Exact Series Expansions Research Articles

Effective stiffness tensor of composite media—I. Exact series expansions

The problem of determining exact expressions for the effective stiffness tensor macroscopically anisotropic two-phase composite media of arbitrary microstructure in arbitrary space dimension d is considered. We depart from previous treatments by introducing an integral equation for the “cavity” strain field. This leads to new, exact series expansions for the effective stiffness tensor of macroscopically anisotropic, d-dimensional, two-phase composite media in powers of the “elastic polarizabilities”. The nth-order tensor coefficients of these expansions are explicitly expressed as absolutely convergent integrals over products of certain tensor fields and a determinant involving n-point correlation functions that characterize the microstructure. For the special case of macroscopically isotropic media, these series expressions may be regarded as expansions that perturb about the optimal structures that realize the Hashin-Shtrikman bounds (e.g. coated-inclusion assemblages or finite-rank laminates). Similarly, for macroscopically anisotropic media, the series expressions may be regarded as expansions that perturb about optimal structures that realize Willis' bounds. For isotropic multiphase composites, we remark on the behavior of the effective moduli as the space dimension d tends to infinity.

Exact Expression for the Effective Elastic Tensor of Disordered Composites

We derive new, exact series expansions for the effective elastic tensor of anisotropic, $d$-dimensional, two-phase disordered composites whose nth-order tensor coefficients are integrals involving n-point correlation functions that characterize the structure. These series expansions, valid for any structure, perturb about certain optimal dispersions. Third-order truncation of the expansions results in formulas for the elastic moduli of isotropic dispersions that are in very good agreement with benchmark data, always lie within rigorous bounds, and are superior to popular self-consistent approximations.

Approximate, saturated and blurred self-affinity of random processes with finite domain power-law power spectrum

We study the (approximate) self-affine regimes of random processes with finite domain power-law power spectra for arbitrary values of the spectral exponent α. In particular, we focus on regimes that are imprinted in the structure function of the process. For this function we obtain exact series and asymptotic expansions. We also provide expressions for the usual statistical parameters: variance, correlation time, and introduce a specific parameter, called chronothesy which measures the intensity of the self-affine fluctuations. There are essentially three qualitatively different self-affine regimes, corresponding to values of α in the following intervals: 1 < α < 3, α ≥ 3, and 0 < α <- 1. When in the first regime, the process exhibits a statistical, however, only approximate self-affinity. The self-affinity is embodied in the leading asymptotic term which represents the familiar ideal fractal behavior. If α > 3, the process shows again approximate self-affinity, which with increasing α saturates and leads to a self-affinity exponent of ≈ 2. For 0 < α < 1, the approximate self-affine behavior, which in this case is characterized by a negative exponent, is blurred by intense oscillations to the extent that, in practice, it cannot be observed. As an application, it is shown that the time series generated by sampling X and Y coordinates of the Lorenz system are characterized by blurred self-affinity.

London equation of state for quantum hard disks

For a two-dimensional hard-disk boson system we propose an analytical London-type expression for the ground-state energy as a function of density ranging from zero to close packing. Such an equation interpolates between the leading term of the exact low-density series expansion and the behavior expected at close-packing based on uncertainty-principle arguments and determined by the correct value of the residue which is a dimensionless constant whose value depends on the space dimension as well as on the kind of close packing configuration involved. As for any reasonably accurate equation of state, variational Monte Carlo calculations for both fluid and crystalline energy branches turn out to be close upper bounds to the resulting so-called London–Hubbard energy which includes pair-correlation effects.

Quantum-Hard-Sphere System Equations of State Revisited

Analytical equations of state for boson and fermion hard-sphere fluids ranging from very low to very high densities are constructed. Such equations of state serve as a zero-order (reference) state upon which to build so-called quantum-thermodynamic-perturbation corrections in describing real but simple quantum fluids at zero temperature. The fluid branch extrapolations from the exact low-density series expansions for the energy are carried out by incorporating various physical arguments, such as close packing densities and residues. Modified London equations of state for the high-density crystalline branch agree very well with computer simulations, and at close packing with certain experimental results at high pressure.

An exact formulation for the electromagnetic fields of a cylindrical antenna with a triangular current distribution

A mathematically exact formulation for the vector potential and corresponding electromagnetic fields of a triangular current cylindrical dipole are presented for the first time in this paper. These exact expressions converge rapidly in the near‐field region of the antenna allowing them to be used for the efficient and accurate computational modeling of electrically short cylindrical antennas. The exact series expansion of the triangular current vector potential is shown to contain two fundamental exponential integrals and their higher‐order associated integrals. Numerically stable forward recurrence relations have been derived which may be used for the efficient evaluation of these higher‐order integrals in addition to the cylindrical wire kernel. These recursions may also be employed in the computation of the electric and magnetic fields. It is demonstrated that the classical thin wire forms of the vector potential and electromagnetic fields are actually special cases of the more general exact expansions. Finally, the exact formulation was used to investigate the near‐field behavior of traditional thin wire as well as moderately thick wire dipoles. Several near‐field plots are presented including a vector plot of the total electric field in the vicinity of a moderately thick quarter‐wavelength dipole.

Series expansions for excited states of quantum lattice models

We show that by means of connected-graph expansions one can effectively generate exact high-order series expansions which are informative of low-lying excited states for quantum many-body systems defined on a lattice. In particular, the Fourier series coefficients of elementary excitation spectra are directly obtained. The numerical calculations involved are straightforward extensions of those which have already been used to calculate series expansions for ground-state correlations and T = 0 susceptibilities in a wide variety of models.

Series expansions from the finite lattice method

The finite lattice method has proved a powerful technique for obtaining exact series expansions for problems in lattice statistics. The requirements of the method are reviewed and used as a basis for describing the range of applications.

An exact integration procedure for vector potentials of thin circular loop antennas

A direct integration procedure for far-zone vector potentials of thin circular loop antennas has been known for many years. This method is general in the sense that it leads to simple integrals which have closed form solutions for most commonly assumed loop current distributions. However, a comparable integration technique has not been available for evaluating the more complicated near-zone vector potentials. This paper introduces a systematic approach for the exact integration of general near-zone vector potentials associated with current-carrying circular loop antennas. A particular example is considered where this new integration technique is used to find exact solutions to the vector potential and electromagnetic field integrals for loops with a Fourier cosine-series expansion of the current. The observation is made that degenerate forms of these exact representations lead to simplified expressions for the important special cases of a uniform and cosinusoidal current loop. Two equivalent forms of exact series expansions are derived for the uniform current vector potential and field integrals. It is shown that the familiar small-loop approximations, as well as the classical far-field expressions, may be obtained as limiting cases of the more general exact series representations for the uniform current loop obtained in this paper. Convenient asymptotic far-field expansions are derived for the loop with a cosinusoidal current distribution. Finally, the far-field analysis for the cosinusoidal loop is generalized to loops having an arbitrary current represented by a Fourier cosine series.

Accurate numerical solution of the radial Schrödinger and Dirac wave equations

A FORTRAN 77 subroutine package for the numerical solution of the Schrödinger and Dirac wave equations for central fields is presented. The considered fields are such that the function ν( r) ≡ rV( r) is finite for all r and tends to constant values for r → 0 and r → ∞. This includes finite-range fields as well as combinations of Coulomb and short-range fields. The potential energy function V( r) used in the calculation is the natural cubic spline that interpolates a table of values provided by the user. The radial wave equations are solved by using piecewise exact power series expansions of the radial functions, which are summed up to the prescribed accuracy so that truncation errors can be completely avoided. Normalized radial wave functions, eigenvalues for bound states and phase shifts for free states are evaluated.

Coherent spontaneous emission in FELIX

Broadband continuous and high-intensity coherent synchrotron radiation (CSR) emitted from 4 ps electron bunches provided by the 30 MeV RF linac of Beijing FEL is analyzed and numerically calculated using an exact series expansion for the infinite integral of fractional modified Bessel function. CSR in the mm-wave and far-IR to mm-wave regions can be respectively generated by directly using these bunches and by applying those ones compressed to ≤=1 ps. The CSR powers, approximately as 108–109 times as the SR ones, in the range from several hundred microwatts to milliwatts are dependent on chosen electron density distribution, wavelength range, and gathering angle. The power produced by rectangular bunches is greater than that generated by Gaussian ones. The shorter the bunch, the stronger the produced CSR, the greater the energy concentrated to the far-IR end. Experiments to generate CSR and measure the bunch length are designed.

Two-dimensional lattice tree exponents and amplitudes: Simulation algorithms versus series

We use a local Monte Carlo algorithm to simulate lattice trees in two dimensions for the site and bond problem. We investigate the properties of radius of gyration, perimeter-to-site ratio, and vertex degree in a tree, adding some new results in the site problem, compare our results on their noncritical properties with those obtained from earlier reversible and slightly nonreversible algorithms, and combine our determinations with new exact series expansion data. On the controversy surrounding the possible lack of universality for the first confluent singularity for the gyration radius, we feel that conclusions must be guarded.

VISUALIZATION OF GRAPHICAL METHODS OF SERIES ANALYSIS

Exact power series expansions provide a powerful method for studying the critical behaviour of many systems. Efficient analysis methods are essential to fully exploit the series approach. A particularly challenging problem in the analysis of these expansions is the elucidation of the critical behaviour in cases where critical temperature (or threshold) as well as dominant and correction exponent is unknown. We describe a scheme (which we call Visualization of Graphical methods for Series analysis, or VGS) developed explicitly for this situation and discuss its high precision implementation in a workstation environment. Our approach involves visualization of multiple approximants in a three-dimensional space. Several examples from Ising models, percolation and exactly solved systems are given.

Critical behavior of self-avoiding walks on fractals

Using a new graph counting technique suitable for self-similar fractals, exact 18th-order series expansions for SAWs on some Sierpinski carpets are generated. From them, the critical fugacityxc and critical exponents ΝSAW and γSAW are obtained. The results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied. We find for some carpets with low lacunarity that ΝSAW ΝSAW (d) for SAWs on the fractals which are embedded in ad-dimensional Euclidean space.

Surface critical exponents for models of polymer collapse and adsorption: the universality of the Theta and Theta ' points

The surface critical exponents are estimated both in ordinary and in special regimes from exact series expansions of up to 28 terms for a model of the Theta point. They are found to be in agreement with those derived for the Theta ' point, confirming the conjecture that the Theta and Theta ' points lie in the same universality class. Some essential features of the phase diagram for a self-interacting, self-avoiding polymer in solution are also calculated to a higher degree of accuracy than previously obtained.

Exact series expansions for systems on regular fractals.

We present an analytical technique for counting exactly the number of ways of embedding a graph in a fairly general family of self-similar deterministic fractals. As a result, series expansion for statistical systems can be obtained that are exact order by order in the expansion parameter. As an illustration, series expansions for self-avoiding walks on a Sierpriaanski carpet are given. Analytical results for systems on infinitely ramified fractals are obtained in the thermodynamic limit without making any approximation on the lattice.

Numerical method for unsteady compressible boundary layers driven by compression or expansion wave

AbstractA numerical analysis is presented for the unsteady compressible laminar boundary layer driven by a compression or expansion wave. Approximate or series expansion methods have been used for the problems because of the characteristics of the governing equations, such as non‐linearity, coupling with the thermal boundary layer equation and initial conditions. Here a transformation of the governing equations and the numerical linearization technique are introduced to deal with the difficulties. First, the governing equations are transformed for the initial conditions by Howarth and semisimilarity variables. These transformations reduce the number of independent variables from three to two and the governing equations from partial to ordinary differential equations at the initial point. Next, the numerical linearization technique is introduced for the non‐linearity and the coupling with the thermal boundary layer equation. Because the non‐linear terms are linearized without sacrifice of numerical accuracy, the solutions can be obtained without numerical iterations. Therefore the exact numerical solution, not approximate or series expansion, can be obtained. Compared with the approximate or series expansion method, this method is much improved. Results are compared with the series expansion solutions.

Variational approach to the stationary states of the double-oscillator potential

A variational approach to the calculation of the two lowest eigenvalues of Schrödinger’s equation for the double-oscillator potential is presented. A family of two-parameter double-Gaussian trial functions provides an alternative to the exact series expansions for the energy eigenfunctions and yields energy estimates of high accuracy.

The shortest path of SAWs with bridges

The shortest paths of self-avoiding walks (SAWs) with bridge lengths b=1, 2, 3 and 2 are studied by exact series expansions and by Monte Carlo simulation methods on the square and simple cubic lattices. In the series work, SAW configurations of up to 24 and 14 steps are generated on the square and simple cubic lattices respectively. Assuming the shortest path SN between two sites, which are separated by N steps along the chain, has the scaling form SN approximately AN+BN1- Delta , it is found that Delta approximately=1 and Delta approximately=1/2 on the square and simple cubic lattices respectively, independent of the bridge length. The problem is also investigated by the Mellin-Pade approximation method using Monte Carlo data on the square lattice. The latter method gives results consistent with those of the series expansion study. The authors' results are not consistent with earlier predictions.

The one-dimensional kinetic Ising model: A series expansion study

Exact power series expansions (through eight terms) in the time are derived for relaxation in the one-dimensional Ising model with nearest-neighbor interactions for a general rate parameter where the activation energy is a variable fraction of the energy required to break nearest-neighbor bonds. It is found that the qualitative nature of the relaxation is very dependent on this parameter, varying from nearly simple exponential decay (as with Glauber dynamics) for an intermediate value of this parameter, to an initial rate of change that is either much slower or faster than a simple exponential at the extremes of the range of variation of the parameter. The rate equations for the limit of rapid internal diffusion (internal equilibration) are integrated for several special values of the rate parameter. In general the internal equilibration approximation is not a good representation of the relaxation except when the relaxation is similar to Glauber dynamics.