Some properties of the moshinsky coefficients

Moshinsky has suggested a method of calculating Talmi transformation coefficients in which the coefficients with two radial quantum numbers equal to zero are calculated first and then recursion formulas over radial quantum numbers are applied. The coefficients with two radial quantum numbers equal to zero are shown to be expressed as factorials, in terms of a particular type of coefficient with all radial quantum numbers equal to zero, and a simple formula for these coefficients is given. Some interesting properties of the Moshinsky coefficient sums are found and a simple formula expressing the coefficients of the “three-body” hyperspherical function transformation in terms of generalized Moshinsky coefficients is obtained. Recursion formulas and symmetry relations for the Moshinsky coefficients are also discussed.     

Raynal–Revai coefficients for a general kinematic rotation

In a three-body system, transitions between different sets of normalized Jacobi coordinates are described as general kinematic transformations that include an orthogonal or a pseudoorthogonal rotation. For such rotations, the Raynal–Revai coefficients execute a unitary transformation between three-body hyperspherical functions. Recurrence relations that make it possible to calculate the Raynal–Revai coefficients for arbitrary angular momenta are derived on the basis of linearized representations of products of hyperspherical functions.     

Infinitesimal transformation for hyperspherical functions and Moshinsky coefficients

In order to obtain coefficients for a finite transformation (such as a change of basis for hyperspherical functions) the matrix of an infinitesimal transformation is written in a pseudodiagonal form by means of sets of homogeneous linear equations.     

Generalized entropy and transport coefficients of hadronic matter

We use the generalized entropy four-current of the Müller-Israel-Stewart (MIS) theories of relativistic dissipative fluids to obtain information about fluctuations around equilibrium states. This allows one to compute the non-classical coefficients of the entropy 4-flux in terms of the equilibrium distribution functions. The Green-Kubo formulae are used to compute the standard transport coefficients from the fluctuations of entropy due to dissipative fluxes.     

Continuum limits and exact finite-size-scaling functions for one-dimensionalO(N)-invariant spin models

We solve exactly the general one-dimensionalO(N)-invariant spin model taking values in the sphereS ^N–1, with nearest-neighbor interactions, in finite volume with periodic boundary conditions, by an expansion in hyperspherical harmonics. The possible continuum limits are discussed for a general one-parameter family of interactions and an infinite number of universality classes is found. For these classes we compute the finite-size-scaling functions and the leading corrections to finite-size scaling. A special two-parameter family of interactions (which includes the mixed isovector/isotensor model) is also treated and no additional universality classes appear. In the appendices we give new formulae for the Clebsch-Gordan coefficients and 6–j symbols of theO(N) group, and some new generalizations of the Poisson summation formula; these may be of independent interest.     

Calculation of geometrical coupling coefficients for the hyperspherical harmonics approach

An elegant and fast method for the calculation of geometrical structure coefficients needed for an expansion of a few-body wavefunction and interaction in hyperspherical harmonics has been proposed. A sum rule for the GSC has also been derived, which is useful for an independent check of the coefficients. The proposed method of computation is many orders of magnitude faster than conventional methods.     

Transformation Coefficients of Hyperspherical Harmonic Functions of an -Body System

Two algorithms are presented for calculating the transformation coefficients between hyperspherical harmonic functions constructed with different sets of Jacobi vectors. They have been tested in the case , where the transformation coefficients of states with grand angular quantum number up to have been studied. The applicability of the two algorithms to larger systems is discussed. The numbers of independent hyperspherical-spin-isospin states with given values, entering the expansion of the alpha-particle ground-state wave function, are also evaluated. The use of complete non-redundant bases is important for future accurate applications of the hyperspherical harmonic technique. Received December 23, 1997; revised May 25, 1998; accepted for publication May 30, 1998     

Modelling radiative mean absorption coefficients

We define and compute mean absorption coefficients for the macroscopic models of radiative transfer. These coefficients take into account the anisotropic form of the photon emission and lead to a better computation of a photonic flow far from the radiative equilibrium. They are deduced by averaging a specific radiative intensity on the space of frequency and are generalized versions of the Planck means. This intensity is obtained by minimizing the mathematical entropy with the constraint of the reconstruction of radiative moments and constitutes the closure of the M ₁ radiative model. We discuss the influence of these coefficients, extend them to the case of multi-frequency problems and perform a numerical comparison with the former Planck mean.     

The Clebsch-Gordan coefficients of SU(4)

A set of recurrence relations connecting the matrix elements of finite transformation belonging to the same irreducible representation of SU(4) is used to obtain a wide class of matrix elements. An expression for the Clebsch-Gordan coefficient is obtained by integrating the product of three matrix elements belonging to three different irreducible representations of the group. The symmetry properties of the matrix elements and the Clebsch-Gordan coefficients are discussed.The author is grateful to Professors S.Datta Majumdar and G.Bandyopadhyay of the Department of Physics, I.I.T., Kharagpur, for many helpful discussions. This work was supported by the C.S.I.R., Government of India.     

Nonequilibrium thermodynamics in field theory: Transport coefficients

Transport coefficients are expressed by real time correlation functions of energy-momentum tensor in the linear response approximation. We establish field theoretical method to compute them in perturbation theory, which is demonstrated in λ?⁴ theory.     

The potential harmonic expansion method

Various properties of the hyperspherical potential basis are investigated. The expansion of any two-body function, in particular the two-body potential, is given. The matrix elements with two and three potential harmonics needed for the construction of the potential matrix are calculated. Useful recurrence formulae are derived. The concept of potential basis is extended to systems with any number of fermions. A method for improving the accuracy of the expansion of the wavefunction by taking into account more than the two-body correlations is suggested.     

Renormalization Group Maps for Ising Models in Lattice-Gas Variables

Real-space renormalization group maps, e.g., the majority rule transformation, map Ising-type models to Ising-type models on a coarser lattice. We show that each coefficient in the renormalized Hamiltonian in the lattice-gas variables depends on only a finite number of values of the renormalized Hamiltonian. We introduce a method which computes the values of the renormalized Hamiltonian with high accuracy and so computes the coefficients in the lattice-gas variables with high accuracy. For the critical nearest neighbor Ising model on the square lattice with the majority rule transformation, we compute over 1,000 different coefficients in the lattice-gas variable representation of the renormalized Hamiltonian and study the decay of these coefficients. We find that they decay exponentially in some sense but with a slow decay rate. We also show that the coefficients in the spin variables are sensitive to the truncation method used to compute them.     

Reliable Recurrence Algorithm for High-Order Krawtchouk Polynomials

Krawtchouk polynomials (KPs) and their moments are promising techniques for applications of information theory, coding theory, and signal processing. This is due to the special capabilities of KPs in feature extraction and classification processes. The main challenge in existing KPs recurrence algorithms is that of numerical errors, which occur during the computation of the coefficients in large polynomial sizes, particularly when the KP parameter (p) values deviate away from 0.5 to 0 and 1. To this end, this paper proposes a new recurrence relation in order to compute the coefficients of KPs in high orders. In particular, this paper discusses the development of a new algorithm and presents a new mathematical model for computing the initial value of the KP parameter. In addition, a new diagonal recurrence relation is introduced and used in the proposed algorithm. The diagonal recurrence algorithm was derived from the existing n direction and x direction recurrence algorithms. The diagonal and existing recurrence algorithms were subsequently exploited to compute the KP coefficients. First, the KP coefficients were computed for one partition after dividing the KP plane into four. To compute the KP coefficients in the other partitions, the symmetry relations were exploited. The performance evaluation of the proposed recurrence algorithm was determined through different comparisons which were carried out in state-of-the-art works in terms of reconstruction error, polynomial size, and computation cost. The obtained results indicate that the proposed algorithm is reliable and computes lesser coefficients when compared to the existing algorithms across wide ranges of parameter values of p and polynomial sizes N. The results also show that the improvement ratio of the computed coefficients ranges from 18.64% to 81.55% in comparison to the existing algorithms. Besides this, the proposed algorithm can generate polynomials of an order ∼8.5 times larger than those generated using state-of-the-art algorithms.     

Trace Formulae of Characteristic Polynomial and Cayley-Hamilton＇s Theorem, and Applications to Chiral Perturbation Theory and General Relativity

By using combinatorics, we give a new proof for the recurrence relations of the characteristic polynomial coefficients, and we further obtain an explicit expression for the generic term of the coefficient sequence, which yields the trace formulae of the Cayley-Hamilton＇s theorem with all coefficients explicitly given. This implies a byproduct, a complete expression for the determinant of any finite-dimensional matrix in terms of the traces of its successive powers. And we discuss some of their applications to ehiral perturbation theory and general relativity.     

Kubo formulas for second-order hydrodynamic coefficients

At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity η and on five additional "second-order" hydrodynamical coefficients τ(Π), κ, λ?, λ?, and λ?. We derive Kubo relations for these coefficients, relating them to equilibrium, fully retarded three-point correlation functions of the stress tensor. We show that the coefficient λ? can be evaluated directly by Euclidean means and does not in general vanish.     

Optical conductivity by simple non-adiabatic model Hamiltonian

Two simple nonadiabatic model Hamiltonian, a generalized Rabi Hamiltonian and a caricature of the Fröhlich Hamiltonian are presented. The eigenvalue problem defined by the first is solved in Bargmann Hilbert space of analytical functions to demonstrate the methods. Results for the second are reported. Eigenvalues are given in terms of continued fractions, the eigenvectors as power series or Neumann series. The coefficients of the series are determined by simple three term recurrence relations. Using these results, the optical conductivity is calculated with Kubo's formulae. We correlate our results with experimental observations and the conventional small polaron theory.     

Gluon operator coefficients in electroproduction

We compute the Wilson coefficients of the twist-two gluon operators which are relevant to the W₂ tensor structure in electroproduction. We find that the simple Green functions used to define the renormalization of the operators each have two independent tensor structures. This gives rise to varying renormalization conditions. We choose conditions which are different from those of other authors, and which simplify the computations. Taking these differences into account, we show that almost all of the published results are equivalent.     

Evaluating virial coefficients for multicomponent mixtures: hard sphere mixtures and flexible chains

A new algorithm to compute the virial coefficients of multicomponent mixtures is proposed. The number of graphs that must be evaluated increases dramatically in a multicomponent mixture so that it becomes difficult to enumerate and compute all possible graphs. However, once all of them are known and evaluated, the virial coefficient of the mixture can be evaluated for any composition. If one is interested in the virial coefficient of a mixture of a certain composition, then a simpler approach can be followed. Starting from the graphs of a pure fluid, we assign a random chemical identity to each of the molecules of the graph. The probability of assigning a given chemical identity is taken from the composition of the mixture. In this way composition is treated as a random variable within the Monte Carlo procedure which determines the virial coefficient. The algorithm is checked by comparison with the virial coefficients of binary hard spheres mixtures which are well known. Good agreement is found. The procedure is then extended to multicomponent mixtures of hard spheres. Finally the procedure is applied to the determination of the virial coefficients of a flexible molecule. For flexible molecules the possible configurations of the molecules are treated as different components of the mixture. In this way we present what appears to be the first determination of the third and fourth virial coefficients of polymers in the continuum.