Journal of Statistical Physics

, Volume 34, Issue 3–4, pp 477–495 | Cite as

Long time tails in stationary random media. I. Theory

  • M. H. Ernst
  • J. Machta
  • J. R. Dorfman
  • H. van Beijeren


Diffusion of moving particles in stationary disordered media is studied using a phenomenological mode-coupling theory. The presence of disorder leads to a generalized diffusion equation, with memory kernels having power law long time tails. The velocity autocorrelation function is found to decay like t−(d/2+1), while the time correlation function associated with the super-Burnett coefficient decays liket−d/2 for long times. The theory is applicable to a wide variety of dynamical and stochastic systems including the Lorentz gas and hopping models. We find new, general expressions for the coefficients of the long time tails which agree with previous results for exactly solvable hopping models and with the low-density results obtained for the Lorentz gas. Finally we mention that if the moving particles are charged, then the long time tails imply that there is an ωd/2 contribution to the low-frequency part of the frequency-dependent electrical conductivity.

Key words

Diffusion random media fluctuations long time tails Lorentz model hopping models velocity correlation functions mode coupling theory diffusion coefficients Burnett coefficients 


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Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • M. H. Ernst
    • 1
  • J. Machta
    • 2
  • J. R. Dorfman
    • 3
  • H. van Beijeren
    • 4
  1. 1.Institute for Theoretical PhysicsState UniversityTA UtrechtThe Netherlands
  2. 2.Department of Physics and AstronomyUniversity of MassachusettsAmherst
  3. 3.Institute for Physical Science and Technology and Department of Physics and AstronomyUniversity of MarylandCollege Park
  4. 4.Institute for Theoretical PhysicsRhein-Westf. Techn. HochschuleAachenWest Germany

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