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Analytically solvable model in fractional kinetic theory.

R E Robson1, A Blumen

  • 1Research School of Physical Sciences and Engineering, Australian National University, Canberra 2600, Australia.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 11, 2005
PubMed
Summary

This study introduces a method for memory effects in phase space kinetic equations, specifically using a fractional relaxation time model. It analyzes charge carrier transport, revealing limits for Fick's law and fractional diffusion equations.

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Area of Science:

  • Physics
  • Physical Chemistry
  • Non-equilibrium statistical mechanics

Background:

  • Kinetic equations describe particle behavior.
  • Memory effects and fractional calculus are advanced concepts.
  • Time-of-flight experiments probe charge carrier dynamics.

Purpose of the Study:

  • To develop a general method for incorporating memory effects into phase space kinetic equations.
  • To analyze charge carrier transport using a generalized fractional relaxation time model.
  • To determine the validity limits of Fick's law and fractional diffusion equations.

Main Methods:

  • Incorporating memory effects into phase space kinetic equations.
  • Solving the generalized fractional relaxation time model for charge carriers.

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  • Using the Chapman-Enskog scheme for approximate solutions in the hydrodynamic regime.
  • Obtaining exact solutions without gradient limitations.

    • Main Results:

    • Derived fractional forms of Fick's law and diffusion equations.
    • Obtained exact expressions for observable quantities using generalized Mittag-Leffler functions.
    • Detailed the transition from nonhydrodynamic to hydrodynamic regimes.
    • Established the limits of validity for Fick's law and fractional diffusion equations.

    Conclusions:

    • The generalized fractional relaxation time model accurately describes charge carrier transport with memory effects.
    • Exact solutions provide a comprehensive understanding of transport dynamics.
    • The study clarifies the applicability of Fick's law and fractional diffusion equations in different regimes.