Asymmetric Random Walk in a One-Dimensional Multizone Environment

  • A. V. Nazarenko Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
  • V. Blavatska Institute for Condensed Matter Physics, Nat. Acad. of Sci. of Ukraine
Keywords: random walk, inhomogeneous environment, diffusion, advection


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.


A.V. Nazarenko, V. Blavatska. A one-dimensional random walk in a multi-zone environment. J. Phys. A: Math. Theor. 50, 185002 (2017).

M. Ascher. Explicit solutions of the one-dimensional heat equation for a composite wall. Math. Comp. 14, 346 (1960).

G. Lehner. One-dimensional random walk with a partially reflecting barrier. Ann. Math. Statist. 34, 405 (1963).

H.S. Gupta. ???????. J. Math. Sci. 1, 18 (1966).

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).

O.E. Percus, J.K. Percus. One-dimensional random walk with phase transition. SIAM J. Appl. Math. 40, 485 (1981);

O.E. Percus. Phase transition in one-dimensional random walk with partially reflecting boundaries . Adv. Appl. Prob. 17, 594 (1985).

P.S. Burada, P. H¨anggi, F. Marchesoni, G. Schmid, P. Talkner. Diffusion in confined geometries. Chem. Phys. Chem. 10, 45 (2009).

D.S. Novikov, E. Fieremans, J.H. Jensen, J.A. Helpern. Random walk with barriers. Nat. Phys. 7, 508 (2011).

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).

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).

H.C. Berg. Random Walks in Biology (Princeton University Press, 1983).

E.A. Codling, M.J. Plank, S. Benhamou. Random walk models in biology. J. R. Soc. Interface 5, 813 (2008).

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).

S.C. Ferreira, jr., M.L. Martins, M.J. Vilela. Reactiondiffusion model for the growth of avascular tumor. Phys. Rev. E 65, 021907 (2002).

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).

M.J. Simpson, K.A. Landman, B.D. Hughes. Cell invasion with proliferation mechanisms motivated by timelapse data. Physica A 389, 3779 (2010).

S. Havlin, D. Ben Abraham. Diffusion in disordered media. Phys. Adv. 36, 695 (1987).

R. Metzler, J.-H. Jeon, A.G. Cherstvy. Non-Brownian diffusion in lipid membranes: Experiments and simulations. Acta BBA-Biomembr. 1858, 2451 (2016).

S.V. Patankar. Numerical Heat Transfer and Fluid Flow (McGraw-Hill, 1980).

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).

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);

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).

R.N. Singh. Advection-diffusion equation models in nearsurface geophysical and environmental sciences. J. Ind. Geophys. Union 17, 117 (2013).

How to Cite
Nazarenko, A., & Blavatska, V. (2018). Asymmetric Random Walk in a One-Dimensional Multizone Environment. Ukrainian Journal of Physics, 62(6), 508.
Soft matter