Nonclassical Terms Research Articles

Applications of regime-switching in the nonlinear double-diffusivity (D-D) model

The linear double-diffusivity (D-D) model of Aifantis, comprising two coupled Fick-type partial differential equations and a mass exchange term connecting the diffusivities, is a paradigm in modeling mass transport in inhomogeneous media, e.g., fissures or fractures. Uncoupling of these equations led to a higher order partial differential equation that reproduced the non-classical transport terms, analyzed independently through Barenblatt’s pseudoparabolic equation and the Cahn–Hilliard spinodal decomposition equation. In the present article, we study transport in a nonlinearly coupled D-D model and determine the regime-switching of the associated diffusive processes using a revised formulation of the celebrated Lux method that combines forward Fourier transform with a Laplace transform followed by an Inverse Fourier transform of the governing reaction–diffusion (R–D) equations. This new formulation has key application possibilities in a wide range of non-equilibrium biological and financial systems by approximating closed-form analytical solutions of nonlinear models.

Smoothed particle hydrodynamics simulations of the evaporation of suspended liquid droplets

The ordinary evaporation and explosive vaporization of equilibrium, van der Waals, liquid drops subjected to uniform heating at supercritical temperatures are investigated by means of numerical simulations with the aid of a modified version of the DualSPHysics code. The models include the effects of surface tension, thermocapillary forces, mass transfer across the interface, and liquid–vapor interface dynamics by means of a diffuse-interface description. In contrast to previous simulations in this line, a new non-classical source term has been added to the internal energy equation to deal with the vaporization rate through the diffuse interface. This term is related to the diffusion of the latent heat in the interface zone and is, therefore, necessary for a correct physical description of the liquid–vapor interface structure. As the heating temperature increases the drops undergo surface evaporation, nucleation of an interior vapor bubble, nucleation followed by fragmentation of the liquid, and explosive vaporization. Heating at supercritical temperatures brings the drop out of equilibrium and forces it to rapid quenching into either the miscibility gap, where it undergoes surface evaporation by spinodal decomposition, or the metastable region bounded by the binodal and spinodal curves, where it nucleates a vapor bubble. The results also indicate that at comparable heating, drops of lower density experience faster evaporation rates than drops of higher density.

Stress recovery of laminated non-prismatic beams under layerwise traction and body forces

Emerging manufacturing technologies, including 3D printing and additive layer manufacturing, offer scope for making slender heterogeneous structures with complex geometry. Modern applications include tapered sandwich beams employed in the aeronautical industry, wind turbine blades and concrete beams used in construction. It is noteworthy that state-of-the-art closed form solutions for stresses are often excessively simple to be representative of real laminated tapered beams. For example, centroidal variation with respect to the neutral axis is neglected, and the transverse direct stress component is disregarded. Also, non-classical terms arise due to interactions between stiffness and external load distributions. Another drawback is that the external load is assumed to react uniformly through the cross-section in classical beam formulations, which is an inaccurate assumption for slender structures loaded on only a sub-section of the entire cross-section. To address these limitations, a simple and efficient yet accurate analytical stress recovery method is presented for laminated non-prismatic beams with arbitrary cross-sectional shapes under layerwise body forces and traction loads. Moreover, closed-form solutions are deduced for rectangular cross-sections. The proposed method invokes Cauchy stress equilibrium followed by implementing appropriate interfacial boundary conditions. The main novelties comprise the 2D transverse stress field recovery considering centroidal variation with respect to the neutral axis, application of layerwise external loads, and consideration of effects where stiffness and external load distributions differ. A state of plane stress under small linear-elastic strains is assumed, for cases where beam thickness taper is restricted to 15^{circ }. The model is validated by comparison with finite element analysis and relevant analytical formulations.

О развитии терминологии жанра джаз в первой половине XX века в англоязычном дискурсе СМИ

The article is devoted to the formation of the terminology of jazz as a musical genre in the English-language media discourse in the first half of the 20th century. Modern studies of terminology show that terms are associated with special vocabulary and professional communication, and can transfer to terminology from the common language. Professional vocabulary, in turn, belongs to special vocabulary along with terms, but the former can also become the latter in the absence of a terminological equivalent. In art terminology, including musical terminology, a large number of professionalisms are noted. Of musical terminology, the terminology of jazz is of particular interest since its formation took place at the junction of classical musical terminology and English-language professional vocabulary. The article compares three periods in the development of jazz music as presented in the media and describes the formation of jazz terminology in media discourse. The material under study is archival texts of American newspapers of the first half of the 20th century devoted to jazz, which were divided into three sub-corpuses in accordance with the period of publication (1900-1920, 1920-1945, 1945-1950). The contexts of the use of jazz music nominations were extracted from this material, and the nominations were divided into groups: classical and non-classical terms, borrowed and English-language terms. The research methods used include content analysis, component analysis, analysis of vocabulary and contextual definitions. The study revealed that the terminology of jazz of the first half of the 20th century is mainly represented by English-language non-classical nominations and, to a lesser extent, by classical terms from the European academic musical tradition.

Non-linear anisotropic damage rheology model: Theory and experimental verification

We extend the isotropic non-linear damage rheology model with a scalar damage parameter to a more complex formulation that accounts for anisotropic damage growth under true triaxial loading. The model takes account of both the anisotropy of elastic properties (associated with textural rock structure) and the stress- and damage-induced anisotropy (associated with loading). The scalar, isotropic model is modified by assuming orthotropic symmetry and introducing a second-order damage tensor, the principal values of which describe damage in three orthogonal directions associated with the orientations of the principal loading axes. Different damage components, accumulated under true triaxial loading conditions, allows us to reproduce both stress-strain curves and damage- and stress-induced seismic wave velocity anisotropy. The suggested model generalization includes a non-classical energy term similar to the isotropic non-linear scalar damage model, which allows accounting for the abrupt change in the effective elastic moduli upon stress reversal.For calibration and verification of the model parameters, we use experimental stress-strain curves from deformation of dry sandstone under both conventional and true triaxial stress conditions. Cubic samples were deformed in three orthogonal directions with independently controlled stress paths. To characterize crack damage, changes in ultrasonic P-wave velocities in the three principal directions were measured, together with the bulk acoustic emission output. The parameters of the developed model were constrained using the conventional triaxial test data, and provided good fits to the stress-strain curves and P-wave velocity variations in the three orthogonal directions. Numerical simulation of the true triaxial test data demonstrates that the anisotropic damage rheology model adequately describes both non-linear stress-strain behavior and P-wave velocity variations in the tested Darley Dale sandstone.

Nonadiabatic dynamics in the semiclassical Liouville representation: Locality, transformation theory, and the energy budget

In this paper, we revisit the semiclassical Liouville approach to describing molecular dynamics with electronic transitions using classical trajectories. Key features of the formalism are highlighted. The locality in phase space and presence of nonclassical terms in the generalized Liouville equations are emphasized and discussed in light of trajectory surface hopping methodology. The representation dependence of the coupled semiclassical Liouville equations in the diabatic and adiabatic bases are discussed and new results for the transformation theory of the Wigner functions representing the corresponding density matrix elements given. We show that the diagonal energies of the state populations are not conserved during electronic transitions, as energy is stored in the electronic coherence. We discuss the implications of this observation for the validity of imposing strict energy conservation in trajectory based methods for simulating nonadiabatic processes.

Nature of beryllium bonds in view of interacting quantum atoms and natural energy decomposition analysis

In the light of interacting quantum atoms (IQA) and natural energy decomposition analysis (NEDA) the nature of different types of beryllium bonds is studied. IQA intra-atomic (self) energies indicate that, in most of the traditional beryllium bonds (in which the Lewis base is a lone pair donor), π-Be bonds (Lewis base is an unsaturated molecule) and also the Be–Ng (noble gases) interactions, the beryllium atom is stabilized upon complex formations. According to IQA inter-atomic energy components, the Be⋯Y bonds (Y is an atom in the Be-bond acceptor that is connected directly to the beryllium atom) have a dominant classical electrostatic character. However, in the IQA partitioning of total inter-molecular interactions, the contributions of non-classical (exchange–correlation) terms are non-negligible. Natural energy decomposition analysis of traditional and π-Be bonds indicate that the attractive interactions come from both the electrical and charge transfer terms. The electrical term in the Be–Ng bonds is negligibly small and the charge transfer is the most dominant contributor in the interaction energy.

Elastic Behavior of 2D Grain Packing Modeled as Micromorphic Media Based on Granular Micromechanics

AbstractDiscrete simulation and continuum modeling are two competing methods for analyzing the behavior of complex material systems and granular assemblies. In this paper, a two-dimensional (2D) granular micromechanics model for micromorphic media is presented. The granular micromechanics method is enhanced by incorporating nonclassical terms in the kinematics. These include, in addition to the usual macroscale displacement gradient, the fluctuations in displacement gradient and also the second gradient of displacement. Microscopic intergranular force vectors and macroscopic stress tensors conjugate to the kinematic measures are defined, and the macroscopic strain energy density function is set equal to the volume average of grain-pair energies. As a result, continuum stiffness tensors are formulated on the basis of intergranular stiffness coefficients and fabric parameters defining the geometry of grains and their contacts. To demonstrate the applicability of the continuum method, a random granular assem...

Resultant entropy/information, phase/entropy continuity and bond multiplicities in degenerate electronic states

The probability and phase/current facets of electronic states generate the classical and nonclassical information terms, respectively. The current-related supplements of the classical information measures and continuity equations for these degrees-of-freedom are summarized. The continuity of the resultant quantum entropy is also explored. This thermodynamic-like description is applied to discuss the temporal aspect of the promolecule-to-molecule transition in $$\hbox {H}_{2}^{+}$$ . The Wiberg-type bond multiplicity concept is extended to cover the degenerate electronic states. They generally exhibit finite spatial phases and hence nonvanishing electronic currents, and thus also nonzero nonclassical contributions to the resultant content of the state entropy/information. Illustrative example of the excited configurations in the $$\pi $$ -electron ring of benzene is investigated using the complex framework of the (ground-state equivalent) molecular orbitals in Huckel approximation. To validate these generalized concepts, correlations between the $$\pi $$ -bond orders/multiplicities and orbital excitation energies are explored.

Entropic descriptors of quantum communications in molecules

The classical Information Theory (IT) deals with entropic descriptors of the probability distributions and probability-propagation (communication) systems, e.g., the electronic channels in molecules reflecting the information scattering via the system chemical bonds. The quantum IT additionally accounts for the non-classical (current/phase)-related contributions in the resultant information content of electronic states. The classical and non-classical terms in the quantum Shannon entropy and Fisher information are reexamined. The associated probability-propagation and current-scattering networks are introduced and their Fisher- and Shannon-type descriptors are identified. The non-additive and additive information descriptors of the probability channels in both the Atomic Orbital and local resolution levels are related to the network conditional-entropy and mutual-information, which represent the IT covalency and ionicity components in the classical communication theory of the chemical bond. A similar partition identifies the associated bond indices in the molecular current/phase channels. The resultant bond descriptors combining the classical and non-classical terms, due to the probability and current distributions, respectively, are proposed as generalized communication-noise (covalency) and information-flow (iconicity) concepts in the quantum IT.

Phase/current information descriptors and equilibrium states in molecules

Quantum‐generalized entropic descriptors of the complex electronic states and their information distances are reexamined and applied to the phase‐equilibria in molecules. The relation between densities of the ordinary Fisher and Shannon measures of information content is used in determining their supplements due to phases/currents. These nonclassical terms complement the familiar classical (probability) functionals of information theory in the resultant information descriptors. The nonclassical Shannon entropy measures the average magnitude of the system phase distribution, while the current term in the related Fisher measure accounts for the gradient content of the state phase. The density constrained (vertical) and unconstrained (horizontal) equilibria in molecules are distinguished. The consistency requirement that the extreme entropic principles in terms of both these resultant measures have common solutions calls for the modified, negative sign of the nonclassical Fisher indeterminicity term. The equilibrium criteria are shown to give rise to the unitary phase‐transformation of molecular states in a “thermodynamic” representation of quantum‐mechanical description. Possible applications of this generalized description are discussed and thermodynamical analogies are commented upon. A separation of the density (modulus) and current (phase) factors of general many‐electron states is effected using the Harriman–Zumbach–Maschke construction of antisymmetric states yielding the specified electron density. A phenomenological description of molecular subsystems is outlined, which accounts for both the density and phase degrees‐of‐freedom of electronic states, and the current promotion of molecular fragments is explored. © 2014 Wiley Periodicals, Inc.

NUMERICAL METHOD OF VIBRATIONS RESEARCH IN LARGE FLEXIBLE SYSTEMS

The mathematical model of the real flexible elastic system is considered with the distributed and discrete parameters, which presents the equation at derivative parts with non-classical maximum terms. Complication of maximum terms makes impossible finding of exact analytical decision of such maximum task, in connection with what for researches the numerical method of the normal fundamental systems of decisions is used. Dependence of frequencies of possible vibrations is explored on different physical parameters of system. It is shown, that introduction to reverse communication network after speed with the defined values of the reverse communication coefficient allows controling the frequency spectrum in which excitation of vibrations are possible.

Method of negative saturations for flow with variable number of phases in porous media: extension to three-phase multi-component case

Various EOR methods lead to the appearance of specific macroscopic surfaces called interfaces of phase transition (IPT) such that the number of phases on two sides of an IPT is different, and fluids separated by an IPT are in non-equilibrium. Therefore, the flow equations are also different on two sides of an IPT and cannot be deduced from each other by a continuous degeneration, which imposes difficulties in numerical modelling. To describe such systems, we developed a new conceptual mathematical method based on the replacement of the real single-phase fluid by an imaginary multi-phase multi-component continuum having fictitious properties. As the result, the fluid over all zones becomes multi-phase and can be described by uniform multi-phase hydro- and thermodynamic equations, which allows applying the direct numerical simulation. The equivalence principle determines the physical properties of the fictitious multi-phase fluid, as well as the structure of the uniform multi-phase equations. It also proves that the saturation of each phase becomes an extended function negative or higher than unity in non-equilibrium zones, which becomes the efficient method of tracking the interfaces, the number of phases at any point, and their degree of disequilibrium. The method was developed in [1, 2] for the two-phase case. In the present paper, the new version of the method is developed for the three-phase case with gravity, diffusion, and capillarity. We have obtained the new equivalent uniform multi-phase equations which contain additional non-classical terms responsible for the diffusion and gravity across an IPT. The comparison with classical method is presented. The presentation is illustrated by several examples of simulation by means of the code developed by the research group; their concern: EOR by miscible methods and CO 2 bubble raising in aquifer.

On phase/current components of entropy/information descriptors of molecular states

Quantum-generalised descriptors of the information content of electronic states in molecules are proposed, in which non-classical (current) terms complement classical (probability) functionals of the ordinary information theory. The relation between densities of the familiar classical Fisher and Shannon information/entropy measures is applied to determine their non-classical complements. The quantum supplement of the classical Shannon entropy describes the average magnitude of the phase distribution, while the current term in the Fisher measure accounts for the gradient content of the state phase function. Illustrative applications of these quantum information concepts are presented and thermodynamical analogies are commented upon. The particle-density-constrained (vertical) and -unconstrained (horizontal) equilibria in molecules and their fragments are explored and the corresponding equilibrium ‘thermodynamic’ phases are determined. A separation of the density (modulus) and current (phase) factors of general many-electron states is effected using the Harriman–Zumbach–Maschke construction of antisymmetric states yielding the specified electron density. The phenomenological framework in spirit of the non-equilibrium thermodynamical description is proposed. It accounts for both the density and current degrees of freedom of molecular states. The associated entropy source in the information continuity equation is derived.

Use of Harriman’s construction in determining molecular equilibrium states

Information-theoretic description of the electron probabilities and currents in molecules is extended to cover the complex amplitudes (wave functions) of quantum mechanics. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. The previous one-electron development in such an entropic perspective on the molecular electronic structure is extended to cover N-electron states by adopting the Harriman-type framework of equidensity orbitals. This analysis emphasizes the phase part of electronic states, which generates the probability-current density and the associated non-classical entropy contributions, which allow one to distinguish the information content of states generating the same electron density and differing in their current composition. A complementary character of the Fisher and Shannon information measures is explored in the associated vertical (density-constrained) information principles, for determining the equilibrium state corresponding to the fixed ground-state electron density. It is argued that the lowest “thermodynamic” state generally differs from the true ground state of the system, by exhibiting the space-dependent phase and hence also the non-vanishing probability current, linked to the system electron distribution.

Entropic representation in the theory of molecular electronic structure

The entropic perspective on the molecular electronic structure is investigated. Information-theoretic description of electron probabilities is extended to cover the complex amplitudes (wave functions) of quantum mechanics. This analysis emphasizes the entropic concepts due to the phase part of electronic states, which generates the probability current density, thus allowing one to distinguish the information content of states generating the same electron density and differing in their current densities. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. A complementary character of the Fisher and Shannon information measures is explored and the relationship between these classical information densities is derived. It is postulated to characterize also their nonclassical (phase/current-dependent) contributions. The continuity equations of the generalized information densities are examined and the associated nonclassical information sources are identified. The variational rules involving the quantum-generalized Shannon entropy, which generate the stationary and time-dependent Schrodinger equations from the relevant maximum entropy principles, are discussed and their implications for the system “thermodynamic” equilibrium states are examined. It is demonstrated that the lowest, stationary “thermodynamic” state differs from the true ground state of the system, by exhibiting the space-dependent phase, linked to the modulus part of the wave function, and hence also a nonvanishing probability current.

Error Analysis of Chebyshev-Legendre Pseudo-spectral Method for a Class of Nonclassical Parabolic Equation

Many physical phenomena are modeled by nonclassical parabolic initial boundary value problems which involve a nonclassical term uxxt in the governed equation. Combining with the Crank-Nicolson/leapfrog scheme in time discretization, Chebyshev-Legendre pseudo-spectral method is applied to space discretization for numerically solving the nonclassical parabolic equation. The proposed approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the proposed method, the computational complexity is reduced and both accuracy and efficiency are achieved. The stability and convergence are rigorously set up. The convergence rate shows ‘spectral accuracy’. Numerical experiments are presented to demonstrate the effectiveness of the method and to confirm the theoretical results.

Momentum budget of the migrating diurnal tide in the mesosphere and lower thermosphere

The leading terms of the momentum budget of the migrating diurnal tide are diagnosed from temperatures and geopotential height provided by Thermosphere‐Ionosphere‐Mesosphere Energetics and Dynamics (TIMED)/Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) and the horizontal wind vector provided by TIMED Doppler Interferometer (TIDI). The “wave drag” upon the tide is inferred as a residual of the classical and linear advective terms in the zonal and meridional momentum equations. Between 85 and 100 km, the migrating diurnal tide is generally far from the classical description of Chapman and Lindzen (1970). The leading nonclassical terms are the meridional advection of zonally averaged momentum and wave drag effects encapsulated in the momentum residuals. The magnitudes of the momentum residuals range between 50 and 150 m s−1 d−1 for the zonal component and 100–250 m s−1 d−1 for the meridional component. The zonal momentum residual is generally in quadrature with the zonal wind tide, while the meridional momentum residual is in antiphase with the meridional wind above 95 km. Winds and temperatures exhibit maximum amplitudes during the vernal equinox season, but only a minor reduction in wave damping is observed. Our results suggest that Rayleigh damping is inadequate to describe the relationship between the tide and the forces acting upon it.

How Accurate Can a Force Field Become? A Polarizable Multipole Model Combined with Fragment-wise Quantum-Mechanical Calculations

A new method to accurately estimate the interaction energy between a large molecule and a smaller ligand is presented. The method approximates the electrostatic and induction contributions classically by multipole and polarizability expansions, but uses explicit quantum-mechanical fragment calculations for the remaining (nonclassical) contributions, mainly dispersion and exchange repulsion. Thus, it represents a limit of how accurate a force field can ever become for interaction energies if pairwise additivity of the nonclassical term is assumed (e.g., all general-purpose force fields). The accuracy is tested by considering protein-ligand model systems for which the true MP2/6-31G* interaction energies can be computed. The method is shown to be more accurate than related fragmentation approaches. The remaining error (2-5 and approximately10 kJ/mol for neutral and charged ligands, respectively) can be decreased by including the polarizing effect from surrounding fragments in the quantum-mechanical calculations.

Multi-dimensional mapping and folding properties for non-classical helix-stabilizing interactions of proteins

Abstract By using a dynamical systems approach we investigate a model for helix-coil phase transition for polypeptides and proteins in the thermodynamic limit on recursive zigzag ladder with three-site interaction. We explicitly take into account a non-classical helix-stabilizing term (hydrogen bonds between C α H⋯O). We obtain folding and quasi-unfolding of the order parameter (degree of helicity Θ ).