Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle

Area of Science:

• Statistical Physics
• Condensed Matter Physics
• Mathematical Physics

Background:

• Master equations describe time evolution on lattices, leading to Fokker-Planck equations in the continuum limit.
• Nonlinear Fokker-Planck equations are widely used but their numerical solutions depend heavily on discretization schemes.
• Existing schemes often yield discrete equations different from the original master equation.

Purpose of the Study:

• To establish a unique and physically motivated discretization scheme for nonlinear Fokker-Planck equations.
• To demonstrate the utility of the Kinetic Interaction Principle (KIP) in this context.
• To provide a method for the correct numerical integration of Fokker-Planck equations.

Main Methods:

• Utilizing the Kinetic Interaction Principle (KIP) to define particle kinetics in many-body systems.
• Deriving a unique master equation from the KIP.
• Analyzing the continuum limit of the derived master equation to recover the general form of nonlinear Fokker-Planck equations.

Main Results:

• The KIP univocally defines a simple master equation.
• This master equation, in the continuum limit, yields the most general form of nonlinear Fokker-Planck equations.
• The derived master equation offers a physically motivated discretization scheme.

Conclusions:

• The Kinetic Interaction Principle provides a direct link between master equations and nonlinear Fokker-Planck equations.
• This principle enables a unique and accurate discretization method for numerical simulations.
• The findings are crucial for advancing research in statistical and condensed matter physics.