P. L. Leath

We review the methods which have been developed over the past several years to determine the behavior of solids whose properties vary randomly at the microscopic level, with principal attention to systems having composition variation on a well-defined structure (random "alloys"). We begin with a survey of the various elementary excitations and put the dynamics of electrons, phonons, magnons, and excitons into one common descriptive Hamiltonian; we then review the use of double-time thermodynamic Green's functions to determine the experimental properties of systems. Next we discuss these aspects of the problem which derive from the statistical specification of the microscopic parameters; we examine what information can and cannot be obtained from averaged Green's functions. The central portion of the review concerns methods for calculating the averaged Green's function to successively better approximation, including various self-consistent methods, and higher-order cluster effects. The last part of the review presents a comparison of theory with the experimental results of a variety of properties---optical, electronic, magnetic, and neutron scattering. An epilogue calls attention to the similarity between these problems and those of other fields where random material heterogeneity has played an essential role.

A new percolation problem is posed which can exhibit a first-order transition. In bootstrap percolation, sites on an empty lattice are first randomly occupied, and then all occupied sites with less than a given number m of occupied neighbours are successively removed until a stable configuration is reached. On any lattice for sufficiently large m, the ensuing clusters can only be infinite. On a Bethe lattice for m>or=3, the fraction of the lattice occupied by infinite clusters discontinuously jumps from zero at the percolation threshold. From an analysis of stable and metastable ground states of the dilute Blume-Capel model (1966), it is concluded that effects like bootstrap percolation may occur in some real magnets.

It is shown that the shape of the large, random clusters, near the critical percolation concentration ${c}_{0}$, is such that their mean boundary $〈b〉$ is proportional to their mean bulk $〈n〉$ and this is illustrated by an argument which shows that the dimension of the boundary is the same as that of the bulk. The resulting ratio $\frac{〈b〉}{〈n〉}$ is simply related to the critical concentration ${c}_{0}$. The detailed results of a Monte Carlo calculation, previously reported, are given for $c<{c}_{0}$ on a simple square lattice; they yield an empirical formula for the probability distribution $\mathcal{P}(n,b)$, for finding a cluster of size $n$ and boundary $b$, that is proportional to a Gaussian in $\frac{b}{n}$, which is independent of concentration and which narrows to a $\ensuremath{\delta}$ function at $\frac{b}{n}={\ensuremath{\alpha}}_{0}$, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. The asymptotic behavior of the Gaussian form gives the critical exponents $\ensuremath{\beta}=0.19\ifmmode\pm\else\textpm\fi{}0.16$, and $\ensuremath{\gamma}=2.34\ifmmode\pm\else\textpm\fi{}0.3$, and ${\ensuremath{\alpha}}_{0}$, gives the critical concentration ${c}_{0}=0.587\ifmmode\pm\else\textpm\fi{}0.14$, in agreement with previous determinations.

An analysis of a prototypical percolation model (the fuse network) for breakdown in quenched random systems is given. The breakdown voltage and the topology of the eventual breakdown path are studied analytically and numerically. New scaling concepts, based on the most critical defect in the network, combined with standard percolation scaling ideas, lead to a complete picture of the strength of the network. The mean breakdown strength and the distribution of breakdown strengths are derived in the different concentration regimes. The breakdown path is described by new order parameters on approach to ${p}_{c}$. One, the number of bonds broken in the breakdown process, is studied in detail. Many models and physical systems should show an analogous behavior and simplified models for two of these problems, brittle fracture and dielectric breakdown in solids, are discussed.

Two percolation models for electrical breakdown in quenched random media, a fuse-wire network and a dielectric network, are introduced and studied. A combination of Lifshitz and scaling arguments leads to a size dependence given by $\frac{{V}_{b}}{L}\ensuremath{\sim}\frac{a(p)}{[1+b(p){(\mathrm{ln}L)}^{\ensuremath{\beta}}]}$, where $\frac{\ensuremath{\beta}=1}{(d\ensuremath{-}1)}$ for the fuse network and $\ensuremath{\beta}=1$ for the dielectric network. Simulations support this hypothesis in the 2D fuse network. We argue that any finite fraction of quenched defects qualitatively reduces the breakdown strength of a wide variety of electrical and mechanical systems in both two and three dimensions.