Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-11T00:44:21.741Z Has data issue: false hasContentIssue false
  • English
  • Français

On the general random walk formulation for diffusion in media with Diffusivities

Published online by Cambridge University Press:  17 February 2009

James M. Hill  and
Barry D. Hughes
James M. Hill
Affiliation:
Department of Mathematics, University of Wollongong, Wollongong, N.S.W.2500.
Barry D. Hughes
Affiliation:
Department of Mathematics, Faculty of Military Studies, University of New South Wales, RMC Duntroon, A.C.T. 2600. Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, A.C.T.2600.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

A general discrete multi-dimensional and multi-state random walk model is proposed to describe the phenomena of diffusion in media with multiple diffusivities. The model is a generalization of a two-state one-dimensional discrete random walk model (Hill [8]) which gives rise to the partial differential equations of double diffusion. The same partial differential equations are shown to emerge as a special case of the continuous version of the present general model. For two states a particular generalization of the model given in [8] is presented which is not restricted to nearest neighbour transitions. Under appropriate circumstances this two-state model still yields the partial differential equations of double diffusion in the continuum limit, but an example of circumstances leading to a radically different continuum limit is presented.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 27 , Issue 1 , July 1985 , pp. 73 - 87
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Aifantis, E. C., “A new interpretation of diffusion in high diffusivity paths—a continuum approach”, Acta Metall. 27 (1979), 683–691.CrossRefGoogle Scholar
[2]Aifantis, E. C., “Continuum basis for diffusion in regions with multiple diffusivity”, J. Appl. Phys. 50 (1979), 1334–1338.CrossRefGoogle Scholar
[3]Deutscher, G., Zallen, R. and Adler, J. (eds.), Percolation processes and structures, Ann. Israel Phys. Soc. 5 (Hilger, Bristol, 1983).Google Scholar
[4]Feller, W., An introduction to probability theory and its applications, Vol. 2, 2nd ed. (Wiley, New York, 1971).Google Scholar
[5]Gikhman, I. I. and Skorokhod, A. V., Introduction to the theory of random processes (Saunders, Philadelphia, 1969).Google Scholar
[6]Gillis, J. E. and Weiss, G. H., “Expected number of distinct sites visited by a random walk with an infinite variance”, J. Math. Phys. 11 (1970), 1307–1312.CrossRefGoogle Scholar
[7]Gnedenko, B. V. and Kolmogorov, A. N., Limit distributions for sums of independent random variables, 2nd ed. (Addison-Wesley, Reading, Massachusetts, 1968).Google Scholar
[8]Hill, J. M., “A discrete random walk model for diffusion in media with double diffusivity”, J. Austral. Math. Soc. Ser. B 22(1980), 58–74.CrossRefGoogle Scholar
[9]Hill, J. M., “A random walk model for diffusion in the presence of high-diffusivity paths”, Adv. in Molecular Relaxation and Interaction Processes 19 (1981), 261–284.CrossRefGoogle Scholar
[10]Hughes, B. D, Montroll, E. W. and Shlesinger, M. F., “Fractal and lacunary stochastic processes”, J. Statist. Phys. 29 (1983), 273–283.CrossRefGoogle Scholar
[11]Hughes, B. D. and Ninham, B. W. (eds.), The mathematics and physics of disordered media, Lecture Notes in Math. 1035 (Springer, Berlin, 1983).Google Scholar
[12]Hughes, B. D., Sahimi, M. and Davis, H. T., “Random walks on pseudo-lattices”, Physica A 120 (1983), 515–536.CrossRefGoogle Scholar
[13]Hughes, B. D., Shlesinger, M. F. and Montroll, E. W., “Random walks with self-similar clusters”, Proc. Nat. Acad. Sci. U.S.A. 78(1981), 3287–3291.CrossRefGoogle ScholarPubMed
[14]Kline, M., Mathematics: the loss of certainty (Oxford University Press, 1981).Google Scholar
[15]Krámli, A. and Szāz, D., “Random walks with internal degrees of freedom”, Z. Wahrsch. Verw. Gebiete 63 (1983), 85–95.CrossRefGoogle Scholar
[16]Landman, U., Montroll, E. W. and Shlesinger, M. F., “Random walks and master equations with internal degrees of freedom”, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 430–433.CrossRefGoogle ScholarPubMed
[17]Lévy, P., Théorie de l'additon des variables aléatoires (Gauthier-Villars, Paris, 1937).Google Scholar
[18]Liyanage, L. H., Hill, J. M. and Gulati, C. M., “A note on the random walk model arising in double diffusion”, J. Austral. Math. Soc. Ser. B 24 (1982), 121–129.CrossRefGoogle Scholar
[19]Mandl, P., Analytical treatment of one-dimensional Markov processes (Springer, Berlin, 1968).Google Scholar
[20]Montroll, E. W. and West, B. J., “On an enriched collection of stochastic processes”, in Fluctuation phenomena (eds. Montroll, E. W. and Lebowiz, J. L.), (North-Holland, Amsterdam, 1979), 61–175.CrossRefGoogle Scholar
[21]Seshadri, V. and West, B. J., “Fractal dimensionality of Lévy processes”, Proc. Nat. Acad. Sci. U.S.A. 79 (1982), 4501–4505.CrossRefGoogle ScholarPubMed
[22]Shlesinger, M. F. and Landman, U., “Solution of physical stochastic processes via mappings onto ideal and defective random walk lattices”, in Applied stochastic processes (ed. Adomian, G.), (Academic Press, New York, 1980), 151–246.CrossRefGoogle Scholar
[23]Stephen, M. J., “Lectures on disordered systems”, in Critical phenomena (ed. Hahne, F. J. W.), Lecture Notes in Physics 186 (Springer, Berlin, 1983), 259–300.CrossRefGoogle Scholar
[24]Zallen, R., The physics of amorphous solids, (Wiley, New York, 1983).CrossRefGoogle Scholar