Abstract
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.
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References
N. G. van Kampen, Composite stochastic processes,Physica 96A:435–453 (1979).
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).
J. C. Giddings and H. Eyring, A molecular dynamic theory of chromatography,J. Phys. Chem. 59:416–421 (1955).
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).
G. I. Taylor, Diffusion by continuous movements,Proc. Lond. Math. Soc. 20:196–212 (1921).
S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation,Q. J. Mech. Appl. Math. IV:129–156 (1950).
M. Kac, A stochastic model related to the telegrapher's equation,Rocky Mountain J. Math. 4:497–509 (1974).
G. Matheron and G. de Marsily, Is transport in porous media always diffusive? A counterexample,Water Resource Res. 16:901–917 (1980).
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).
B. Gaveau and L. S. Schulman, Anomalous diffusion in a random velocity field,J. Stat. Phys. 66:375–383 (1992).
E. Ben-Naim, S. Redner, and D. ben-Avraham, Bimodal, diffusion in power-law shear flows,Phys. Rev. A 45:7207–7213 (1992).
R. Czech and K. W. Kehr, Depolarization of rotating spins by random walks on lattices,Phys. R. B. 34:261–277 (1986).
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).
G. H. Weiss and R. J. Rubin, Random walks: Theory and selected applications, inAdvances in Chemical Physics, Vol. LII, Section IIA4 (1983).
G. Soto-Campos, Ph.D. thesis, University of Oregon (1995).
G. Soto-Campos, Unpublished.
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.
R. Aris, On the dispersion of a solute in a fluid flowing through a tube,Proc. R. Soc. Lond. A 235:67–77 (1956).
R. M. Mazo, On the Green's function for a one-dimensional random walk,Cell Biophys. 11:19–24 (1987).
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Soto-Campos, G., Mazo, R.M. Asymptotic results for a persistent diffusion model of Taylor dispersion of particles. J Stat Phys 85, 165–177 (1996). https://doi.org/10.1007/BF02175560
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DOI: https://doi.org/10.1007/BF02175560