Convexity and exponent inequalities for conduction near percolation

Self-consistent effective-medium approximations with path integrals

We study effective-medium approximations for linear composite media by means of a path integral formalism with replicas. We show how to recover the Bruggeman and Hori-Yonezawa effective-medium formulas. Using a replica-coupling ansatz, these formulas are extended into ones which have the same percolation thresholds as those of the Bethe lattice and Potts model of percolation, and critical exponents s=0 and t=2 in any space dimension d>/=2. Like the Bruggeman and Hori-Yonezawa formulas, the obtained formulas are exact to second order in the weak-contrast and dilute limits. The dimensional range of validity of the four effective-medium formulas is discussed, and it is argued that out formulas are of better relevance than the classical ones in dimensions d=3,4 for systems obeying the nodes-links-blobs picture, such as random-resistor networks.

The percolation phase transition in sea Ice

Sea ice exhibits a marked transition in its fluid transport properties at a critical brine volume fraction pc of about 5 percent, or temperature Tc of about -5 degreesC for salinity of 5 parts per thousand. For temperatures warmer than Tc, brine carrying heat and nutrients can move through the ice, whereas for colder temperatures the ice is impermeable. This transition plays a key role in the geophysics, biology, and remote sensing of sea ice. Percolation theory can be used to understand this critical behavior of transport in sea ice. The similarity of sea ice microstructure to compressed powders is used to theoretically predict pc of about 5 percent.