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A note on the random walk model arising in double diffusion

Published online by Cambridge University Press:  17 February 2009

L. H. Liyanage ,
J. M. Hill  and
C. M. Gulati
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Abstract

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The discrete random walk problem for the unrestricted particle formulated in the double diffusion model given in Hill [2] is solved explicitly. In this model it is assumed that a particle moves along two distinct horizontal paths, say the upper path I and lower path 2. For i = 1, 2, when the particle is in path i, it can move at each jump in one of four possible ways, one step to the right with probability pi, one step to the left with probability qi, remains in the same position with probability ri, or exchanges paths but remains in the same horizontal position with probability si (pi + qi + ri + si = 1). Using generating functions, the probability distribution of the position of an unrestricted particle is derived. Finally some special cases are discussed to illustrate the general result.

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 2 , October 1982 , pp. 121 - 129
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Cox, D. R. and Miller, H. D., The theory of stochastic processes (Methuen, London, 1965).Google Scholar
[2]Hill, J. M., “A discrete random walk model for diffusion in media with double diffusivity”, J. A ustral. Math. Soc. Ser. B 22 (1980), 58–74.CrossRefGoogle Scholar
[3]Hill, J. M., “A random walk model for diffusion in the presence of high-diffusivity paths”, Advances in Molecular Relaxation and Interaction Processes 19 (1981), 261–284.CrossRefGoogle Scholar