Riesz s-Equilibrium Measures on d-Dimensional Fractal Sets as s Approaches d

Let $A$ be a compact set in \Rp of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $\mu^{s,A}$ is the unique Borel probability measure with support in $A$ that minimizes \Is(\mu):=\iint\Rk{x}{y}{s}d\mu(y)d\mu(x) over all such probability measures. In this paper we show that if $A$ is a strictly self-similar $d$-fractal, then $\mu^{s,A}$ converges in the weak-star topology to normalized $d$-dimensional Hausdorff measure restricted to $A$ as $s$ approaches $d$ from below

A sequence of discrete minimal energy configurations that does not converge in the weak-star topology

We demonstrate a set A and a value of s for which the sequence of N-point discrete minimal Riesz s-energy configurations on A does not have an asymptotic distribution in the weak-star sense as N tends to infinity

Pressure dependence of diffusion coefficient and orientational relaxation time for acetonitrile and methanol in water: DRISM/mode-coupling study

We present results of theoretical description and numerical calculation of the dynamics of molecular liquids based on the Reference Interaction Site Model / Mode-Coupling Theory. They include the temperature-pressure(density) dependence of the translational diffusion coefficients and orientational relaxation times for acetonitrile and methanol in water at infinite dilution. Anomalous behavior, i.e. the increase in mobility with density, is observed for the orientational relaxation time of methanol, while acetonitrile does not show any deviations from the usual. This effect is in qualitative agreement with the recent data of MD simulation and with experimental measurements, which tells us that presented theory is a good candidate to explain such kind of anomalies from the microscopical point of view and with the connection to the structure of the molecules.Comment: 10 pages, 2 eps-figures, 3 table

Diffusion controlled initial recombination

This work addresses nucleation rates in systems with strong initial recombination. Initial (or `geminate') recombination is a process where a dissociated structure (anion, vortex, kink etc.) recombines with its twin brother (cation, anti-vortex, anti-kink) generated in the same nucleation event. Initial recombination is important if there is an asymptotically vanishing interaction force instead of a generic saddle-type activation barrier. At low temperatures, initial recombination strongly dominates homogeneous recombination. In a first part, we discuss the effect in one-, two-, and three-dimensional diffusion controlled systems with spherical symmetry. Since there is no well-defined saddle, we introduce a threshold which is to some extent arbitrary but which is restricted by physically reasonable conditions. We show that the dependence of the nucleation rate on the specific choice of this threshold is strongest for one-dimensional systems and decreases in higher dimensions. We discuss also the influence of a weak driving force and show that the transport current is directly determined by the imbalance of the activation rate in the direction of the field and the rate against this direction. In a second part, we apply the results to the overdamped sine-Gordon system at equilibrium. It turns out that diffusive initial recombination is the essential mechanism which governs the equilibrium kink nucleation rate. We emphasize analogies between the single particle problem with initial recombination and the multi-dimensional kink-antikink nucleation problem.Comment: LaTeX, 11 pages, 1 ps-figures Extended versio

The Role of Diagenesis at Vera Rubin Ridge in Gale Crater, Mars, and the Chemostratigraphy of the Murray Formation as Observed by the Chemcam Instrument

The Mars Science Laboratory (MSL) Curiosity rover explored Vera Rubin ridge (VRR) in Gale crater, Mars, for almost 500 sols (Mars days) between arriving at the ridge on sol 1809 of the mission in September 2017 and leaving it on sol 2302 upon entering the Glen Torridon area south of the ridge. VRR is a topographic ridge on the central mound, Aeolis Mons (Mt. Sharp), in Gale crater that displays a strong hematite spectral signature from orbit. In-situ observations on the ridge led to the recognition that the ridge-forming rocks belong to the Murray formation, the lowermost exposed stratigraphic unit of the Mt. Sharp group, that was first encountered at the Pahrump Hills location. Including VRR rocks, the Murray formation, interpreted to be primarily deposited in an ancient lacustrine environment in Gale crater, is more than 300 m thick. VRR itself is composed of two stratigraphic members within the Murray formation, the Pettegrove Point member overlain by the Jura member. The Pettegrove Point member overlies the Blunts Point member of the Murray formation. Areas of gray coloration are observed in the Jura member predominantly, but also in the Pettegrove Point member. Generally, gray areas are found in local topographic depressions, but contacts between red and gray rocks crosscut stratigraphy. Additionally, cm-scale dark concretions with very high iron-content are commonly observed in gray rocks, typically surrounded by a lighttoned zone that is conversely depleted in iron. A key goal for the VRR campaign was to characterize geochemical variations in the ridge-forming rocks to investigate the role of primary and diagenetic controls on the geochemistry and morphology of VRR. Here, we present observations by the ChemCam instrument on VRR and compare these to the full Murray formation chemostratigraphy. This work was recently submitted to a special issue of JGRPlanets that detail the full VRR campaign