Percolation Theory: Topology and Structure

Abstract

The fundamental concepts of percolation theory are introduced in Chaps. 1 and 2. Chapter 1 includes descriptions of geometrical and topological variables, while the description of such properties as tortuosity and transport are deferred to Chap. 2. Chapter 1 has, generally speaking, two purposes: (1) to be a reference for material that will be applied later in the book, (2) to provide a means for a scientist to gain a working familiarity with percolation theory. The chapter is organized in such a way that those sections that provide essentially all of the expertise required for arbitrary applications of percolation theory to porous media problems can be read in early explorations, while those sections (1.11–1.15) that provide additional detail can be skipped. The role of universal scaling functions and the relevance of power-law behavior are discussed in terms of the understanding of the fractal structure of percolation theory. The chapter concludes with a discussion of the non-universality of the percolation threshold and presentation of what general methods have been proposed to predict it.

Problems

• 1.1 Show that at arbitrary p the largest clusters have cluster radius r _s=s ^σν, and argue then that for arbitrary p and arbitrary s, r _s=s ^σν g[s^σ(p−p _c)], where g is an unknown function. Does your result for r _s imply an effective dimension of the clusters?

• 1.2 Derive Eq. (1.50) from Eq. (1.49).

• 1.3 Derive Eq. (1.42) from Eq. (1.41).

• 1.4 The critical exponent α is defined through the singular contribution to ∑s ⁰ n _s∝(p−p _c)^2−α. Find α in terms of known exponents using the results of development of (1.14) and an analogy to Eq. (1.16).

• 1.5 The critical exponent γ is defined through the singular contribution ∑s ² n _s∝(p−p _c)^−γ. Find γ in terms of known exponents.

• 1.6 Show explicitly that Eq. (1.30) results from Eq. (1.29).

• 1.7 Verify the scaling relationships for the critical exponents in 1-d, 2-d, and 6-d. Do you expect them to be precisely satisfied in 3-d (where the exponents may not ever be represented in terms of rational fractions)?

• 1.8 Show that, for one-dimensional systems, definition of χ as

  $$\chi = \frac{\sum_{r = 1}^{\infty} rp^{r}}{\sum_{r = 1}^{\infty} p^{r}} $$

  leads to χ=p/(1−p) instead of χ=(p+1)/(1−p) as obtained from Eq. (1.5). How would you characterize the sensitivity of the scaling behavior of the correlation length relative to the details of its definition?

• 1.9 The sum in Problem 1.4 has been argued to describe, for a magnetic system, the free energy, while P (the first moment) corresponds to the magnetization, and the sum in Problem 1.5 (the second moment) to the susceptibility. Find an argument for why an increase in the moment of the cluster distribution by one corresponds to a derivative with respect to the applied field.