Dedicated to George Papanicolaou on the occasion of his 70th birthday.Abstract. The analytic continuation method of homogenization theory provides Stieltjes integral representations for the effective parameters of composite media. These representations involve the spectral measures of self-adjoint random operators which depend only on the composite geometry. On finite bond lattices, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. Here we provide the mathematical foundation for rigorous computation of spectral measures for such composite media, and develop a numerically efficient projection method to enable such computations. This is accomplished by providing a unified formulation of the analytic continuation method which is equivalent to the original formulation and holds for finite and infinite lattices, as well as in continuum settings. We also introduce a family of bond lattices and directly compute the associated spectral measures and effective parameters. The computed spectral measures are in excellent agreement with known theoretical results. The behavior of the associated effective parameters is consistent with the symmetries and theoretical predictions of models, and the computed values fall within rigorous bounds. Some previous calculations of spectral measures have relied on finding the boundary values of the imaginary part of the effective parameter in the complex plane. Our method instead relies on direct computation of the eigenvalues and eigenvectors which enables, for example, statistical analysis of the spectral data. ). 825 826 SPECTRAL MEASURE COMPUTATIONSThis formulation demonstrated that the measures underlying these integral representations are spectral measures associated with the random operators, which depend only on the composite geometry. These measures contain all the information about the mixture geometry, and provide a link between microgeometry and transport. Local geometry is encoded in "geometric" resonances in the measures [46], while global connectivity is encoded by spectral gaps [58,46] and the presence of δ-components in the measures at the spectral endpoints [58]. A remarkable feature of the method is that once the spectral measures are found for a given composite geometry, by the spectral coupling of the governing equations [13,56,14,18], the effective electrical, magnetic, and thermal transport properties are all completely determined by these measures.The integral representations yield rigorous forward bounds on the effective parameters of composites, given partial information on the microgeometry [7,53,33,8,10]. One can also use the integral representations to obtain inverse bounds, where data on the electromagnetic response of a sample, for example, is used to bound its structural parameters, such as the volume fractions of the components [51,52,16,13,17,75,9,15,21,32], and even the separation of the inclusions in matrix particle composites [59]. Furthermore, the spec...

We present a general theory for critical behavior of transport in binary composite media. The theory holds for lattice and continuum percolation models in both the static case with real parameters and the quasi–static case (frequency dependent) 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 at 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.

The enhancement in diffusive transport of passive tracer particles by incompressible, turbulent flow fields is a challenging problem with theoretical and practical importance in many areas of science and engineering, ranging from the transport of mass, heat, and pollutants in geophysical flows to sea ice dynamics and turbulent combustion. The long time, large scale behavior of such systems is equivalent to an enhanced diffusive process with an effective diffusivity tensor D *. Two different formulations of integral representations for D * were developed for the case of time-independent fluid velocity fields, involving spectral measures of bounded self-adjoint operators acting on vector fields and scalar fields, respectively. Here, we extend both of these approaches to the case of space-time periodic velocity fields, with possibly chaotic dynamics, providing rigorous integral representations for D * involving spectral measures of unbounded self-adjoint operators. We prove that the different formulations are equivalent. Their correspondence follows from a one-to-one isometry between the underlying Hilbert spaces. We also develop a Fourier method for computing D * , which captures the phenomenon of residual diffusion related to Lagrangian chaos of a model flow. This is reflected in the spectral measure by a concentration of mass near the spectral origin.

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The Anderson transition in solids and optics is a wave phenomenon where disorder induces localization of the wavefunctions. We find here that the hallmarks of the Anderson transition are exhibited by classical transport at a percolation threshold -without wave interference or scattering effects. As long range order or connectedness develops, the eigenvalue statistics of a key random matrix governing transport crossover toward universal statistics of the Gaussian orthogonal ensemble, and the field eigenvectors delocalize. The transition is examined in resistor networks and sea ice structures.