Stable equilibrium based on Lévy statistics: Stochastic collision models approach

Area of Science:

• Statistical mechanics
• Non-equilibrium physics
• Kinetic theory

Background:

• The Maxwell-Boltzmann distribution describes equilibrium velocities in classical systems.
• Stochastic collision processes can lead to complex, non-equilibrium dynamics.
• Understanding generalized equilibrium distributions is crucial for statistical physics.

Purpose of the Study:

• To investigate equilibrium properties of distinct stochastic collision models.
• To identify generalized velocity distributions beyond the Maxwell-Boltzmann distribution.
• To explore connections between these models and fractional kinetic equations.

Main Methods:

• Analysis of the Rayleigh particle model.
• Analysis of the driven Maxwell gas model.
• Derivation of equilibrium velocity distributions.
• Examination of relationships to fractional kinetic equations.

Main Results:

• Both models yield Lévy distributions at equilibrium, with Maxwell distribution as a specific instance.
• A stable, power-law equilibrium distribution emerges, independent of model specifics.
• The study establishes links between these stochastic models and fractional kinetic equations.

Conclusions:

• Lévy distributions represent a natural generalization of Maxwell's velocity distribution.
• Power-law equilibrium distributions are robust features of certain stochastic collision processes.
• Fractional kinetic equations provide a framework for understanding these generalized equilibria.