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Generalized second law for a simple chaotic system.

Hiroshi H Hasegawa1, Tomomi Nakamura1, Dean J Driebe2

  • 1Department of Mathematical Sciences, Ibaraki University, Mito 310-8512, Japan.

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Summary

This study derives the generalized second law for chaotic systems, showing invariant densities reduce maximum extractable work. Unique equilibrium states emerge when densities match canonical or Tsallis distributions, preventing work extraction.

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Area of Science:

  • Thermodynamics
  • Statistical Mechanics
  • Chaos Theory

Background:

  • The generalized second law governs entropy changes in thermodynamic systems.
  • Chaotic systems exhibit complex dynamics and unique statistical properties.
  • Understanding equilibrium states is crucial for energy extraction and system stability.

Purpose of the Study:

  • To derive the generalized second law for a simple chaotic system using a nonequilibrium maximum work formulation.
  • To investigate the role of invariant probability densities in determining maximum extractable work.
  • To identify conditions for unique equilibrium states in chaotic systems.

Main Methods:

  • Consideration of a probability density that weakly converges to an invariant density.
  • Rewriting the generalized second law for an initial invariant density.
  • Extension of the formulation to power-invariant densities, such as the Tsallis distribution.

Main Results:

  • The generalized second law was derived for a simple chaotic system.
  • Invariant densities, having greater entropy than prepared densities, reduce maximum extractable work.
  • Work is unextractable by cyclic operations if the invariant density is a canonical or Tsallis distribution, defining unique equilibrium states.

Conclusions:

  • The study establishes a connection between invariant densities, entropy, and work extractability in chaotic systems.
  • The findings identify specific distributions (canonical and Tsallis) as unique equilibrium states where no further work can be extracted.
  • This work provides a framework for understanding nonequilibrium thermodynamics in chaotic systems.