Showing papers on "Dissipative system published in 1989"
TL;DR: A detailed survey of perturbation theory for nearly integrable systems, based upon the inverse scattering transform, and a minute account of results obtained by means of that technique and alternative methods are given in this paper.
Abstract: A detailed survey of the technique of perturbation theory for nearly integrable systems, based upon the inverse scattering transform, and a minute account of results obtained by means of that technique and alternative methods are given. Attention is focused on four classical nonlinear equations: the Korteweg-de Vries, nonlinear Schr\"odinger, sine-Gordon, and Landau-Lifshitz equations perturbed by various Hamiltonian and/or dissipative terms; a comprehensive list of physical applications of these perturbed equations is compiled. Systems of weakly coupled equations, which become exactly integrable when decoupled, are also considered in detail. Adiabatic and radiative effects in dynamics of one and several solitons (both simple and compound) are analyzed. Generalizations of the perturbation theory to quasi-one-dimensional and quantum (semiclassical) solitons, as well as to nonsoliton nonlinear wave packets, are also considered.
1,286 citations
Book•
04 Jan 1989
TL;DR: The Dynamics of Differential Equations and Solitons in Semiclassical Mechanics as mentioned in this paper, and the Dynamics of Dissipative Systems in Hamiltonian Systems and Area-Preserving Mappings.
Abstract: The Dynamics of Differential Equations. Hamiltonian Dynamics. Classical Perturbation Theory. Chaos in Hamiltonian Systems and Area--Preserving Mappings. The Dynamics of Dissipative Systems. Chaos and Integrability in Semiclassical Mechanics. Nonlinear Evolution Equations and Solitons. Analytic Structure of Dynamical Systems. Index.
455 citations
TL;DR: In this article, an analytical theory for the thermal (classical mechanical) rate of escape from a metastable state coupled to a dissipative thermal environment is formulated, and the working expressions are given solely in terms of the quantities entering the generalized Langevin equation for the particle dynamics.
Abstract: An analytical theory is formulated for the thermal (classical mechanical) rate of escape from a metastable state coupled to a dissipative thermal environment. The working expressions are given solely in terms of the quantities entering the generalized Langevin equation for the particle dynamics. The theory covers the whole range of damping strength and is applicable to an arbitrary memory friction. This solves what is commonly known as the Kramers turnover problem. The basic idea underlying the approach is the observation that the escape dynamics is governed by the unstable normal mode coordinate—and not the particle system coordinate. An application to the case of a particle moving in a piecewise harmonic potential with an exponentially decaying memory‐friction is presented. The comparison with the numerical simulation data of Straub, Borkovec, and Berne [J. Chem. Phys. 84, 1788 (1986)] exhibits good agreement between theory and simulation.
300 citations
TL;DR: In this article, the authors study the long-time behavior of a class of nonlinear dissipative partial differential equations and show that the equation has an inertial manifold, provided that a certain gap condition in the spectrum of the linear part of the equation is satisfied.
Abstract: In this paper, we study the long-time behavior of a class of nonlinear dissipative partial differential equations. By means of the Lyapunov-Perron method, we show that the equation has an inertial manifold, provided that certain gap condition in the spectrum of the linear part of the equation is satisfied. We verify that the constructed inertial manifold has the property of exponential tracking (i.e., stability with asymptotic phase, or asymptotic completeness), which makes it a faithful representative to the relevant long-time dynamics of the equation. The second feature of this paper is the introduction of a modified Galerkin approximation for analyzing the original PDE. In an illustrative example (which we believe to be typical), we show that this modified Galerkin approximation yields a smaller error than the standard Galerkin approximation.
233 citations
TL;DR: In this paper, the authors studied the large-time behavior of solutions to the initial-value problem for the Korteweg-de Vries equation and for the regularized long-wave equation, with a dissipative term appended.
220 citations
TL;DR: A dynamic renormalization-group calculation allows us to determine various critical exponents exactly in all dimensions of dissipative transport in open systems.
Abstract: Motivated by recent models of Bak, Tang, and Wiesenfeld we study dissipative transport in open systems. A simple continuum equation is constructed to describe fluctuations around a steady state in a flowing ``sandpile.'' The principle of scale invariance and self-similarity is understood in terms of a conservation law in dynamics. A dynamic renormalization-group calculation allows us to determine various critical exponents exactly in all dimensions.
183 citations
TL;DR: It is found that for a large parameter range the localized solutions can interact so that they emerge after the collision with a size and shape unchanged compared to that well before the collision.
Abstract: We discuss the interaction of localized solutions as they arise for the subcritical bifurcation to traveling waves. We find that for a large parameter range the localized solutions can interact so that they emerge after the collision with a size and shape unchanged compared to that well before the collision. The mechanism for this behavior, which is unusual for a strongly dissipative system, is qualitatively different from that associated with solitons for completely integrable systems. In accord with this we find that for other parameter values counterpropagating localized solutions can annihilate.
147 citations
TL;DR: In this article, a fundamental thermodynamic equation is derived for thermoelastic martensitic transformations, taking into account internal interaction between domains, the existence of interfaces and dissipative effects associated with motion of boundaries.
144 citations
TL;DR: In this paper, the authors considered second order in time equations of Petrovsky type (wave and plate-like) with nonlinear dissipative boundary conditions and established the results on asymptotic behavior of the solutions.
126 citations
TL;DR: In this article, the authors introduce the notion of a spectral barrier for a nonlinear dissipative partial differential equation and prove the existence of inertial manifolds that require easily verifiable conditions.
Abstract: In recent years, the theory of inertial manifolds for dissipative partial differential equations has emerged as an active area of research. An inertial manifold is an invariant manifold that is finite dimensional, Lipschitz, and attracts exponentially all trajectories. In this paper, we introduce the notion of a spectral barrier for a nonlinear dissipative partial differential equation. Using this notion, we present a proof of existence of inertial manifolds that requires easily verifiable conditions, namely, the existence of large enough spectral barriers.
111 citations
TL;DR: In this article, the long-time behavior of partially dissipative reaction-diffusion systems is studied and bounds of the Hausdorff and fractal dimensions of the universal attractor are derived.
Abstract: The long-time behavior of the solutions of some partly dissipative reaction-diffusion systems is studied. Two types of problems are considered: systems with a polynomial growth nonlinearity, and systems admitting a positively invariant region. It is shown that the long-time behavior can be described by a universal attractor, and bounds of the Hausdorff and fractal dimensions of this attractor are derived. The results are applied to several classical systems borrowed from mathematical biology, physics and chemistry.
TL;DR: It is shown that although the maps are dissipative their trajectories can be derived from a Hamiltonian, which allow us to calculate unstable periodic orbits of arbitrary length finding all points to any desired accuracy.
Abstract: A numerical technique for the characterization of the chaotic regime of dissipative maps through unstable periodic orbits is presented. It is shown that although the maps are dissipative their trajectories can be derived from a Hamiltonian, which allow us to calculate unstable periodic orbits of arbitrary length finding all points to any desired accuracy
TL;DR: The quantum dynamics of the Landau-Zener transition in a dissipative environment is studied and analytical results for the transition probability are given in terms of temperature, coupling strength, and the Landauer-Zner time.
Abstract: The quantum dynamics of the Landau-Zener transition in a dissipative environment is studied and analytical results for the transition probability are given in terms of temperature, coupling strength, and the Landau-Zener time. For short Landau-Zener time there is no effect of dissipation. In the oppotise limit we distinguish various temperature regimes: The adiabatic limit is shown to be restricted to low temperatures and no effect of dissipation is present at zero temperature. At intermediate temperatures thermal transitions dominate for weak coupling and high temperatures correspond to the strong-coupling limit.
TL;DR: The theory generalizes the derivation of the conventional superradiance master equation for molecules in vacuum and yields all local field and screening factors for both the dispersive and the dissipative couplings from a single, unified starting point.
Abstract: A reduced equation of motion that describes the excited-state dynamics of interacting two-level impurity molecules in a dielectric host crystal is derived starting from a microscopic model for the total system. Our theory generalizes the derivation of the conventional superradiance master equation for molecules in vacuum; the role of photons in the conventional theory is played by polaritons (mixed crystal-radiation excitations) in our approach. Our final equation thus contains dispersive and superradiant polariton-mediated intermolecular interactions. The effect of the dielectric host is completely contained within a rescaling of these interactions with the transverse dielectric function \ensuremath{\epsilon}(\ensuremath{\omega}) of the crystal taken at the impurity's transition frequency. Our theory yields all local field and screening factors for both the dispersive and the dissipative couplings from a single, unified starting point. Known scaling laws for the spontaneous-emission rate and the instantaneous dipole-dipole interaction are extended to the frequency region where the dispersion of \ensuremath{\epsilon}(\ensuremath{\omega}) is important.
TL;DR: In this article, the exact equation of motion for the reduced density matrix of a system weakly coupled to a bath is obtained using projection operator techniques, which reduces to a generalized master equation when the bath relaxation is faster than the relaxation of the system induced by the weak interaction with the bath.
Abstract: The equation of motion for the reduced density matrix of a system weakly coupled to a bath is obtained using projection operator techniques. The exact equation of motion reduces to a generalized master equation when the bath relaxation is faster than the relaxation of the system induced by the weak interaction with the bath. The equation separates into streaming or systematic terms and dissipative terms which are separately equal to zero at equilibrium. We find both statistical and dynamical system frequency shifts; the statistical shifts are present in equilibrium but the dynamical shifts affect the time-dependence, only. The general results are applied to the two-level system model for tunneling in condensed phases.
TL;DR: In this paper, a long chain of coupled vibrating strings with stabilizers installed at each internal node and perhaps also at a boundary point is considered and the exponential stability of the stabilizers' arrangement for this large dynamic structure is determined.
Abstract: Consider a long chain of coupled vibrating strings, where a stabilizer is installed at each internal node and perhaps also at a boundary point. The exponential stability of the stabilizers’ arrangement for this large dynamic structure will be determined.Through a careful transformation of the coupled wave equations for this structure into an equivalent hyperbolic system and analysis of the eigendeterminant, it will be proven that the energy of the system decays uniformly exponentially if there is a stabilizer installed at a boundary point. If the stabilizers are installed only at internal nodes, it will be proven that the energy may decay either uniformly exponentially or nonuniformly, or may not decay at all, depending on the different wave speeds and the stabilizers– arrangement. All possible outcomes have been classified.
TL;DR: Theoretical calculations of the basic quantities that characterize the stopping of an interacting electron gas for slow ions are presented and an appropriate low-frequency expansion for the imaginary part of the density response function is used to modify well-known results for the noninteracting electron gas.
Abstract: Theoretical calculations of the basic quantities that characterize the stopping of an interacting electron gas for slow ions are presented. An appropriate low-frequency expansion for the imaginary part of the density response function has been used to modify well-known results for the noninteracting electron gas. The inner dissipative nature of the elementary electron-hole excitation is characterized by a complex local-field correction function. The basic quantities are expressed in terms of the phase shifts determined from a nonlinear density-functional formalism.
TL;DR: This model illustrates that the appearance of dissipative behavior from the back reaction of particle creation in quantum fields is a general feature and suggests that the role of gravity in the display of dissipatives behavior in semiclassical processes is not unique.
Abstract: We discuss the nature and origin of the dissipation of quantum fields due to the back reaction of particle creation. We derive the effective action of a scalar $g{\ensuremath{\varphi}}^{3}$ theory in the closed-time-path-integral formalism. From the real and causal equation of motion for the background field we deduce a dissipative function for this process and for the cosmological anisotropy damping problem studied earlier. This model illustrates that the appearance of dissipative behavior from the back reaction of particle creation in quantum fields is a general feature. It also suggests that the role of gravity in the display of dissipative behavior in semiclassical processes is not unique.
TL;DR: In this article, a new type of open reactor where both chemical constraints and transport rates are controlled is introduced, and a sequence of different spatiotemporal reaction -diffusion structures, and most interestingly the first nontrivial stationary pattern, have been obtained by continuous variation of a control parameter with variants of the chlorite-iodide and the Belousov-Zhabotinskii reactions.
TL;DR: In this paper, the authors describe a hierarchy of dissipative structures traversing a specification hierarchy of increasingly intensive being, called ontogenetic trajectories, the seat of self-organization.
Abstract: Scalar hierarchies represent the structure of the world of matter in motion, the overall extensive system. Subsystems within this undergo changes that can be dichotomized into development and evolution (individuation). At a higher scalar level these modes of change are united under the rubric ‘self-organization’. Development finds a very general, indeed, lawful description in non-equilibrium thermodynamics discourse. Subsystems in the scalar hierarchical world are all dissipative structures. In their development at least living dissipative structures traverse a specification hierarchy of increasingly intensive being. At a higher temporal scalar level, developing dissipative structures form ontogenetic trajectories, the seat of self-organization. At these higher scalar levels ontogenetic trajectories multiply to fill space as the dissipative structures that make them up access more and more geographical coordinates in a thrust outward toward what can be called a virtual thermodynamic equilibrium.
TL;DR: In this paper, a hierarchy of simplified models for the Kuramoto-Takens model is presented. But it is not a complete hierarchy of models, as shown in Figure 1.
TL;DR: Geometrical phases are shown to be relevant to systems of this type, when their Hamiltonian changes slowly with time, and a method for obtaining higher-order corrections to the adiabatic approximation is demonstrated.
Abstract: Berry's [Proc. R. Soc. London, Ser. A 392, 45 (1984)] geometrical phase is discussed in the context of dissipative evolution of an interacting spin system, governed by the stochastic Liouville equation. An analytical treatment is given for a possible ESR experiment on an interacting electron-nucleus system, modulated by two-site jumps. Geometrical phases are shown to be relevant to systems of this type, when their Hamiltonian changes slowly with time. A method for obtaining higher-order corrections to the adiabatic approximation is demonstrated. It is found that if the jumps are slow relative to the rate of change of the Hamiltonian, their effect reduces to familiar line broadening, and the geometrical phases may be observed experimentally. Equations are also set up for a similar ESR experiment on an electron-nucleus system undergoing isotropic rotational diffusion, and a brief discussion of the equations follows.
TL;DR: In this article, the effects of dissipative phenomena, with emphasis on nonlocal effects, on propagation and absorption of sound waves in a rarefied monoatomic gas are investigated within the framework of extended irreversible thermodynamics, which raises the thermodynamic fluxes to the status of independent variables.
TL;DR: The quantum signature of chaos in dissipative systems is cubic repulsion of their generalized energies (eigenvalues of the generators of the dynamics) in the complex plane and, as a useful byproduct, the generalization of detailed balance for periodically driven systems with damping is found.
Abstract: The quantum signature of chaos in dissipative systems is cubic repulsion of their generalized energies (eigenvalues of the generators of the dynamics) in the complex plane. As in the Hamiltonian case the degree of repulsion is the same under temporally homogeneous conditions and periodic driving. Somewhat surprisingly, however, cubic repulsion prevails irrespective of whether or not the Hamiltonian embedding of the dissipative system obeys time-reversal invariance. More-over, even antiunitary symmetries of the dissipative generator itself cannot modify the repulsion exponent. In establishing the universality in question we find, as a useful byproduct, the generalization of detailed balance for periodically driven systems with damping.
TL;DR: In this paper, the existence of specific associated nodispersive soliton-like (particle-like) solutions piloted by the surrounding field is established, and these solutions can be considered as the first known possible representations of de Broglie's double solution theory.
TL;DR: In this article, the universal attractor for partially dissipative reaction-diffusion systems is shown to be infinite-dimensional in space dimension n ⩽ 2, where n is the number of vertices in the manifold.
TL;DR: In this article, the problem of obtaining a consistent plasma pinch in the dissipative regime is discussed, and detailed quasilinear trapped electron transport expressions are compared with the simple scaling forms of the extreme dissipative regimes used in recent modeling studies.
Abstract: Detailed quasilinear trapped electron transport expressions are compared with the simple scaling forms of the extreme dissipative regime used in recent modeling studies. The problem of obtaining a consistent plasma pinch in the dissipative regime is discussed.
Journal Article•
TL;DR: In this article, the theoretical and physical meaning of dissipation of background fields due to particle creation and statistical effects in interacting quantum field theories and in semiclassical gravitational theories is discussed.
TL;DR: In this paper, a macroscopic theory of superfluid turbulence is developed in which the vortex tangle is characterised by both its line-length density and its drift velocity.
Abstract: A macroscopic theory of superfluid turbulence is developed in which the vortex tangle is characterised by both its line-length density and its drift velocity. Three velocity fields are accordingly distinguished, viz. the mass velocity υ, the normal-fluid velocity υn and the drift velocity υe of the vortex tangle. The introduction of the drift velocity of the tangle as an additional variable seems to be new. The non-dissipative equations of motion are derived from a generalised form of Lin's variational principle. Both the energy and the impulse of the vortex tangle into account. It is shown that the effective mass density of the vortex tangle vanishes. In equilibrium the relative velocity of the vortex tangle is given by the derivative of the energy of the tangle with respect to the tangle impulse. A similar relation holds for a ring vortex in a fluid of infinite extent.
When dissipative terms are added to the equations according to the thermodynamics of irreversible processes the Vinen equation follows immediately. The derivation suggests a new interpretation of the right-hand member of the Vinen equation in terms of the derivative of a potential energy. A similar potential energy has been used in investigations on vortex nucleation and critical velocities. The theory is extended by including the effects of large gradients of the line-length density. The corresponding generalisation of the Vinen equation is presented.