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Quantum Bounds on the Generalized Lyapunov Exponents.

Silvia Pappalardi1, Jorge Kurchan1

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Summary

We introduce generalized quantum Lyapunov exponents (Lq) to quantify chaos. These exponents obey a generalized bound to chaos, with stronger bounds for larger q, limiting chaotic properties.

Keywords:
generalized Lyapunov exponentsquantum bound to chaosquantum chaos

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Area of Science:

  • Quantum mechanics
  • Statistical mechanics
  • Chaos theory

Background:

  • Quantum Lyapunov exponents (Lq) quantify chaos via commutator growth.
  • These exponents relate to the commutator spectrum's thermodynamic limit.
  • A fluctuation-dissipation theorem provides a generalized bound to chaos.

Purpose of the Study:

  • To discuss generalized quantum Lyapunov exponents (Lq).
  • To explore their relation to the commutator spectrum and large deviation functions.
  • To investigate generalized bounds to chaos and their implications for large deviations.

Main Methods:

  • Definition of generalized quantum Lyapunov exponents (Lq) from commutator power growth.
  • Relating Lq to the thermodynamic limit of the commutator spectrum.
  • Applying a Legendre transform to obtain large deviation functions from Lq.
  • Utilizing the fluctuation-dissipation theorem for generalized bounds.
  • Numerical study of the kicked top model at infinite temperature.

Main Results:

  • Generalized quantum Lyapunov exponents (Lq) are defined and discussed.
  • A connection is established between Lq, the commutator spectrum, and large deviation functions.
  • A generalized bound to chaos is shown to be obeyed by Lq, consistent with the fluctuation-dissipation theorem.
  • Stronger bounds for larger q values are demonstrated, limiting chaotic properties.
  • The findings are exemplified by numerical results from the kicked top model.

Conclusions:

  • Generalized quantum Lyapunov exponents provide a robust framework for quantifying quantum chaos.
  • The derived bounds offer insights into the limits of chaotic behavior in quantum systems.
  • The study highlights the interplay between quantum dynamics, thermodynamics, and chaos theory.