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Nonlinear Ehrenfest's urn model.

G A Casas1, F D Nobre1, E 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.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 15, 2015
PubMed
Summary
This summary is machine-generated.

This study modifies Ehrenfest

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Area of Science:

  • Statistical Mechanics
  • Nonlinear Dynamics
  • Complex Systems

Background:

  • The Ehrenfest urn model is a fundamental concept in statistical mechanics for understanding particle distribution.
  • Traditional models often rely on linear transition probabilities, limiting their application to certain complex systems.
  • Anomalous diffusion phenomena require advanced modeling beyond standard linear approaches.

Purpose of the Study:

  • To introduce nonlinear terms into the Ehrenfest urn model's transition probabilities.
  • To investigate the resulting mathematical and physical implications, particularly concerning diffusion and entropy.
  • To analyze the behavior of entropy production in the modified model.

Main Methods:

  • Modification of Ehrenfest urn model transition probabilities with nonlinear terms.
  • Derivation of the continuous limit Fokker-Planck equation.
  • Application of a generalized H theorem for entropy calculation.
  • Analysis of stationary states and entropy production.

Main Results:

  • The modified model yields a Fokker-Planck equation with competing linear and nonlinear diffusion terms.
  • The calculated entropy is a composite form, combining Boltzmann-Gibbs and Tsallis entropies.
  • The stationary state of the Fokker-Planck equation aligns with the entropy extremization condition.
  • Nonlinear contributions significantly impact the entropy production dynamics.

Conclusions:

  • Nonlinear modifications to the Ehrenfest urn model introduce anomalous diffusion characteristics.
  • The study unifies concepts of Boltzmann-Gibbs and Tsallis entropy within a single framework.
  • The findings provide a new perspective on entropy production in systems exhibiting anomalous diffusion.