Showing papers on "Dissipative system published in 1994"
TL;DR: It is proved that the measure of those phases that generate second law violating phase space trajectories vanishes exponentially with time.
Abstract: For reversible deterministic N-particle thermostatted systems, we examine the question of why it is so difficult to find initial microstates that will, at long times under the influence of an external dissipative field and a thermostat, lead to second law violating nonequilibrium steady states. We prove that the measure of those phases that generate second law violating phase space trajectories vanishes exponentially with time.
720 citations
TL;DR: In this paper, the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms was studied and the convergence to the reduced dynamics for the 2 × 2 case was studied.
Abstract: We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N × N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 × 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 × 2 case. © 1994 John Wiley & Sons, Inc.
696 citations
TL;DR: It is shown how the interaction with the environment limits distances over which quantum coherence can persist, and therefore reconciles quantum dynamics with classical Hamiltonian chaos.
Abstract: Quantum wave function of a chaotic system spreads rapidly over distances on which the potential is significantly nonlinear. As a result, the effective force is no longer just a gradient of the potential, and predictions of classical and quantum dynamics begin to differ. We show how the interaction with the environment limits distances over which quantum coherence can persist, and therefore reconciles quantum dynamics with classical Hamiltonian chaos. The entropy production rate for such open chaotic systems exhibits a sharp transition between reversible and dissipative regimes, where it is set by the chaotic dynamics.
353 citations
TL;DR: In this article, the authors propose an iterative scheme for the evaluation of path integrals by stepwise multiplication of a propagator tensor, thereby making exact quantum dynamics calculations in condensed phase systems feasible for arbitrarily long times.
286 citations
TL;DR: In this paper, a new decomposition of the Redfield relaxation tensor is proposed for the density matrix of a multilevel quantum-mechanical system interacting with a thermal bath.
Abstract: We present a new method for solving the Redfield equation, which describes the evolution of the reduced density matrix of a multilevel quantum‐mechanical system interacting with a thermal bath. The method is based on a new decomposition of the Redfield relaxation tensor that makes possible its direct application to the density matrix without explicit construction of the full tensor. In the resulting expressions, only ordinary matrices are involved and so any quantum system whose Hamiltonian can be diagonalized can be treated with the full Redfield theory. To efficiently solve the equation of motion for the density matrix, we introduce a generalization of the short‐iterative‐Lanczos propagator. Together, these contributions allow the complete Redfield theory to be applied to significantly larger systems than was previously possible. Several model calculations are presented to illustrate the methodology, including one example with 172 quantum states.
273 citations
TL;DR: In this paper, the authors used the quasiadiabatic propagator path integral (QUAPI) methodology to evaluate the flux-flux correlation function whose time integral determines the rate coefficient.
Abstract: We present accurate fully quantum calculations of thermal rate constants for a symmetric double well system coupled to a dissipative bath. The calculations are performed using the quasiadiabatic propagator path integral (QUAPI) methodology to evaluate the flux–flux correlation function whose time integral determines the rate coefficient. The discretized path integral converges very rapidly in the QUAPI representation, allowing efficient calculation of quantum correlation functions for sufficiently long times. No ad hoc assumption is introduced and thus these calculations yield the true quantum mechanical rate constants. The results presented in the paper demonstrate the applicability of the QUAPI methodology to practically all regimes of chemical interest, from thermal activation to deep tunneling, and the quantum transmission factor exhibits a Kramers turnover. Our calculations reveal an unusual step structure of the integrated reactive flux in the weak friction regime as well as quantum dynamical enhancement of the rate above the quantum transition state theory value at low temperatures, which is largely due to vibrational coherence effects. The quantum rates are compared to those obtained from classical trajectory simulations. We also use the numerically exact classical and quantum results to establish the degree of accuracy of several analytic and numerical approximations, including classical and quantum Grote–Hynes theories, semiclassical transition state theory (periodic orbit) estimates, classical and quantum turnover theories, and the centroid density approximation.
268 citations
TL;DR: The generalized complex Ginzburg-Landau equation (GGLE) is a model for fluid turbulence described by the incompressible Navier-Stokes equations as mentioned in this paper, which is a dissipative version of the Hamiltonian nonlinear Schrodinger equation possessing solutions that form localized singularities.
158 citations
TL;DR: In this article, generalizations of the decomposition method are discussed and results for the theory and applications of the method are presented for the application of decomposition methods to a wide range of problems.
Abstract: Recent generalizations are discussed and results are presented for the theory and applications of the decomposition method. Application is made to the Duffing equation with an error of 0.0001% in only four terms and less than 10−16 in thirteen terms of the decomposition series. Application is also made to a dissipative wave equation, a matrix Riccati equation, and advection-diffusion nonlinear transport.
150 citations
TL;DR: A subclass of algorithms which retain these strong notions of nonlinear stability and long-term dissipative behavior is identified which, in addition, has the remarkable property of being linear within the time step.
144 citations
TL;DR: In this article, it was shown that chaos can occur in a sinusoidally driven second-order circuit made of three linear elements and a Chua's diode, which is simpler from a circuit-theoretic point of view.
Abstract: We show by computer simulation that chaos can occur in a sinusoidally driven second-order circuit made of three linear elements and a Chua's diode. Unlike many other nonautonomous second-order chaotic circuits whose nonlinear element is a nonlinear capacitor, the Chua's diode is a nonlinear resistor, and is therefore simpler from a circuit-theoretic point of view. >
134 citations
TL;DR: For a dissipative system with Ohmic friction, this work obtains a simple and exact solution for the wave function of the system plus the bath, described by the direct product in two independent Hilbert space.
Abstract: For a dissipative system with Ohmic friction, we obtain a simple and exact solution for the wave function of the system plus the bath. It is described by the direct product in two independent Hilbert spaces. One of them is described by an effective Hamiltonian, the other represents the effect of the bath, i.e., the Brownian motion, thus clarifying the structure of the wave function of the system whose energy is dissipated by its interaction with the bath. No path-integral technology is needed in this treatment. The derivation of the Weisskopf-Wigner linewidth theory follows easily.
TL;DR: In this paper, the numerical approximation of dissipative initial value problems by fixed time-stepping Runge-Kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared.
Abstract: The numerical approximation of dissipative initial value problems by fixed time-stepping Runge–Kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared. A general class of ordinary differential equations, for which dissipativity is induced through an inner product, is studied throughout. This class arises naturally in many finite dimensional applications (such as the Lorenz equations) and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics.
It is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step-size h > 0. The numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor A_h exists; upper-semicontinuity of A_h at h = 0 is established, which shows that, for h small, every point on the numerical attractor is close to a point on the true global attractor A. Under the additional assumption that the problem is globally Lipschitz, it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of A_h at h = 0 is again established.
For gradient systems with globally Lipschitz vector fields it is shown that any Runge–Kutta method preserves the gradient structure for h sufficiently small. For general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small. Convergence of the numerical attractor is studied and, for a dissipative gradient system with hyperbolic equilibria, lower semicontinuity at h = 0 is established. Thus, for such a system, A_h converges to A in the Hausdorff metric as h → 0.
TL;DR: A microscopic many-body QED theory for dipole-dipole resonance energy transfer has been developed from first principles and addresses cases where the surrounding medium is either absorbing or lossless over the range of energies transferred.
Abstract: A microscopic many-body QED theory for dipole-dipole resonance energy transfer has been developed from first principles. A distinctive feature of the theory is full incorporation of the dielectric effects of the supporting medium, in the context of an interplay between what have traditionally been regarded as radiationless and radiative excitation transfer. The approach employs the concept of bath polaritons mediating the energy transfer. The transfer rate is derived in terms of the Green's operator corresponding to the polariton matrix Hamiltonian. In contrast to the more common lossless polariton models, the present theory accommodates an arbitrary number of energy levels for each molecule of the medium. This includes, in particular, a case of special interest, where the excitation energy spectrum of the bath molecules is sufficiently dense that it can be treated as a quasicontinuum in the energy region in question, as in the condensed phase normally results from homogeneous and inhomogeneous line broadening. In such a situation, the photon ``dressed'' by the medium polarization (the polariton) acquires a finite lifetime, the role of the dissipative subsystem being played by bath molecules. It is this which leads to the appearance of the exponential decay factor in the microscopically derived pair transfer rates. Accordingly, the problem associated with potentially infinite total ensemble rates, due to the divergent ${\mathit{R}}^{\mathrm{\ensuremath{-}}2}$ contribution, is solved from first principles. In addition, the medium modifies the distance dependence of the energy transfer function A(R) and also produces extra modifications due to screening contributions and local field effects. The formalism addresses cases where the surrounding medium is either absorbing or lossless over the range of energies transferred. In the latter case the exponential factor does not appear and the dielectric medium effect in the near zone reduces to that which is familiar from the theory of radiationless (F\"orster) energy transfer.
TL;DR: In this article, it was shown that in deterministic dynamical systems any orbit is associated with an invariant spectrum of stretching numbers, i.e. numbers expressing the logarithmic divergences of neighbouring orbits within one period.
Abstract: We show that in deterministic dynamical systems any orbit is associated with an invariant spectrum of stretching numbers, i.e. numbers expressing the logarithmic divergences of neighbouring orbits within one period. The first moment of this invariant spectrum is the maximal Lyapunov characteristic number (LCN). In the case of a chaotic domain, a single invariant spectrum characterizes the whole domain. The invariance of this spectrum allows the estimation of the LCN by calculating, for short times, many orbits with initial conditions in the same chaotic region instead of calculating one orbit for extremely long times. However, if part of the initial conditions are in an ordered region, the average of the short-time calculations may deviate considerably from the LCN. Invariant spectra appear not only for conservative but also for dissipative systems. A few examples are given.
Abstract: Nonclassical states of light may be generated by processes involving the creation or annihilation of photons in pairs. A quadratic coupling, characteristic of a parametric amplifier, generates a squeezed vacuum from a normal vacuum, and a two-photon absorber can also generate a squeezed state (though not a minimum-uncertainty state) even though it is a purely dissipative process. We consider here the simultaneous action of a quadratic pump on a two-photon absorber and demonstrate how superpositions of distinct coherent states may be generated by their combined effects. We use standard master equations to describe the time development, employing split operators and direct numerical integrations to determine the field density-matrix elements and quasiprobabilities. The purities of the nonclassical states are determined by evaluating the field entropy. Provided one-photon dissipative processes may be ignored, a pure superposition state is formed in the steady state. This superposition is destroyed if one-photon loss processes are important.
TL;DR: In this paper, dissipative types of stable soliton structures can exist in nonlinear optical media with broadband gain and group-velocity dispersion (GVD), which resemble ionization or combustion waves and are essentially self-accelerating pulses with a stationary envelope form and a permanently shifting wave spectrum.
Abstract: It is found that dissipative types of stable soliton structures can exist in nonlinear optical media with broadband gain and group-velocity dispersion (GVD). These structures resemble ionization or combustion waves and are essentially self-accelerating pulses with a stationary-envelope form and a permanently shifting wave spectrum. Contrary to the conservative solitons, the dissipative ones exist for any sign of GVD. Being an attractor in the development of arbitrary initial distributions, the dissipative structures cause the fundamental Schr\"odinger solitons to disappear in the course of evolution in weakly nonconservative systems.
TL;DR: Nonlinear chemically dissipative mechanisms have been proposed as providing a possible underlying process for some aspects of biological self-organization, pattern formation, and morphogenesis.
Abstract: Nonlinear chemically dissipative mechanisms have been proposed as providing a possible underlying process for some aspects of biological self-organization, pattern formation, and morphogenesis. Nonlinearities during the formation of microtubular solutions result in a chemical instability and bifurcation between pathways leading to macroscopically self-organized states of different morphology. The self-organizing process, which contains reactive and diffusive contributions, involves chemical waves and differences in microtubule concentration in the sample. Patterns of similar appearance are observed at different distance scales. This behavior is in agreement with theories of chemically dissipative systems.
TL;DR: In this article, the density matrix theory is utilized for the description of ultra fast optical properties and related vibrational wave packet dynamics of molecular systems in condensed media, and the complete theoretical description has been carried out in a representation of the vibration wave functions of the diabatic states which refer to the two coupled vibrational surfaces.
Abstract: The density matrix theory is utilized for the description of ultra fast optical properties and related vibrational wave packet dynamics of molecular systems in condensed media. As an example, optically induced vibrational wave packets in the so‐called curve–crossing system are considered. Such a system goes beyond the standard treatment of optical phenomena since the vibrational wave packet moves in a double well potential and is subject to environmental influences like wave function dephasing and relaxation. The complete theoretical description has been carried out in a representation of the vibrational wave functions of the diabatic states which refer to the two coupled vibrational surfaces. Solving the corresponding density matrix equations by numerical methods allows us to incorporate the static coupling between the crossed surfaces in a nonperturbative manner. Standard projection operator technique is used to treat environmental contributions up to the second order. For the case of a bilinear couplin...
TL;DR: In this paper, the authors examine a system in which both coherent driving and dissipative damping involve the simultaneous creation and annihilation of pairs of photons, and compare the results of the simulation methods with each other and with density-matrix calculations.
Abstract: Recent work on the dynamics of open systems has shown how the density operator may be unraveled into component state-vector trajectories using quantum-state diffusion or the quantum-jump model. In traditional dissipative environments the coherent evolution is stochastically perturbed by the action of the reservoir or environment, so that superposition states are dephased. We examine a system in which both coherent driving and dissipative damping involve the simultaneous creation and annihilation of pairs of photons. This has unusual consequences for the creation and decay of coherences. We analyze this problem using the two recently proposed simulation methods and compare the results of the simulation methods with each other and with density-matrix calculations. We also demonstrate the formation of Schr\"odinger ``cat'' states of the field through the action of dissipation and depict them using the Wigner and Husimi quasiprobability functions.
TL;DR: Based on the numerical solution of the Liouville-von Neumann equation for dissipative systems, the photodesorption dynamics of NO/Pt(111) are studied in this article.
01 Jan 1994
TL;DR: In this article, Lagrangian and Hamiltonian formalism for reversible nonequilibrium fluids with heat flow is presented. But it does not address the problem of minimum dissipation in presence of convection and chemical reactions.
Abstract: Preface. Introduction: aims and scope. 1. Physical significance of Noether's symmetries and extremum principles. 2. Lagrangian and Eulerian descriptions of perfect fluids. 3. Conservation laws for given system of equations. 4. Thermodynamics and kinetics of nonequilibrium fluids. 5. Lagrangian and Hamiltonian formalism for reversible nonequilibrium fluids with heat flow. 6. Extended reversible problem involving mass diffusion, heat flow and thermal inertia. 7. A generalized action with dissipative potentials. 8. Thermohydrodynamic potentials and geometries: the union of thermodynamics and hydromechanics. 9. Intrinsic symmetries and conservation of mass in chemically reacting systems. 10. Conservation laws as given constraints for processes at mechanical equilibrium. 11. Generalized minimum dissipation in presence of convection and chemical reactions. 12. Some associated relativistic results. References. Glossary of principal symbols. Index.
TL;DR: In this article, a detailed investigation of the bifurcation and chaos phenomenon associated with the simplest sinusoidally driven dissipative second-order circuit made up of three linear circuit elements and a Chua's diode is presented.
Abstract: We present a detailed investigation of the bifurcation and chaos phenomenon associated with the simplest sinusoidally driven dissipative second-order circuit made up of three linear circuit elements and a Chua’s diode. Unlike other non-autonomous second-order chaotic circuits whose nonlinear element is a nonlinear capacitor/inductor, the Chua’s diode of this circuit is a nonlinear resistor, and is therefore simpler from a circuit theoretic point of view. The chaotic dynamics of this circuit is confirmed both by experiments and by computer simulation of the circuit model.
TL;DR: In this paper, it was shown that for a dissipative, three dimensional, competitive, and irreducible system of ordinary differential equations having a unique equilibrium point, at which point the Jacobian matrix has negative determinant, either the equilibrium point is stable or there exists an orbitally stable periodic orbit.
TL;DR: In this paper, the Kolmogorov relation for the third order structure function is used to derive the energy spectrum in the far dissipation range (k\ensuremath{\rightarrow}\ensureMath{\infty) which contains no unspecified constants.
Abstract: The Kolmogorov relation for the third order structure function is used to derive the energy spectrum in the far dissipation range (k\ensuremath{\rightarrow}\ensuremath{\infty}). This contains no unspecified constants. Using methods from matched asymptotic expansions and mild analyticity assumptions, a uniformly valid form for the inertial through the dissipative ranges is obtained. An analogous energy spectrum is presented. This is compared with the results of physical and numerical experiments on the energy spectra E(k). The theoretical predictions are found to deviate by not more than a few percent from the measured data in the entire range of wave numbers where the energy spectrum E(k) varies by more than 30 orders of magnitude.
01 Jan 1994
TL;DR: In this article, the authors considered the problem of strain wave propagation in nonlinearly elastic wave guides and solved the porblem of wave motion in a rod, having slowly varying cross-section or elastic moduli.
Abstract: Strain wave propagation in nonlinearly elastic wave guides is considered. The general idea is how, starting from the first principles, to reduce the initial highly nonlinear elastic wave problem governed by coupled p.d.e. to the only one “double dispersion” equation, describing longitudinal strain waves in a one-dimensional wave guide, e.g., in a rod. This study is aimed to arrange real physical experiments including generation, detection and observation of strain solitary waves in solids. The porblem of wave motion in a rod, having slowly varying cross-section or elastic moduli is solved. Special attention is paid to nonlinear dissipative waves, corresponding to the wave motion in a rod embedded into an active or a dissipative medium.
TL;DR: A systematic method of deriving the equation of motion for interacting fronts or pulses in one dimension is developed, applicable to both dissipative and dispersive systems.
Abstract: We develop a systematic method of deriving the equation of motion for interacting fronts or pulses in one dimension. The theory is applicable to both dissipative and dispersive systems. In the case of the time-dependent Ginzburg-Landau equation, which is a typical example of a dissipative system, the front equation obtained is the same as has been obtained previously. The pulse interaction is also derived for the Korteweg\char21{}de Vries equation, emphasizing the difference between the cases with and without dissipative terms.
TL;DR: Pattern formation exhibited by a two-dimensional reaction-diffusion system in the fast inhibitor limit is considered from the point of view of interface motion and a dissipative nonlocal equation of motion for the boundary between high and low concentrations of the slow species is derived heuristically.
Abstract: Pattern formation exhibited by a two-dimensional reaction-diffusion system in the fast inhibitor limit is considered from the point of view of interface motion. A dissipative nonlocal equation of motion for the boundary between high and low concentrations of the slow species is derived heuristically. Under these dynamics, a compact domain of high concentration may develop into a space-filling labyrinthine structure in which nearby fronts repel. Similar patterns have been observed recently by Lee, McCormick, Ouyang, and Swinney in a reacting chemical system.
TL;DR: In this article, a method is proposed whereby the full state vector of a chaotic system can be reconstructed and tracked using only the time series of a single observed scalar, assuming that an accurate mathematical description of the system is available.
Abstract: A method is proposed whereby the full state vector of a chaotic system can be reconstructed and tracked using only the time series of a single observed scalar. It is assumed that an accurate mathematical description of the system is available. Noise effects on the procedure are investigated using as an example a kicked mechanical system which results in a four-dimensional dissipative map.
TL;DR: In this article, an order-parameter model for pattern fields is proposed, which is a regularization of the Cross-Newell phase diffusion equation obtained by averaging over the local periodicity of the pattern.
TL;DR: Using a form of linear feedback the authors call dissipative feedback control, it is shown how to use external forcing to control a chaotic dynamical system to a fixed point or an unstable periodic orbit when the location of the fixed point may change slowly with time.
Abstract: Using a form of linear feedback we call dissipative feedback control, we show how to use external forcing to control a chaotic dynamical system to a fixed point or an unstable periodic orbit when the location of the fixed point or unstable periodic orbit may change slowly with time. The ability to follow a desired state of the system by an external control even when that state is slowly varying in time we call tracking. This slow ``drift'' of states is the usual situation in actual experimental realizations of chaotic systems in nonlinear circuits and other physical manifestations, and this drift can be accounted for by providing a slow dynamics for the location of the fixed point or periodic orbit. We discuss the theoretical aspects of this idea and show its feasibility in some experiments with nonlinear circuits with chaotic behavior.