Nonlinear Ehrenfest's urn model

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.