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Entropy evolution for the Baker map.

Ronald F. Fox1

  • 1School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430.

Chaos (Woodbury, N.Y.)
|June 5, 2003
PubMed
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Gibbs entropy remains constant for the Baker map. Spectral decomposition reveals that initial densities evolve to a maximal entropy state, resolving an entropy conundrum through convergence analysis.

Area of Science:

  • Thermodynamics
  • Dynamical Systems Theory
  • Information Theory

Background:

  • The Baker map is a dynamical system known for its unique properties.
  • Gibbs entropy is a fundamental concept in statistical mechanics and information theory.
  • The Frobenius-Peron operator describes the evolution of probability densities in dynamical systems.

Purpose of the Study:

  • To investigate the behavior of Gibbs entropy under the Baker map.
  • To analyze the evolution of initial densities using spectral decomposition.
  • To resolve the apparent entropy conundrum observed in the Baker map.

Main Methods:

  • Jordan basis spectral decomposition of the Baker Frobenius-Peron operator.
  • Analysis of weak and strong convergence of probability densities.

Related Experiment Videos

  • Utilizing a binary representation for clarity.

    • Main Results:

    • Gibbs entropy is demonstrated to be invariant for the Baker map.
    • Any initial density was shown to evolve towards a stationary density with maximal entropy.
    • The entropy conundrum was resolved by distinguishing between weak and strong convergence.

    Conclusions:

    • The study clarifies the behavior of entropy in the Baker map, reconciling invariant entropy with the tendency towards maximal entropy states.
    • The findings highlight the importance of convergence types in understanding dynamical systems.
    • Binary representation offers a transparent method for illustrating complex concepts in dynamical systems.