Showing papers on "Dissipative system published in 1990"
01 Jan 1990
TL;DR: In this article, a brief overview of the current understanding of temporal and spatio-temporal chaos, both termed weak turbulence according to the context, is presented, and the process which allows one to reduce the primitive problem to a low-dimensional dynamical system is discussed.
Abstract: We present a brief overview of the current understanding of temporal and spatio-temporal chaos, both termed weak turbulence according to the context [1]. The process which allows one to reduce the primitive problem to a low-dimensional dynamical system is discussed. It turns out to be appropriate as long as confinement effects are sufficiently strong to freeze the space dependence of unstable modes, hence temporal chaos only. Otherwise modulated patterns arise, yielding genuine space-time chaos. The corresponding theory rests on envelope equations providing a useful framework for weak turbulence in a globally super-critical setting. spatio-temporal intermittency analyzed next is the relevant scenario in the sub-critical case. Finally, the connection with hydrodynamic turbulence and the more general relevance of some of the ideas developed here are examined.
473 citations
TL;DR: In this article, the formalism of internal state variables is established when gradients of these variables are involved, thus allowing for a spatial localization of dissipative effects that give rise to dissipative structures.
Abstract: The formalism of internal state variables is established when gradients of these variables are involved, thus allowing for a spatial localization of dissipative effects that give rise to dissipative structures. Three illustrative cases are briefly sketched out: nematic liquid crystals, localization of damage coupled to elasticity or plasticity, and localization of plastic strains in plasticity with hardening
286 citations
TL;DR: It is shown that stable localized waves can be generated in the vicinity of an inverted Hopf bifurcation, and the size of the localized wave envelope perturbatively in the case of slightly dissipative systems is computed.
Abstract: We show that stable localized waves can be generated in the vicinity of an inverted Hopf bifurcation. We compute the size of the localized wave envelope perturbatively in the case of slightly dissipative systems. The size selection traces back to the broken scale invariance by the dissipative terms. This mechanism is a possible explanation for the localized structures, widely observed in various hydrodynamic flows in dissipative systems driven far from equilibrium.
235 citations
TL;DR: This work investigates the theories of dissipative relativistic fluids in which all of the dynamical equations can be written as total-divergence equations and finds the general theory of this type.
Abstract: We investigate the theories of dissipative relativistic fluids in which all of the dynamical equations can be written as total-divergence equations. Extending the analysis of Liu, Mueller, and Ruggeri, we find the general theory of this type. We discuss various features of these theories, including the causality of the full nonlinear evolution equations and the nature and stability of the equilibrium states.
170 citations
TL;DR: In this article, the authors assume a model in which phase-breaking and dissipation are caused by the interaction of electrons with a reservoir of oscillators through a delta potential, leading to a kinetic equation with a simple physical interpretation.
Abstract: An important problem in quantum transport is to understand the role of dissipative processes. In this paper the author assume a model in which phase-breaking and dissipation are caused by the interaction of electrons with a reservoir of oscillators through a delta potential. In this model the self-energy is a delta function in space, leading to a kinetic equation with a simple physical interpretation. A novel treatment of the contacts is used to introduce the external current into the kinetic equation. One specializing to linear response the author obtains an integral equation that looks like the Buttiker formula (1961) extended to a continuous distribution of probes. The author show that this equation can be reduced to the usual Buttiker formula which involves only the actual physical probes. Dissipation modifies the transmission coefficients, and the author presents explicit expressions derived from this model. Also, in a homogeneous medium the integral equation reduces to the diffusion equation, it the electrochemical potential is assumed to vary slowly. This paper serves to establish a bridge between the quantum kinetic approach which rigorously accounts for the exclusion principle and the one-particle approach which is intuitively appealing.
140 citations
TL;DR: In this paper, the formation, structure, and stability of intermediate shocks in dissipative MHD are considered in detail, and the differences between the conventional theory and the present one are pointed out and clarified.
Abstract: It was recently shown by Wu (1987) that intermediate shocks are admissible and can be formed through nonlinear wave steepening from continuous waves. In this paper, the formation, structure, and stability of intermediate shocks in dissipative MHD are considered in detail. The differences between the conventional theory and the present one are pointed out and clarified. It is shown that all four types of intermediate shocks can be formed from smooth waves. It is also shown that there are free parameters in the structure of the intermediate shocks, and that these parameters are related to the shock stability. In addition, the paper shows that a rotational discontinuity can not exist with finite width, indicates how this is related to the existence of time-dependent intermediate shocks, and shows why the conventional theory is not a good approximation to dissipative MHD solutions whenever there is rotation in magnetic field.
134 citations
TL;DR: In this paper, the intramolecular non-Born-Oppenheimer quantum dynamics on conically intersecting potential energy surfaces is analyzed on the basis of exact (numerical) time-dependent quantum calculations for two representative two-state three-mode vibronic coupling models.
Abstract: The intramolecular non–Born–Oppenheimer quantum dynamics on conically intersecting potential‐energy surfaces is analyzed on the basis of exact (numerical) time‐dependent quantum calculations for two representative two‐state three‐mode vibronic‐coupling models. A compact description of the time‐dependent dynamics in terms of reduced density matrices of the electronic and vibrational subsystems is introduced. Results are presented for the time evolution of electronic and vibrational coherences, populations, as well as subsystem entropies. It is found that such simple two‐state three‐mode vibronic coupling models exhibit a rich variety of dissipative phenomena on femtosecond time scales. The numerical results reveal an interesting interplay of driven electronic surface‐hopping processes and dephasing of coherent vibrational motion which is presumably a generic feature of ultrafast internal conversion processes in polyatomic molecules.
132 citations
TL;DR: In this article, a nonlinear quantum transport theory for many-body systems arbitrarily far away from equilibrium, based on the nonequilibrium statistical operator method, is discussed, and an iterative process is described that allows for the calculation of the collision operator in a series of instantaneous in time partial contributions of ever increasing power in the interaction strengths.
Abstract: A nonlinear quantum transport theory for many-body systems arbitrarily far away from equilibrium, based on the nonequilibrium statistical operator method, is discussed. An iterative process is described that allows for the calculation of the collision operator in a series of instantaneous in time partial contributions of ever increasing power in the interaction strengths. These partial collision operators are shown to be composed of three contributions to the dissipative processes: one is a direct result of the collisions, another is accounted for in the internal state variables, and the third arises from memory effects. In the lowest order, the so-called linear theory of relaxation, the equations become Markovian.
119 citations
01 Jan 1990
TL;DR: For degenerate (1.1) equations, the existence of global solutions has been established by Nishida, Arosio and Spagnolo as discussed by the authors, and for non-degenerate equations, it has been shown that there is a local solvability for nonanalytic initial data which belong to suitable Sobolev spaces.
Abstract: $L$ is the length, $E$ is the Young’s modulus and $P_{0}$ is the initial axial tension. We are interested in the case $P_{0}=0$, which is corresponding to $a(0)=0$ in (1.1). When $a(r)$ has zeros, we say that (1.1) is degenerate. For degenerate equations (1.1) without dissipative terms $(¥gamma=0)$ , the existence of global solutions has been established by Nishida [11], Arosio and Spagnolo [1] when initial data $¥mathrm{u}_{0}$ and $u_{1}$ are analytic, while Yamazaki [18] and Yamada [21] have recently proved the local solvability for non-analytic initial data which belong to suitable Sobolev spaces. See also Ebihara, Medeiros and Milla Miranda [5]. (For non-degenerate equations of the form (1.1), there is a lot of literature; see Dickey [4], Lions [8], Nishihara [12], Perla Menzala [15], Pohozaev [16], Rivera [17], Yamada [20] and the references therein.)
98 citations
01 Jan 1990
TL;DR: In this paper, the authors introduce the concepts of integrability and singularity analysis for a set of nonlinear PDE's and apply them to the model of ferromagnetic inhomogeneities.
Abstract: I. Waves in Physical Systems.- Competing interactions and complexity in condensed matter.- Exact solutions of the Boltzmann equation.- Nonlinear interaction between short and long waves.- Nonlinear optics.- The phase diffusion and mean drift equations for convection at finite Rayleigh numbers in large containers I.- Nonlinear evolution equations, quasi-solitons and their experimental manifestation.- The initial-boundary value problem for the Davey-Stewartson 1 equation how to generate and drive localized coherent structures in multidimensions.- II. Instabilities and Defects.- Ginzburg-Landau models of non-equilibrium.- Nonlinear Ginzburg-Landau equation and its application to fluid-mechanics.- Defects and disorder of nonlinear waves in convection.- III. Concepts Of Integrability and Singularity Analysis.- An introduction to Kowalevski's exponents.- Singularity analysis and its relation to complete, partial and non-integrability.- A concept of integrability based on the symmetry approach.- Backlund transformations and the Painleve property.- IV. Mathematical Methods.- Differential geometry techniques for sets of nonlinear partial differential equations.- Inertial manifolds and attractors of partial differential equations.- Hirota's bilinear method and partial integrability.- Generalized symmetries, recursion operators and bihamiltonian systems.- Nonlinear dispersive equations without inverse scattering.- Group theory and exact solutions of partially integrable differential systems.- Contributed Papers.- The nonlinear evolution equation for the order parameter in superfluid Helium-Four.- Quasimonomial transformations and integrability.- Partial integrability of the damped kink equation.- New similarity reductions of Boussinesq-type equations.- The homographic invariance of PDE Painleve analysis.- The nonlocal amplitude equation.- Pressure waves in fluid-filled nonlinear viscoelastic tubes.- Elliptic function solutions for Landau-Ginzburg equation.- Application of a Macsyma program for the Painleve test to the Fitzhugh-Nagumo Equation.- The strongly dissipative Toda lattice.- A perturbative approach to Hirota's bilinear equations of KdV-type.- Construction of two dimensional super potentials for classical super systems.- Applications of nonlinear PDE's to the modelling of ferromagnetic inhomogeneities.- Author Index.
97 citations
TL;DR: In this article, the authors considered the quantum-mechanical version of the Kramers turnover problem and derived an expression for the quantum escape rate in the thermally activated tunneling regime.
Abstract: The quantum-mechanical version of the Kramers turnover problem is considered. The multidimensional character of the problem is taken into account via transformation to normal modes. This eliminates the coupling to the bath near the barrier top allowing the use of a simple harmonic transmission coefficient for the barrier dynamics. The well dynamics is described by a continuum form of a master equation for the energy in the unstable normal mode. Within first-order perturbation theory, the equations of motion for the stable normal modes have the form of a forced oscillator. The transition probability kernel is found using the known solution for the quantum forced oscillator problem. An expression for the quantum escape rate is derived. It encompasses all previously known limiting results in the thermally activated tunneling regime. The depopulation factor, which accounts for the nonequilibrium energy distribution is evaluated. The quantum transition probability kernel is broader than the classical and is skewed towards lower energies. Interplay between these two effects, together with a positive tunneling contribution, determines the relative magnitude of the quantum rate compared to the classical one. The theory is valid for arbitrary dissipation. Its use is illustrated for the case of a cubic potential with Ohmic (Markovian) dissipation.
TL;DR: In this paper, the authors argue that the vacuum configuration of open strings correspond to the Caldeira-Leggett models of dissipative quantum mechanics (DQM) evaluated at a delocalization critical point.
TL;DR: In this paper, a continuous model for a non-demolition observation of an atom is given and a stochastic dissipative Schrodinger equation for the unnormalized posterior wave function of the atom is derived.
Abstract: A continuous model for a nondemolition observation of an atom is given. An equation for the corresponding instrument is found and a stochastic dissipative Schrodinger equation for the unnormalized posterior wave function of the atom is derived. It is shown that the continuously observed isolated atom relaxes to the ground state without mixing.
TL;DR: In this paper, the applicability of the Nonequilibrium Statistical Operator Method (NSOM) for the study of dissipative dynamic systems far from equilibrium is discussed, which can be encompassed by a unifying variational principle, which produces a large family of NSO that contains existing examples as particular cases.
Abstract: We describe the large applicability of the Nonequilibrium Statistical Operator Method (NSOM) for the study of dissipative dynamic systems far from equilibrium. It is shown that the NSOM can be encompassed by a unifying variational principle, which produces a large family of NSO that contains existing examples as particular cases. Further, we review the application of the NSOM for the construction of a nonlinear quantum theory of large scope, and for the generation of a response function theory, for far-from-equilibrium Hamiltonian systems. An accompanying non-equilibrium thermodynamic Green's function theory is briefly described. Also it is shown that the NSOM provides mechano-statistical foundations for phenomenological irreversible thermodynamics, and for the important question of stability of far-from-equilibrium steady states and the emergence of self-organized dissipative structures in condensed matter.
TL;DR: In this article, asymptotically exact amplitude equations for a dissipative system near a Hopf bifurcation were derived for O(1) group velocities.
Abstract: New asymptotically exact amplitude equations are derived for a dissipative system near a Hopf bifurcation. Unlike the usual coupled complex Ginzburg-Landau equations these are valid for O(1) group velocities.
TL;DR: In this paper, a class of dissipative, stratified, parallel shear flows which, as a consequence of linear supercritical instability, evolve directly into three-dimensional flows without the requirement for an intermediate two-dimensional finite-amplitude state was demonstrated.
Abstract: We demonstrate the existence of a class of dissipative, stratified, parallel shear flows which, as a consequence of linear supercritical instability, evolve directly into three-dimensional flows without the requirement for an intermediate two-dimensional finite-amplitude state. This represents a counter-example to a common misinterpretation of Squire's theorm, namely that the fastest-growing unstable mode of a dissipative parallel shear flow must be two-dimensional.
TL;DR: In this article, the authors consider the case of a non-equilibrium structure which is only periodic in space and show that the same kinetic instabilities which lead to a periodic variation of the system variables in time may also induce a periodic variations in space.
Abstract: The term “dissipative structures” which was introduced by Prigogihe [17] describes a broad class of non-equilibrium systems where a constant flow of energy and/or matter leads to structures ordered in space or time, e.g. causes kinetic oscillations or spatial pattern formation. Temporal oscillations and spatial pattern formation are closely related since the same kinetic instabilities which lead to a periodic variation of the system variables in time may also induce a periodic variation in space. Consequently one often observes spatio-temporal structures, but one may also consider the case of a non-equilibrium structure which is only periodic in space. Such structures which rarely have been observed in chemical reaction systems were first discussed by Turing [20] and have been termed accordingly as “Turing structures”.
TL;DR: In this article, it was shown that the Stokes eigenfunctions and their corresponding spectra, frequently used in mathematical investigations of the Navier-Stokes equations, provide estimates on the spectrum of the two-point spatial covariance tensor.
Abstract: It is shown that the Stokes eigenfunctions and their corresponding spectra, frequently used in mathematical investigations of the Navier–Stokes equations, provide estimates on the spectrum of the two‐point spatial covariance tensor. This, in turn, is used to estimate the far‐dissipative turbulent spectrum. An exponential falloff is predicted and evidence given which implies that this is a sharp estimate.
TL;DR: In this article, various scenarios for self-organization in a wide range of nonequilibrium physical, chemical, and biological systems are examined, and the conditions under which turbulence arises in such systems in the absence of a flow are discussed.
Abstract: Various scenarios for self-organization in a wide range of nonequilibrium physical, chemical, and biological systems are examined. It is stressed that in many systems small-amplitude dissipative structures never form: At the instant at which the homogeneous state stratifies, large-amplitude dissipative structures appear abruptly. These large-amplitude structures are striations, spots, or blobs. Methods for the construction of such dissipative structures and for studying their stability are discussed for an arbitrary departure of the system from equilibrium. Many self-organization scenarios do not involve an instability of the dissipative structures of a given type and are instead determined by a local breakdown effect. In real systems, self-organization is determined by the spontaneous appearance and subsequent evolution of autosolitons (localized dissipative structures). The conditions under which turbulence arises in such systems in the absence of a flow are discussed. This turbulence is a complicated picture of the random appearance and disappearance of interacting autosolitons in various parts of the system. In gaseous and semiconductor plasmas, dissipative structures can arise in the form of numerous current filaments or electric-field domains. Their formation is unrelated to the shape of the current-voltage characteristic of the system. There is a discussion of certain self-organization phenomena in systems in which static dissipative structures may be accompanied by pulsating dissipative structures and autowaves. Corresponding effects in systems with flows (or fluxes of material) are also discussed. The general results of self-organization theory are used to explain the properties of the dissipative structures which have been observed and studied in numerical and experimental studies of physical systems of various types in recent years.
TL;DR: The low-temperature features of the free energy, the static susceptibility, and the equilibrium correlation function are presented for Ohmic and super-Ohmic dissipation, and various universal properties are deduced.
Abstract: The thermodynamics and dynamics of the spin-boson model are formulated in terms of the exact formal solutions. The low-temperature features of the free energy, the static susceptibility, and the equilibrium correlation function are presented for Ohmic and super-Ohmic dissipation, and various universal properties are deduced. In the Ohmic case the Wilson ratio and the spectral properties at low frequencies (Shiba's relation) are calculation in the range 01 of the coupling strength. The corresponding universal relations are determined also for super-Ohmic dissipation.
TL;DR: In this article, a dissipative version of the quantized standard map is constructed by analytical means and iterated numerically to study the long time behavior in various regions of the damping rate.
TL;DR: In this paper, the authors derived the escape rate of a particle trapped in a metastable well and interacting with a dissipative medium, leading to a reduced two-degrees-of-freedom Hamiltonian involving the unstable normal mode and a newly defined collective bath mode.
Abstract: Variational upper bounds are derived for the escape rate of a particle trapped in a metastable well and interacting with a dissipative medium. The theory leads to a reduced two-degrees-of-freedom Hamiltonian involving the unstable normal mode and a newly defined collective bath mode. Explicit treatment of strong nonlinearities or low-barrier systems presents no special problem
01 Jul 1990
TL;DR: In this article, a dissipative compact two-four scheme (second-order time, fourth-order space similar to the original MacCormack scheme) has been developed, that exhibits greater accuracy than conventional fourthorder schemes.
Abstract: A dissipative compact two-four scheme (second-order time, fourth-order space similar to the original MacCormack scheme has been developed, that exhibits greater accuracy than conventional fourth-order schemes. The dissipative nature of the scheme allows it to resolve weak discontinuities without artificial damping. A derivation of the scheme is presented, as well as the theoretical stability characteristics. The temporal scheme is then generalized into a steady-state formulation which achieves fourth-order spatial accuracy at steady-state. Several test problems are used to show that the scheme is more accurate than the traditional MacCormack scheme, and is nearly as efficient.
TL;DR: The statistical properties of the third-order nonlinear dissipative oscillator, which evolves from any state, are derived on the basis of the exact solution to the master equation.
Abstract: The statistical properties of the third-order nonlinear dissipative oscillator, which evolves from any state, are derived on the basis of the exact solution to the master equation. Some important features of the nonlinear oscillator model, such as the recurrences of the initial state, are related to the properties of quasidistributions connected with the phase of the complex field amplitude.
TL;DR: In this paper, the authors characterized the chaos in a one-dimensional system, which would be nonlinear stationary Alfven waves in the absence of an external driver, and numerically integrated the evolution equations for the transverse wave magnetic field amplitude and phase using the derivative nonlinear Schrodinger equation (DNLS).
Abstract: The chaos in a one‐dimensional system, which would be nonlinear stationary Alfven waves in the absence of an external driver, is characterized. The evolution equations are numerically integrated for the transverse wave magnetic field amplitude and phase using the derivative nonlinear Schrodinger equation (DNLS), including resistive wave damping and a long‐wavelength monochromatic, circularly polarized driver. A Poincare map analysis shows that, for the nondissipative (Hamiltonian) case, the solutions near the phase space (soliton) separatrices of this system become chaotic as the driver amplitude increases, and ‘‘strong’’ chaos appears when the driver amplitude is large. The dissipative system exhibits a wealth of dynamical behavior, including quasiperiodic orbits, period‐doubling bifurcations leading to chaos, sudden transitions to chaos, and several types of strange attractors.
TL;DR: In this paper, the authors examined the linearized Israel-Stewart theory of relativistic dissipative fluids with the fluid four-velocity chosen to be a timelike eigenvector of the stress-energy tensor.
TL;DR: In this paper, it was shown that a modified version of the entropy rate admissibility criterion can be described by a kinetic relation of the form f = φ(s) relating the driving traction f at a phase boundary to the phase boundary velocity s that corresponds to a notion of maximum dissipation analogous to the concept of maximum plastic work.
Abstract: This paper is concerned with the kinetics of propagating phase boundaries
in a bar made of a special nonlinearly elastic material. First, it is shown that
there is a kinetic law of the form f = φ(s) relating the driving traction f at a phase
boundary to the phase boundary velocity s that corresponds to a notion of maximum
dissipation analogous to the concept of maximum plastic work. Second, it is shown
that a modified version of the entropy rate admissibility criterion can be described by
a kinetic relation of the above form, but with a different φ. Both kinetic relations
are applied to the Riemann problem for longitudinal waves in the bar.
TL;DR: In this paper, the influence of system-bath correlations in the initial state on the dynamics of biased two-state systems was studied and the equilibrium correlation function in the form of a power series in the number of transitions between the system states was determined.
Abstract: We study the influence of system-bath correlations in the initial state on the dynamics of biased two-state systems. The formal solution for the equilibrium correlation function in the form of a power series in the number of transitions between the system states is determined. For Ohmic damping and a special value of the coupling strength (K=1/2), the exact solution for all times and temperatures is presented. We find that at low temperatures the long-time tails of the decay into equilibrium are strongly affected by the initial preparation. Correspondingly, the low-frequency behavior of the neutron-scattering function is qualitatively changed by the presence or absence of correlations in the initial state.