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Nonextensive random matrix theory approach to mixed regular-chaotic dynamics.

A Y Abul-Magd1

  • 1Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 11, 2005
PubMed
Summary

We introduce a new nonextensive random matrix theory using Tsallis

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Area of Science:

  • Statistical Mechanics
  • Quantum Chaos
  • Random Matrix Theory

Background:

  • Conventional random matrix theory (RMT) describes systems with complex dynamics.
  • Systems exhibiting mixed regular-chaotic dynamics present challenges for standard RMT.
  • Tsallis entropy offers a framework for nonextensive statistical mechanics.

Purpose of the Study:

  • To formulate a nonextensive random matrix theory applicable to mixed regular-chaotic systems.
  • To investigate the statistical properties of matrix elements and energy levels in such systems.
  • To explore the transition from Wigner to Poisson statistics using Tsallis entropy.

Main Methods:

  • Application of Tsallis' q-indexed entropy to develop a nonextensive random matrix theory.
  • Joint distribution of matrix elements derived by folding conventional RMT with inverse variance distribution.
  • Calculation of level density and spacing distribution for varying entropic index (q).

Main Results:

  • The proposed theory maintains basis invariance but violates matrix element independence.
  • Calculated level densities align with previous findings by Tsallis and collaborators.
  • The nonextensive model accurately describes the initial stage of the transition to Poisson statistics in mixed systems.

Conclusions:

  • The developed nonextensive random matrix theory provides a suitable framework for systems with mixed regular-chaotic dynamics.
  • The model successfully captures the transition from Wigner to Poisson statistics.
  • Results are validated against numerical experiments, confirming the model's descriptive power.

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