Scenery reconstruction in two dimensions with many colors

Kesten has observed that the known reconstruction methods of random sceneries seem to strongly depend on the one-dimensional setting of the problem and asked whether a construction still is possible in two dimensions. In this paper we answer this question in the affirmative under the condition that the number of colors in the scenery is large enough

Rheology of gelling polymers in the Zimm model

In order to study rheological properties of gelling systems in dilute solution, we investigate the viscosity and the normal stresses in the Zimm model for randomly crosslinked monomers. The distribution of cluster topologies and sizes is assumed to be given either by Erd\H os-R\'enyi random graphs or three-dimensional bond percolation. Within this model the critical behaviour of the viscosity and of the first normal stress coefficient is determined by the power-law scaling of their averages over clusters of a given size $n$ with $n$. We investigate these Mark--Houwink like scaling relations numerically and conclude that the scaling exponents are independent of the hydrodynamic interaction strength. The numerically determined exponents agree well with experimental data for branched polymers. However, we show that this traditional model of polymer physics is not able to yield a critical divergence at the gel point of the viscosity for a polydisperse dilute solution of gelation clusters. A generally accepted scaling relation for the Zimm exponent of the viscosity is thereby disproved.Comment: 9 pages, 2 figure

A general treatment of snow microstructure exemplified by an improved relation for thermal conductivity

Finding relevant microstructural parameters beyond density is a longstanding problem which hinders the formulation of accurate parameterizations of physical properties of snow. Towards a remedy, we address the effective thermal conductivity tensor of snow via anisotropic, second-order bounds. The bound provides an explicit expression for the thermal conductivity and predicts the relevance of a microstructural anisotropy parameter <i>Q</i>, which is given by an integral over the two-point correlation function and unambiguously defined for arbitrary snow structures. For validation we compiled a comprehensive data set of 167 snow samples. The set comprises individual samples of various snow types and entire time series of metamorphism experiments under isothermal and temperature gradient conditions. All samples were digitally reconstructed by micro-computed tomography to perform microstructure-based simulations of heat transport. The incorporation of anisotropy via <i>Q</i> considerably reduces the root mean square error over the usual density-based parameterization. The systematic quantification of anisotropy via the two-point correlation function suggests a generalizable route to incorporate microstructure into snowpack models. We indicate the inter-relation of the conductivity to other properties and outline a potential impact of <i>Q</i> on dielectric constant, permeability and adsorption rate of diffusing species in the pore space