Generalized Kinetic Equations with Fractional Time-Derivative and Nonlinear Diffusion: H-Theorem and Entropy

Area of Science:

• Mathematical Physics
• Nonlinear Dynamics
• Statistical Mechanics

Background:

• Generalized kinetic equations are crucial for modeling complex systems.
• Understanding entropy production is fundamental in thermodynamics and statistical mechanics.
• Nonlinear diffusion and fractional time-derivatives introduce unique behaviors in kinetic models.

Purpose of the Study:

• To investigate the H-theorem for generalized kinetic equations with fractional time-derivatives and nonlinear diffusion.
• To demonstrate the emergence of different entropic forms due to nonlinearity.
• To analyze the invariance of entropy production and explore anomalous diffusion behaviors.

Main Methods:

• Analytical investigation of the H-theorem.
• Derivation of entropic forms and entropy production.
• Numerical and analytical exploration of equation behaviors.
• Analysis of anomalous diffusion phenomena.

Main Results:

• The H-theorem is satisfied for the considered class of equations.
• Nonlinearity in the equations leads to the emergence of diverse entropic forms.
• The form of entropy production remains invariant despite varying entropic forms.
• A wide range of anomalous diffusion behaviors and their impact on entropy were identified.

Conclusions:

• Generalized kinetic equations with fractional time-derivatives and nonlinear diffusion exhibit rich thermodynamic properties.
• The study highlights the interplay between nonlinearity, anomalous diffusion, and entropy in kinetic theory.
• The findings contribute to a deeper understanding of complex systems described by generalized kinetic equations.