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Lyapunov exponents in one-dimensional disordered systems with long-range memory.

Alexander Iomin1

  • 1Department of Physics, Technion, Haifa 32000, Israel.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 8, 2009
PubMed
Summary

Lyapunov exponents reveal Anderson localization in 1D disordered systems. Positive exponents indicate exponential growth of eigenfunctions, confirming localization.

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Area of Science:

  • Condensed matter physics
  • Statistical mechanics

Background:

  • Anderson localization describes the suppression of wave function motion in disordered systems.
  • Lyapunov exponents are crucial for quantifying the stability and localization properties of dynamical systems.

Purpose of the Study:

  • To investigate Lyapunov exponents in a 1D disordered system with a specific correlation function.
  • To analyze the behavior of eigenfunctions and their derivatives under Anderson localization conditions.

Main Methods:

  • Studied a one-dimensional disordered system with a Gaussian random potential.
  • Considered a correlation function with power-law decay (1/|x|q).
  • Analyzed the moments of eigenfunctions and their derivatives.

Main Results:

  • Obtained exponential growth for the moments of eigenfunctions and their derivatives.
  • Found positive Lyapunov exponents, indicating asymptotic growth rates.
  • Confirmed Anderson localization in the considered system.

Conclusions:

  • The study confirms Anderson localization in a 1D disordered system.
  • Positive Lyapunov exponents are directly linked to the exponential growth of eigenfunctions.
  • The findings provide insights into wave function behavior in disordered media.