Journal of Statistical Physics

, Volume 85, Issue 1–2, pp 165–177 | Cite as

Asymptotic results for a persistent diffusion model of Taylor dispersion of particles

  • Gerardo Soto-Campos
  • Robert M. Mazo


We study Taylor diffusion for the case when the diffusion transverse to the bulk motion is a persistent random walk on a one-dimensional lattice. This is mapped onto a Markovian walk where each lattice site has two internal states. For such a model we find the effective diffusion coefficient which depends on the rate of transition among internal states of the lattice. The Markovian limit is recovered in the limit of infinite rate of transitions among internal states; the initial conditions have no role in the leading-order time-dependent term of the effective dispersion, but a strong effect on the constant term. We derive a continuum limit of the problem presented and study the asymptotic behavior of such limit.

Key Words

Taylor diffusion non-Markovian processes composite stochastic processes 


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  1. 1.
    N. G. van Kampen, Composite stochastic processes,Physica 96A:435–453 (1979).Google Scholar
  2. 2.
    G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube,Proc. R. Soc. Lond. A 219:186–203 (1953); Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion,Proc. R. Soc. Lond. A 225:473–477 (1954).Google Scholar
  3. 3.
    J. C. Giddings and H. Eyring, A molecular dynamic theory of chromatography,J. Phys. Chem. 59:416–421 (1955).Google Scholar
  4. 4.
    C. Van den Broeck and R. M. Mazo, Exact results for the asymptotic dispersion of particles inn-layer systems,phys. Rev. Lett. 51:1309–1312 (1983); The asymptotic dispersion of particles inN-layer systems,J. Chem. Phys. 81:3624–3634 (1984).Google Scholar
  5. 5.
    G. I. Taylor, Diffusion by continuous movements,Proc. Lond. Math. Soc. 20:196–212 (1921).Google Scholar
  6. 6.
    S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation,Q. J. Mech. Appl. Math. IV:129–156 (1950).Google Scholar
  7. 7.
    M. Kac, A stochastic model related to the telegrapher's equation,Rocky Mountain J. Math. 4:497–509 (1974).Google Scholar
  8. 8.
    G. Matheron and G. de Marsily, Is transport in porous media always diffusive? A counterexample,Water Resource Res. 16:901–917 (1980).Google Scholar
  9. 9.
    R. M. Mazo and C. Van den Broeck, The asymptotic dispersion of particles inN-layered systems: Periodic boundary conditions,J. Chem. Phys. 86:454–459 (1987).Google Scholar
  10. 10.
    B. Gaveau and L. S. Schulman, Anomalous diffusion in a random velocity field,J. Stat. Phys. 66:375–383 (1992).Google Scholar
  11. 11.
    E. Ben-Naim, S. Redner, and D. ben-Avraham, Bimodal, diffusion in power-law shear flows,Phys. Rev. A 45:7207–7213 (1992).Google Scholar
  12. 12.
    R. Czech and K. W. Kehr, Depolarization of rotating spins by random walks on lattices,Phys. R. B. 34:261–277 (1986).Google Scholar
  13. 13.
    R. M. Mazo and C. Van den Broeck, Spectrum of an oscillator making a random walk in frequency,Phys. Rev. B 34:2364–2374 (1986).Google Scholar
  14. 14.
    G. H. Weiss and R. J. Rubin, Random walks: Theory and selected applications, inAdvances in Chemical Physics, Vol. LII, Section IIA4 (1983).Google Scholar
  15. 15.
    G. Soto-Campos, Ph.D. thesis, University of Oregon (1995).Google Scholar
  16. 16.
    G. Soto-Campos, Unpublished.Google Scholar
  17. 17.
    N. N. Bogoliubov, InStudies in Statistical Mechanics, Vol. I, J. de Boer and G. E. Uhlenbeck, eds. (North-Holland Amsterdam 1962) N. N. Bogolubov, Jr., B. I. Sadovnikov, and A. S. Shumovsky,Mathematical Methods of Statistical Mechanics of Model Systems (CRC, Boca Raton, Florida, 1994), p. 172.Google Scholar
  18. 18.
    R. Aris, On the dispersion of a solute in a fluid flowing through a tube,Proc. R. Soc. Lond. A 235:67–77 (1956).Google Scholar
  19. 19.
    R. M. Mazo, On the Green's function for a one-dimensional random walk,Cell Biophys. 11:19–24 (1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Gerardo Soto-Campos
    • 1
  • Robert M. Mazo
    • 1
  1. 1.Chemistry Department and Institute of Theoretical ScienceUniversity of OregonEugene

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