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Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle.

Giorgio Kaniadakis1, Dionissios T Hristopulos2,3

  • 1Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.

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Summary
This summary is machine-generated.

This study introduces the Kinetic Interaction Principle (KIP) to derive master equations for nonlinear Fokker-Planck equations. This provides a unique, physically motivated discretization scheme for accurate numerical solutions in physics.

Keywords:
Fokker–Planck currentFokker–Planck equationsHaldane statisticsKinetic interaction principleanomalous diffusionboson statisticsfermion statistics

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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.