Dissipative dynamics

Baer R and Kosloff R 1997 Quantum dissipative dynamics of adsorbates near metal surfaces a surrogate Hamiltonian theory applied to hydrogen on nickel J. Chem. Rhys. 106 8862... 

Mandelshtam V A and Taylor H S 1997 Spectral analysis of time correlation function for a dissipative dynamical system using filter diagonalization application to calculation of unimolecular decay rates Phys. Rev. Lett. 78 3274... 

But a computer simulation is more than a few clever data structures. We need algorithms to manipulate our system. In some way, we have to invent ways to let the big computer in our hands do things with the model that is useful for our needs. There are a number of ways for such a time evolution of the system the most prominent is the Monte Carlo procedure that follows an appropriate random path through configuration space in order to investigate equilibrium properties. Then there is molecular dynamics, which follows classical mechanical trajectories. There is a variety of dissipative dynamical methods, such as Brownian dynamics. All these techniques operate on the fundamental degrees of freedom of what we define to be our model. This is the common feature of computer simulations as opposed to other numerical approaches. 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

In contrast to dissipative dynamical systems, conservative systems preserve phase-space volumes and hence cannot display any attracting regions in phase space there can be no fixed points, no limit cycles and no strange attractors. There can nonetheless be chaotic motion in the sense that points along particular trajectories may show sensitivity to initial conditions. A familiar example of a conservative system from classical mechanics is that of a Hamiltonian system. 

In many ways, May s sentiment echoes the basic philosophy behind the study of CA, the elementary versions of which, as we have seen, are among the simplest conceivable dynamical systems. There are indeed many parallels and similarities between the behaviors of discrete-time dissipative dynamical systems and generic irreversible CA, not the least of which is the ability of both to give rise to enormously complicated behavior in an attractive fashion. In the subsections below, we introduce a variety of concepts and terminology in the context of two prototypical discrete-time mapping systems the one-dimensional Logistic map, and the two-dimensional Henon map. 

The typical strategy employed in studying the behavior of nonlinear dissipative dynamical systems consists of first identifying all of the periodic solutions of the system, followed by a detailed characterization of the chaotic motion on the attractors. 

Since the phase space of a dissipative dynamical system contracts with time, we know that, in the long time limit, t oo, the motion will be confined to some fixed attractor, A. Moreover, becaust of the contraction, the dimension, D, of A, must be lower than that of the actual phase space. While D adds little information in the case of a noiichaotic attractor (we know immediately, and trivially, for example, that all fixed-points have D = 0, limit cycles have D = 1, 2-tori have D = 2, etc.), it is of significant interest for strange attractors, whose dimension is typically non-integer valued. Three of the most common measures of D are the fractal dimension, information dimension and correlation dimension. 

These equations are formally identical to the equations (19) for the diagonal coordinates of the Langevin equation. The diagonal coordinates are in both cases determined only by the deterministic part of the dynamics. In the Hamiltonian setting they can naturally be identified as coordinates or momenta, which is impossible for the dissipative dynamics. 

Time Correlation Function for a Dissipative Dynamical Systems Using Filter Diagonalization Application to Calculation of Unimolecular Decay Rates. 

In dissipative dynamics, the backward-propagating target operator decays into a stationary operator, and therefore, L = 0. This leads to loss of... 

Models for the dissipative dynamics can frequently be based on the assumption of fast decay of memory effects, due to the presence of many degrees of freedom in the s-region. This is the usual Markoff assumption of instantaneous dissipation. Two such models give the Lindblad form of dissipative rates, and rates from dissipative potentials. The Lindblad-type expression was originally derived using semigroup properties of time-evolution operators in dissipative systems. [45, 46] It has been rederived in a variety of ways and implemented in applications. [47, 48] It is given in our notation by... 

The powerful mathematical tools of linear algebra and superoperators in Li-ouville space can be used to proceed from the identification of molecular phenomena, to modelling and calculation of physical properties to interpret or predict experimental results. The present overview of our work shows a possible approach to the dissipative dynamics of a many-atom system undergoing localized electronic transitions. The density operator and its Liouville-von Neumann equation play a central role in its mathematical treatments. 

D. A. Micha and B. Thorndyke. Dissipative dynamics in many-atom systems a density matrix treatment. Intern. J. Quantum Chem., vv ppp, 2001. submitted. 

Time-Local Quantum Master Equations and their Applications to Dissipative Dynamics and Molecular Wires... 

Fast dissipation is treated numerically within the Markoff approximation, which leads to differential equations in time, and dissipative rates most commonly written in the Redfield [9,10] or Lindblad [11,12] forms. Several numerical procedures have been introduced for dissipative dynamics within the Markoff approximation. The differential equations have been solved using a pseudospectral method [13], expansions of the Liouville propagator in terms of polynomials, [14-16] and continued fractions. [17]... 

We present in Section 2 the formalism giving the equations for the reduced density operator and for competing instantaneous and delayed dissipation. Section 3 presents matrix equations in a form suitable for numerical work, and the details of the numerical procedure used to solve the integrodiffer-ential equations with the two types of dissipative processes. In Section 4 on applications to adsorbates, results are shown for quantum state populations versus time for the dissipative dynamics of CO/Cu(001). The fast electronic relaxation to the ground electronic state is shown first without the slow relaxation of the frustrated translation mode of CO vibrations, for comparison with previous work, and this is followed by results with both fast and slow relaxation. In Section 5 we comment on the general conclusions that can be reached in problems involving both vibrational and electronic relaxation at surfaces. 

G. Stock. A semiclassical self-consistent-field approach to dissipative dynamics - the spin-boson problem. J. Chem. Phys., 103(4) 1561-1573, Jul 1995. 

Y. Pomeau and P. Manneville Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 1980, 74 189-197. 

The trajectories of dissipative dynamic systems, in the long run, are confined in a subset of the phase space, which is called an attractor [32], i.e., the set of points in phase space where the trajectories converge. An attractor is usually an object of lower dimension than the entire phase space (a point, a circle, a torus, etc.). For example, a multidimensional phase space may have a point attractor (dimension 0), which means that the asymptotic behavior of the system is an equilibrium point, or a limit cycle (dimension 1), which corresponds to periodic behavior, i.e., an oscillation. Schematic representations for the point, the limit cycle, and the torus attractors, are depicted in Figure 3.2. The point attractor is pictured on the left regardless of the initial conditions, the system ends up in the same equilibrium point. In the middle, a limit cycle is shown the system always ends up doing a specific oscillation. The torus attractor on the right is the 2-dimensional equivalent of a circle. In fact, a circle can be called a 1-torus,... 

The simplest example of a classical or quantum dissipative system is a particle evolving in a potential V(x) and coupled linearly to a fluctuating dynamical reservoir or bath. If the bath is only weakly perturbed by the system, it can be considered as linear, described by an ensemble of harmonic oscillators. Starting from the corresponding system-plus-bath Hamiltonian and using some convenient approximations, it is possible to get a description of the dissipative dynamics of the system. 

Willems, J. C. Dissipative Dynamical Systems, Part I General Theory," Arch, Rational Mech. Anal., 45, 321-350 (1974). 

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