Nucleus-Electron Model for States Changing from a Liquid Metal to a Plasma and the Saha Equation

We extend the quantal hypernetted-chain (QHNC) method, which has been proved to yield accurate results for liquid metals, to treat a partially ionized plasma. In a plasma, the electrons change from a quantum to a classical fluid gradually with increasing temperature; the QHNC method applied to the electron gas is in fact able to provide the electron-electron correlation at arbitrary temperature. As an illustrating example of this approach, we investigate how liquid rubidium becomes a plasma by increasing the temperature from 0 to 30 eV at a fixed normal ion-density $1.03 \times 10^{22}/cm^3$. The electron-ion radial distribution function (RDF) in liquid Rb has distinct inner-core and outer-core parts. Even at a temperature of 1 eV, this clear distinction remains as a characteristic of a liquid metal. At a temperature of 3 eV, this distinction disappears, and rubidium becomes a plasma with the ionization 1.21. The temperature variations of bound levels in each ion and the average ionization are calculated in Rb plasmas at the same time. Using the density-functional theory, we also derive the Saha equation applicable even to a high-density plasma at low temperatures. The QHNC method provides a procedure to solve this Saha equation with ease by using a recursive formula; the charge population of differently ionized species are obtained in Rb plasmas at several temperatures. In this way, it is shown that, with the atomic number as the only input, the QHNC method produces the average ionization, the electron-ion and ion-ion RDF's, and the charge population which are consistent with the atomic structure of each ion for a partially ionized plasma.Comment: 28 pages(TeX) and 11 figures (PS

Pressure formulas for liquid metals and plasmas based on the density-functional theory

At first, pressure formulas for the electrons under the external potential produced by fixed nuclei are derived both in the surface integral and volume integral forms concerning an arbitrary volume chosen in the system; the surface integral form is described by a pressure tensor consisting of a sum of the kinetic and exchange-correlation parts in the density-functional theory, and the volume integral form represents the virial theorem with subtraction of the nuclear virial. Secondly on the basis of these formulas, the thermodynamical pressure of liquid metals and plasmas is represented in the forms of the surface integral and the volume integral including the nuclear contribution. From these results, we obtain a virial pressure formula for liquid metals, which is more accurate and simpler than the standard representation. From the view point of our formulation, some comments are made on pressure formulas derived previously and on a definition of pressure widely used.Comment: 18 pages, no figur

A high order qq-difference equation for qq-Hahn multiple orthogonal polynomials

A high order linear $q$-difference equation with polynomial coefficients having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation is related to the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when $q\to1$ are studied. Indeed, the difference equation for Hahn multiple orthogonal polynomials given in \cite{Lee} is corrected and obtained as a limiting case

Probing Ion-Ion and Electron-Ion Correlations in Liquid Metals within the Quantum Hypernetted Chain Approximation

We use the Quantum Hypernetted Chain Approximation (QHNC) to calculate the ion-ion and electron-ion correlations for liquid metallic Li, Be, Na, Mg, Al, K, Ca, and Ga. We discuss trends in electron-ion structure factors and radial distribution functions, and also calculate the free-atom and metallic-atom form-factors, focusing on how bonding effects affect the interpretation of X-ray scattering experiments, especially experimental measurements of the ion-ion structure factor in the liquid metallic phase.Comment: RevTeX, 19 pages, 7 figure

Wigner quantization of some one-dimensional Hamiltonians

Recently, several papers have been dedicated to the Wigner quantization of different Hamiltonians. In these examples, many interesting mathematical and physical properties have been shown. Among those we have the ubiquitous relation with Lie superalgebras and their representations. In this paper, we study two one-dimensional Hamiltonians for which the Wigner quantization is related with the orthosymplectic Lie superalgebra osp(1|2). One of them, the Hamiltonian H = xp, is popular due to its connection with the Riemann zeros, discovered by Berry and Keating on the one hand and Connes on the other. The Hamiltonian of the free particle, H_f = p^2/2, is the second Hamiltonian we will examine. Wigner quantization introduces an extra representation parameter for both of these Hamiltonians. Canonical quantization is recovered by restricting to a specific representation of the Lie superalgebra osp(1|2)

Escort mean values and the characterization of power-law-decaying probability densities

Escort mean values (or $q$-moments) constitute useful theoretical tools for describing basic features of some probability densities such as those which asymptotically decay like {\it power laws}. They naturally appear in the study of many complex dynamical systems, particularly those obeying nonextensive statistical mechanics, a current generalization of the Boltzmann-Gibbs theory. They recover standard mean values (or moments) for $q=1$. Here we discuss the characterization of a (non-negative) probability density by a suitable set of all its escort mean values together with the set of all associated normalizing quantities, provided that all of them converge. This opens the door to a natural extension of the well known characterization, for the $q=1$ instance, of a distribution in terms of the standard moments, provided that {\it all} of them have {\it finite} values. This question would be specially relevant in connection with probability densities having {\it divergent} values for all nonvanishing standard moments higher than a given one (e.g., probability densities asymptotically decaying as power-laws), for which the standard approach is not applicable. The Cauchy-Lorentz distribution, whose second and higher even order moments diverge, constitutes a simple illustration of the interest of this investigation. In this context, we also address some mathematical subtleties with the aim of clarifying some aspects of an interesting non-linear generalization of the Fourier Transform, namely, the so-called $q$-Fourier Transform.Comment: 20 pages (2 Appendices have been added

Vector Continued Fractions using a Generalised Inverse

A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse permits construction of vector analogues of the Jacobi continued fraction. These vector Jacobi fractions are related to vector and scalar-valued polynomial functions of the vectors, which satisfy recurrence relations similar to those of orthogonal polynomials. The vector Jacobi fraction has strong convergence properties which are demonstrated analytically, and illustrated numerically.Comment: Published form - minor change

The smallest eigenvalue of Hankel matrices

Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of lambda_N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where lambda_N is known to be bounded below by a positive constant, we prove that the limit of the n'th smallest eigenvalue of H_N for N tending to infinity tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed

Weighted norm inequalities for polynomial expansions associated to some measures with mass points

Fourier series in orthogonal polynomials with respect to a measure $\nu$ on $[-1,1]$ are studied when $\nu$ is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in $[-1,1]$. We prove some weighted norm inequalities for the partial sum operators $S_n$, their maximal operator $S^*$ and the commutator $[M_b, S_n]$, where $M_b$ denotes the operator of pointwise multiplication by b \in \BMO. We also prove some norm inequalities for $S_n$ when $\nu$ is a sum of a Laguerre weight on $\R^+$ and a positive mass on $0$

Block orthogonal polynomials: I. Definition and properties

Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg-Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean scalar product, to a given $i$-dimensional subspace ${\cal E}_i$ of polynomials associated with the constraints. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. We recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an Euclidean vector space endowed with distinct scalar products. An explicit two step Gram-Schmidt orthogonalization (G-SO) procedure to determine these bases is given. By definition, the standard block orthogonal (SBO) polynomials are associated with a choice of ${\cal E}_i$ equal to the subspace of polynomials of degree less than $i$. We investigate their properties, emphasizing similarities to and differences from the standard orthogonal polynomials. Applications to classical orthogonal polynomials will be given in forthcoming papers.Comment: This is a reduced version of the initial manuscript, the number of pages being reduced from 34 to 2