Closed-form evaluation of two-dimensional static lattice sums.

Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sums S2 (for conductivity) and T2 (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between S2 and T2 for different lattices are established. In particular, the value S2=π for the square and hexagonal arrays is discussed and T2=π/2 for the hexagonal is deduced.

Rayleigh approximation to ground state of the Bose and Coulomb glasses.

Glasses are rigid systems in which competing interactions prevent simultaneous minimization of local energies. This leads to frustration and highly degenerate ground states the nature and properties of which are still far from being thoroughly understood. We report an analytical approach based on the method of functional equations that allows us to construct the Rayleigh approximation to the ground state of a two-dimensional (2D) random Coulomb system with logarithmic interactions. We realize a model for 2D Coulomb glass as a cylindrical type II superconductor containing randomly located columnar defects (CD) which trap superconducting vortices induced by applied magnetic field. Our findings break ground for analytical studies of glassy systems, marking an important step towards understanding their properties.

Resurgence flows in porous media.

Porous media with resurgences can be described by a double structure, namely, a continuous porous medium and capillaries with impermeable walls which relate distant points of the continuous medium. The resurgences can be either punctual or extended. The equations for flow in such media are derived; some general properties of the resulting system, which involves nonlocal aspects, are deduced. A "dilute" approximation is detailed for punctual resurgences in two-dimensional media and is illustrated by a few examples.

Macroscopic diffusion on rough surfaces.

We consider diffusion on rough and spatially periodic surfaces. The macroscopic diffusion tensor D is determined by averaging the local fluxes over the unit cell. D is proved to be the unit tensor for macroscopically isotropic surfaces. For general surfaces, an asymptotic analysis is applied, when the ratio of the oscillation amplitude to the size of the unit cell is a small parameter epsilon. The microscopic field is determined up to O(epsilon(6)) in analytical form and an algorithm is derived to calculate higher order terms. We also deduce general analytical formulas for D up to O(epsilon(6)) and derive an algorithm to compute D as a series in epsilon(2).