Showing papers on "Dissipative system published in 1980"

Geometry from a Time Series

Abstract: It is shown how the existence of low-dimensional chaotic dynamical systems describing turbulent fluid flow might be determined experimentally. Techniques are outlined for reconstructing phase-space pictures from the observation of a single coordinate of any dissipative dynamical system, and for determining the dimensionality of the system's attractor. These techniques are applied to a well-known simple three-dimensional chaotic dynamical system.

Dissipative Dynamical Systems: Basic Input-Output and State Properties

Abstract: A complete account is given of the theory of so-called dissipative dynamical systems. The concept of dissipativeness is defined as a general input-output property which includes, as notable special cases, passivity and other properties related to finite-gain. The aim is to treat input-output and state properties side-by-side with emphasis on exploring connections between them. The key connection is that a dissipative system in general possesses a set of energy-like functions of the state. The properties of these functions are studied in some detail. It is demonstrated that this connection represents a direct generalization of the well-known Kalman-Yakubovich lemma to arbitrary dynamical systems. Applications to stability theory and passive system synthesis are briefly discussed for non-linear systems.

Attractor Sets and Quasi-Geostrophic Equilibrium

Abstract: The attractor set of a forced dissipative dynamical system is for practical purposes the set of points in phase-space which continue to be encountered by an arbitrary orbit after an arbitrary long time. For a reasonably realistic atmospheric model the attractor should be a bounded set, and most of its points should represent states of approximate geostrophic equilibrium. A low-order primitive-equation (PE) model consisting of nine ordinary differential equations is derived from the shallow-water equations with bottom topography. A low-order quasi-geostrophic (QG) model with three equations is derived from the PE model by dropping the time derivatives in the divergence equations. For the chosen parameter values, gravity waves which are initially present in the PE model nearly disappear after a few weeks, while the quasi-geostrophic oscillations continue undiminished. The states which are free of gravity waves form a three-dimensional stable invariant manifold within the nine-dimensional phase spac...

On the physics and modeling of small semiconductor devices—II: The very small device

Abstract: In a previous paper, we attempted to lay a conceptual framework for an ultimate physics of small semiconductor devices and concentrated on the medium small device. Here we treat the very small device (VSD), characterized by an effective channel length of 250 A. We demonstrate how such a device could conceivably be fabricated using two side processing of the wafer. In treating the transport, however, it is found that the time and distance scales are such that the Boltzmann transport equation is completely invalidated. Here we develop the appropriate quantum transport equations based upon the density matrix for the entire system, device plus boundaries plus environment. It is found that the boundaries and environment can lead to renormalization of the energy spectrum as well as long range dissipative interactions. Two special cases of the transport equations are treated. If the transport is dominantly stochastic, an exact Langevin equation is found for the various transport parameters. In a second case, a parameterized density matrix is used in analogy to the displaced Maxwellian. In this latter case, a hierarchy of moment equations can be developed to yield, e.g. energy and momentum balance equations.

On Mode Interactions in Reaction Diffusion Equation with Nearly Degenerate Bifurcations

Abstract: Chemical reactions as an open system have been investigated both theoretically and experimentally since nonlinear and non-equilibrium physics became one of current branches of physics. A typical example of these is the Belousov-Zhabotinsky reaction; a cerium ion catalyzes oxidation reaction of malonic acid by acidic bromate. In this reaction it is observed that the ratio of concentrations of Ce'+ and Ce3 '· may temporally oscillate, form a spatially periodic pattern or move as a wave under appropriate conditions. These phenomena are called dissipative structures. The concept of dissipative structures may help us to understand from a unified viewpoint various phenomena in open systems such as the Benard convection, 11 the laser oscillation,21 the temporal rhythm in living systems besides chemical reacting systems.3J Several nonlinear reaction diffusion equations4l have already been studied both theoretically and numerically. It is shown that they have generally the desired solutions to explain those dissipative structures which appear in the B-Z reaction system except the wave mode. The characteristic features of these equations are as follows. First, they exhibit the bifurcation of solution. Secondly their solutions will approach finally to certain states irrespective of initial conditions. Thirdly, bifurcating modes may mutually interact in vanous ·ways. The first and the second have already been studied. Therefore, we study the third in this paper.

Feedback Kinetics in Physicochemical Oscillators

Abstract: Analogously to feedback mechanisms in electrical networks kinetical self-influence of chemical reactions and physicochemical transportation processes may lead under certain conditions to oscillations, bistability, triggerability, reaction propagation and other temporal coupling phenomena. It is shown that all of these phenomena including dissipative structures can be interpreted and classified kinetically on the basis of positive, negative and antagonistic feedback.

Hydrodynamical fluctuations in extended irreversible thermodynamics

Abstract: Starting from the generalized Gibbs equation of extended irreversible thermodynamics, we define a thermodynamic potential that provides a suitable description of the fluctuations of the hydrodynamical dissipative fluxes when it is used in an expression analogous to the classical Einstein formula for the probability of fluctuations. In the limit of vanishing relaxation times, our results coincide with those of Landau-Lifshitz. The effect of the rapid normal modes is taken into account as a stochastic noise in the evolution equations of the dissipative fluxes, and their covariance matrix is found from a fluctuation-dissipation theorem.

Dissipative structures in reaction-diffusion systems: Numerical determination of bifurcations in the sphere

Abstract: The steady state spatial patterns arising spontaneously in open nonlinear reaction–diffusion systems beyond an instability point of the thermodynamic branch are studied numerically for a simple kinetic scheme. The set of nonlinear partial differential equations, describing the system, is converted to a (large) set of ordinary differential equations. It is stressed that the resulting system is stiff, and must be solved accordingly. An efficient algorithm is outlined, based on Stiff predictor–corrector formulas and sparse matrix techniques, which yield a gain of a factor 470 in computing time over nonstiff methods. The developed algorithm is used to determine quantitatively the primary and first few secondary bifurcations in the sphere, thus simulating a biological cell or early blastula. Spontaneous gradient formation and ’Chemical hysteresis’, connected to the occurrence of multiple steady states, is encountered. The succession of stable patterns found for increasing size of the sphere is suggested to act as an ideal mechanism underlying the process of mitosis.

Dynamical systems and models for reversals of the earth's magnetic field

Abstract: We give an elementary introduction to some ideas and methods in the qualitative theory of differentiable dynamical systems, emphasizing the geometrical description of certain simple bifurcations. As an example of the use of such methods we review two models for the reversal phenomenon exhibited by the earth's magnetic field. The second model displays surprisingly rich dynamical behavior that has only recently been studied in detail. In closing we show that recent work on periodically forced weakly dissipative systems occurring as models of magneto-elastic interactions may be relevant to the geomagnetic reversal question.

Absolute dissipative drift-wave instabilities in tokamaks

Abstract: Contrary to previous theoretical predictions, it is shown that the dissipative drift-wave instabilities are absolute in tokamak plasmas. The existence of unstable eigenmodes is shown to be associated with a new eigenmode branch induced by the finite toroidal couplings.

General theory unifying and extending the Landau-Khalatnikov, Ginzburg-Pitaevskii, and Hills-Roberts theories of superfluid He 4

Abstract: A general phenomenological theory of superfluid $^{4}\mathrm{He}$ is presented which takes account of the effects of healing and relaxation. The Landau-Khalatnikov theory and the Hills-Roberts theory are treated as special cases. The starting point for the derivation of the general theory is an extension of Zilsel's variational principle that includes the gradient of the superfluid mass density as an additional independent variable. Among the resulting equations an essential part is played by a partial differential equation that expresses the equilibrium of the superfluid density. The extension of the general theory to dissipative situations is characterized by a 3\ifmmode\times\else\texttimes\fi{}3 matrix of kinetic coefficients. Introduction of the complex order parameter in the variational principle leads to a nondissipative version of the Ginzburg-Pitaevskii (GP) equation. When extended to the dissipative case, it contains three complex relaxation coefficients which are simply related to the kinetic coefficients. The $\ensuremath{\Psi}$ theory of Ginzburg and Pitaevskii and its modification given by Khalatnikov are both formally included in the general time-dependent GP equation as special cases.

Chemical Waves in the Oscillatory Zhabotinskii System. A Transition from Temporal to Spatio-temporal Organization

Abstract: The Zhabotinskii system [1], [2], is an excellent example of chemical synergetics [3]. When four or five chemical compounds are mixed in the appropriate concentration ranges and at the appropriate temperature, the Zhabotinskii system spontaneously organizes itself into temporal or spatio-temporal dissipative structures of macroscopic dimensions [4]. In this chemical reaction, at least twenty intermediates are formed. The chemical mechanism involved is so complex that almost all theoretical work is performed on models rather than on the best rate equations available today [5]. A first type of model involves “macrokinetic” steps rather than elementary ones. This type includes the model of ZHABOTINSKII and his collaborators [1] which attempts to reproduce both waveforms and periods of oscillations. It includes also the many versions of the Oregonator [6] which are designed to reproduce the waveforms of a few intermediates. Other models are of the heuristic-topological type, according to the PACAULT [7] classification. Two of them are the well-known PRIGOGINE-LEFEVER model [8] (or Brusselator) and the analytic BAUTIN system [9], [10] (or DREITLEIN-SMOES model). It is unnecessary to emphasize here the role played by the PRIGOGINE-LEFEVER model as a research tool in the theory of dissipative structures. The BAUTIN system is less well known in spite of several attractive features: this system is solvable in closed form; it exhibits a limit cycle and bistability; there is a saddle-node transition between steady-state and finite amplitude oscillations [2].

Stability and Nonlinear Evolution of Plasma Clouds via Regularized Contour Dynamics

Abstract: A model is introduced of an ionospheric plasma cloud (deformable dielectric) with a piecewise-constant ion density and a diffusive-regularized boundary. The linear stability of a single-contour circular cloud is studied and a new evolution equation for modal amplitudes is obtained which has the property that the wave number of maximum amplitude decreases with time (downward cascade). Analytic expressions show that large clouds evolve more slowly and appear more dissipative.

A class of dissipative evolutions with applications in thermodynamics of fermion systems

Abstract: We construct explicitly a special class of semigroups of completely positive maps on the CAR algebra and give an application on a model of thermal contact.

Dissipative structures, broken symmetry, and the theory of equilibrium phase transitions

Abstract: The claim that nonlinear instabilities leading to ’’dissipative structures’’ far from thermodynamic equilibrium are analogous to equilibrium phase transitions is investigated. As a representative model, the spin wave instability occurring in a driven ferromagnetic sample is studied in arbitrary dimension. Broken symmetry of the type occurring in equilibrium phase transitions is not found in any dimension, nor does there appear to be an upper critical dimension beyond which mean field theory is correct. The Benard instability and ’’Brusselator,’’ which are known to disobey mean field theory in three dimensions, are also studied in higher dimensions; it is found here that although broken symmetry may occur in higher dimensions, again no upper critical dimension exists. Finally, we speculate under what conditions a ’’dissipative structure’’ may exhibit true broken symmetry, and a consequent generalized rigidity, in three dimensions.

What Can we Learn about Dissipative Structures by Model Reactions

Abstract: Some questions will be listed which could be answered by the use of simple reaction models, developed in recent years and showing typical phenomena of nonlinear thermodynamics like chemical instabilities, bifurcations, and nonequilibrium phase transitions. A comparison will be made of different forms of descriptions. These are a deterministic theory and stochastic theories based on the use of Langevin forces or of a master equation. The role of certain general stochastic measures as thermodynamic quantities will be discussed. An outlook on possible extensions to limit cycle models will be given.

Stability and dissipation: non-equilibrium phase transition in semiconductors

Abstract: On driving open systems far from equilibrium, the kinetics satisfy non-linear equations and several steady-state solutions can become available with transitions ('non-equilibrium phase transitions') between them. These transitions depend on the values of the so-called control parameters, and these systems are sometimes called dissipative structures. Self-organising systems, as in living materials, oscillating chemical reactions, etc are encompassed by these phenomena. In this article the basis of this behaviour, including the notion of critical exponents, is illustrated using simple semiconductor models which have recently been proposed. The reaction constants which enter are already familiar from standard semiconductor applications. Impact ionisation emerges as a key process in these effects, for it is autocatalytic. A theory of threshold switching based on these ideas is suggested.

Propagation of guided electron plasma waves on a plasma cylinder in the presence of dissipative processes

Abstract: It is shown analytically that the presence of dissipative processes limits the propagation coefficient of guided electron plasma waves propagating along a plasma cylinder and that the main dipole and multiple resonances are limits of respective mode of guided electron plasma waves.

Coherent Informational Energy and Entropy.

Abstract: Aiming to provide a common theoretical foundation for all known biblio‐metric laws, the author starts from a systemic view of the information transfer process and assimilates it with a physical diffusion process, in particular the conduction of heat in solids. Previous literature induces in the properly ranked space of new authors an interest potential (temperature) confirmed by exchange of reference‐citation pairs, and driving a controlled information flow. The model gives its distribution for given initial and borderline conditions, allowing at the same time the establishment of new definitions for informational energy and entropy, which are coherent with the corresponding physical ones. The theory shows that impulsory energy supply is bound to negative entropy inflow which brakes the normal entropy ‘production’ in the ‘dissipative’ structures of the considered system. In this way the introduction of information into concrete thermodynamic systems analysis can hopefully be expected.

Variational principle for perfect and imperfect fluids in general relativity

Abstract: The Einstein equations and the relativistic Navier-Stokes equations for a simple fluid characterized by an arbitrary flux tensor are obtained by an action principle to which the equation for the entropy production is added as a constraint The procedure is a generalization of the classical Herivel variational principle to relativistic and dissipative systems The inclusion of dissipative processes requires a reformulation of the action integral to refer to a proper-time slice (tau/sub 1/,tau/sub 2/) in the limit tau/sub 2/ --> tau/sub 1/, otherwise their nonconservative nature gives rise to non-Markoffian effects Such a procedure is called a differential variational principle (DVP) The principle of least dissipation of energy can be incorporated into the DVP so that a linear form for the flux tensor can be produced as well as the above-mentioned equations

The Fundamental Basis for Fiber Formation and Fiber Structure

Abstract: The basic fiber-forming process is treated as a dissipative process in which the structures formed depend upon both the stochastic nature of the fluctuation that initiates the process and the minimum dissipation of energy that determines its path. In contrast with equilibrium structures, which are homogeneous and unlimited in size, dissipative structures are inhomogeneous and have characteristic sizes. Such structures are formed when a steady-state irreversible process far from equilibrium becomes unstable to a space-dependent fluctuation. In effect, an inhomogeneous structure becomes a more efficient means of storing the excess energy--given quantitatively by the temperature times the entropy production. What would appear to be a highly improbable molecular arrangement accord ing to equilibrium thermodynamics becomes the arrangement that minimizes the dissipation of energy as well as the thermodynamic forces on the system.

Time-dependent--S-matrix Hartree-Fock theory of complex reactions

Abstract: Some limitations of the conventional time-dependent Hartree-Fock method for describing complex reactions are noted, and one particular ubiquitous defect is discussed in detail: the post-breakup spurious cross channel correlations which arise whenever several asymptotic reaction channels must be simultaneously described by a single determinant. A reformulated time-dependent--S-matrix Hartree-Fock theory is proposed, which obviates this difficulty. Axiomatic requirements minimal to assure that the time-dependent--S-matrix Hartree-Fock theory represents an unambiguous and physically interpretable asymptotic reaction theory are utilized to prescribe conditions upon the definition of acceptable asymptotic channels. That definition, in turn, defines the physical range of the time-dependent--S-matrix Hartree-Fock theory to encompass the collisions of mathematically well-defined ''time-dependent Hartree-Fock droplets.'' The physical properties of these objects then circumscribe the content of the Hartree-Fock single determinantal description. If their periodic vibrations occur for continuous ranges of energy then the resulting ''classical'' time-dependent Hartree-Fock droplets are seen to be intrinsically dissipative, and the single determinantal description of their collisions reduces to a ''trajectory'' theory which can describe the masses and relative motions of the fragments but can provide no information about specific asymptotic excited states beyond their constants of motion, or the average properties of the limit, if it exists, of their equilibrizationmore » process. If, on the other hand, the periodic vibrations of the time-dependent Hartree-Fock droplets are discrete in energy, then the time-dependent--S-matrix Hartree-Fock theory can describe asymptotically the time-average properties of the whole spectrum of such periodic vibrations.« less