Classical chaos

Silver, K.E., Chaos Classicism. Art in France, Italy, and Germany, igi8-igs6 (New York, Guggenheim Museum Pubhcations, 2010). 

In coimection with the energy transfer modes, an important question, to which we now turn, is the significance of classical chaos in the long-time energy flow process, in particnlar the relative importance of chaotic classical dynamics, versus classically forbidden processes involving dynamical tuimelling . 

The question of non-classical manifestations is particularly important in view of the chaos that we have seen is present in the classical dynamics of a multimode system, such as a polyatomic molecule, with more than one resonance coupling. Chaotic classical dynamics is expected to introduce its own peculiarities into quantum spectra [29, 77]. In Fl20, we noted that chaotic regions of phase space are readily seen in the classical dynamics corresponding to the spectroscopic Flamiltonian. Flow important are the effects of chaos in the observed spectrum, and in the wavefiinctions of tire molecule In FI2O, there were some states whose wavefiinctions appeared very disordered, in the region of the... 

Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York Springer)... 

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

Stehle, P. (1994) Order, Chaos, Order The Transition from Classical to Quantum Physics (Oxford University Press, Oxford) pp. 55, 123. 

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

I.Guarneri, Relevance of classical chaos in quantum mechanics the hydrogen atom in a monochromatic field , Physics Reports 154 (1987) 77. 

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. 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

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. 

The results of the previous section have already established that classical chaos and quantum mechanics are not incompatible in the macroscopic limit. The question then naturally arises whether observed quantum mechanical systems can be chaotic far from the classical limit This question is particularly significant as closed quantum mechanical systems are not chaotic, at least in the conventional sense of dynamical systems theory (R. Kosloff et.al., 1981 1989). In the case of observed systems it has recently been shown, by defining and computing a maximal Lyapunov exponent applicable to quantum trajectories, that the answer is in the affirmative (S. Habib et.al., 1998). Thus, realistic quantum dynamical systems are chaotic in the conventional sense and there is no fundamental conflict between quantum mechanics and the existence of dynamical chaos. 

Similarities with classical waves are considered. In particular we propose that the networks of electric resonance RLC circuits may be used to study wave chaos. However, being different from quantum billiards there is a resistance from the inductors which gives rise to heat power and decoherence. 

The nature of quantum chaos in a specific system is traditionally inferred from its classical counterpart. It is an interdisciplinary field that extends into, for example, atomic and molecular physics, condensed matter physics, nuclear physics, and subatomic physics (H.-... 

Gutzwiller, M. C. Chaos in Classical and Quantum Mechanics Springer-Verlag, New York, 1990. 

For the past three decades deterministic classical systems with chaotic dynamics have been the subject of extensive study (Chirikov, 1979)-(Sagdeev et. al., 1988). Dynamical chaos is a phenomenon peculiar to the deterministic systems, i.e. the systems whose motion in some state space is completely determined by a given interaction and the initial conditions. Under certain initial conditions the behaviour of these systems is unpredictable. 

It should be noted that there is a limited number of works on classical relativistic dynamical chaos (Chernikov et.al., 1989 Drake and et.al., 1996 Matrasulov, 2001). However, the study of the relativistic systems is important both from fundamental as well as from practical viewpoints. Such systems as electrons accelerating in laser-plasma accelerators (Mora, 1993), heavy and superheavy atoms (Matrasulov, 2001) and many other systems in nuclear and particle physics are essentially relativistic systems which can exhibit chaotic dynamics and need to be treated by taking into account relativistic dynamics. Besides that interaction with magnetic field can also strengthen the role of the relativistic effects since the electron gains additional velocity in a magnetic field. 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

Reichl, L. E. (1992), The Transition to Chaos in Conservative Classical Systems Quantum Manifestations, Springer-Verlag, New York. 

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