Chirikov

In the classical case, the evolution of the kicked rotor dynamics is described by the well-known standard map (Chirikov, 1979). This map greatly facilitates the qualitative treatment of the system. A map describing the evolution of the wave function can be obtained in the quantum case, too (Casati et.al., 1979). In spite of the fact, that the first work with detailed treatment of the quantum kicked rotor appeared 23 years ago (Casati et.al., 1979), this system is still studied extensively (Casati et.al., 1987 Izrailev, 1990). 

As is well known (Chirikov, 1979 Izrailev, 1990), the phase-space evolution of the norelativistic classical kicked rotor is described by nonrelativistic standard map. The analysis of this map shows that the motion of the nonrelativistic kicked rotor is accompanied by unlimited diffusion in the energy and momentum. However, this diffusion is suppressed in the quantum case (Casati et.al., 1979 Izrailev, 1990). Such a suppression of diffusive growth of the energy can be observed when one considers the (classical) relativistic extention of the classical standard map (Nomura et.al., 1992) which was obtained recently by considering the motion of the relativistic electron in the field of an electrostatic wave packet. The relativistic generalization of the standard map is obtained recently (Nomura et.al., 1992)... 

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. 

Cvitanovic P. and Eckhardt B. Phys. Rev. Lett. 63, 823 (1989) Eckhardt B. et al Pinball scattering Quantum chaos between order and disorder, eds G. Casati and B. Chirikov (Cambridge University press, Cambridge, 1995) P. 405. 

Abstract. Classical regular and chaotic dynamics of the particle bound in the Coulomb plus linear potential under the influence of time-periodical perturbations is treated using resonace analysis. Critical value of the external field at which chaotization will occur is evaluated analytically based on the Chirikov criterion of stochasticity. 

In this paper we consider the QCD counterpart of this problem. Namely, we address the problem of regular and chaotic motion in periodically driven quarkonium. Using resonance analysis based on the Chirikov criterion of stochasticity we estimate critical values of the external field strength at which quarkonium motion enters into chaotic regime. 

To estimate the critical value of the external filed strength ecr we use Chirikov s resonance overlap criterion (Zaslavsky, 1988 Jensen, 1984), which can be written as ... 

Thus we have treated the chaotic dynamics of the quarkonium in a time periodic field. Using the Chirikov s resonance overlap criterion we obtain estimates for the critical value of the external field strength at which chaotization of the quarkonium motion will occur. The experimental realization of the quarkonium motion under time periodic perturbation could be performed in several cases in laser driven mesons and in quarkonia in the hadronic or quark-gluon matter. 

Chirikov, B. V. (1979), A Universal Instability of Many-Dimensional Oscillator Systems, Phys. Rep. 52, 263. 

G. Casati, B. V. Chirikov, J. Ford, and F. M. Izrailev, Lectures Notes in Physics (Springer) 93,334 (1979) G. Casati and B. V. Chirikov, Quantum Chaos, Cambridge University Press, Cambridge, 1995. 

G. Casati and B. Chirikov, Quantum Chaos Between Order and Disorder, Cambridge University Press, Cambridge, 1995. 

The plan of Chapter 5 is the following. In order to get a feeUng for the dynamics of the kicked molecule, we approximate it by a one-dimensional schematic model by restricting its motion to rotation in the x, z) plane and ignoring motion of the centre of mass. In this approximation the kicked molecule becomes the kicked rotor, probably the most widely studied model in quantum chaology. This model was introduced by Casati et al. in 1979. The classical mechanics of the kicked rotor is discussed in Section 5.1. Section 5.2 presents Chirikov s overlap criterion, which can be applied generally to estimate analytically the critical control parameter necessary for the onset of chaos. We use it here to estimate the onset of chaos in the kicked rotor model. The quantum mechanics of the kicked rotor is discussed in Section 5.3. In Section 5.4 we show that the results obtained for the quantum kicked rotor model are of immediate... 

The magnitude of the critical control parameter Kc can be estimated analytically with the help of Chirikov s criterion. This criterion is naturally derived in the context of the kicked rotor, and is introduced in the following section. The Chirikov criterion is also the basis for estimating the onset of chaos in many other chaotic atomic physics systems. Examples are presented in Chapters 6 and 7. 

Thus, the resonance angular momenta are given by = 2mTV. The resonance angular momenta correspond exactly to the large elliptical structures seen in the phase-space plots shown in Fig. 5.4. These phase-space structures, by inference called resonances too, are of prime importance for Chirikov s criterion. 

In the case of the kicked rotor we were able to predict the critical perturbation strength by applying the Chirikov overlap criterion. This criterion does two things for us. First, it provides us with an excellent physical picture which explains qualitatively the mechanism of the chaos transition secondly, it provides us with an analytical estimate for the critical field. 

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