Showing papers on "Dissipative system published in 1979"

Three Dimensional Beach-Stage Model

Abstract: A three dimensional beach model applicable to open sandy coastal environments is presented. The model consists of ten beach stages incorporating erosional and accretionary sequences of beach-surfzone morphody-namic conditions. Each stage is associated with a particular level of breaker wave power. Decreasing breaker wave power produces spatially controlled onshore bar migration, eventual bar welding, beach accretion and reflective surfzone conditions ($$beach stages 6 \rightarrow 5 \rightarrow 4 \rightarrow 3 \rightarrow 2 \rightarrow 1$$). Increasing wave power generates beach erosion, dynamically controlled bar-channel formation and dissipative surfzone conditions (stages $$1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 5 \rightarrow 6$$). Temporal variations in wave power cause predictable movement through and within the model. The effect of beach gradient (tan $$tan \beta$$) is considered with regard to the influence on the degree of dissipativeness, edge wave length and cut-off modes, and st...

Preturbulence: A Regime Observed in a Fluid Flow Model of Lorenz*

Abstract: This paper studies a forced, dissipative system of three ordinary differential equations. The behavior of this system, first studied by Lorenz, has been interpreted as providing a mathematical mechanism for understanding turbulence. It is demonstrated that prior to the onset of chaotic behavior there exists a preturbulent state where turbulent orbits exist but represent a set of measure zero of initial conditions. The methodology of the paper is to postulate the short term behavior of the system, as observed numerically, to establish rigorously the behavior of particular orbits for all future time. Chaotic behavior first occurs when a parameter exceeds some critical value which is the first value for which the system possesses a homoclinic orbit. The arguments are similar to Smale's “horseshoe”.

Climate and thermodynamic systems of maximum dissipation

Abstract: THE Earth–atmosphere is a classic example of a closed, dissipative and nonlinear thermodynamic system which is subject to both regular and irregular impulses causing significant departure from steady state. It is closed because it exchanges energy (solar and thermal radiant energy) but not mass with its environment. It is dissipative because the net input of radiant energy occurs mainly in regions of high temperature towards the Equator and the net output occurs mainly in regions of low temperature towards the poles. It is nonlinear basically because of the multiplicity of internal feedbacks and because of the importance of advective processes. It has steady-state character in the sense that the annual mean radiant energy input is very close to the annual mean output, and parameters such as the annual mean temperature do not vary significantly from one period to another. The regular seasonal variation in solar position ensures significant departure from the steady state so defined, and there are also significant irregular departures arising (for instance) from variations of solar input and IR output caused by variations in the amount and distribution of cloud. Recently I have shown1 that the overall Earth–atmosphere climate system seems to have adopted a format whereby the total thermodynamic dissipation associated with the horizontal energy flows in the atmosphere and ocean is a maximum. ‘Format’ in this context refers to the annual average geographic distribution of cloud, surface temperature and the horizontal energy flows. The practical significance of this is that, if one could accept it as a general principle governing climate behaviour, one could use it directly as a means of a priori prediction of climate and climate change without needing detailed analysis of the internal workings of the system. I could not explain previously why the Earth–atmosphere system should be so constrained. This note points out that the Earth–atmosphere has characteristics such that it might be expected to obey such a constraint. Furthermore, these characteristics are sufficiently general that the same principle of selection of steady-state mode of maximum dissipation may apply to a broad class of non-linear systems.

Dynamics of domain walls in ferrodistortive materials. I. Theory

Abstract: A theoretical study of domain walls in uniaxial displacive ferrodistortive systems is presented. We start from a generalized Langevin equation of motion for the movements of the ions, which includes dissipative terms and external fields, in addition to anharmonic and strain-force terms. We obtain large- and small-amplitude solutions corresponding to domain walls and the usual soft-mode phonons, respectively. We show that apart from translation the domain walls are absolutely stable solutions of our equation and that in external fields they reach a unique terminal velocity. The linear dependence of the velocity on the field allows us to define a temperature-dependent mobility which is related to the diffusion coefficient for the wall. Furthermore, we calculate analytically the dynamic structure factor due to domain walls and soft-mode phonons. We find that the Brownian motion of the domain walls leads to a very narrow Rayleigh peak. As we show in the second paper of this series, our model is useful in correlating and interpreting experiments in this field.

Possible inverse cascade behavior for drift-wave turbulence

Abstract: The turbulent spectral properties of the dynamical equation of Hasegawa and Mima (1978) governing the evolution of the electrostatic potential in drift-wave turbulence is investigated for two formulations of the problem: (1) as a nondissipative initial value problem, with the potential represented by a truncated Fourier series with large number of terms, and (2) as a dissipative problem with a small viscous dissipation at very short spatial scales, and a long wavelength forcing term at longer wavelengths It is found that Hasegawa and Mima's prediction for the nondissipative, truncated initial value modal problem is accurate, but substantial differences exist for the forced dissipative case between computer results and analytical predictions based on a wave kinetic equation of Kadomtsev Much better agreement is found with a simple dual-cascade model based on Kraichnan's generalization of Kolmogorov's cascade arguments

Relaxation times in dissipative heavy-ion collisions

Abstract: The classical model introduced earlier for analyzing experimental data on dissipative heavy-ion collisions, is generalized to include effects from the gradual dissipation of radial kinetic energy and from the development of fragment deformations during the collision. Relaxation times for the dissipation of radial kinetic energy (τ R ) and relative angular momentum (τ l ) as well as for the development of deformations (τα) are fitted to the reaction86Kr (8.18 MeV/u) +166Er and applied to three other reactions. A consistent set of relaxation times isτ R = 0.3 · 10−21 s,τ l =1.5 · 10−21 s andτ α = 5 · 10−21 s. Empirical mass transport coefficients are deduced from comparisons with experimental element distributions. Effects from fluctuations in the deflection function are discussed. Evidence is found for the existence of a relaxation time of the order 10−21 s in the mass-drift coefficient.

Linear response theory revisited. II. The master equation approach

Abstract: We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iL0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.

Self-energy of a fast-moving charge near a surface

Abstract: The self-energy of an external moving charged particle near a surface is studied within the fast-charge approximation. For a reflecting trajectory, a formula for the complex self-energy, valid for an arbitrary dielectric function of the medium and angle of incidence, is first derived by means of a semiclassical analysis in which the imaginary part is obtained from a classical evaluation of the work done by the moving charge. This work is discussed in terms of conservative and dissipative contributions. The formula for the complex self-energy is then recovered from a quantum mechanical formulation in which the passage to the semiclassical limit is readily made. The quantum approach is used later to obtain a semiclassical formula for the complex self-energy when the charged particle penetrates the medium.

Diagrammatic theory of transition of pendulum like systems

Abstract: Orbit-orbit and spin-orbit gravitational resonances are analyzed using the model of a rigid pendulum subject to both a time-dependent periodic torque and a constant applied torque. First, a descriptive model of passage through resonance is developed from an examination of the polynomial equation that determines the extremes of the momentum variable. From this study, a probability estimate for capture into libration is derived. Second, a lowest order solution is constructed and compared with the solution obtained from numerical integration. The steps necessary to systematically improve this solution are also discussed. Finally, the effect of a dissipative term in the pendulum equation is analyzed.

Role of Trapped Particles and Waves in Plasma Solitons-Theory and Application

Abstract: In this tutorial and review paper, we investigate the influence of trapped particles on the propagation of ion acoustic solitons and the role of trapped waves on the propagation of Langmuir solitons. The classical potential method allows us to construct finite amplitude soliton solutions, and trapping phenomena are found to retard the motion of solitons. Thus, ion acoustic solitons have minimum speed when they are based on an isothermal electron equation of state since this corresponds to maximum electron trapping. In the small amplitude regime, deviations from the Boltzmann law lead to a new nonlinear term. Subsequently, the dynamics of the soliton is governed by a modified KdV-equation [e.g., (23)]. For Langmuir solitons an existence diagram is found and exhibits the comparison between experimentally observed localized structures and theory. The connection with the various small amplitude soliton solutions is also pointed out. Moreover, the inclusion of a pump and of dissipative terms in the coupled nonlinear Schrodinger-ion equation gives rise to a transient phenomenon called soliton flash, whose implication to the laboratory experiments is discussed. Finally, as an application in numerical analysis the propagation of solitons is followed by solving the KdV-equation in Fourier-space. This turns out to be an excellent tool to test stability and accuracy of numerical schemes. Our results reveal that the so-called aliasing interactions lead to erroneous solutions. The conservation of invariants does not guarantee the accuracy of a numerical algorithm, although it is needed for its stability.

On the phase space quantization of the linearly damped harmonic oscillator

Abstract: The quantal form of the classical Liouville equation will be investigated on the basis of a recently developed generalized Hamiltonian theory. The essential novelty inthat theory comprises the use of complex classical coordinates and momenta. We first show how for the nondissipative harmonic oscillator driven by an external classical force, the theory leads to the correct well-known quantum analogue of the classical Liouville equation. We then generalize this procedure to include frictional phenomena for which the novel theory has been observed to be particularly suited. The resulting quantal master equation for the simple linearly damped harmonic oscillator demonstrates that one cannot expect to find a proper quantum mechanical description of dissipative systems in terms of a single Schrodinger wave function. The master equation will then be transformed into its Wigner representation, providing a convinient form for discussion. The diffusion coefficients occuring in the resultant Fokker-Planck equation will be seen to be intimately connected with the survival of Heisenberg's uncertainty principle for dissipative systems. Apart from conceptual elegance, the present approach has superiority to a previous one in at least three aspects: i) there is no need to introduce ad-hoc quantal noise operators, ii) the above mentioned diffusion coefficients are specified and emerge in a natural way, and iii) the present approach has the important advantage of easy extension to more general systems.

Time-dependent harmonic oscillators

Abstract: A formalism for a time-dependent harmonic oscillator is presented. The quantum-mechanical solution is developed and the Green's function is derived. Particular examples of runaway and dissipative behavior are considered.

Nonlinear transport processes: Hydrodynamics

Abstract: Exact equations of motion for the distribution function and for dynamical variables in systems which are nonlinearly displaced from equilibrium are derived and examined. A projection operator is introduced to resolve these equations into a local equilibrium contribution and correction terms. These are of two types: dissipative and fluctuating and are related by a generalized fluctuation-dissipation theorem. The dissipative terms are essential for a valid description of transport processes. Simplifications are introduced for systems where the local thermodynamic potentials are slowly varying on the scale of the molecular correlation length. This leads to local transport equations. For the hydrodynamical variables these are precisely the Navier-Stokes equations. The entropy production for a system described by such nonlinear equations is positive semidefinite and vanishes if and only if the system is in equilibrium.

Dissipative structures and nonlinear kinetics of the Marangoni-instability

Abstract: The Marangoni-instability with the driving force of heat- or mass transfer across fluid interphases causes a self-amplification and self-organisation of movements at a fluid interphase which develop a spectrum of multiform dissipative structures There exist manifold substructured spatial periodic systems of a hydrodynamic kind with a time depending behaviour which can degenerate to a typical relaxation oscillation The latter differs completely from the spatial and temporal period structure of a two-parameter-oscillation with a wave-like behaviour,

Nature of 1f frequency fluctuations in quartz crystal resonators

Abstract: The frequency fluctations measured in quartz crystal resonators of quality factor Q are proportional to Q −4.3 . The quantum approach to 1 f noise predicts fundamental fluctuations of the cross sections of elementary dissipative processes. Starting therefore from fluctuations of the total dissipative coefficient, a Q −4 -law is derived. Consequently, the noise is caused by dissipation fluctuations, rather than fluctuations in the density, or the real part of the Young modulus of the crystal.

A new equation for quantum dissipative systems

Abstract: In this paper, we propose a new approach to the description of quantum dissipative systems. It is based on a finite-difference retarded equation governing the time evolution of the state vector. We exhibit the fundamental features of the equation, as well as the properties of its solutions, with particular emphasis to the case of the free particle. Moreover, a detailed analysis shows that our equation provides a natural way to understand the origin of important special functions of mathematical physics. Finally, we discuss the physical interpretation of our results and we examine with care the decay of the state vector belonging to excited energy levels.

Nonassociative, real algebras and quadratic differential equations

Abstract: THE PURPOSE of this paper is to present a theorem of a general nature for quadratic differential systems with bounded solutions. Requiring that all solutions of a quadratic differential equation are bounded is a great restriction algebraically, but one that is necessary in many physically motivated systems. The theorem we prove suggests a conjecture about differential systems whose right hand side contains both linear and quadratic terms. Our results were motivated by our work on a fascinating mathematical model of a turbulent system, the forced dissipative system of Lorenz [ 11,

Bifurcation and Catastrophe in a Simple, Forced, Dissipative Quasi-Geostrophic Flow

Abstract: The steady solutions and their stability properties are investigated for a low-order spectral model of a forced, dissipative, nonlinear, quasi-geostrophic flow. A zonal flow is modified by two smaller scale disturbances in the model. If only the zonal component (or only the smallest scale component) is forced, then the stationary solution is unique, always locally stable, and globally stable for weak forcing. There is also a unique locally stable stationary solution for weak forcing of only the middle component. But as this forcing exceeds a critical value, a supercritical bifurcation to new solutions appears. The entire solution surface for forcing of the zonal and middle components can be displayed graphically and is a form of the well-known cusp catastrophe surface. For forcing of all three components, the morphogenesis set is more complex, containing regions in which there are one, three or five solutions. Numerical integrations of the phase-sparce trajectories of the solutions reveal that fo...

Experiment on Solitons in the Dissipative Toda Lattice Using Nonlinear Transmission Line

Abstract: Adding dissipative elements to a nonlinear transmission line equivalent to the Toda lattice, we have studied the behavior of a soliton propagating in the dissipative system. The damping rate is found to be explained to large extent by the theory derived from the K-dV equation with a loss term, while its slight dependence on the initial amplitude of the soliton is explained by the theory of the dissipative Toda lattice given by Kako. For a relatively small conductance parallel to the nonlinear capacitor, a somewhat distorted shape of the soliton followed by the long `dip on tail' was observed, which is compared with the theory given by Karpman and Maslov. The `bump on tail' behind the soliton, which is considered as one of the features of the dissipative Toda lattice, was also observed only for the case of connecting a relatively large resistance in series to each inductance.

Irreversibility and randomness

Abstract: Some recent development in the theory of irreversible processes and nonequilibrium thermodynamics are reviewed. At the macroscopic level, the concept of irreversibility as a source of order, as developed during the last decade, is presented in the context of nonequilibrium phase transitions and dissipative structures. Next, a stochastic theory of rate processes is introduced to study fluctuations, the mechanisms of instability and transition. The paper concludes with a brief discussion of quite recent results on the relation between the deterministic laws of dynamics and the probabilistic description of physical processes.

Analytic form of the simplest limit cycle in the Lorenz model

Abstract: A method to obtain the analytic form of limit cycles is presented from a viewpoint of constants of motion in dissipative systems The method is applied to the Lorenz equation in the high-Rayleigh-number limit The analytic form of the simplest limit cycle is obtained, which is in a good agreement with the result of computer simulation

An Integral Hydrodynamic Model of Upper Ocean Frontal Dynamics: Part I. Development and Analysis

Abstract: The paper develops and analyzes a model of frontal-scale dynamics applicable to established, persistent upper ocean density fronts. The effects of interfacial friction and mass entrainment arising from turbulent dissipative processes are incorporated as well as the effects of earth rotation and wind stress. The model is of hydrodynamic character in that the circulation is not permitted to do its own mixing. The equations of motion are solved after their integration over the vertical from the pycnocline bottom to the sea surface. Two independent frontal length scales are found. one is Lt, the dissipative length scale, defined as the ratio of the asymptotic pycnocline depth to the magnitude of the interfacial entrainment coefficient; the other is the baroclinic Rossby radius, the internal wave phase speed divided by the Coriolis parameter. The ratio of these length scales forms the fundamental parameter of the model dynamics, Pr, called the rotation parameter. For large values of Pr the frontal len...

The decomposition into a detailed balance part and a circulation part of an irreversible stationary markov chain

Abstract: In the present paper, we study a construction of the stationary Markovian chain, and a mathematical expression suitable for the dissipative system introduced by H. Haken, I. Prigogine and the others is given.The result shows the stationarity can only be reached in two cases: the detailed balance and the circulation balance.

Gibbsian Distribution on the Lorenz Attractor

Abstract: A representation of the Lorentz attractor by a 1-dimensional Ising system is constructed. Gibbsian distribution function which describes the irregular motion of this dissipative dynamical system is investigated with the help of this representation. Relation between measure-theoretical entropy and positive Lyapunov characteristic exponent is also investigated. The following conclusions are obtained: (1) The statistical properties of the Lorenz system can be reduced to those of 1-dimensional Ising system with short-range interaction, in other words, the time correlation function of the Lorenz system shows no singular long-time behaviour. (2) The positive Lyapunov characteristic exponent of the Lorenz system is almost equal to its measure-theoretical entropy.

Effects of fluctuation on dissipative structures. II. Appearance of optically active states as dissipative structures

Abstract: Two chemical systems which fall into chiral states after starting from an optically inactive state are proposed. These optically active states are realized as dissipative structures and consequently disappear when the chemical system comes close to the equilibrium state. These chemical systems involve stereospecific and autocatalytic processes. The stereospecificity is characteristic of enzymatic processes in living systems and the asymmetric syntheses which take place universally in all living systems. The idea proposed here would open a way to an explanation for asymmetric syntheses in living systems.

Propagation of Solitons in a Dissipative Nonlinear Transmission Line

Abstract: The perturbation theory for the inverse scattering transform applies to study a dissipative nonlinear transmission line, which is equivalent to the Toda lattice in the limit of vanishing dissipations. The time evolution of a one-soliton solution is calculated. It is found that the weak dissipations lead to change in the soliton parameters, the amplitude and the velocity, the creation of small solitons and the formation of a tail behind the initial soliton.