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Random processes and random systems: An introduction

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The Mathematics and Physics of Disordered Media: Percolation, Random Walk, Modeling, and Simulation

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1035))

Abstract

We introduce and review a number of topics drawn from the theories of random processes and random systems. In particular we address the following subjects: random walks in continuous spaces and on lattices; continuum limits of random walks and stable distributions; master equations, generalized master equations and continuous-time random walks; self-avoiding walks on lattices; percolation theory; steady-state and transient transport in random lattices; and diffusion and conduction in heterogeneous continua.

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  137. Applications oriented reviews include: H.L. Frisch and J.M. Hammersley, "Percolation processes and related topics", J. Soc. Indust. Appl. Math. 4 (1963) 894–918

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  139. Applications oriented reviews include: G. Deutscher, "Experimental relevance of percolation", Lecture Notes in Physics 149 (1981) 26–40.

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  140. The article by V.K.S. Shante and S. Kirkpatrick, "An introduction to percolation theory", Adv. Phys. 20 (1971) 325–357 remains a good introduction to the subject, and addresses applications of interest in condensed matter physics. For more recent developments in this area see the following articles in Ill-Condensed Matter (ed. R. Balian, R. Maynard and G. Toulouse: Amsterdam, North-Holland, 1979): D.J. Thouless, "Percolation and localization" (pp. 1–62); S. Kirkpatrick, "Models of disordered materials" (pp. 321–403); T.C. Lubensky, "Thermal and geometrical critical phenomena in random systems" (pp. 405–475).

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  141. The structure of the connected component as p → p +c is examined by S. Kirkpatrick, "The geometry of the percolation threshold", in AIP Conference Proceedings Vol. 40, ed. J.C. Garland and D.B. Tanner, pp. 99–117 (New York, American Institute of Physics, 1978); see also Redner's first article in this volume. Illustrations from a motion picture recording the growth of connectivity as p increases are given by C. Domb, E. Stoll and T. Schneider, "Percolation clusters", Contemp. Phys. 21 (1980) 577–592.

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  142. The problem of polymer gelation, interpreted as a percolation process, is reviewed by D. Stauffer, A. Coniglio and M. Adam, "Gelation and critical phenomena", Adv. Polymer Sci. 44 (1982) 103–158; this article also contains a useful survey of variants of the basic percolation model.

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  151. In principle one should prove that the critical value of p defined via the conductivity coincides with the topologically defined bond percolation threshold of section 6. Kesten [116] has given a proof of this for the square lattice.

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  176. Ya. G. Sinai, “The limiting behavior of a one-dimensional random walk in a random environment“, Theory Prob. Appl. 27 (1982) 256–268. Stated precisely Sinai's key result is as follows. Assume that <log σ>=0. Let ɛ>0 and δ>0 be given. For all sufficiently large n there exist a set Cn in the space of all realizations ω and a point m(n)=m(n;ω) for each ω ɛ Cn such that (i) the probability that a given realization belongs to Cn exceeds 1-ɛ; and (ii) with Pr denoting probability measure over all walks in a given environment ω, if ω ɛ Cn, then Pr(|Xn/log2n-m(n)| ⩽ δ) → 1 as n → ∞. The convergence is uniform in ω ɛ Cn, and as n → ∞ the probability distributions for m(n) converge weakly to some limit distribution.

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  178. A distribution is called ‘arithmetic’ (see Ref. 20, p. 138) if it is concentrated on the set of points 0,±λ,±2λ,... for some positive λ called the 'span'. In the example (8.7), log σ has probability density function ψ(η)=(1−p) δ (η−log{a/[1−a]})+pδ(η+log{a/[1−a]}), i.e. an arithmetic distribution of span log{a/[1−a]}, and so is not covered by the results of Kesten et al.

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  189. Alternative models with continuous time-dependence are easily defined but not considered here. See, for example, G. Ritter, "A continuous-time analogue of random walk in a random environment", J. Appl. Prob. 17 (1980) 259–264, and B.D. Hughes, M. Sahimi, L.E. Scriven and H.T. Davis, "Transport and conduction in random systems", to appear in Int. J. Engng. Sci.

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  202. V.V. Anshelevich and A.V. Vologodskii, "Laplace operator and random walk on a one-dimensional nonhomogeneous lattice", J. Stat. Phys. 25 (1981) 419–430. These authors consider the master equation \((\partial /\partial t)P_\ell (t){\text{ }} = {\text{ }}[1 - \delta _{\ell ,1} ]a_{\ell - 1} {\text{ }}P_{\ell - 1} (t){\text{ }} - {\text{ }}(a_{\ell - 1} + a_\ell )P_\ell (t){\text{ }} + {\text{ }}[1 - \delta _{\ell ,{\rm N} - 1} ]a_\ell P_{\ell - 1} (t)\) which describes motion on a finite linear chain, with the zeroth and Nth sites absorbing boundaries. The lth site is assigned the coordinate l/N. When the limit N → ∞ is taken in an appropriate manner, the solution of the master equation is shown to approach the solution of the diffusion equation with absorbing boundaries, \((\partial /\partial t){\text{ }}p(x,t) = a(\partial ^2 /\partial x^2 )p(x,t){\text{ }},{\text{ }}p(0,t) = p(l,t) = 1\), so long as a \(a = \mathop {\lim }\limits_{N \to \infty } N(\sum\nolimits_{\ell = 0}^{N - 1} {a_\ell ^{ - 1} } )\) exists and is non-zero. For independent, randomly distributed coefficients al, this implies that if a −1l has finite mean <a −1l >, then the effective diffusion coefficient is a=<a −1l >−1 with probability 1.

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  212. For the Kirkpatrick random resister problem, the Green function formalism has been developed by J. A. Blackman, "A theory of conductivity in disordered resistor networks", J. Phys. C 9 (1976) 2049–2071, and G. Ahmed and J. A. Blackman, "On theories of transport in disordered media" ibid. 12 (1976) 837–853. Its extension to the present problem may be found in Ref. 201.

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  218. J. Bernasconi and W.R. Schneider, "Diffusion on a one-dimensional lattice with random asymmetric transition rates", J. Phys. A 15 (1983) L729–L734.

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  219. M. Barma and D. Dhar, "Directed diffusion in a percolation network", J. Phys. C 16 (1983) 1451–1458.

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  220. See also B.D. Hughes, M. Sahimi, L.E. Scriven and H.T. Davis, "Transport and conduction in random systems", Int. J. Eng. Sci, in press. Such models are lattice versions of a classic colloid problem of M. Smoluchowski, ref 11; important recent papers with a physical chemistry orientation include B.U. Felderhof and J.M. Deutch, "Concentration dependence of the rate of diffusion-controlled reactions, J. Chem. Phys. 64 (1976) 4551–4558, P. Grassberger and I. Procaccia, "The long-time properties of diffusion in a medium with static traps", ibid. 77 (1982) 6281–6284, M. Muthukumar, "Concentration dependence of diffusion controlled processes among static traps", ibid. 76 (1982) 2667–2671, S. Prager and H. L. Frisch, "Diffusion-controlled reactions on a two-dimensional lattice", ibid. 72 (1980) 2941-, and R.F. Keyser and J.B. Hubbard, "Diffusion in a medium with a random distribution of static traps", Phys. Rev. Lett. 51 (1983) 79–82.

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  222. J. Klafter and R. Silbey, “Derivation of the continuous-time random walk equation”, Phys. Rev. Lett. 44 (1980) 55-.

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  223. We apologize to Charles Darwin for borrowing the structure, and the tone, of his concluding sentence in The Origin of Species, as our closing remark.

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Authors and Affiliations

  1. Department of Chemical Engineering and Materials Science, University of Minnesota, 55455, Minneapolis, Minnesota

    Barry D. Hughes

  2. Department of Chemistry, University of Minnesota, 55455, Minneapolis, Minnesota

    Stephen Prager

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B. D. Hughes B. W. Ninham

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Hughes, B.D., Prager, S. (1983). Random processes and random systems: An introduction. In: Hughes, B.D., Ninham, B.W. (eds) The Mathematics and Physics of Disordered Media: Percolation, Random Walk, Modeling, and Simulation. Lecture Notes in Mathematics, vol 1035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073255

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