Anomalous diffusion and ballistic peaks: A quantum perspective
| Autor: | Bruce J. West, Marco Stefancich, Paolo Grigolini, Paolo Allegrini, Luca Bonci |
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| Rok vydání: | 1998 |
| Předmět: | |
| Zdroj: | Physical Review E. 57:6625-6633 |
| ISSN: | 1095-3787 1063-651X |
| DOI: | 10.1103/physreve.57.6625 |
| Popis: | The quantum kicked rotor and the classical kicked rotor are both shown to have truncated Levy distributions in momentum space, when the classical phase space has accelerator modes embedded in a chaotic sea. The survival probability for classical particles at the interface of an accelerator mode and the chaotic sea has an inverse power-law structure, whereas that for quantum particles has a periodically modulated inverse power law, with the period of oscillation being dependent on Planck's constant. These logarithmic oscillations are a renormalization group property that disappears as \!0 in agreement with the correspondence principle. In the past two decades classical mechanics has emerged as an area of fundamental study in three distinct domains. The most familiar is that of regular, predictable motion, the so-called integrable Hamiltonian systems with Kolmogorov- Arnold-Moser ~KAM! tori on the energy shell. The most exotic are the completely nonintegrable Hamiltonian systems in which trajectories exponentially separate from one an- other. If a dynamical system such as a standard map is fully chaotic, meaning that all the KAM tori have become globally unstable and disintegrated, producing a chaotic sea in phase space then the mean-square momentum of the system in- creases linearly in time. Such classical systems are said to be diffusive and this relation between statistics and dynamics has been understood for nearly two decades @1#. The third and largest category of motion is called weakly chaotic and contains aspects of both regular and chaotic motion in that there are islands of KAM tori in a sea of chaos. The dynami- cal orbits can, rather than exponentially separating as they do in the case of strong chaos, stick to the cantori at the phase space boundary between stable islands and the chaotic sea in weakly chaotic systems @2# with a resulting anomalous dif- fusion, i.e., diffusion that can be either faster or slower than normal. Here we investigate the connection between a noninte- grable classical Hamiltonian system, in the weak chaos case, and its corresponding quantum system. We wish to under- stand the sense in which a classical chaotic solution is the limit of the corresponding quantum solution to the Schro ¨- dinger equation as \!0. The resolution of this question bears on how good any semiclassical approximation is to the solution of quantum problems. For the sake of generality we study the paradigm of how chaos arises in simple Hamil- tonian systems, namely, the standard map @1-3# |
| Databáze: | OpenAIRE |
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