Commutator and anticommutator green functions in the theory of disordered spin systems

For the Heisenberg model with random exchange integrals J_ij or local fields H_i the application of the commutator and anticommutator Green function is discussed. Because of the different algebraic structure of both Green functions the Tjablikov-decoupling procedure and an approximated structure averaging yield different results. For a Lorentzian distribution of J_ij and H_i, where the low energy excitations are localized, the density of states, the magnetization, the susceptibility and the specific heat are investigated by the anticommutator Green function. The commutator Green function fails to work in this case which is adequate for amorphous unsaturated ferromagnets with an inhomogeneous spin structure. We have also discussed the opposite case of an amorphous unsaturated ferromagnet in which the spin structure is homogenized due to extended zero-point fluctuations. The commutator Green function appropriately describes this case while the approximate results of the anticommutator Green function become inadequate.     

New microscopic wave function of ��-condensation

We explain how to treat a microscopic wave function of ??-condensation taking a 3??-nucleus as a typical example. The wave function has been originally proposed ten years before by Horiuchi, R?pke, Schuck and the present author (Phys. Rev. Lett., 2001, 87: 192501). The microscopic model, which fully takes into account the Pauli principle between all the constituent nucleons, effective internucleon forces and the Coulomb force, can play an important role in reproducing an ??-gas nature thanks to ??-condensation as an excited state of ??-like nuclei. An essential point of the wave function is to describe their ground state simultaneously. We study its typical features by giving an analytical formula of the norm kernel and the kernel concerning the one-body operator for 3??-condensation.     

A Wigner-function formulation of finite-state quantum mechanics

For a non-relativistic system with only continous degrees of freedom (no spin, for example), the original Wigner function can be used as an alternative to the density matrix to represent an arbitrary quantum state. Indeed, the quantum mechanics of such systems can be formulated entirely in terms of the Wigner function and other functions on phase space, with no mention of state vectors or operators. In the present paper this Wigner-function formulation is extended to systems having only a finite number of orthogonal states. The “phase space” for such a system is taken to be not continuous but discrete. In the simplest cases it can be pictured as an N×N array of points, where N is the number of orthogonal states. The Wigner function is a real function on this phase space, defined so that its properties are closely analogous to those of the original Wigner function. In this formulation, observables, like states, are represented by real functions on the discrete phase space. The complex numbers still play an important role: they appear in an essential way in the rule for forming composite systems.     

Ingoing and outgoing waves in the non-relativistic theory of photoionization

A well-known formula for the differential cross section for photoionization contains a so-called ‘final state’ wave function which asymptotically behaves as a Coulomb-modified plane wave plus an ingoing Coulomb-modified radial wave. We explain the reason of appearance of that function on the ground of an analysis of an asymptotic form of a relevant outgoing Green's function. The reasoning is carried out without referring to an intuitive argumentation of Breit and Bethe [Phys. Rev. 93 (1954) 888].     

Probability density function of the sum of N partially correlated speckle patterns

A method is developed which allows exact calculation of the probability density function of the sum of N correlated speckle patterns. To find the density function, it is only necessary to first find the eigenvalues of an N × N coherence matrix. When the eigenvalues are distinct, the density function can be expressed as a simple sum of N exponential terms.     

Comments on the method of moments applied to the calculation of optical absorption line shapes

In order to check reliability of the method of moments proposed by Sati et al., the line shape for an A_1g → T_1μ transition (plus T_2g-mode) is investigated in the high temperature limit. This method is applicable even when the original function has a singular point. It is essential to adopt an appropriate generating function which has the same asymptotic behavior as the original function.     

Self-organization analysis for a nonlocal convective Fisher equation

Using both an analytical method and a numerical approach we have investigated pattern formation for a nonlocal convective Fisher equation with constant and spatial velocity fields. We analyze the limits of the influence function due to nonlocal interaction and we obtain the phase diagram of critical velocities vc as function of the width μ of the influence function, which characterize the self-organization of a finite system.     

Pairing in multiparticle amplitudes: Inclusive distributions

A model for multi-pion production in the central region in high-energy collisions is studied which describes factorizable emission of pion pairs. A mathematical identification between the exclusive cross section for pion emission in our model (with all interference terms) and the configurational probability distribution function for a classical system of interacting molecules in equilibrium is exploited to obtain an expansion for the asymptotic single-particle inclusive distribution, the two-particle inclusive correlation function, and the exponent of s in the total cross section by means of cluster diagrams. An integral equation is exhibited for summing the terms corresponding to the cluster diagrams.A specific model is then considered, which we call “s-channel pole dominance”. In this model the amplitude is assumed to be large only when the subenergies of pairs of pions are near the mass of a low-lying two-pion resonance, and the transverse momentum of each resonance is small. The dependence of the amplitude on other variables is ignored, so that we effectively have independent emission of two-pion resonances with non-zero width. It is seen that an I = 0 or I = 1 resonance results in a positive two-particle inclusive I = 2 correlation function at small rapidity separations, as s → ∞, and that the correlation function can have an exponential “tail” in rapidity of qualitatively longer range than the resonance. A crude numerical simulation of a broad I = 0 spinless resonance is discussed, and the resulting I = 2 inclusive correlation function is seen to be quite large at small rapidity separations, and to have the same exponential “tail” as the I = 0 correlation function.     

Small-angle neutron scattering at fractal objects

The calculation of the correlation function of an isotropic fractal particle with the finite size ξ and the dimension D is presented. It is shown that the correlation function γ(r) of volume and surface fractals is described by a generalized expression and is proportional to the Macdonald function (D–3)/2 of the second order multiplied by the power function r ^(D–3)/2. For volume and surface fractals, the asymptotics of the correlation function at the limit r/ξ < 1 coincides with the corresponding correlation functions of unlimited fractals. The one-dimensional correlation function G(z), which, for an isotropic fractal particle, is described by an analogous expression with a shift of the index of the Macdonald function and the exponent of the power function by 1/2, is measured using spin-echo small-angle neutron scattering. The boundary case of the transition from a volume to a surface fractal corresponding to the cubic dependence of the neutron scattering cross section Q ^?3 leads to an exact analytical expression for the one-dimensional correlation function G(z) = exp(?z/ξ), and the asymptotics of the correlation function in the range of fractal behavior for r/ξ < 1 is proportional to ln(ξ/r). This corresponds to a special type of self-similarity with the additive law of scaling rather than the multiplicative one, as in the case of a volume fractal.     

The autocorrelation function of polychromatic laser speckle patterns near the image plane

The autocorrelation function of polychromatic speckle patterns produced near the image plane of a double diffraction imaging system is experimentally studied. The condition under which the polychromatic speckle field obeys Gaussian statistics is further investigated as a function of the numberN of scattering cells, using the average contrast of the speckle intensity fluctuations. The profile of the autocorrelation function is next investigated as a function ofN and the defocused distance from the image plane. It is found that the average contrast and autocorrelation function of the polychromatic speckle intensity fluctuations converge less rapidly with an increase inN than those of the monochromatic ones.     

Formal analytical solutions for the Gross-Pitaevskii equation

Considering the Gross-Pitaevskii integral equation we are able to formally obtain an analytical solution for the order parameter Φ(x) and for the chemical potential μ as a function of a unique dimensionless non-linear parameter Λ. We report solutions for different ranges of values for the repulsive and the attractive non-linear interactions in the condensate. Also, we study a bright soliton-like variational solution for the order parameter for positive and negative values of Λ. Introducing an accumulated error function we have performed a quantitative analysis with respect to other well-established methods as: the perturbation theory, the Thomas-Fermi approximation, and the numerical solution. This study gives a very useful result establishing the universal range of the Λ-values where each solution can be easily implemented. In particular, we showed that for Λ<−9, the bright soliton function reproduces the exact solution of GPE wave function.     

Proximity effects in a superconductor/ferromagnet junction

A proximity effect in an s-wave superconductor/ferromagnet (SC/F) junction is theoretically studied using the second order perturbation theory for the tunneling Hamiltonian and Green's function method. We calculate a pair amplitude induced by the proximity effect in a weak ferromagnetic metal (FM) and a half-metal (HM). In the SC/FM junction, it is found that a spin-singlet pair amplitude (Ψ_s) and spin-triplet pair amplitude (Ψ_t) are induced in FM and both amplitudes depend on the frequency in the Matsubara representation. Ψ_s is an even function and Ψ_t is an odd function with respect to the Matsubara frequency (ω_n). In the SC/HM junction, we examine the proximity effects by taking account of magnon excitations in HM. It is found that the triplet-pair correlation is induced in HM. The induced pair amplitude in HM shows a damped oscillation as a function of the position and contains the terms of even and odd functions of ω_n as in the case of the SC/FM junction. We discuss that in our tunneling model the pair amplitude of even function of ω_n only contributes to a Josephson current.     

Correlation between the pionization region and the fragmentation region in high energy proton-proton collisions

Measurements are reported of two-particle correlations in high energy proton-proton collisions with one particle in the pionization region and the other a proton in the fragmentation region. The correlation function is independent of x of the fragmentation proton for 0.55 ? x ? 0.9 and rises at small x. There is an indication that the correlation is an energy-independent function of x. The measurements for two values of the rapidity of the pionization particle give similar results.     

Fractal basin boundary in dynamical systems and the Weierstrass-Takagi functions

It is shown that the basin boundary of the complex maps Z_n+1 = Z^q_n + C (q⩾2 is an integer and |C| ⪡ 1) is expressible with the Weierstrass function which is continuous but nowhere differentiable. The relation between the Weierstrass function and the Takagi function is discussed, and these functions are extended in a general situation. The fractal basin boundaries expressed by the generalized Weierstrass-Takagi functions are investigated.     

A note on the plasma dispersion function

In the gas laser theory for non-doppler broadening, the plasma dispersion function is often considered to be an antisymmetric function: Z(-μ) = -Z(μ). We shown here that this property is not correct, because the usual integral definition for the plasma dispersion function is not entire in the complex plane.     

Tunneling of an energy eigenstate through a parabolic barrier viewed from Wigner phase space

We analyze the tunneling of a particle through a repulsive potential resulting from an inverted harmonic oscillator in the quantum mechanical phase space described by the Wigner function. In particular, we solve the partial differential equations in phase space determining the Wigner function of an energy eigenstate of the inverted oscillator. The reflection or transmission coefficients R or T are then given by the total weight of all classical phase-space trajectories corresponding to energies below, or above the top of the barrier given by the Wigner function.     

Autocorrelation function formulations and the turbulence dissipation rate: Application to dispersion models

The classical statistical diffusion theory and the binomial autocorrelation function are used to obtain a new formulation for the turbulence dissipation rate ε. The approach employs the Maclaurin series expansion of a logarithm function contained in the dispersion parameter formulation. The numerical coefficient of this new relation for ε is 100% larger than the numerical coefficient of the classical relation derived from the exponential autocorrelation function. A similar approach shows that the dispersion parameter obtained from the even exponential autocorrelation function does not result in a relation for ε and, therefore, is not suitable for application in dispersion models. In addition, a statistical comparison to experimental ground-level concentration data demonstrates that this newly derived relation for ε as well as other formulations for the turbulence dissipation rate are suitable for application in Lagrangian stochastic dispersion models. Therefore, the analysis shows that there is an uncertainty regarding the turbulence dissipation rate function form and the autocorrelation function form.     

On the Landau diamagnetism

The grand-canonical partition function of an assembly of free spinless electrons in a magnetic field enclosed in a box (Dirichlet boundary conditions) is shown to be an entire function of the fugacityz and the magnetic fieldH, as a consequence of the trace-norm convergence of the perturbation series for the statistical semigroup. This allows to derive analyticity properties of the pressure as a function ofz andH, and to express the coefficients of its power series expansion aroundz=H=0 by means of the unperturbed semigroup. Hence, the magnetic susceptibility at zero field and fixed density is expressed in terms of Green functions of the heat equation. Its asymptotic expansion for Λ→∞ (Fisher) along parallelepipedic domains is obtained up to 0 (S(Λ)/V(Λ)). The volume term of this expansion is the Landau diamagnetism.     

Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions

The modified Boussinesq hierarchy associated with the 3×3 matrix spectral problem is derived with the help of Lenard recursion equations. Based on the characteristic polynomial of Lax matrix for the modified Boussinesq hierarchy, we introduce an algebraic curve K_m−1 of arithmetic genus m−1, from which we establish the associated Baker-Akhiezer function, meromorphic function and Dubrovin-type equations. The straightening out of various flows is exactly given through the Abel map. Using these results and the theory of algebraic curve, we obtain the explicit theta function representations of the Baker-Akhiezer function, the meromorphic function, and in particular, that of solutions for the entire modified Boussinesq hierarchy.     

Sensitivity to point-spread function parameters in medical ultrasound image deconvolution

Clinical ultrasound images are often perceived as difficult to interpret due to image blurring and speckle inherent in the ultrasound imaging. But the image quality can be improved by deconvolution using an estimate of the point-spread function. However, it is difficult to obtain a sufficiently accurate estimate of the point-spread function in vivo because of the unknown properties of the soft tissue in clinical applications. Local variations in the speed of sound and attenuation change the pulse and beam shape. These in turn affect the point-spread function. The purpose and novelty of this paper is therefore to explore the sensitivity of a state-of-the-art deconvolution algorithm to uncertainty in the point-spread function. The point-spread function in our restoration algorithm is made shift invariant in the lateral dimension but shift dependent in the axial direction, and is modelled to match a 128-element 1D linear array often found in clinical use. We present simulated and in vitro sensitivity analyses of two-dimensional deconvolution while varying six parameters on which the point-spread function depends. Uncertainty in the ultrasound machine is analysed by varying the axial depths of lateral and elevational foci alongside height and width of transducer elements. Sensitivity to tissue influence is investigated by varying the speed of sound and frequency-dependent attenuation of the electro-mechanical impulse response. The results are analysed both quantitatively and in terms of the perceived image quality. First, the assessment of deconvolution using the logarithmic image amplitude is found to be a better indicator of the perceived improvement in the restoration. Secondly, the two most critical parameters for two-dimensional deconvolution are discovered to be the lateral focus and the speed of sound, because the success of deconvolution is perceived primarily in terms of deblurring. We also observed similar patterns for the simulation and in vitro experiment. Finally, we show that it is possible to restore in vivo ultrasound images using an assumed point-spread function and hence conclude that an exact point-spread function is not necessary for enhancing ultrasound image quality by deconvolution.