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Related Experiment Videos

On dynamical zeta function.

Eugene Bogomolny1

  • 1Division de Physique Theorique,(b)) Institut de Physique Nucleaire, 91406 Orsay Cedex, France.

Chaos (Woodbury, N.Y.)
|January 1, 1992
PubMed
Summary
This summary is machine-generated.

The dynamical zeta function, crucial for chaotic systems, can be simplified using a semiclassical approach. This method approximates the function using a select set of periodic orbits, reducing complexity in quantum chaos studies.

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Area of Science:

  • Quantum chaos
  • Dynamical systems theory
  • Statistical mechanics

Background:

  • The dynamical zeta function is typically an infinite product over primitive periodic orbits.
  • Its semiclassical limit (Planck's constant over 2pi -> 0) offers a potential simplification.

Purpose of the Study:

  • To explore consequences of representing the dynamical zeta function as det(1-T) in chaotic systems.
  • To show the zeta function can be approximated by a subset of selected orbits.

Main Methods:

  • Semiclassical approximation of the dynamical zeta function.
  • Representation using the determinant of the semiclassical Poincaré map operator T(q,q').
  • Analysis of chaotic systems dynamics.

Main Results:

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  • The dynamical zeta function is shown to be equivalent to det(1-T) in the semiclassical limit.
  • An approximation of the zeta function using a specific subset of orbits is derived.
  • The error of this approximation is demonstrated to be small in the semiclassical limit.

Conclusions:

  • The det(1-T) representation provides a powerful tool for studying chaotic systems.
  • A reduced set of orbits can effectively approximate the dynamical zeta function.
  • This work offers insights into the quantum-classical correspondence in chaotic dynamics.