Showing papers on "Dissipative system published in 1996"

The Euler-Poincaré equations and double bracket dissipation

Abstract: This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincare) system by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements our earlier work (Bloch, Krishnaprasad, Marsden and Ratiu [1991, 1994]) in which we studied the corresponding problem for systems with symmetry with the dissipation added to the internal variables; here it is added directly to the group or Lie algebra variables. The mechanisms discussed here include a number of interesting examples of physical interest such as the Landau-Lifschitz equations for ferromagnetism, certain models for dissipative rigid body dynamics and geophysical fluids, and certain relative equilibria in plasma physics and stellar dynamics.

Nonlinear Oscillations and Waves in Dynamical Systems

Abstract: Preface. Introduction. Part I: Basic Notions and Definitions. 1. Dynamical Systems. Phase Space. Stochastic and chaotic Systems. The Number of Degrees of Freedom. 2. Hamiltonian Systems Close to Integrable. Appearance of Stochastic Motions in Hamiltonian Systems. 3. Attractors and Repellers. Reconstruction of Attractors from an Experimental Time Series. Quantitative Characteristics of Attractors. 4. Natural and Forced Oscillations and Waves. Self-Oscillations and Auto-Waves. Part II: Basic Dynamical Models of the Theory of Oscillations and Waves. 5. Conservative Systems. 6. Non-Conservative Hamiltonian Systems and Dissipative Systems. Part III: Natural (Free) Oscillations and Waves in Linear and Non-Linear Systems. 7. Natural Oscillations of Non-Linear Oscillators. 8. Natural Oscillations in Systems of Coupled Oscillators. 9. Natural Waves in Bounded and Unbounded Continuous Media. Solitons. Part IV: Forced Oscillations and Waves in Passive Systems. 10. Oscillations of a Non-Linear Oscillator Excited by an External Force. 11. Oscillations of Coupled Non-linear Oscillators Excited by an External Periodic Force. 12. Parametric Oscillations. 13. Waves in Semibounded Media Excited by Perturbations Applied to their Boundaries. Part V: Oscillations and Waves in Active Systems. Self-Oscillations and Auto-Waves. 14. Forced Oscillations and Waves in Active Non-Self-Oscillatory Systems. Turbulence. Burst Instability. Excitation of Waves with Negative Energy. 15. Mechanisms of Excitation and Amplitude Limitation of Self-Oscillations and Auto-Waves. Classification of Self-Oscillatory Systems. 16. Examples of Self-Oscillatory Systems with Lumped Parameters. I. 17. Examples of Self-Oscillatory Systems with Lumped Parameters. II. 18. Examples of self-oscillatory Systems with High Frequency Power Sources. 19. Examples of Self-Oscillatory Systems with Time Delay. 20. Examples of Continuous Self-Oscillatory Systems with Lumped Active Elements. 21. Examples of Self-Oscillatory Systems with Distributed Active Elements. 22. Periodic Actions on Self-Oscillatory Systems. Synchronization and Chaotization of Self-Oscillations. 23. Interaction between Self-Oscillatory Systems. 24. Examples of Auto- Waves and Dissipative Structures. 25. Convective Structures and Self-Oscillations in Fluid. The Onset of Turbulence. 26. Hydrodynamic and Acoustic Waves in Subsonic Jet and Separated Flows. Appendix A: Approximate Methods for Solving Linear Differential Equations with Slowly Varying Parameters. Appendix B: The Whitham Method and the Stability of Periodic Running Waves for the Klein--Gordon Equation. Bibliography. Index.

H∞ nonlinear filtering

Abstract: This paper investigates the problem of H∞ estimation of nonlinear processes. An estimator, which may be nonlinear, is looked for so that a given bound on the ratio between the energy of the estimation error and the energy of the oxogeneous inputs to the estimated process is achieved. Conditions for the existence of such an estimator and formulas for its derivation are obtained using both the game theory approach and the theory of dissipative systems. The results of the paper extend the recent results on H∞ nonlinear control. They are demonstrated by a simple example of a linear system with a nonlinear measurement rule and compared with corresponding results that are obtained by the extended Kalman filter.

Morphological characterization of patterns in reaction-diffusion systems

Abstract: ~Received 24 October 1995! Morphological measures for spatial patterns occuring as dissipative structures in systems driven far from equilibrium are introduced. They characterize the geometry and topology of the patterns and are capable to distinguish irregular structures with respect to the morphology. In particular, we analyze turbulent and regular patterns ~Turing patterns! in chemical reaction-diffusion systems observed in a two-dimensional open gel reactor with a chlorite-iodide-malonic acid reaction. Introducing the concept of level contours, the measures turn out to be polynomials of low order ~cubic and fourth degree! in the grey-scale level of the images. Thus the dependence on the experimental conditions is reflected only in a finite number of coefficients, which can be used as order parameters for the morphology of patterns. We observe a symmetry breaking of the polynomials when the type of the pattern changes from hexagons to turbulence or stripes. Therefore it is possible to describe the pattern transitions quantitatively and it may be possible to classify them in a similar way like thermodynamic phase transitions.

Simulation models of a dissipative transmission line above a lossy ground for a wide-frequency range. I. Single conductor configuration

Abstract: For pt. I see ibid., vol.38, no.2, p.127, 1996. The simulation model of a multiconductor dissipative line above a lossy ground, based on the exact formulation of the Maxwell equations, is proposed for a wide frequency range. The procedure is an extension of the analysis of single conductor configurations. The exact expression of the matrix modal equation of the line is first proposed, assuming that in the system there are as many dominant discrete modes of propagation as there are conductors. New expressions of the distributed series-impedance and shunt-admittance matrices are proposed, with reference to the definition of the wire-to-ground voltage. Moreover, an easy-to-implement simulation model is proposed for use in computer codes, based on the logarithmic approximation of the Sommerfeld integrals and Bessel functions. Applications are carried out in order to compare the results of the proposed procedure and of the Carson (1926) theory, with reference to a three-conductor line above a lossy ground.

Quantum tunneling in a dissipative system.

Abstract: Applying a technique developed recently [L. H. Yu and C. P. Sun, Phys. Rev. A {bold 49}, 592 (1994); L. H. Yu, Phys. Lett. {bold 202}, 167 (1995)] for a harmonic oscillator coupled to a bath of harmonic oscillators, we present an exact solution for the tunneling problem in an Ohmic dissipative system with inverted harmonic potential. The result shows that while the dissipation tends to suppress the tunneling, the Brownian motion tends to enhance the tunneling. Whether the tunneling rate increases or not would then depend on the initial conditions. We give a specific formula to calculate the tunneling probability determined by various parameters and the initial conditions. {copyright} {ital 1996 The American Physical Society.}

Laser control processes in solids

Abstract: The large amount of information carried by the energy, spectral, and space–time characteristics of a laser beam makes it feasible to use laser radiation to control the processes that occur in solids. The influence of laser radiation with a wide range of deposited energies on a variety of processes is considered: these processes include crystal growth, formation of dissipative (spatial and temporal) defect structures on the surface and in the bulk of a solid, instabilities in melts, materials fracture considered from the point of view of both selectivity and self-organisation. An analysis is made of the relationship between the nature and parameters of such structures, on the one hand, and the characteristics of laser radiation, on the other.

Quasi-Modes as Dissipative Magnetohydrodynamic Eigenmodes: Results for One-Dimensional Equilibrium States

Abstract: Quasi-modes, which are important for understanding the MHD wave behavior of solar and astro-physical magnetic plasmas, are computed as eigenmodes of the linear dissipative MHD equations. This eigenmode computation is carried out with a simple numerical scheme, which is based on analytical solutions to the dissipative MHD equations in the quasi-singular resonance layer. Nonuniformity in magnetic field and plasma density gives rise to a continuous spectrum of resonant frequencies. Global discrete eigenmodes with characteristic frequencies lying within the range of the continuous spectrum may couple to localized resonant Alfven waves. In ideal MHD, these modes are not eigenmodes of the Hermitian ideal MHD operator, but are found as a temporal dominant, global, exponentially decaying response to an initial perturbation. In dissipative MHD, they are really eigenmodes with damping becoming independent of the dissipation mechanism in the limit of vanishing dissipation. An analytical solution of these global modes is found in the dissipative layer around the resonant Alfvenic position. Using the analytical solution to cross the quasi-singular resonance layer, the required numerical effort of the eigenvalue scheme is limited to the integration of the ideal MHD equations in regions away from any singularity. The presented scheme allows for a straightforward parametric study. The method is checked with known ideal quasi-mode frequencies found for a one-dimensional box model for the Earthis magnetosphere (Zhu & Kivelson). The agreement is excellent. The dependence of the oscillation frequency on the wavenumbers for a one-dimensional slab model for coronal loops found by Ofman, Davila, & Steinolfson is also easily recovered.

Dissipative solitary states in driven surface waves.

Abstract: We present an experimental study of highly localized, solitonlike structures that propagate on the two-dimensional surface of highly dissipative fluids. Like the well-known Faraday instability, these highly dissipative structures are driven by means of the spatially uniform, vertical acceleration of a thin fluid layer. These structures, harmonically coupled to the external driving frequency, are observed above a critical ratio of the viscous boundary layer height to the depth of the fluid layer for a wide range of fluid viscosities and system parameters. [S0031-9007(96)00186-X]

Regularity of the attractor for a weakly damped nonlinear schrödinger equation

Abstract: In this article we prove that the global attractor for a weakly damped nonlinear Schrodinger equation is smooth i.e. it is made of smooth functions when the forcing term is smooth. The proof of this result which is well-known for other dissipative equations does not apply to dispersive equations for which the dissipation is on the low order term. A new simpler proof of existence of this gloabl attractor is given as well.

Stable and unstable sets for evolution equations of parabolic and hyperbolic type

Abstract: Some roles in the global dynamics of so called stable and unstable sets will be given for semilinear heat equations and semilinear wave equations with dissipative terms

On the Korteweg-de Vries-Kuramoto-Sivashinsky equation

Abstract: Considered herein is the Korteweg-de Vries equation with a Kuramoto-Sivashinsky dissipative term appended. This evolution equation, which arises as a model for a number of interesting physical phenomena, has been extensively investigated in a recent paper of Ercolani, McLaughlin and Roitner. The numerical simulations of the initial-value problem reported in the just-mentioned study showed solutions to possess a more complex range of behavior than the unadorned Korteweg-de Vries equation. The present work contributes some basic analytical facts relevant to the initial-value problem and to some of the conclusions drawn by Ercolanet al. In addition to showing the initial-value problem is well posed, we determine the limiting behavior of solutions as the dissipative or the dispersive parameter tends to zero.

On the Discretization of van Roosbroeck’s Equations with Magnetic Field

Abstract: We investigate qualitative properties of the drift-diffusion model of carrier transport in semiconductors when a magnetic field is present. At first the spatially continuous problem is studied. Essentially, global stability of the thermal equilibrium is shown using the free energy as a Lyapunov function. This result implies exponential decay of any perturbation of the thermal equilibrium. Next, we introduce a time discretization that preserves the dissipative properties of the continuous system and assumes no more than the naturally available smoothness of the solution. Finally, we present a space discretization scheme based on weak and consistent definitions of discrete gradients and currents. Starting with a fundamental result on global stability (dissipativity) of the classical Scharfetter-Gummel scheme (without magnetic field), we adapt this scheme with respect to magnetic fields and study the M-property of the associated matrix. For two dimensional applications we formulate sufficient conditions in terms of the grid geometry and the modulus of the magnetic field such that our scheme is dissipative and yields positive solutions. These conditions cover fields up to |b|µv ≈ 0.5 for very fine grids. This means approximately 200 Tesla for Silicon. Sufficient for some typical semiconductor sensor applications. The grid requirements might become prohibitive for large magnetic fields and complex three dimensional structures. Our techniques of defining discrete currents can be applied to similar situations, especially if projections of currents are involved in model parameters.

Enforcing Coherent Evolution in Dissipative Quantum Dynamics

Abstract: The major obstacle to the preparation and manipulation of many-particle entangled states is decoherence due to the coupling of the system to the environment. A scheme to correct for the effects of decoherence and enforce coherent evolution in the system dynamics is described and illustrated for the particular case of the ion-trap quantum computer.

Equilibrium dynamics of the dissipative two-state system.

Abstract: Wilson's momentum shell renormalization group method is used to solve for the dynamics of the dissipative two--state system. We utilize the mapping of the spin--boson model onto the anisotropic Kondo model (AKM) and solve for the dynamics of the latter. We find that the AKM captures the physics of the dissipative two--state system for dissipation strength $0\leq\alpha\leq 4$ corresponding to $\infty\geq J_{\parallel}\geq -\infty$ in the AKM. The dynamics of the AKM shows a smooth crossover between two strong coupling regimes corresponding to weak and strong dissipation in the spin--boson model.

Poincaré resonances and the extension of classical dynamics

Abstract: Classical dynamics can be formulated in terms of trajectories or in terms of statistical ensembles whose time evolution is described by the Liouville equation It is shown that for the class of large non-integrable Poincare systems (LPS) the two descriptions are not equivalent Practically all dynamical systems studied in statistical mechanics belong to the class of LPS The basic step is the extension of the Liouville operator LH outside the Hilbert space to functions singular in their Fourier transforms This generalized function space plays an important role in statistical mechanics as functions of the Hamiltonian, and therefore equilibrium distribution functions belong to this class Physically, these functions correspond to situations characterized by ‘persistent interactions’ as realized in macroscopic physics Persistent interactions are introduced in contrast to ‘transient interactions’ studied in quantum mechanics by the S-matrix approach (asymptotically free in and out states) The eigenvalue problem for the Liouville operator LH is solved in this generalized function space for LPS We obtain a complex, irreducible spectral representation Complex means that the eigenvalues are complex numbers, whose imaginary part refers to the various irreversible processes such as relaxation times, diffusion etc Irreducible means that these representations cannot be implemented by trajectory theory As a result, the dynamical group of evolution splits into two semi-groups Moreover, the laws of classical dynamics take a new form as they have to be formulated on the statistical level They express ‘possibilities’ and no more ‘certitudes’ The reason for the new features is the appearance of new, non-Newtonian effects due to Poincare resonances The resonances couple dynamical events and lead to ‘collision operators’ (such as the Fokker-Planck operator) well-known from various phenomenological approaches to non-equilibrium physics These ‘collision operators’ represent diffusive processes and mark the breakdown of the deterministic description which was always associated with classical mechanics ‘Subdynamics’ as discussed in previous publications, is derived from the spectral representation The eigenfunctions of the Liouville operator have remarkable properties as they lead to long-range correlations due to resonances even if the interactions as included in the Hamiltonian are short-range (only equilibrium correlations remain short-range) This is in agreement with the results of non-equilibrium thermodynamics as the appearance of dissipative structures is connected to long-range correlations In agreement with previous results, it is shown that there exists an intertwining relation between LH and the collision operator Θ as defined in the text Both have the same eigenvalues and are connected by a non-unitary similitude ΛLHΛ−1 = Θ The various forms of Λ and their symmetry properties are discussed A consequence of the intertwining relation are ‘non-linear Lippmann-Schwinger’ equations which reduce to the classical linear Lippmann-Schwinger equations when the dissipative effects due to the Poincare resonances can be neglected Using the transformation operator Λ, we can define new distribution functions and new observables whose evolution equations take a specially simple form (they are ‘bloc diagonalized’) Dynamics is transformed in an infinite set of kinetic equations Starting with these equations, we can derive H -functions which present a monotonous time behavior and reach their minimum at equilibrium This requires no extra-dynamical assumptions (such as coarse graining, environment effects …) Moreover, our formulation is valid for strong coupling (beyond the so-called Van Hove's λ2t limit) We then study the conditions under which our new non-Newtonian effects are observable For a finite number N of particles and transient interactions (such as realized in the usual scattering experiments) we recover traditional trajectory theory To observe our new effects we need persistent interactions associated to singular distribution functions We have studied in detail two examples, both analytically and by computer simulations These examples are persistent scattering in which test particles are continuously interacting with a scattering center, and the Lorentz model in which a ‘light’ particle is scattered by a large number of ‘heavy’ particles The agreement between our theoretical predictions and the numerical simulations is excellent The new results are also essential in the thermodynamic limit as introduced in statistical mechanics We recover also, the results of non-equilibrium statistical mechanics obtained by various phenomenological approximations Of special interest is the domain of validity of the trajectory description as a trajectory is traditionally considered as a primitive, irreducible concept In the Liouville description the natural variables are wave vectors k which are constants in free motion and modified by interactions and resonances A trajectory can be considered as a coherent superposition of plane waves corresponding to wave vectors k Resonances correspond to non-local processes in space-time They threaten therefore the persistence of trajectories In fact, we show that whenever the thermodynamic limit exists, trajectories are destroyed and transformed into singular distribution functions We have a ‘collapse’ of trajectories, to borrow the terminology from quantum mechanics The trajectory becomes a stochastic object as in Brownian motion theory In conclusion, we obtain a unified formulation of dynamics and of thermodynamics This involves the introduction of LPS which leads to dissipation together with the consideration of delocalized situations From this point of view, there is a strong analogy with phase transitions which are also defined in the thermodynamic limit Irreversibility is, in this sense, an ‘emergent’ property which could not be included in classical dynamics as long as its study was limited to local, transient situations

Qualitative viscous cosmology.

Abstract: The Full (non--truncated) Israel--Stewart theory of bulk viscosity is applied to dissipative FRW spacetimes. Dimensionless variables and dimensionless equations of state are used to write the Einstein--thermodynamic equations as a plane autonomous system and the qualitative behaviour of this system is determined. Entropy production in these models is also discussed.

Solitary waves in active-dissipative dispersive media

Abstract: Solitary waves in active-dissipative dispersive media are considered. The exact solutions in the form of solitary waves and kink-shaped waves are presented. The difference equation for numerical simulation of nonlinear waves is given. Numerical results of the interaction of solitary waves are discussed. It is shown that there is a solitary wave in active- dissipative dispersive media that has the soliton property.

Micromechanics and homogenization of inelastic composite materials with growing cracks

Abstract: A homogenization scheme is employed to derive the effective constitutive equations of an elastoplastic composite system with growing damage. The homogenization procedure followed herein is based on the thermodynamics of dissipative media. It is shown that when damage consists of sharps microcracks the macroscopic constitutive behavior is that of a so-called generalized standard material. The latter is a general dissipative medium whose constitutive equations are completely characterized by a single scalar convex potential function of the chosen state variables and whose evolution is completely characterized by a single convex dissipation potential function of the thermodynamic forces conjugate to the chosen internal state variables. The analysis presented is valid under the assumption that the evolution of the representative volume element at hand is unique and stable. The results of the theoretical analysis are then employed for formulating an approximate method for practically deriving the macroscopic constitutive equations. Computer software development for the application of said method is currently ongoing. A simple example of the numerical results obtained so far is presented.

The short‐time intramolecular dynamics of solutes in liquids. I. An instantaneous‐normal‐mode theory for friction

Abstract: It is sometimes useful to be able to think of the energy relaxation of a solute dissolved in a liquid as being caused by some sort of solvent‐inspired friction. This intuitive association can, in fact, be made literal and quantitative in classical mechanics by casting the dynamics into a solute‐centered equation of motion, a generalized Langevin equation, in which the dissipative character of the solvent is embodied in a (generally time delayed) friction force. An exact prescription is available for finding this friction, but the process is formal and the connection with microscopic degrees of freedom is rather indirect. An alternate approach due to Zwanzig, which portrays the solvent as a harmonic bath, makes explicit use of a set of solvent coordinates, but these coordinates have no immediate relationship with any of the real solvent degrees of freedom. We show here that by taking a short‐time perspective on solute relaxation we can derive a generalized Langevin equation, and hence a friction kernel, which is both exact (at least at short times) and has a completely transparent connection with solvent motion at the molecular level. We find, in particular, that under these conditions the instantaneous normal modes of the solution fill the role of the Zwanzig harmonic oscillators precisely, meaning that one can analyze friction in molecular terms by appealing to the explicitly microscopic definitions of the instantaneous modes. One of the implications of this perspective is that fluctuations of the solvent are automatically divided into configuration‐ to‐configuration fluctuations and dynamics resulting from a given liquid configuration. It is the latter, instantaneous, friction that we shall want to decompose into molecular ingredients in subsequent papers. However, even here we note that it is the character of this instantaneous friction that leads to the fluctuating force on a solute having slightly, but measurably, non‐Gaussian statistics. Our basic approach to liquid‐state friction and a number of results are illustrated for the special case of the vibrational relaxation of a diatomic molecule in an atomic liquid.

Dissipative Chaos in Semiconductor Superlattices

Abstract: We consider the motion of ballistic electrons in a miniband of a semiconductor superlattice (SSL) under the influence of an external, time-periodic electric field. We use a semiclassical, balance-equation approach, which incorporates elastic and inelastic scattering (as dissipation) and the self-consistent field generated by the electron motion. The coupling of electrons in the miniband to the self-consistent field produces a cooperative nonlinear oscillatory mode which, when interacting with the oscillatory external field and the intrinsic Bloch-type oscillatory mode, can lead to complicated dynamics, including dissipative chaos. For a range of values of the dissipation parameters we determine the regions in the amplitude-frequency plane of the external field in which chaos can occur. Our results suggest that for terahertz external fields of the amplitudes achieved by present-day free-electron lasers, chaos may be observable in SSL{close_quote}s. We clarify the nature of this interesting nonlinear dynamics in the superlattice{endash}external-field system by exploring analogies to the Dicke model of an ensemble of two-level atoms coupled with a resonant cavity field, and to Josephson junctions. {copyright} {ital 1996 The American Physical Society.}

Finite-element simulation of a new two-dissipative mechanisms model for bulk medium-density polyethylene

Abstract: A two-dissipative mechanisms model, associating a Maxwell and an elastoplastic model in parallel, is discussed in order to account for the non-linear viscoelasticity of bulk medium-density polyethylene. On the one hand, the experimental determination of the constitutive equations coefficients is described from a tensile specimen machined from gas pipes. On the other hand, finite-element simulation of the stress relaxation experiment, proposed by Sweeney and Ward, is achieved, which yields a complete analysis of the dissipative mechanisms interaction during the test. The finite-element code built upon this modelling is finally used in a tentative simulation of a cyclic pressure test on a pipe specimen.

Exact Stationary Solutions of Stochastically Excited and Dissipated Integrable Hamiltonian Systems

Abstract: It is shown that the structure and property of the exact stationary solution of a stochastically excited and dissipated n-degree-of-freedom Hamiltonian system depend upon the integrability and resonant property of the Hamiltonian system modified by the Wong-Zakai correct terms. For a stochastically excited and dissipated nonintegrable Hamiltonian system, the exact stationary solution is a functional of the Hamiltonian and has the property of equipartition of energy. For a stochastically excited and dissipated integrable Hamiltonian system, the exact stationary solution is a functional of n independent integrals of motion or n action variables of the modified Hamiltonian system in nonresonant case, or a functional of both n action variables and α combinations of phase angles in resonant case with α (1 ≤ α ≤ n - 1) resonant relations, and has the property that the partition of the energy among n degrees-of-freedom can be adjusted by the magnitudes and distributions of dampings and stochastic excitations. All the exact stationary solutions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonintegrable Hamiltonian systems, which are further generalized to account for the modification of the Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. The above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissipated Hamiltonian systems as special cases. Examples are given to illustrate the applications of the procedures.