluid Mechanics.- 3. Fluctuation Dissipation Relations.- 3.1. Examples of Fluctuation Dissipation Relations.- 3.2. FDR for N-Particle Distribution Functions.- 3.3. Thermodynamic Form of the FDR. The Callen-Welton Formula.- 3.4. FDR for a Boltzmann gas. The Fluctuative Representation of Boltzmann Collision Integral.- 3.5. FDR for Large-Scale (Kinetic) Fluctuations.- 3.6. Examples of FDR for Large-Scale Fluctuations.- 3.7. System of Quantum Atoms Oscillators.- 3.8. Fluctuation Dissipation Relations in Hydrodynamics.- 3.9. Two Ways of Defining Kinetic Coefficients.- 3.10. The Molecular Langevin Source in the Difffusion Equation.- 3.11. Connection between the Intensities of Langevin Sources and the Correlator of of Phase Density Fluctuations.- 3.12. Natural Flicker Noise ("1/f Noise"). FDR for Flicker Noise.- 3.13. Natural Flicker Noise and Superconductivity.- 4. Brownian Motion.- 4.1. Fokker-Planck and Langevin Equations.- 4.2. Three Definitions of the Langevin and Fokker-Planck Equations.- 4.3. The Fokker-Planck Equations in the Statistical Theory of Nonequilibrium Processes.- 4.4. Transition to the Fokker-Planck Equation from the Smoluchowski equation (the Chapman-Kolmogorov Equation) and from the Master Equation.- 4.5. Langevin.Sources in Kinetic Equations.- 4.6. Langevin Sources in Fokker-Planck and Einstein-Smoluchowski Equations.- 4.7. Turbulent Langevin Sources and Fluctuation Dissipation Relations in Hydrodynamics.- 4.8. Brownian Motion in Systems with a Variable Number of Particles..- 5. The Boltzmann-Gibbs-Shannon Entropy As Measure of the Relative Degree of Order in Open Systems.- 5.1. Van der Pol Generator.- 5.2. Generator with Inertial Nonlinearity.- 5.3. Invariant Measures. Examples of Gibbs Distributions for Open Systems.- 5.4. Generalized Van Der Pol Generators. Bifurcations of the Limiting Cycle Energy and the Period of Oscillations.- 5.5. Dynamic and Statistical Distributions.- 5.6. Comparison Between the Degrees of Order in the Bifurcation Points and in the State of Dynamic Chaos.- 5.7. Evolution of Entropy in Systems with Two Controlling Parameters.- 5.8. A Medium of Linked Generators. The Kinetic Approach in the Theory of Self-Organization.- 5.9. A System of Van der Pol Generators with Common Feedback. Associative Memory and Pattern Recognition.- 5.10. Kinetic Description of Chemically Reacting Systems.- 5.11. A Medium of Bistable Elements. The Kinetic Approach in the Theory of Phase Transitions.- 6. Turbulent Motion. The Structure of Chaos.- 6.1. Characteristic Features of Turbulent Motion. The Main Problems.- 6.2. Incompressible Fluid. Reynolds Equations. Reynolds Stresses.- 6.3. Well-Developed Turbulence. Turbulent Viscosity.- 6.4. Semiempirical Prandtl-Karman Theory of Turbulence.- 6.5. Onset of Turbulence in Steady Couette and Poiseuille Flows.- 6.6. Entropy Production in Laminar and Turbulent Flows.- 6.7. The Principle of Least Dissipation and the Principle of Minimum Entropy Production in Self-Organization Processes.- 6.8. Evolution of Entropy in the Transition from Laminar to Turbulent Flow.- 6.9. Kinetic Description of Hydrodynamic Motion.- Conclusion.- References.

Spontaneous formation of space-time structures and criticality

Abstract: Scaling and universality in statistical physics.- Self-organized criticality and the perception of large events.- Dynamical aspects of sandpile cellular automata.- Steady state selection in driven diffusive systems.- Earthquakes and faulting: Self-organized crtical phenomena with a characteristic dimension.- Self-organized criticality in plate tectonics.- Experiments and simulations modeling earthquakes.- Fractal time-series and fractional Brownian motion.- 1/f noise, lattice gases, and diffusion.- Fluctuations in a Levy flight gas.- Spatio-temporal correlations in semiconductors.- New travelling and stationary chemical patterns in open spatial reactors.- Structure of layered systems under reactions.- Localized and "blinking" traveling wave patterns in a convective binary mixture.- Localized solutions of generalized Swift-Hohenberg equation.- Experiments on the Couette-Taylor flow with an axial flow.- Convection in microemulsions.- Routes to chaos in a dissipative chain of strongly coupled driven spins.- Localized structures and solitary waves excited by interfacial stresses.- Subcritical bifurcations and spatiotemporal intermittency.- Spatiotemporal intermittency in coupled maps.- Multifractals, multiscaling and the energy cascade of turbulence.- Smooth and rough turbulence.- Structure functions in a "forest-fire" model of turbulence.- Magnetic flux tubes as coherent structures.- Helium in a big box.- Scaling laws in weak turbulence.- Selforganization and instabilities in a system of magnetic hole pairs.- Universality in fully developed chaos, and statistics at small scales in turbulence.- Does dimension grow in flow systems.- Turbulent flows and coupled maps.- New Monte Carlo renormalization group method for phase transitions of lattice systems.

NND Schemes and Their Applications to Numerical Simulation of Two- and Three-Dimensional Flows

Abstract: Publisher Summary This chapter presents the nonoscillatory nonfree dissipative (NND) schemes and their applications to numerical simulation of two- and three-dimensional flows. An NND scheme, which is nonoscillatory, contains no free parameters, and is dissipative, is examined. The problem of one-dimensional shock wave is studied using a time-dependent method. It has been found that the spurious oscillations occurring near the shock with the second-order finite difference equations are related to the dispersion term in the corresponding modified differential equations. It is observed that based on the semi-discrete NND scheme, the numerical solution of the scalar equation can be computed by the following five different explicit schemes discretized in time and space. The numerical results show that the explicit NND schemes possess good convergence accuracy and a high resolution for capturing shock waves, vortices, and shear layers. It is observed that for simplicity, the mesh is generated using an algebraic grid generator that has the capability of clustering in the body normal direction to resolve the boundary layers, as well as to reach a specified outer boundary. The hypersonic flow around space-shuttle-like geometry is also elaborated.

Quantum dynamics of a two-state system in a dissipative environment

Abstract: An analytical study of the dissipative Landau-Zener model is presented. The model where two energy levels at constant speed are brought to cross is a standard model used to describe a large variety of phenomena. In many cases of interest the presence of coupling of the two-state system to an environment is of importance, the accounting for which shall be done here from first principles. Analytical results for the excitation transition from the ground state of the two-state system at large negative times to the excited state at large positive times (as well as the opposite, decay, transition) are obtained in terms of the speed by which the two levels approach each other, the energy gap between the adiabatic energies, the coupling strength to the environment, and its temperature. For the excitation transition we find the following results: In the slow-passage limit of small sweeping speed it is shown that adiabaticity is limited to low temperatures and the quantitative adiabatic criterion is established. Particularly, at zero temperature there is no influence of the environment on the transition probability as a consequence of a compensation property shown to be peculiar to the linear-sweep model. The transition is at low temperatures due to quantum tunneling and, with an increase in temperature, an intermediate region appears where the transition is dominated by thermally assisted transitions across the energy gap before finally at high temperatures a saturated regime is reached with equal population of the levels. In contrast to the dependence on the temperature the dependence of the transition probability as a function of coupling strength is nonmonotonic with maximum influence at intermediate strength. In the fast-passage limit with rapid sweep speed there is no influence of the environment on the transition probability. For the decay transition the adiabatic limit does not exist for the linear-sweep model for physically relevant spectra of the environment and the decay transition is dominated by spontaneous emission, except in the fast-passage limit and the high-temperature limit where the decay transition probability equals the excitation probability.

Convective variational approach to relativistic thermodynamics of dissipative fluids

Abstract: Using guidelines provided by Noether identities arising from a generalized variation procedure of convective type, a new (nonlinear and exactly self-consistent) category of relativistic thermodynamic models is developed for the systematic representation of viscous conducting fluid media (allowing for several independent charged or neutral chemical constituents). Apart from the provision of a set of dissipation coefficients of the usual (reactivity, resistivity, and viscosity) type, the specification of a particular model is determined just by giving the algebraic dependence a single ‘master function’, Λ say, on the relevant dynamical variables, which are supposed here to consist of an entropy current 4-vector and a set of particle current 4-vectors corresponding to the various chemical constituents, together with a set of symmetric (rank 3) viscosity tensors, which are considered as being dynamically independent of the corresponding current vectors except in the degenerate limit of linear viscosity. The master function is set up as a generalization of an ordinary lagrangian function, to which it reduces in the relevant non - dissipative limit, and, as in the conservative case, it is used for the construction of derived quantities in such a way that appropriate self-consistency conditions are satisfied as identities. In particular the relevant stress-momentum-energy tensor is obtained directly in terms of the independent variables and of their dynamical conjugates (whose role is hidden in the traditional approach as developed by Israel & Stewart), which are set of ordinary 4-momentum (not 4-momentum density) covectors associated with the independent currents, and a set of generalised Cauchy type strain (not strain - rate) tensors associated with the independent viscous stress contributions. The range of application of the category obtained in this way is intended to include that of the standard (Israel-Stewart) formalism to which it is expected to be effectively equivalent in the limit of sufficiently small deviations from thermodynamic equilibrium.

Nonclassical Dynamics of Classical Gases

Abstract: In the present article we examine the dynamics of single-phase, equilibrium, i.e., classical, fluids in the dense gas regime. The behavior of fluids of moderately large molecular weight is seen to differ significantly from that of air and water under normal conditions. New phenomena include the formation and propagation of expansion shocks, sonic shocks, double sonic shocks, and shock-splitting. The more complicated existence conditions for shock waves are described and related to the dissipative structure. We also give a brief description of transonic flows and show that the critical Mach number for conventional blade shapes can be increased by a factor of 30–50% for these fluids.

The Oscillations of Rapidly Rotating Newtonian Stellar Models. II. Dissipative Effects

Abstract: A method is described for determining the dissipative effects of viscosity and gravitational radiation on the modes of rapidly rotating Newtonian stellar models. Integral formulae for the dissipative imaginary parts of the frequencies (i.e., the damping or growth times) of these modes are derived. These expressions are evaluated numerically to determine the angular-velocity dependence of these dissipative effects on the l = m f-modes of uniformly rotating polytropes. The importance of the gravitational-radiation driven secular instability in limiting the rotation rate of neutron stars is estimated using these results. 13 refs.

Spherical accretion onto black holes : a complete analysis of stationary solutions

Abstract: The problem of stationary, spherical accretion onto a Schwarzshild hole is reinvestigated by the construction of a self-consistent model which incorporates all relevant physical processes taking place in an astrohpysical plasma, apart from the presence of magnetic fields and dissipative processes. In particular, transfer of radiation through the accreting gas is treated in full generality using a completely relativistic formalism. A careful analysis of critical points and boundary conditions for radiation hydrodynamics equations is performed

Superfluid Hydrodynamics in Rotating Neutron Stars. II. Dissipative Effects

Abstract: The Newtonian superfluid hydrodynamic equations describing the outer-core regions of neutron stars are generalized to include dissipation. The effects of viscosity, thermal conductivity, and mutual friction (due to the scattering of electrons off the neutron and proton vortices) are included. The low-frequency-long-wavelength limit is taken to obtain a set of equations suitable for studies of p-modes in rapidly rotating neutron stars. An energy functional is constructed which determines the damping times due to the various forms of dissipation.

Dissipative Korteweg–de Vries description of Marangoni–Bénard oscillatory convection

Abstract: For a horizontal liquid layer open to air and heated either from the air side or the liquid side, threshold values for the onset of oscillatory interfacial instability are provided, as well as the nonlinear evolution equation describing the deformable open surface. The latter equation is the dissipation‐modified Korteweg–de Vries evolution equation for solitary excitations in the liquid layer.

A Mixed Finite Element Formulation for Microwave Devices Problems. Application to MIS Structure

Abstract: A mixed finite element formulation for analyzing the propagation characteristics of lossy or dissipative waveguides is presented. Maxwell's equations are expressed in terms of the transverse electr...

Nonlinear magnetohydrodynamics by Galerkin-method computation

Abstract: A fully spectral numerical code is used to explore the properties of voltage-driven dissipative magnetofluids inside a periodic cylinder with circular cross section. The trial functions are orthonormal eigenfunctions of the curl (Chandrasekhar-Kendall functions). Transitions are observed from axisymmetric resistive equilibria without flow to helically deformed laminar states with flow, and between pairs of helical laminar states with different pairs of poloidal and toroidal m and n numbers. States of minimum energy dissipation rate seem to be preferred. At high values of the pinch ratio, fully developed magnetohydrodynamic turbulence is observed.

Dissipativity of numerical schemes

Abstract: The authors show that the way in which finite differences are applied to the nonlinear term in certain partial differential equations (PDES) can mean the difference between dissipation and blow up. For fixed parameter values and arbitrarily fine discretizations they construct solutions which blow up in finite time for two semi-discrete schemes. They also show the existence of spurious steady states whose unstable manifolds, in some cases, contain solutions which explode. This connection between the blow-up phenomenon and spurious steady states is also explored for Galerkin and nonlinear Galerkin semi-discrete approximations. Two fully discrete finite difference schemes derived from a third semi-discrete scheme, reported to be dissipative, are analysed. Both latter schemes are shown to have a stability condition which is independent of the initial data.

Chaotic behaviour of deterministic dissipative systems

Abstract: Introduction 1. Differential equations, maps and asymptotic behaviour 2. Transition from order to chaos 3. Numerical methods for studies of parametric dependences, bifurcations and chaos 4. Chaotic dynamics in experiments 5. Forced and coupled chemical oscillators: a case study of chaos 6. Chaos in distributed systems Appendices Bibliography Index.

Non-canonical Poisson bracket for nonlinear elasticity with extensions to viscoelasticity

Abstract: The authors introduce a generalized bracket which is capable of generating the dynamical equations governing the flow of both elastic and viscoelastic media. This generalized bracket is divided into two parts: a noncanonical Poisson bracket and a new dissipation bracket. The non-canonical Poisson bracket is the Eulerian equivalent of the canonical Lagrangian Poisson bracket corresponding to an ideal (non-dissipative) continuum. It is derived for a nonlinear elastic medium, and then it is used to obtain the Eulerian equations of motion in nonlinear elasticity valid for large deformations. It is shown that the proposed non-canonical bracket naturally leads to a materially objective relation involving the upper-convected time derivative of a strain tensor, as suggested by Oldroyd 40 years ago. The dissipation bracket for linear irreversible thermodynamics is next proposed in a general, phenomenological circumstance, so that the dissipation processes occurring in real systems (viscous and relaxation phenomena) can be incorporated into the Hamiltonian formalism. This bracket diverges from previously proposed dissipation brackets in that it uses the same generating functional (i.e. the Hamiltonian) as the Poisson bracket, rather than an entropy functional or a dissipative potential. It is shown that, in combination with the choice of an appropriate Hamiltonian functional, the generalized bracket proposed here can generate the governing equations for many viscoelastic media, including the Voigt solid and the Maxwell viscoelastic fluid.

Scaling in open dissipative systems.

Abstract: The scaling behavior of open dissipative systems including the Kim-Kosterlitz exponents for interfacial growth, and the current fluctuations in a flowing sandpile are derived using a generalization of the arguments applied by Kolmogorov to the inertial range of turbulence. The approach may be considered a nonequilibrium equivalent to Flory theory.

Change in the Small Group: A Dissipative Structure Perspective

Abstract: In living systems that are experiencing highly turbulent conditions, dissipative self-organization sometimes takes place and results in greater viability. The potential applications of the paradigm of dissipative self-organization to the study of change dynamics in small groups are explored. A certain type of change in groups is likened to change in dissipative structures found in the physical world, and group effectiveness amid complex and turbulent environments is seen to require the key elements of dissipative self-organization: an ongoing tolerance for error and for deviation from an established order, a breaking of existing system relationships so that new ones may emerge, a reflective, self-referencing mode, and a creative process of boundary reparation and movement into new configurations. Lewin's model of change in social systems is extended here through the application of the self-organization paradigm. Instead of utilizing Lewin's formulation of a discrete movement through phases, the emphasis i...

Solitons and pattern formation in liquid crystals in a rotating magnetic field.

Abstract: We have discovered novel nonlinear dissipative dynamic patterns in nematic liquid crystals under the influence of a continuously rotating magnetic field. We present a state diagram of the soliton structures as a function of field and rotation rate and discuss physical models describing their growths and propagation and transitions.

Comparison of bifurcation structures of driven dissipative nonlinear oscillators

Abstract: The bifurcation sets of driven strictly dissipative nonlinear oscillators are compared in terms of phase diagrams and fixed-point diagrams. The comparison reveals distinctive bifurcation patterns that occur for all models. In particular, two subpatterns of bifurcation curves within the period-doubling hierarchy can be identified. The results suggest that there exists a universal bifurcation structure for oscillators of the type investigated.