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Stationary solution and H theorem for a generalized Fokker-Planck equation.

Max Jauregui1, Anna L F Lucchi2,3, Jean H Y Passos2,3

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This study explores generalized Fokker-Planck equations, revealing stationary solutions linked to effective potentials and a novel Tsallis entropy form. These findings advance understanding of statistical mechanics and probability distributions.

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Area of Science:

  • Statistical Mechanics
  • Non-linear Dynamics
  • Mathematical Physics

Background:

  • Generalized Fokker-Planck equations model complex systems.
  • Richardson and porous media equations are specific examples.
  • Understanding stationary solutions and thermodynamic properties is crucial.

Purpose of the Study:

  • Investigate a family of generalized Fokker-Planck equations.
  • Analyze stationary solutions and their relation to effective potentials.
  • Verify an H theorem using novel free-energy-like functionals.

Main Methods:

  • Analysis of generalized Fokker-Planck equations with confining drift terms.
  • Derivation of stationary solutions dependent on effective potentials.
  • Construction and verification of free-energy-like functionals for H theorem.

Main Results:

  • Each equation possesses a potential-dependent stationary solution, encompassing known probability distributions.
  • A generalized Tsallis entropic form, dependent on position, is identified.
  • Optimization of this entropy under constraints recovers the stationary solution.

Conclusions:

  • The study establishes a connection between effective potentials, generalized Fokker-Planck equations, and probability distributions.
  • A novel H theorem is demonstrated using a generalized Tsallis entropy.
  • The findings offer new insights into the behavior of complex systems and statistical mechanics.