Three-point interpolation approximation for the macroscopic properties of isotropic two-component materials

A simple unified interpolation scheme, which involves certain three-point correlation information about the microgeometry of an isotropic two-component material, is suggested to approximate the macroscopic conductivity and elastic properties of the composite. The approximation satisfies Hashin–Shtrikman bounds as well as the three-point bounds on the effective properties, and reduces to the well-established self-consistent and matrix-mixture (Maxwell) approximations for the respective microgeometries. Applications to some high-contrast property composites illustrate the results.     

A terahertz time-domain study on the estimation of opto-mechanical properties of pharmaceutical tablets using the Hashin–Shtrikman bounds for refractive index: a case study of microcrystalline cellulose and starch acetate compacts

This work highlights the use of Hashin–Shtrikman (H–S) bounds in the prediction and verification of the effective refractive index, the height and the Young’s modulus of given training sets of pharmaceutical compacts using the measured time delay of a THz pulse traversing the compacts. Set A consisted of 13 microcrystalline cellulose (MCC) compacts whereas set B was made up of 5 starch acetate (SA) compacts. MCC is a typical ingredient of many pharmaceutical tablets. In the case of the MCC compacts, tight and closely matched bounds were obtained between the experimental, the calculated upper, lower bound values for the effective refractive index, and the height values. This promising outcome has shown the high possibility of utilizing H–S bounds in the verification and prediction of the decision level of useful parameters, which can serve as a quality check for pharmaceutical tablets. For the SA compacts, although less tight bounds were observed, the experimental values for the effective refractive index and the Young’s modulus were closely matched with the upper and the lower bounds, respectively. We therefore speculate based on the above observations that the MCC tablets contain an almost evenly distributed spherically shaped air voids whereas in the SA compacts, this assumption might not necessary be true.     

Rigorous Computation of Diffusion Coefficients for Expanding Maps

For real analytic expanding interval maps, a novel method is given for rigorously approximating the diffusion coefficient of real analytic observables. As a theoretical algorithm, our approximation scheme is shown to give quadratic exponential convergence to the diffusion coefficient. The method for converting this rapid convergence into explicit high precision rigorous bounds is illustrated in the setting of Lanford’s map \(x\mapsto 2x +\frac{1}{2}x(1-x) \pmod 1 \).     

Density functionals for nondynamical correlation constructed from an upper bound to the exact exchange energy density

ABSTRACT

Hyper-generalised-gradient approximations (hGGAs) for the exact exchange-correlation functional are increasingly popular in density functional theory. HGGAs model nondynamical correlation using a flexible local combination of exact (Hartree–Fock, HF) exchange and approximate exchange. We present a simplified ‘Rung 3.5’ upper bound to the HF exchange energy density, the essential ingredient of hGGAs. We also present a nonempirical generalised gradient approximation for this upper bound. Both upper bounds go to zero in the high-density and density tail limits, facilitating the construction of hGGAs that recover HF exchange in these limits. The ‘Rung 3.5’ construction enables facile evaluation of analytic derivatives and calculations in periodic boundary conditions. Extensive numerical tests show that the upper bounds capture the critical difference between HF and approximate exchange, showing these ingredients' promise for building simple hGGAs. The tests also indicate a need for more sophisticated semi-local upper bounds.     

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Pinsker’s and Fannes’ type bounds on the Tsallis relative entropy are derived. The monotonicity property of the quantum f -divergence is used fot its estimation from below. For order $\alpha \in (0,1)$ , a family of lower bounds of Pinsker type is obtained. For $\alpha >1$ and the commutative case, upper continuity bounds on the relative entropy in terms of the minimal probability in its second argument are derived. Both the lower and upper bounds presented are reformulated for the case of Rényi’s entropies. The Fano inequality is extended to Tsallis’ entropies for all $\alpha >0$ . The deduced bounds on the Tsallis conditional entropy are used to obtain inequalities of Fannes’ type.     

Study about Light Hypernuclei and YN Interaction Based on Chiral SU(3) Quark Model

In terms of the YN interaction based on the chiral SU(3) quark model,the spectra of light hypernuclei 4ΛH,4ΛHe and 5ΛHe are calculated. The result shows that the effective local ΛN potential,as an approximation,cannot offer a reasonable explanation of the empirical data, while the non-local ΛN potential,also as an approximation,can provide a reasonable range of the binding energy of 5ΛHe and correct ordering of the energy levers of 4ΛH,4ΛHe. This indicates that the non-local character of the effective YN interaction plays an important role in binding behavior. The upper bounds and lower bounds of energy levels are given by employing two sets of model parameters in the calculation. Thus,the experimental data of hypernuclei can provide more information as the constraints of model. In the present stage,YN interaction(no matter phenomenology of theoretical interaction) cannot be finalized due to lack of experimental data. Utilizing experimental data,it is possible to optimize model parameters in the theoretical studies of YN interaction.     

A singularity theorem for Einstein–Klein–Gordon theory

Hawking’s singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a nonnegative effective energy density (EED), thereby guaranteeing the timelike convergence property. However, there are models that do not satisfy the SEC and therefore lie outside the scope of Hawking’s hypotheses, an important example being the massive Klein–Gordon field. Here we derive lower bounds on local averages of the EED for solutions to the Klein–Gordon equation, allowing nonzero mass and nonminimal coupling to the scalar curvature. The averages are taken along timelike geodesics or over spacetime volumes, and our bounds are valid for a range of coupling constants including both minimal and conformal coupling. Using methods developed by Fewster and Galloway, these lower bounds are applied to prove a Hawking-type singularity theorem for solutions to the Einstein–Klein–Gordon theory, asserting that solutions with sufficient initial contraction at a compact Cauchy surface will be future timelike geodesically incomplete. These results remain true in the presence of additional matter obeying both the strong and weak energy conditions.     

Effective permittivity of liquid crystal nanodispersions and composites with different pore structure

The volume filling fraction dependence of the effective permittivity of the nematic liquid crystal 4-n-pentyl-4’-cyanobiphenyl embedded in different porous membranes and dispersed with aerosil nanoparticles was determined using broadband dielectric spectroscopy in the frequency range from 10⁶ to 10⁹ Hz. The experimental data were analyzed and compared with some existing theories based on the effective medium approximation and their modifications. The obtained effective permittivities as a function of the volume filling fraction lie between the lower limits of the Wiener and Hashin–Shtrikman bounds. The observed shift of the experimental points reflects the changes in the structure of the investigated composites.     

Anharmonic phonons and structural phase transitions

Exact results for the order parameter and the meansquare displacement as functions of temperature are given for a quantum interacting phonon Hamiltonian with quartic anharmonicities. Upper bounds for the transition temperature are also derived. Approximate theories including the mean field approximation, the random phase approximation (or quasiharmonic approximation) and the self consistent approach (using Blume-Hubbard scheme) are compared with our exact results. The mean field approximation for the meansquare displacement is found to violate our bounds.The classical value is shown to form a lower bound for the kinetic energy. Upper bounds for the kinetic energy are obtained showing the region of temperature in which the use of the high temperature expansion of the fluctuation-dissipation theorem is justified. Comparison of the Hamiltonian and our results with electron-paramagnetic-resonance measurements are discussed.     

Upper bounds to the susceptibility and the spin pair correlation function of the Ising ferromagnet

Improved upper bounds are presented to the susceptibility and the spin pair correlation function given in the Bethe approximation, for the Ising ferromagnet under zero external field above the critical temperature of the Bethe approximation.     

Nonrelativistic potential model calculations for three-quark systems in the integro-differential equation approach

Employing a recently derived integro-differential equation which is equivalent to the Faddeev equation, in an adiabatic approximation, we obtain accurate lower bounds to the binding energy of three-quark systems close to the upper bounds provided by the hypercentral approximation to the hyperspherical expansion method. Calculations have been performed for the Martin and the spin-dependent Ono-Schöberl potential.     

Development of the self-consistent approximation and its application to the problem of magnetoelastic resonance in an inhomogeneous medium

Our previously proposed approximation involving both the first and second terms of the expansion of the vertex function is generalized to the system of two interacting wavefields of different physical nature. A system of self-consistent equations for the matrix Green’s function and matrix vertex function is derived. On the basis of this matrix generalization of the new self-consistent approximation, a theory of magnetoelastic resonance is developed for a ferromagnetic model, where the magnetoelastic coupling parameter ε(x) is inhomogeneous. Equations for magnetoelastic resonance are analyzed for one-dimensional inhomogeneities of the coupling parameter. The diagonal and off-diagonal elements of the matrix Green’s function of the system of coupled spin and elastic waves are calculated with the change in the ratio between the average value ε and rms fluctuation Δε of the coupling parameter between waves from the homogeneous case (ε ≠ 0, Δε = 0) to the extremely randomized case (ε = 0, Δε ≠ 0) at various correlation wavenumbers of inhomogeneities k _c. For the limiting case of infinite correlation radius (k _c = 0), in addition to approximate expressions, exact analytical expressions corresponding to the summation of all diagrams of elements of the matrix Green’s function are obtained. The results calculated for an arbitrary k _c value in the new self-consistent approximation are compared to the results obtained in the standard self-consistent approximation, where only the first term of the expansion of the vertex function is taken into account. It is shown that the new approximation corrects disadvantages of the Green’s functions calculated in the standard approximation such as the dome shape of resonances and bends on the sides of resonance peaks. The appearance of a fine structure of the spectrum in the form of a narrow resonance on the Green’s function of spin waves and a narrow antiresonance on the Green’s function of elastic waves, which was previously predicted in the standard self-consistent approximation, is confirmed. With an increase in the parameter k _c, the Green’s functions calculated in the standard and new approximations approach each other and almost coincide with each other at k _c/k ≥ 0.5. At the same time, the results of this work indicate that the new self-consistent approximation has a certain advantage for studying the problems of stochastic radiophysics in media with long-wavelength inhomogeneities (small k _c values), because it describes both the shape and width of peaks much better than the standard approximation.     

Bounds for the complex dielectric constant of a two-phase composite

We study the effective dielectric constant of a two-phase composite for the case where the dielectric constants of the two components are complex. Milton has derived a sequence of narrowing bounds in the complex plane limiting the possible values of the effective dielectric constant, assuming that certain geometrical coefficients characterizing the medium are known. We cast Milton's bounds in a different form, making use of auxiliary functions introduced by Bergman. We derive explicit expressions for the bounds up to third order and compare values for the corresponding geometrical coefficient derived for different geometries.     

Ab initio investigation of the structural,elastic and electronic properties of the anti-perovskite TlNCa3

We have investigated the structural, elastic and electronic properties of the anti-perovskite TlNCa₃ using ab initio calculations within the generalized gradient approximation and the local density approximation for the exchange–correlation potential. The lattice constant, bulk modulus, elastic constants and their pressure dependence, energy band structures, density of states and charge density distribution are calculated and analyzed in comparison with the available experimental and theoretical data. The bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, Lamé’s coefficients, average sound velocity and Debye temperature are numerically estimated for ideal polycrystalline TlNCa₃ aggregates in the framework of the Voigt–Reuss–Hill approximation. This is the first theoretical prediction of the elastic constants and their related properties for TlNCa₃ that requires experimental confirmation.     

Escape time of cosmic rays from the galaxy in the anomalous diffusion model

Two versions of the anomalous diffusion model (Lagutin and Uchaikin’s and Lagutin and Tyumentsev’s) based on fractional transport equations are considered within the leaky-box approximation with respect to cosmic ray problems. The distributions of the first passage and escape times are computed via the Monte Carlo method. The observed difference between the results for the two versions is found to be a result of incorrectly choosing the Lagutin-Tyumentsev version’s parameters.     

Dark materials based on graphene sheet stacks

The effective medium properties of graphene sheet stacks are calculated, and it is shown that such stacks can have very low reflectivity and high absorbance. These properties make graphene-sheet-stack-based materials darker than recently studied carbon nanotube materials. Graphene stacks thus hold promise for realizing lower reflectivity coatings and enhanced photodetectors. The bounds of the effective medium approximation and the possible benefits of using graphene sheet stacks in a regime where this approximation does not hold are discussed.     

Free-Energy Bounds for Hierarchical Spin Models

In this paper we study two non-mean-field (NMF) spin models built on a hierarchical lattice: the hierarchical Edward–Anderson model (HEA) of a spin glass, and Dyson’s hierarchical model (DHM) of a ferromagnet. For the HEA, we prove the existence of the thermodynamic limit of the free energy and the replica-symmetry-breaking (RSB) free-energy bounds previously derived for the Sherrington–Kirkpatrick model of a spin glass. These RSB mean-field bounds are exact only if the order-parameter fluctuations (OPF) vanish: given that such fluctuations are not negligible in NMF models, we develop a novel strategy to tackle part of OPF in hierarchical models. The method is based on absorbing part of OPF of a block of spins into an effective Hamiltonian of the underlying spin blocks. We illustrate this method for DHM and show that, compared to the mean-field bound for the free energy, it provides a tighter NMF bound, with a critical temperature closer to the exact one. To extend this method to the HEA model, a suitable generalization of Griffith’s correlation inequalities for Ising ferromagnets is needed: since correlation inequalities for spin glasses are still an open topic, we leave the extension of this method to hierarchical spin glasses as a future perspective.     

Simple inequality for the free energy of disordered systems

It is proved that the free energy of a disordered system described by a quadratic form in Bose or Fermi operators with random coefficients, calculated in the simplest approximation for the associated eigenvalue problem, gives the upper (Bose case) and lower (Fermi case) bounds for the exact free energy.     

Quantum Entanglement and Spin Squeezing of Two Species Bose-Einstein Condensates

We investigate quantum entanglement and spin squeezing of two species Bose-Einstein condensates. By the rotating-wave approximation, we obtain the effective Hamiltonian and the wave function of the system. It’s shown that more entanglement and squeezing may be achieved by increasing the population difference of particles.     

A non-adapted sparse approximation of PDEs with stochastic inputs

We propose a method for the approximation of solutions of PDEs with stochastic coefficients based on the direct, i.e., non-adapted, sampling of solutions. This sampling can be done by using any legacy code for the deterministic problem as a black box. The method converges in probability (with probabilistic error bounds) as a consequence of sparsity and a concentration of measure phenomenon on the empirical correlation between samples. We show that the method is well suited for truly high-dimensional problems.