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Geometric phases in dissipative systems.

Thomas B. Kepler1, Michael L. Kagan, Irving R. Epstein

  • 1Department of Biology and Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02254Department of Chemistry and Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02254.

Chaos (Woodbury, N.Y.)
|December 1, 1991
PubMed
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Geometric phase shifts, analogous to Berry and Hannay effects, are observed in dissipative oscillatory systems. This phenomenon can be detected in numerical simulations of chemical oscillators, offering a unified perspective on geometric phases.

Area of Science:

  • Physics
  • Chemistry
  • Mathematics

Background:

  • Geometric phases, such as Berry and Hannay phases, are fundamental concepts in quantum and classical mechanics.
  • Previous studies primarily focused on conservative systems, leaving the behavior in dissipative systems less explored.

Purpose of the Study:

  • To investigate the occurrence and detection of geometric phase shifts in dissipative oscillatory systems.
  • To develop a unified theoretical framework for understanding geometric phases across different system complexities.

Main Methods:

  • The study employs a theoretical approach starting with simple first-order differential equations on the circle (circle dynamics).
  • Complex systems exhibiting geometric phases are analyzed through reduction to this fundamental circle dynamics.

Related Experiment Videos

  • Numerical simulations are conducted on various model systems, including analytically solvable and realistic chemical oscillators.

    • Main Results:

    • A phenomenon analogous to geometric phase shifts is demonstrated in dissipative oscillatory systems.
    • The occurrence of these phases is confirmed through numerical simulations of chemical oscillators.
    • A unified perspective on geometric phases is achieved by reducing complex systems to circle dynamics.

    Conclusions:

    • Geometric phase shifts are not limited to conservative systems and manifest in dissipative oscillatory systems.
    • The reduction to circle dynamics provides a unifying framework for understanding diverse geometric phase phenomena.
    • Numerical experiments validate the theoretical findings in both simplified and realistic chemical oscillator models.