Exact result for the effective conductivity of a continuum percolation model

Journal ArticleA random two-dimensional checkerboard of squares of conductivities 1 and 8 in proportions p and 1 - p is considered. Classical duality implies that the effective conductivity obeys o* = V8 at p = 1/2. It is rigorously found here that to leading order as 8--0, this exact result holds for all p in the interval (1- pc,pc), where pc=0.59 is the site percolation probability, not just at p = 1/2. In particular, o*(p,8)=78+O (8), as 8 -- 0. which is argued to hold for complex 8 as well. The analysis is based on the identification of a "symmetric" backbone, which is statistically invariant under interchange of the components for any pE(1--pc,pc), like the entire checkerboard at p =1/2. This backbone is defined in terms of "choke points" for the current, which have been observed in an experiment

Heat balance of the Earth

Results of improved calculations of the heat balance components of Earth's surface are reported for yearly average conditions. The technique used to determine the heat-balance components from land- and sea-based actinometric observations as well as from satellite data on the radiation balance of the Earth-atmosphere system is described, with special attention given to short-wavelength solar radiation on the continents, effective radiation from the land surface, the radiation balance of the ocean surface, heat expended by both evaporation from the ocean surface, and turbulent heat transfer between the ocean surface and the atmosphere. World maps of heat-balance components show yearly average values of total radiation, radiation balance, heat expended by evaporation, the turbulent heat flow between Earth's surface and atmosphere, and heat transfer between the ocean surface and underlying waters. The global surface heat balance is estimated along with global values of the various components and the heat-balance components for different latitude zones

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

Ginzburg-Landau model with small pinning domains

We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the {\it pinning domains}. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\v\to0$; here, \v is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $\Omega \subset \mathbb{C}$ with Dirichlet boundary condition $g$ on \d \O, with topological degree {\rm deg}_{\d \O} (g) = d >0. Our main result is that, for small \v, minimizers have $d$ distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to 1. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by {\it local renormalized energy} which does not depend on the external boundary conditions.Comment: 39 page

Are the Tails of Percolation Thresholds Gaussians ?

The probability distribution of percolation thresholds in finite lattices were first believed to follow a normal Gaussian behaviour. With increasing computer power and more efficient simulational techniques, this belief turned to a stretched exponential behaviour, instead. Here, based on a further improvement of Monte Carlo data, we show evidences that this question is not yet answered at all.Comment: 7 pages including 3 figure

Efficient Monte Carlo algorithm and high-precision results for percolation

We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run which takes an amount of time scaling linearly with the number of sites on the lattice. We use our algorithm to determine that the percolation transition occurs at occupation probability 0.59274621(13) for site percolation on the square lattice and to provide clear numerical confirmation of the conjectured 4/3-power stretched-exponential tails in the spanning probability functions.Comment: 8 pages, including 3 postscript figures, minor corrections in this version, plus updated figures for the position of the percolation transitio

Effective Viscosity of Dilute Bacterial Suspensions: A Two-Dimensional Model

Suspensions of self-propelled particles are studied in the framework of two-dimensional (2D) Stokesean hydrodynamics. A formula is obtained for the effective viscosity of such suspensions in the limit of small concentrations. This formula includes the two terms that are found in the 2D version of Einstein's classical result for passive suspensions. To this, the main result of the paper is added, an additional term due to self-propulsion which depends on the physical and geometric properties of the active suspension. This term explains the experimental observation of a decrease in effective viscosity in active suspensions.Comment: 15 pages, 3 figures, submitted to Physical Biolog

Chiral tunneling in single and bilayer graphene

We review chiral (Klein) tunneling in single-layer and bilayer graphene and present its semiclassical theory, including the Berry phase and the Maslov index. Peculiarities of the chiral tunneling are naturally explained in terms of classical phase space. In a one-dimensional geometry we reduced the original Dirac equation, describing the dynamics of charge carriers in the single layer graphene, to an effective Schr\"odinger equation with a complex potential. This allowed us to study tunneling in details and obtain analytic formulas. Our predictions are compared with numerical results. We have also demonstrated that, for the case of asymmetric n-p-n junction in single layer graphene, there is total transmission for normal incidence only, side resonances are suppressed.Comment: submitted to Proceedings of Nobel Symposium on graphene, May 201