Entropy evolution for the Baker map.

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.
• 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.