• Українська
  • English

< | >

Current issue   Ukr. J. Phys. 2017, Vol. 62, N 6, p.508-517
https://doi.org/10.15407/ujpe62.06.0508    Paper

Nazarenko A.V.1, Blavatska V.2

1 Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(4-b, Metrologichna Str., Kyiv 03680, Ukraine; e-mail: nazarenko@bitp.kiev.ua)
2 Institute for Condensed Matter Physics, Nat. Acad. of Sci. of Ukraine
(1, Svientsitskii Str., Lviv 79011, Ukraine; e-mail: viktoria@icmp.lviv.ua)

Asymmetric Random Walk in a One-Dimensional Multizone Environment

Section: Soft Matter
Original Author's Text:  English

Abstract: We consider a random walk model in a one-dimensional environment formed by several zones of finite widths with fixed transition probabilities. It is assumed that the transitions to the left and right neighboring points have unequal probabilities. In the continuous limit, we derive analytically the probability distribution function, which is mainly determined by a walker diffusion and a drift and takes perturbatively the interface effects between zones into account. It is used for computing the probability to find a walker at a given space-time point and the time dependence of the mean squared displacement of a walker trajectory, which reveals the transient anomalous diffusion. To justify our approach, the probability function is compared with the results of numerical simulations for the case of three-zone environment.

Key words: random walk, inhomogeneous environment, diffusion, advection.

References:

  1. A.V. Nazarenko, V. Blavatska. A one-dimensional random walk in a multi-zone environment. J. Phys. A: Math. Theor. 50, 185002 (2017).
    https://doi.org/10.1088/1751-8121/aa6466
  2. M. Ascher. Explicit solutions of the one-dimensional heat equation for a composite wall. Math. Comp. 14, 346 (1960).
    https://doi.org/10.1090/S0025-5718-60-99228-0
  3. G. Lehner. One-dimensional random walk with a partially reflecting barrier. Ann. Math. Statist. 34, 405 (1963).
    https://doi.org/10.1214/aoms/1177704151
  4. H.S. Gupta. ???????. J. Math. Sci. 1, 18 (1966).
  5. J.E. Tanner. Transient diffusion in a system partitioned by permeable barriers. Application to NMR measurements with a pulsed field gradient. J. Chem. Phys. 69, 1748 (1978).
    https://doi.org/10.1063/1.436751
  6. O.E. Percus, J.K. Percus. One-dimensional random walk with phase transition. SIAM J. Appl. Math. 40, 485 (1981);
    https://doi.org/10.1137/0140041
    O. E. Percus. Phase transition in one-dimensional random walk with partially reflecting boundaries . Adv. Appl. Prob. 17, 594 (1985).
    https://doi.org/10.1017/S000186780001524X
  7. P.S. Burada, P. H¨anggi, F. Marchesoni, G. Schmid, P. Talkner. Diffusion in confined geometries. Chem. Phys. Chem. 10, 45 (2009).
    https://doi.org/10.1002/cphc.200800526
  8. D.S. Novikov, E. Fieremans, J.H. Jensen, J.A. Helpern. Random walk with barriers. Nat. Phys. 7, 508 (2011).
  9. J.G. Powels, M.J.D. Mallett, G. Rickayzen, W.A.B. Evans. Exact analytic solutions for diffusion impeded by an infinite array of partially permeable barriers. Proc. R. Soc. Lond. A 436, 391 (1992).
    https://doi.org/10.1098/rspa.1992.0025
  10. See, e.g., M.F. Shlesinger, B. West (ed.) Random Walks and their Applications in the Physical and Biological Sciences (AIP Conf. Proc., vol. 109) (AIP, 1984); F. Spitzer. Principles of Random Walk (Springer, 1976).
  11. H.C. Berg. Random Walks in Biology (Princeton University Press, 1983).
  12. E.A. Codling, M.J. Plank, S. Benhamou. Random walk models in biology. J. R. Soc. Interface 5, 813 (2008).
    https://doi.org/10.1098/rsif.2008.0014
  13. A.R.A. Anderson, M.A.J. Chaplain, E.L. Newman, R.J.C. Steele, A.M. Thompson. Mathematical modelling of tumour invasion and metastasis. J. Theor. Med. 2, 129 (2000).
    https://doi.org/10.1080/10273660008833042
  14. S.C. Ferreira, jr., M.L. Martins, M.J. Vilela. Reactiondiffusion model for the growth of avascular tumor. Phys. Rev. E 65, 021907 (2002).
    https://doi.org/10.1103/PhysRevE.65.021907
  15. P.J. Murray, C.M. Edwards, M.J. Tindall, P. K. Maini. From a discrete to a continuum model of cell dynamics in one dimension. Phys. Rev. E 80, 031912 (2009).
    https://doi.org/10.1103/PhysRevE.80.031912
  16. M.J. Simpson, K.A. Landman, B.D. Hughes. Cell invasion with proliferation mechanisms motivated by timelapse data. Physica A 389, 3779 (2010).
    https://doi.org/10.1016/j.physa.2010.05.020
  17. S. Havlin, D. Ben Abraham. Diffusion in disordered media. Phys. Adv. 36, 695 (1987).
    https://doi.org/10.1080/00018738700101072
  18. R. Metzler, J.-H. Jeon, A.G. Cherstvy. Non-Brownian diffusion in lipid membranes: Experiments and simulations. Acta BBA-Biomembr. 1858, 2451 (2016).
    https://doi.org/10.1016/j.bbamem.2016.01.022
  19. S.V. Patankar. Numerical Heat Transfer and Fluid Flow (McGraw-Hill, 1980).
  20. J.S. P’erez Guerro, L.C.G. Pimentel, T.H. Skaggs, M.Th. van Genuchten. Analytical solution for the advection–dispersion transport equation in layered media. Int. J. Heat Mass Trans. 52, 3297 (2009).
  21. A. Kumar, D. Kumar Jaiswal, N. Kumar. Analytical solutions to one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media. J. Hydrol. 380, 330 (2010);
    https://doi.org/10.1016/j.jhydrol.2009.11.008
    A. Kumar, D. Kumar Jaiswal, R.R Yadav. Analytical solutions of one-dimensional temporally dependent advection-diffusion equation along longitudinal semiinfinite inhomogeneous porous domain for uniform flow. IOSR J. Math. 2, 1 (2012).
    https://doi.org/10.9790/5728-0210111
  22. R.N. Singh. Advection-diffusion equation models in nearsurface geophysical and environmental sciences. J. Ind. Geophys. Union 17, 117 (2013).