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Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...
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H theorem for generalized entropic forms within a master-equation framework.

Gabriela A Casas1, Fernando D Nobre1, Evaldo M F Curado1

  • 1Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil.

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Summary
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The H theorem is proven for generalized entropic forms, extending statistical mechanics to complex systems. This research introduces a novel equation governing system evolution towards equilibrium, applicable to diverse natural phenomena.

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Area of Science:

  • Statistical Mechanics
  • Complex Systems Theory
  • Information Theory

Background:

  • The H theorem is a fundamental concept in statistical mechanics, describing the tendency of systems to evolve towards equilibrium.
  • Generalized entropic forms offer a broader framework than traditional Boltzmann-Gibbs entropy for describing physical systems.
  • Understanding the dynamics of complex systems with non-symmetric transition rates remains a challenge.

Purpose of the Study:

  • To prove the H theorem for generalized entropic forms in discrete systems.
  • To derive a master equation governing the time evolution of probabilities and transition rates.
  • To explore the applicability of generalized entropies to complex systems.

Main Methods:

  • Mathematical derivation of the H theorem for generalized entropies.
  • Development of a master equation for time evolution of transition rates and probabilities.
  • Analysis of the microcanonical ensemble for Boltzmann-Gibbs entropy and symmetric transition rates.

Main Results:

  • The H theorem is successfully proven for generalized entropic forms.
  • A novel equation is identified that drives systems towards equilibrium.
  • For Boltzmann-Gibbs entropy, equilibrium is reached via a single path only with symmetric transition rates.
  • The derived equation supports the H theorem proof for complex systems with multiple equilibrium paths.

Conclusions:

  • Generalized entropic forms are applicable to systems with complex dynamics, including non-symmetric transition rates.
  • The findings extend the applicability of the H theorem to a wider range of natural phenomena.
  • This work provides a theoretical foundation for studying complex systems in statistical mechanics.