Journal of Statistical Physics

, Volume 76, Issue 5–6, pp 1267–1285 | Cite as

Dynamic structure factor in a random diffusion model

  • Frank den Hollander
  • Jan Naudts
  • Frank Redig


Let {Xt:≥0} denote random walk in the random waiting time model, i.e., simple random walk with jump ratew−1(Xt), where {w(x):x∈ℤd} is an i.i.d. random field. We show that (under some mild conditions) theintermediate scattering functionF(q,t)=E0\(e^{iqX_l } \) (q∈ℝd) is completely monotonic int (E0 denotes double expectation w.r.t. walk and field). We also show that thedynamic structure factorS(q, w)=2∫ 0 cos(ωt)F(q, t) exists for ω≠0 and is strictly positive. Ind=1, 2 it diverges as 1/|ω|1/2, resp. −ln(|ω|), in the limit ω→0; ind≥3 its limit value is strictly larger than expected from hydrodynamics. This and further results support the conclusion that the hydrodynamic region is limited to smallq and small ω such that |ω|≫D |q|2, whereD is the diffusion constant.

Key Words

Random walk in random environment dynamic structure factor hydrodynamic limit long-time tail 


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  1. 1.
    L. Van Hove, Correlations in space and time and Born approximation scattering in systems of interacting particles,Phys. Rev. 95:249–262 (1954).Google Scholar
  2. 2.
    G. H. Vineyard, Scattering of slow neutrons by a liquid,Phys. Rev. 110:999–1010 (1958).Google Scholar
  3. 3.
    K. Sköld, J. M. Rowe, G. Ostrowski, and P. D. Randolph, Coherent- and incoherent-scattering laws of liquid argon,Phys. Rev. A 6:1107–1131 (1972).Google Scholar
  4. 4.
    P. E. Egelstaff,Introduction to the Liquid State (Academic Press, New York, 1967).Google Scholar
  5. 5.
    B. J. Alder and T. E. Wainwright, Velocity autocorrelations for hard spheres,Phys. Rev. Lett. 18:988–990 (1967); Decay of the velocity autocorrelation function,Phys. Rev. A 1:18–21 (1970).Google Scholar
  6. 6.
    W. Montfrooij and I. de Schepper, Velocity autocorrelation function of simple dense fluids from neutron scattering experiments.Phys. Rev. A 39:2731–2733 (1989).Google Scholar
  7. 7.
    P. Denteneer and M. H. Ernst, Diffusion in systems with static disorder,Phys. Rev. B 29:1755–1768 (1989).Google Scholar
  8. 8.
    F. den Hollander, J. Naudts, and F. Redig, Long-time tails in a random diffusion model,J. Stat. Phys. 69:731–762 (1992).Google Scholar
  9. 9.
    D. Forster,Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Benjamin, London, 1975).Google Scholar
  10. 10.
    J. P. Boon and S. Yip,Molecular Hydrodynamics (McGraw-Hill, New York, 1980).Google Scholar
  11. 11.
    H. Spohn,Large Scale Dynamics of Interacting Particle Systems (Springer, Berlin, 1991).Google Scholar
  12. 12.
    D. V. Widder,The Laplace Transform (Princeton University Press, Princeton, New Jersey, 1941).Google Scholar
  13. 13.
    C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion,Commun. Math. Phys. 104:1–19 (1986).Google Scholar
  14. 14.
    A. De Masi, P. A. Ferrari, S. Goldstein, and D. W. Wick, An invariance principle for reversible Markov processes. Applications to random motions in random environments,J. Stat. Phys. 55:787–855 (1989).Google Scholar
  15. 15.
    G. Doetsch,Handbuch der Laplace-Transformation, Vol. 1 (Birkhauser, Basel, 1971).Google Scholar
  16. 16.
    N. H. Bingham, C. M. Goldie, and J. L. Teugels,Regular Variation (Cambridge University Press, Cambridge, 1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Frank den Hollander
    • 1
  • Jan Naudts
    • 2
  • Frank Redig
    • 2
    • 3
  1. 1.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands
  2. 2.Departement FysicaUniversiteit AntwerpenAntwerpenBelgium
  3. 3.Gemeenschap “Ster van David”Gooreind-WuustwezelBelgium

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