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Deformed Fokker-Planck equation: Inhomogeneous medium with a position-dependent mass.

Bruno G da Costa1, Ignacio S Gomez2, Ernesto P Borges2

  • 1Instituto Federal de Educação, Ciência e Tecnologia do Sertão Pernambucano, Rua Maria Luiza de Araújo Gomes Cabral s/n, 56316-686 Petrolina, Pernambuco, Brazil.

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Area of Science:

  • Statistical Mechanics
  • Theoretical Physics
  • Nonlinear Dynamics

Background:

  • The Fokker-Planck equation (FPE) describes diffusion processes in various physical systems.
  • Standard FPE models often assume homogeneous media and constant parameters, limiting their applicability.
  • Inhomogeneous media with position-dependent properties present significant challenges for traditional FPE formulations.

Purpose of the Study:

  • To develop a generalized Fokker-Planck equation for inhomogeneous media with position-dependent particle mass.
  • To establish an equivalence between FPE in inhomogeneous media and a deformed FPE in a deformed space.
  • To explore the implications of this formalism for statistical mechanics and diffusion phenomena.

Main Methods:

  • Utilized the Langevin equation to derive the Fokker-Planck equation.
  • Introduced a generalized deformed derivative for arbitrary deformation spaces.
  • Analyzed the consistency with the diffusion equation and nonlinear Langevin approaches.
  • Applied the deformed H-theorem to analyze entropic functionals.

Main Results:

  • Presented a deformed Fokker-Planck equation applicable to inhomogeneous media with position-dependent mass and diffusion coefficients.
  • Demonstrated that the deformed FPE in a deformed space is equivalent to the FPE in an inhomogeneous medium with a constant diffusion coefficient.
  • Showed that the deformed H-theorem decomposes the Boltzmann-Gibbs entropy into particle and medium contributions.
  • Illustrated the formalism using the infinite square well and confining potential models.

Conclusions:

  • The developed deformed Fokker-Planck equation provides a powerful framework for studying diffusion in complex, inhomogeneous media.
  • The formalism offers a unified approach connecting concepts from superstatistics and position-dependent Langevin equations.
  • The results have implications for understanding statistical properties and entropic behavior in non-uniform environments.