| Description |
Transport in disordered composite media is a problem that arises throughout the sciences and engineering and has attracted significant theoretical, computational, and experimental interest. One of the key features of these types of problems is the critical dependence of the effective transport properties on system parameters, such as volume fraction, component contrast ratio, applied field strength, etc. In recent years a broad range of mathematical techniques have been developed to study phase transitions exhibited by such composites, revealing features which are virtually ubiquitous in disordered systems. Here we construct a multifaceted mathematical framework describing phase transitions exhibited by two phase random media, using techniques from: statistical mechanics, percolation theory, random matrix theory, and a critical theory for Stieltjes functions of a complex variable involving the spectral measure of a self-adjoint random operator (or matrix). In particular, we present a general theory for critical behavior of transport in two phase random media. The theory holds for lattice and continuum percolation models in both the static case with real parameters and the frequency dependent quasi-static case with complex parameters. Through a direct, analytic correspondence between the magnetization of the Ising model and the effective parameter problem of two phase random media, we show that the critical exponents of the transport coefficients satisfy the standard scaling relations for phase transitions in statistical mechanics. Our work also shows that delta components form in the underlying spectral measures at the spectral endpoints precisely at the percolation threshold pc and 1 − pc. This is analogous to the Lee-Yang-Ruelle characterization of the Ising model phase transition, and identifies these transport transitions with the collapse of spectral gaps in these measures. Using random matrix theory, we also characterize these transport transitions via transitions in the eigenvalue statistics of the underlying random matrix. Finally, we construct a canonical ensemble statistical mechanics framework for general transport models of two phase random dielectric media, which parallels the Ising model. Our physically consistent model is formulated from first principles in physics, and is both physically transparent and mathematically tractable. |
