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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
1Departamento de Matemática, Universidade Estadual de Maringá, Av. Colombo, 5790 CEP 87020-900 - Maringá - PR - Brazil.
Physical Review. E
|October 16, 2021
Summary
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.
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.
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