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Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations.

A V Chechkin1, R Gorenflo, I M Sokolov

  • 1Institute for Theoretical Physics, National Science Center, Kharkov Institute of Physics and Technology, Akademicheskaya Street 1, 61108 Kharkov, Ukraine. achechkin@kipt.kharkov.ua

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 22, 2002
PubMed
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We introduce novel fractional diffusion equations to model anomalous diffusion and relaxation. These models capture time-varying diffusion exponents, describing phenomena from superslow diffusion to accelerating superdiffusion.

Area of Science:

  • Physics
  • Mathematical Physics
  • Statistical Mechanics

Background:

  • Anomalous diffusion and relaxation phenomena deviate from classical Brownian motion.
  • Existing models often assume constant diffusion exponents, limiting their applicability to complex systems.
  • Self-affine random processes with a unique Hurst exponent do not fully capture time-dependent diffusion behaviors.

Purpose of the Study:

  • To propose and analyze novel diffusionlike equations with distributed-order fractional derivatives.
  • To provide a kinetic description for anomalous diffusion and relaxation phenomena with time-varying diffusion exponents.
  • To establish the connection between these fractional equations and continuous-time random walk (CTRW) theory.

Main Methods:

  • Formulation of time and space fractional diffusion equations with distributed-order derivatives.

Related Experiment Videos

  • Mathematical proof of the positivity of solutions for the proposed equations.
  • Establishment of the relationship between the fractional diffusion equations and CTRW theory.

    • Main Results:

    • The distributed-order time fractional diffusion equation models retarding subdiffusion, where the diffusion exponent decreases over time, potentially leading to superslow diffusion.
    • The mean square displacement in retarding subdiffusion can grow logarithmically with time.
    • The distributed-order space fractional diffusion equation describes accelerating superdiffusion, characterized by an increasing diffusion exponent over time.

    Conclusions:

    • The proposed distributed-order fractional diffusion equations offer a versatile framework for modeling anomalous transport phenomena.
    • These models successfully describe time-dependent diffusion behaviors, including retarding subdiffusion and accelerating superdiffusion.
    • The established link to CTRW theory provides a deeper understanding of the underlying stochastic processes governing anomalous diffusion.