Holtsmark distribution

Abstract: The distribution $\Psi(\mathbf{x}; Z, M) = \operatorname{const}. \exp(\mathrm{tr} (ZM^T \mathbf{xx}^T M))$ on the unit sphere in three-space is discussed. It is parametrized by the diagonal shape and concentration matrix $Z$ and the orthogonal orientation matrix $M. \Psi$ is applicable in the statistical analysis of measurements of random undirected axes. Exact and asymptotic sampling distributions are derived. Maximum likelihood estimators for $Z$ and $M$ are found and their asymptotic properties elucidated. Inference procedures, including tests for isotropy and circular symmetry, are proposed. The application of $\Psi$ is illustrated by a numerical example.

Abstract: Atomic data for opacity calculations A fast and accurate method for evaluating the nonrelativistic free-free gaunt factor The equation of state for stellar envelopes Statistical mechanics of partially ionized stellar plasmas Rational approximations for the Holtsmark distribution The atomic internal partition function Opacities for stellar envelopes Tables: Energy levels and subset of gf-values for He to Si.

Abstract: We discuss the distribution of the gravitational force created by a Poissonian distribution of field sources (stars, galaxies,...) in different dimensions of space d. In d = 3, when the particle number N →+∞, it is given by a Levy law called the Holtsmark distribution. It presents an algebraic tail for large fluctuations due to the contribution of the nearest neighbor. In d = 2, for large but finite values of N, it is given by a marginal Gaussian distribution intermediate between Gaussian and Levy laws. It presents a Gaussian core and an algebraic tail. In d = 1, it is exactly given by the Bernouilli distribution (for any particle number N) which becomes Gaussian for N ≫ 1. Therefore, the dimension d = 2 is critical regarding the statistics of the gravitational force. We generalize these results for inhomogeneous systems with arbitrary power-law density profile and arbitrary power-law force in a d-dimensional universe.

Abstract: Random tree-type partitions for finite sets are used as a model of a chemical polymerization process when ring formation is forbidden. Technically, our series of three papers studies the asymptotic behavior (in the thermodynamic limit as $n \rightarrow \infty$) of a particular probability distribution on the set of all forests of trees on a set of $n$ elements (monomers). The study rigorously establishes the existence of three stages of polymerization dependent upon the ratio of association and dissociation rates of monomers. The subcritical stage has been analyzed in the other two papers of this series. The present paper, second in the series, concentrates on the analysis of the near-critical and supercritical stages. In the supercritical stage we discover that the molecular weight of the largest connected component (gel) has the Holtsmark distribution. Our study combines elements of a classical Flory-Stockmayer polymerization theory with the spirit of more recent developments in the Erdos-Renyi theory of random graphs. Although this paper has a chemical motivation, conceptually similar mathematical models have been found useful in other disciplines, such as computer science and biology, etc.