Nonlinear classical dynamics

Finally, new mathematical developments in the study of nonlinear classical dynamics came to be appreciated by molecular scientists, with applications such as the bifiircation approaches stressed in this section. 

The above is a comprehensive, readable introduction to modern nonlinear classical dynamics, with quantum applications. 

The ar tide is organized as follows. We will begin with a discussion of the various possibilities of dynamical description, clarify what is meant by nonlinear quantum dynamics , discuss its connection to nonlinear classical dynamics, and then study two experimentally relevant examples of quantum nonlinearity - (i) the existence of chaos in quantum dynamical systems far from the classical regime, and (ii) real-time quantum feedback control. 

In nonlinear classical dynamics it is convenient to express the Hamiltonian in action-angle variables. The total Hamiltonian H can then be resolved as... 

A classical mechanical % provides insights into intramolecular dynamics, guided by the powerful diagnostic tools of nonlinear classical dynamics. The action-angle representation of a classical mechanical problem is easily derived from the spectroscopic Heff. In fact, the action-angle picture lies at the core of Heisenberg s version of the Correspondence Principle (Heisenberg, 1925). The prescription is that... 

Analysis of this 7feff using the techniques of nonlinear classical dynamics reveals the structure of phase space (mapped as a continuous function of the conserved quantities E, Ka, and Kb) and the qualitative nature of the classical trajectory that corresponds to every eigenstate in every polyad. This analysis reveals qualitative changes, or bifurcations, in the dynamics, the onset of classical chaos, and the fraction of phase space associated with each qualitatively distinct class of regular (quasiperiodic) and chaotic trajectories. 

To illustrate an application of nonlinear quantum dynamics, we now consider real-time control of quantum dynamical systems. Feedback control is essential for the operation of complex engineered systems, such as aircraft and industrial plants. As active manipulation and engineering of quantum systems becomes routine, quantum feedback control is expected to play a key role in applications such as precision measurement and quantum information processing. The primary difference between the quantum and classical situations, aside from dynamical differences, is the active nature of quantum measurements. As an example, in classical theory the more information one extracts from a system, the better one is potentially able to control it, but, due to backaction, this no longer holds true quantum mechanically. 

After an overview of the main papers devoted to chaos in lasers (Section I.A) and in nonlinear optical processes (Section I.B), we present a more detailed analysis of dynamics in a process of second-harmonic generation of light (Section II) as well as in Kerr oscillators (Section III). The last case we consider particularly in the context of coupled nonlinear systems. Finally, we present a cumulant approach to the problem of quantum corrections to the classical dynamics in second-harmonic generation and Kerr processes (Section IV). 

The prototype potential surface invoked in chemical kinetics is a two-dimensional surface with a saddle equilibrium point and two exit channels at lower energies. The classical and quantal dynamics of such surfaces has been the object of many studies since the pioneering works by Wigner and Polanyi. Recent advances in nonlinear dynamical systems theory have provided powerful tools, such as the concepts of bifurcations and chaos, to investigate the classical dynamics from a new point of view and to perform the semiclassical... 

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978 A. J. Lieberman and A. J. Lichienberg, Regular and Stochastic Motion, Springer, New York, 1983 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. 

Another of Poincare s new methods was the reduction of the continuous phase-space flow of a classical dynamical system to a discrete mapping. This is certainly one of the most useful techniques ever introduced into the theory of dynamical systems. Modem journals on nonlinear dynamics abound with graphical representations of Poincare mappings. A quick glance into any one of these journals will attest to this fact. Because of the usefulness and the formal simplicity of mappings, this topic is introduced and discussed in Section 2.2. 

On the theoretical physics side, the Kolmogorov-Arnold-Moser (KAM) theory for conservative dynamical systems describes how the continuous trajectories of a particle break up into a chaotic sea of randomly disconnected points. Furthermore, the strange attractors of dissipative dynamical systems have a fractal dimension in phase space. Both these developments in classical dynamics—KAM theory and strange attractors—emphasize the importance of nonanalytic functions in the description of the evolution of deterministic nonlinear dynamical systems. We do not discuss the details of such dynamical systems herein, but refer the reader to a number of excellent books on the... 

Two dependent mathematical variables are needed for a network to be capable of periodic behavior. A third is required to permit chaos. It has become quite a sport among the apostles of nonlinear chemical dynamics to invent ever new simple, if not exactly realistic networks that can admit chaos. A classical example, and one of the simplest, is the Hudson-Rossler model [45]. The core of the network is... 

This approach allows for a fully quantum mechanical treatment of the dynamics, avoiding the nse of quantum correction factors used to denote classical dynamical approaches, with the concession that the potential energy surface must be expanded, ignoring higher order nonlinearity in the mode coupling. The potential energy surface is expanded with respect to the normal coordinates of the system, q, and bath, 01, and their freqnencies up to third and fourth order nonlinear conpling ... 

A great deal of attention has been focused in recent years by workers in classical dynamics on the geometric properties of phase space structures and their manifestation on Poincare maps (also referred to as surfaces of section). The result has been the blossoming of a huge literature on the subject of nonlinear dynamics (quasiperiodicity and dynamical chaos), which is discussed in a number of recent textbooks and articles. - ... 

The classical dynamics of molecular models is generated by Hamilton s (or Newton s) equations of motion. In the absence of external, time-dependent forces, and within the Born-Oppenheimer approximation, the dynamics of molecular vibrations, rotations, and reactions conserves the total energy . We therefore restrict our attention in the nonlinear dynamics literature to energy-conserving systems, which are technically referred to as Hamiltonian systems. For the purposes of the present discussion, we restrict our attention to Hamiltonian systems with two degrees of freedom ... 

W L Hase (ed) 1992 Advances in Classical Trajectory Methods. 1. Intramolecular and Nonlinear Dynamics (London JAI)... 

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

Since Laplace transform can only be applied to a linear differential equation, we must "fix" a nonlinear equation. The goal of control is to keep a process running at a specified condition (the steady state). For the most part, if we do a good job, the system should only be slightly perturbed from the steady state such that the dynamics of returning to the steady state is a first order decay, i.e., a linear process. This is the cornerstone of classical control theory. 

To model this highly complex and nonlinear dynamics properly, we need the heat and mass balances. In classical control, however, we would replace them with a linearized model that is the sum of two functions in parallel ... 

Abstract. The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples. 

Keeping the lesson of the above example in mind, we will explore three different dynamical possibilities below isolated evolution, where the system evolves without any coupling to the external world, unconditioned open ev olution, where the system evolves coupled to an external environment but where no information regarding the system is extracted from the environment, and conditioned open evolution where such information is extracted. In the third case, the evolution of the physical state is driven by the system evolution, the coupling to the external world, and by the fact that observational information regarding the state has been obtained. This last aspect - system evolution conditioned on the measurement results via Bayesian inference - leads to an intrinsically nonlinear evolution for the system state. The conditioned evolution provides, in principle, the most realistic possible description of an experiment. To the extent that quantum and classical mechanics are eventually just methodological tools to explain and predict the results of experiments, this is the proper context in which to compare them. 

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