Statistical mechanics of conducting phase transitions

Journal ArticleThe critical behavior of the effective conductivity o* of the random resistor network in Zd, near its percolation threshold, is considered. The network has bonds assigned the conductivities 1 and E >_0 in the volume fractions p and 1 -p. Motivated by the statistical mechanics of an Ising ferromagnet at temperature T in a field H, we introduce a partition function and free-energy for the resistor network, which establishes a direct correspondence between the two problems. In particular, we show that the free energies for the resistor network and the Ising model both have the same type of integral representation, which has the interpretation of the complex potential due to a charge distribution on [0, l] in the s = l/( 1 -e) plane for the resistor network, and on the unit circle in the z=exp(-2BH) plane for the ferromagnet. Based on this correspondence, we develop a Yang-Lee picture of the onset of nonanalytic behavior of the effective conductivity o*, so that the percolation threshold p=pc is characterized as an accumulation point of zeros of the partition function in the complex p-plane as E -- 0. A scheme is developed to find the locations of a certain sequence of zeros in the p-plane, which is based on Pade approximation of a perturbation expansion of o*(p,e) around a homogeneous medium (E= 1). Furthermore, for E > 0, we construct a domain De containing [0, l] in the p-plane in which o*i(p,e) is analytic, and which collapses as E -- 0. The explicit construction of this domain allows us to obtain a lower bound on the size of the gap in zeros of the partition function around the percolation threshold p =pc , which leads to the gap exponent inequality A<_l. 0 1995 American Institute of Physics

Critical behavior of transport in lattice and continuum percolation models

Journal ArticleIt has been observed that the critical exponents of transport in the continuum, such as in the Swiss cheese and random checkerboard models, can exhibit nonuniversal behavior, with values different than the lattice case. Nevertheless, it is shown here that the transport exponents for both lattice and continuum percolation models satisfy the standard scaling relations for phase transitions in statistical mechanics. The results are established through a direct, analytic correspondence between transport coefficients for two component random media and the magnetization of the Ising model, which is based on the observation we made previously that both problems share the Lee-Yang property

Critical behavior of transport in percolation-controlled smart composites

Journal ArticleThe electrical transport properties of many smart composites and other technologically important materials are dominated by the connectedness, or percolation properties of a particular component. Predicting the critical behavior of such media near their percolation threshold, where one typically obtains the most interesting and useful material properties, is a formidable challenge, and not well understood from a modeling perspective

Brine percolation and the transport properties of sea ice

Journal ArticleSea ice is distinguished from many other porous composites, such as sandstones or bone, in that its microstructure and bulk material properties can vary dramatically over a small temperature range. For brine-volume fractions below a critical value of about 5%, which corresponds to a critical temperature of about -5oC for salinity of 5 ppt, columnar sea ice is effectively impermeable to fluid transport

Convexity in random resistor networks

Journal ArticleThe bulk conductivity o*(p) of the bond lattice in Zd is considered, where the conductivity of the bonds is either 1 with probability p or e > 0 with probability 1 - p. Rigorous and non-rigorous results demonstrating convexity of o*(p) near the percolation threshold pc are presented

Bounds on the complex permittivity of sea ice

Journal ArticleAn analytic method for obtaining bounds on effective properties of composites is applied to the complex permittivity e* of sea ice. The sea ice is assumed to be a two-component random medium consisting of pure ice of permittivity e1, and brine of permittivity e2. The method exploits the properties of e* as an analytic function of the ratio e1/e2. Two types of bounds on e * are obtained. The first bound R1 is a region in the complex e* plane which assumes only that the relative volume fractions pl and p2 = 1 - pl of the ice and brine are known. The region Rl is bounded by circular arcs and e* for any microgeometry with the given volume fractions must lie inside it. In addition to the volume fractions, the second bound R2 assumes that the sea ice is statistically isotropic within the horizontal plane

Convexity and exponent inequalities for conduction near percolation

Journal ArticleThe bulk conductivity o*(p) of the bond lattice in Zd with a fraction p of conducting bonds is analyzed. Assuming a hierarchical node-link-blob (NLB) model of the conducting backbone, it is shown that o*(p) (for this model) is convex in p near the percolation threshold pc, and that its critical exponent t obeys the inequalities 1 _ 4. The upper bound t = 2 in d = 3, which is realizable in the NLB class, virtually coincides with two very recent numerical estimates obtained from simulation and series expansion

Classical transport in quasiperiodic media

Journal ArticleAbstract. Classical transport coefficients such as the effective conductivity or diffusivity of a quasiperiodic medium were observed [1] to depend discontinuously on the frequencies of the quasiperiodicity. For example, for a one-dimensional medium with a potential V(x) = cosx + coskx , the effective diffusion coefficient D*(k) has the same value D for all irrational k , but differs from D and depends on k for k rational, where it is thus discontinuous. Here we review some recent progress [2-4] in understanding this discontinuous behavior. In particular, a class of examples which explicitly exhibit the discontinuity in dimensions d >_ 2 is constructed. In addition, we examine some rather surprising consequences of the discontinuity for the rate of approach to limiting behavior of diffusion or conduction in quasiperiodic media as time or volume becomes infinite. It is found that these rates can be "arbitrarily slow," which contrasts with the power laws observed for random media

Broad spectral, interdisciplinary investigation of the electromagnetic properties of sea ice

Journal ArticleThis paper highlights the interrelationship of research completed by a team of investigators and presented in the several individual papers comprising this Special Section on the Office of Naval Research (ONR), Arlington, VA, Sponsored Sea Ice Electromagnetics Accelerated Research Initiative (ARI)