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Quantifying X-Ray Fluorescence Data Using MAPS
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Incommensurate standard map.

Leonardo Ermann1,2, Dima L Shepelyansky3

  • 1Departamento de Física Teórica, GIyA, Comisión Nacional de Energía Atómica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina.

Physical Review. E
|February 21, 2019
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Summary
This summary is machine-generated.

We introduce the incommensurate standard map, revealing bounded dynamics at low amplitudes and diffusive behavior at higher strengths. Quantum analysis shows spatial metal-insulator transitions and momentum localization, akin to Anderson localization.

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Area of Science:

  • Nonlinear dynamics
  • Quantum chaos
  • Statistical physics

Background:

  • The Chirikov standard map is a fundamental model in nonlinear dynamics and chaos theory.
  • Incommensurate potentials introduce complex behaviors in classical and quantum systems.
  • Previous studies explored similar models with cold atoms in static potentials.

Purpose of the Study:

  • To introduce and analyze the incommensurate standard map with two and three spatial harmonics.
  • To investigate the transition from bounded to diffusive dynamics in the classical system.
  • To explore quantum mechanical behavior, including spatial and momentum localization.

Main Methods:

  • Extension of the Chirikov standard map to include incommensurate spatial harmonics.
  • Analysis of classical dynamics based on kick amplitude.
  • Study of quantum evolution and localization phenomena.

Main Results:

  • Bounded dynamics by Kolmogorov-Arnold-Moser surfaces at small kick amplitudes.
  • Unbounded, diffusive dynamics above a critical kick strength.
  • Quantum evolution exhibits spatial metal-insulator transitions and momentum localization.
  • Three-frequency potential leads to Anderson transition, analogous to 3D disordered systems.

Conclusions:

  • The incommensurate standard map provides a rich model for studying complex dynamics.
  • Quantum behavior mirrors Anderson localization in disordered systems.
  • Potential applications include modeling comet dynamics and dark matter in planetary systems.