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Memory-controlled diffusion.

Steffen Trimper1, Knud Zabrocki, Michael Schulz

  • 1Fachbereich Physik, Martin-Luther-Universität, D-06099 Halle, Germany. trimper@physik.uni-halle.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2004
PubMed
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This study introduces memory effects into random-walk models, generalizing the Fokker-Planck equation. Nonlinear memory can halt particle transport, with memory strength influencing diffusion behavior.

Area of Science:

  • Physics
  • Statistical Mechanics
  • Nonlinear Dynamics

Background:

  • Conventional random-walk models lack memory effects.
  • The Fokker-Planck equation describes particle density evolution.
  • Incorporating memory requires extending standard equations.

Purpose of the Study:

  • Generalize the Fokker-Planck equation to include spatial-temporal memory effects.
  • Investigate the impact of nonlinear and nonlocal memory on particle transport.
  • Determine criteria for stationary solutions and analyze long-time diffusion behavior.

Main Methods:

  • Extended the linear Fokker-Planck equation with nonlinear and nonlocal memory kernels.
  • Analyzed the conservation of probability density.
  • Derived criteria for stationary solutions.

Related Experiment Videos

  • Obtained an exact solution for diffusion with nonlinear memory in the long-time limit.

    • Main Results:

    • Nonlinear memory kernels can prevent particle transport by localizing particles due to delay effects.
    • A general criterion for the existence of stationary solutions was established.
    • For nonlinear memory, the mean square displacement shows diffusive behavior.
    • Higher-order cumulants reveal deviations from standard diffusion, dependent on memory strength.

    Conclusions:

    • Spatial-temporal memory effects significantly alter random-walk dynamics.
    • Nonlinear memory introduces novel transport phenomena, including potential localization and termination of transport.
    • The generalized model provides a framework for studying complex diffusion processes with memory.