Coherent view of crystal chemistry and ab initio analyses of Pb(II) and Bi(III) Lone Pair in square planar coordination

The stereochemistry of 6s2 (E) lone pair of divalent Pb and trivalent Bi (PbII and BiIII designated by M*) in structurally related PbO, PbFX (X= Cl, Br, I), BiOX (X= F, Cl, Br, I) and Bi2NbO5F is rationalized. The lone pair LP presence determined by its sphere of influence E, equal to those of oxygen or fluorine anions, was settled by its center then giving M*-E directions and distances. Detailed description of structural features of both elements in the title compounds characterized by [PbEO]n and [BiEO]n layers allowed to show the evolution of M*-E distance versus the changes with the square pyramidal SP coordination polyhedra. All are different, in red PbO one finds {PbEO4E4} square antiprism, a {[Bi.E]O4X4Xapical} monocapped square antiprism in PbFX and BiOX and {BiEO4F4}square antiprism in Bi2NbO5F. To analyze the crystal chemistry results, the electronic structures of these compounds were calculated within density functional theory DFT. Real space analyses of electron localization illustrate a full volume development of the lone pair on PbII within {PbEO4E4} in PbOE, {PbEF4X4} in PbFXE and Bi(III) within {BiEO4X4} square antiprisms, contrary to Bi(III) within {[Bi.E]O4F4Fapical} monocapped square antiprism. Larger hardness (larger bulk modules B0) and band gap characterize BiOF versus PbO due to the presence of F which brings antibonding Bi-F interactions oppositely to mainly bonding Bi-O. In PbFX and BiOX series there is a systematic decrease of B0 with the increasing volume following the nature and size of X which is decreasingly electronegative and increasingly large. The electronic densities of states mirror these effects through the relative energy position and relative electronegativities of F/X and O/X leading to decrease the band gap.Comment: 20 text pages/ 10 multifigures/large review article, J. Progress Solid State Chemistry, under production MAY 201

Non-Singular Bounces Catalysed by Dark Energy

We investigate classically non-singular bounces caused by dark energy. In the presence of positive spatial curvature, vacuum energy, either in the form of a cosmological constant or a scalar field potential, allows for an open set of initial conditions leading to non-singular bounces, without any violation of the null energy condition. We study anisotropic Bianchi IX cosmologies, and demonstrate that they can even have multiple bounces, accompanied by intricate evolutions of the anisotropies that provide a non-singular analogue of mixmaster crunches. The relation of these solutions to more complete cosmological models, as well as to the recently proposed swampland criteria, are briefly discussed.Comment: 30 pages, 25 figure files; v2: references adde

Generalized diffusion equation

Modern analyses of diffusion processes have proposed nonlinear versions of the Fokker-Planck equation to account for non-classical diffusion. These nonlinear equations are usually constructed on a phenomenological basis. Here we introduce a nonlinear transformation by defining the $q$-generating function which, when applied to the intermediate scattering function of classical statistical mechanics, yields, in a mathematically systematic derivation, a generalized form of the advection-diffusion equation in Fourier space. Its solutions are discussed and suggest that the $q$-generating function approach should be a useful tool to generalize classical diffusive transport formulations.Comment: 5 pages with 3 figure

Introduction to special section: Active Fault-Related Folding: Structural Evolution, Geomorphologic Expression, Paleoseismology, and Seismic Hazards

N/

Molecular theory of anomalous diffusion

We present a Master Equation formulation based on a Markovian random walk model that exhibits sub-diffusion, classical diffusion and super-diffusion as a function of a single parameter. The non-classical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean-squared displacement ($\sim t^{\gamma}$ with $0 <\gamma \leq 1.5$) but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub- and super-diffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent $\gamma$. Simulations based on the Master Equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.Comment: Published text with additional comment

Clustering properties of rectangular Macdonald polynomials

The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the $(q,t)$-deformed problem involving Macdonald polynomials. The present paper is devoted to the proof of this formula. To this aim we use four families of Jack/Macdonald polynomials: symmetric homogeneous, nonsymmetric homogeneous, shifted symmetric and shifted nonsymmetric.Comment: 43 pages, 2 figure