688 research outputs found
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 . 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 -difference equation for -Hahn multiple orthogonal polynomials
A high order linear -difference equation with polynomial coefficients
having -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 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 -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 . 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 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 -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 on
are studied when is a linear combination of a generalized Jacobi
weight and finitely many Dirac deltas in . We prove some weighted norm
inequalities for the partial sum operators , their maximal operator
and the commutator , where denotes the operator of pointwise
multiplication by b \in \BMO. We also prove some norm inequalities for
when is a sum of a Laguerre weight on and a positive mass on
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 -dimensional subspace 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 equal
to the subspace of polynomials of degree less than . 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
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