Leading-order Approximation Research Articles

A model of micro lubrication between two walls with unequal temperature distribution based on kinetic theory

Micro lubrication of a gas between two walls with arbitrary and independent temperature distributions is studied on the basis of the Bhatnagar–Gross–Krook–Welander (BGKW) model of the Boltzmann equation. The BGKW equation is studied analytically using the slowly varying approximation. Following the author's previous study [T. Doi, “A model of micro lubrication between two walls with an arbitrary temperature difference based on kinetic theory,” Phys. Fluids 32, 052005 (2020)], the leading-order approximation, which ought to be the solution of the nonlinear heat transfer problem, is replaced by its free molecular solution. A lubrication model of the Reynolds-type equation is derived in closed form. A direct numerical analysis of the lubrication flow subject to localized heating or cooling of the walls is conducted for an assessment of the lubrication model. The lubrication lift calculated using the model agrees with that of the direct numerical solution within an error of 5% when the Knudsen number based on the gap size lies between 0.1 and 10. The result of the lubrication model agrees also with that of the Boltzmann equation for a variable hard sphere gas. A sharp peak arises in the pressure distribution for large Knudsen numbers owing to the effect of thermal creep flows induced by localized heating.

Mixed QCD-electroweak corrections to W -boson production in hadron collisions

We compute mixed QCD-electroweak corrections to the fully differential production of an on-shell $W$ boson. Decays of $W$ bosons to lepton pairs are included in the leading order approximation. The required two-loop virtual corrections are computed analytically for arbitrary values of the electroweak gauge boson masses. Analytic results for integrated subtraction terms are obtained within a soft-collinear subtraction scheme optimized to accommodate the structural simplicity of infrared singularities of mixed QCD-electroweak contributions. Numerical results for mixed corrections to the fiducial cross section of $pp\ensuremath{\rightarrow}{W}^{+}\ensuremath{\rightarrow}{l}^{+}\ensuremath{\nu}$ and selected kinematic distributions in this process are presented.

Anisotropic fermionic quasiparticles

We have carried out a comprehensive investigation of the quasiparticle properties of a two-dimensional electron gas, interacting via the long-range Coulomb interaction in the presence of bare mass anisotropy (i.e., with an elliptic noninteracting Fermi surface) by calculating the self-energy, the spectral function, the scattering rate, and the effective mass within the leading-order dynamical self-energy approximation. Our theory is exact in the high-density limit. We find anisotropic features of quasiparticle properties that are not captured by the commonly used isotropic approximation where the anisotropic effective mass is replaced by the isotropic averaged density-of-states mass. Some of these interesting results are as follows: (1) The many-body renormalization of the quasiparticle spectrum becomes highly anisotropic as the quasiparticle energy increases away from the Fermi energy; (2) the interaction-induced inelastic-scattering rate features a strong anisotropy, exhibiting an abrupt jump at different injected energies depending on the momentum direction of the injected electron; (3) the effective-mass enhancement is larger (smaller) for the light (heavy) mass, showing that the anisotropy is reduced by interactions. Our results and analysis show that the unjustified neglect of the mass anisotropy can lead to an incorrect description of quasiparticle properties of the anisotropic system and inaccurate estimates of physical quantities of interest although the use of an equivalent isotropic approximation using the density-of-states effective mass as is commonly and uncritically performed in the literature, works as a reasonable approximation in many situations. In addition to the complete random phase approximation theory for the anisotropic quasiparticles, we also provide a theory using the simpler plasmon-pole approximation, commenting on its validity for anisotropic self-energy calculations. We comment also on the interaction effect on the Fermi-surface topology, finding that the elliptic shape of the bare Fermi surface is preserved with suppressed ellipticity in the interacting system to a high degree of accuracy except in the very strongly interacting limit (and for very high bare mass anisotropy). Our theory provides a complete generalization of the existing isotropic many-body theory of interacting electrons to the corresponding anisotropic systems.

On the pseudo hermiticity and q-deformation

The non hermiticity of the Hamiltonian is shown to be generated from the q-deformation of the algebra. As an example the q-deformed harmonic oscillator is investigated within the leading order approximation in the pseudo Hermitian perturbation theory. A generalized form of the metric is constructed. The spectrum as well as eigenstates of the system are derived explicitly.. Some applications and illustrations in q-deformed quantum statistical mechanics are also discussed.

The Generalized Uncertainty Principle

AbstractThe uncertainty principle lies at the heart of quantum physics, and is widely thought of as a fundamental limit of the measurement precision of incompatible observables. Here it is shown that the traditional uncertainty relation in fact belongs to the leading order approximation of a generalized uncertainty relation. That is, the leading order linear dependence of observables gives the Heisenberg type of uncertainty relations, while higher order nonlinear dependence may reveal more different and interesting correlation properties. Applications of the generalized uncertainty relation and the high order nonlinear dependence between observables in quantum information science are also discussed.

The unified treatment of the non-relativistic bound state solutions, thermodynamic properties, and expectation values of exponential-type potentials

In this paper, we obtained the analytical N-dimensional bound state solutions of exponential-type potential functions using the semi-classical Wentzel–Kramers–Brillouin (WKB) approximation method with an improved approximation to the orbital centrifugal barrier. Furthermore, we derived the partition function with the associated thermodynamic properties. We calculated the expectation values (〈1/r2〉) via the Hellmann–Feynman theorem. Our results for the energy levels are in excellent agreement with the best available results obtained by other methods in the existing literature. The variations of the thermodynamic functions with the temperature parameter (β) agree very well with the other results obtained under different potential energy functions. The expectation values conformed to the ones obtained by another approach in previous work. This work demonstrates the exactness of the leading-order WKB approximation.

Deflection of light in time-periodic spherically symmetric gravitational fields

Using the geodesic method and the perturbative approach, we study the deflection of light by time-periodic spherically symmetric gravitational fields. Assuming the weakness of the gravitational field, we derive general formulas that determine the deflection angle in the leading order approximation. The formulas are valid for both time-periodic and static metrics. Using these results, we calculate the deflection angle of a light ray passing through a spherically symmetric oscillating distribution of a self-gravitating scalar field with a logarithmic potential. It turned out that in this case the deflection angle does not depend on time in the leading order.

Asymptotic derivation of refined dynamic equations for a thin elastic annulus

Low-frequency vibrations of a thin elastic annulus are considered. The dynamic equations of plane strain are subjected to asymptotic treatment beyond the leading-order approximation. The main peculiarity of the considered problem is a specific degeneration associated with the effect of the almost inextensible midline of the annulus, resulting in a few unexpected features of the mechanical behaviour. In particular, it is discovered that the leading-order even component of the circumferential stress is not uniform across the thickness, as is usually assumed, and can be determined only at the next order. The derived refined equations also govern vibrations of a cylindrical shell at the lowest cut-off frequencies. The two-term asymptotic formula obtained for the latter fully agrees with the expansion of the transcendental dispersion relation for plane strain but does not coincide in the second term with the prediction of the Kirchhoff–Love theory for thin shells.

Probing the holographic dilaton

Many strongly coupled field theories admit a spectrum of gauge-invariant bound states that includes scalar particles with the same quantum numbers as the vacuum. The challenge naturally arises of how to characterise them. In particular, how can a dilaton — the pseudo-Nambu-Goldstone boson associated with approximate scale invariance — be distinguished from other generic light scalars with the same quantum numbers? We address this problem within the context of gauge-gravity dualities, by analysing the fluctuations of the higher-dimensional gravitational theory. The diagnostic test that we propose consists of comparing the results of the complete calculation, performed by using gauge-invariant fluctuations in the bulk, with the results obtained in the probe approximation. While the former captures the mixing between scalar and metric degrees of freedom, the latter removes by hand the fluctuations that source the dilatation operator of the boundary field- theory. Hence, the probe approximation cannot capture a possible light dilaton, while it should fare well for other scalar particles. We test this idea on a number of holographic models, among which are some of the best known, complete gravity backgrounds constructed within the top-down approach to gauge-gravity dualities. We compute the spectra of scalar and tensor fluctuations, that are interpreted as bound states (glueballs) of the dual field theory, and we highlight those cases in which the probe approximation yields results close to the correct physical ones, as well as those cases where significant discrepancies emerge. We interpret the latter occurrence as an indication that identifying one of the lightest scalar states with the dilaton is legitimate, at least as a leading-order approximation.

A model of micro lubrication between two walls with an arbitrary temperature difference based on kinetic theory

Micro lubrication of a gas between two walls with an arbitrary temperature difference is studied on the basis of the Bhatnagar–Gross–Krook–Welander model of the Boltzmann equation. Applying the slowly varying approximation, the kinetic equation is studied analytically when the Knudsen number based on the gap size is large. The leading order approximation, which ought to be the solution of the nonlinear heat transfer problem, is replaced by its free molecular solution. Due to this crude approximation, a macroscopic lubrication model of Reynolds-type equation is derived in a closed form. For an assessment of the model, a direct numerical analysis of the kinetic equation is also conducted. The lift calculated using our model approximates that of the direct numerical analysis within the error of 4% uniformly in the range of the temperature ratio between 0.75 and 2 and the Knudsen number Kn between 0.1 and 10. A heating of the moving wall reduces the lift acting on the other wall when Kn is sufficiently large, whereas it is enhanced when Kn is sufficiently small.

How to renormalize integral equations with singular potentials in effective field theory

We briefly review general concepts of renormalization in quantum field theory and discuss their application to solutions of integral equations with singular potentials in the few-nucleon sector of the low-energy effective field theory of QCD. We also describe a particular subtractive renormalization scheme and consider a specific application to a toy-model with a singular potential serving as its effective field theoretical leading-order approximation.

Poisson–Nernst–Planck equations with high-order steric effects

The Poisson–Nernst–Planck–Lennard-Jones (PNP–LJ) model is a mathematical model for ionic solutions with Lennard-Jones interactions between the ions. Due to the singular nature of the Lennard-Jones interaction kernel, however, the PNP–LJ model gives rise to an intractable analytic problem and a highly demanding computational problem. Previous works tackled this problem by replacing the LJ potential with a leading order approximation of the LJ potential giving rise to the steric PNP model. The steric PNP proved to be a successful model for the transport of ions in biological ionic channels. However, in other parameter regimes related to high concentrations, it is ill-posed. Presumably, since the steric PNP model is derived using only a leading-order approximation of the LJ potential. In this study, we go beyond the leading order approximation of the Lennard-Jones (LJ) interaction kernel and develop a new class of steric PNP equations that rely on a high-order approximation of the LJ potential. Surprisingly, we show that the introduction of high-order terms does not regularize the steric PNP model. Namely, at the limit of vanishing ionic sizes, high-order steric PNP equations, at all orders including the PNP–LJ model, are ill-defined at certain parameters regimes. Further analysis shows that this is an inherent property of PNP equations that account for steric effects and is related to pattern formation in these systems.

Parton and Valon Distributions in the Nuclei

In the valon model a nucleon is assumed to be a bound state of three valence quark clusters (valons). On this base the nucleus structure function which are made from nucleons are analysed. Here in the frame work of perturbative quantum chromodynamics (pQCD) the structure of the valons are described at next to leading-order (NLO) approximation. Using the constituent-quark model, structure function of nucleon is largely investigated in the nuclear medium. For a series of nuclei with the nuclear mass A such that 2 ≤ A ≤ 40 within the kinetic region 0.0001 ≤ x ≤ 0.9 which covers a wide range of x Bjorken variable, the theoretical predications are in good agreement with the existing lepton-nucleus data. A better and fair description of experimental data can be done, considering the shadowing effect for small values of x i.e. x < 0.01. Using the valon model, we do an analysis at the NLO approximation while shadowing effect is employed. A comparison between our theoretical prediction for nuclear parton distribution functions (nPDFs) and the results from recent collaborations like AT12 and HKN07 sets have been done. The results of our calculations are in good agreement with the experimental data in deep inelastic scattering (DIS) processes which covers an extended region of x-variable, including the small and also large x amounts and also with some theoretical models.

Modelling the inclusion of swelling pressure in a tissue level poroviscoelastic model of cartilage deformation.

Swelling pressure in the interstitial fluid within the pores of cartilage tissue is known to have a significant effect on the rheology of cartilage tissue. The swelling pressure varies rapidly within thin regions inside pores known as Debye layers, caused by the presence of fixed charge, as observed in cartilage. Tissue level calculation of cartilage deformation therefore requires resolution of three distinct spatial scales: the Debye lengthscale within individual pores; the lengthscale of an individual pore; and the tissue lengthscale. We use asymptotics to construct a leading order approximation to the swelling pressure within pores, allowing the swelling pressure to be systematically included within a fluid-solid interaction model at the level of pores in cartilage. We then use homogenization to derive tissue level equations for cartilage deformation that are very similar to those governing the finite deformation of a poroviscoelastic body. The equations derived permit the spatial variations in porosity and electric charge that occur in cartilage tissue. Example solutions are then used to confirm the plausibility of the model derived and to consider the impact of fixed charge heterogeneity, illustrating that local fixed charge loss is predicted to increase deformation gradients under confined compression away from, rather than at, the site of loss.

A study of power suppressed contributions in $$J/\psi \rightarrow p\bar{p}$$ decay

The power suppressed amplitude which describes the Pauli ($\sigma^{\mu\nu}$) coupling in the $J/\psi\to p\bar p$ decay is calculated within the effective field theory framework. It is shown that at the leading-order approximation this contribution is factorisable and the overlap with the hadronic final state can be described by collinear matrix elements. The obtained contribution depends on the nucleon light-cone distribution amplitudes of twist-3 and twist-4. This result is used for a qualitative phenomenological analysis of existing data for $J/\psi\to p\bar p$ decay: branching ratio and the angular distribution in the cross section $e^+e^-\to J/\psi\to p\bar p$.

On the reflection, diffraction and refraction of a spherical wave of parallel polarization by a sphere of electrically long radius

A novel mathematical approach is proposed for the analysis of the reflection, diffraction and refraction of spherical waves from curved material interfaces. As a paradigm, this approach is applied to the radiation of a vertical electric dipole over an electrically homogenous sphere. The material of the sphere can be dielectric or conducting. The radius of the sphere is assumed to be much greater than the wavelength in its less dense exterior. The formulation is developed for any height of the dipole above the sphere and for any radial distance of the point of observation from the center of the sphere. Novel expansions are applied to the integral representations of the resulting fields. It is shown that contributions to the value of the fields can be produced either from stationary-phase points of the integrands that are generated by such expansions or from any of the two strings of poles of these integrands. Contributions from rapidly converging residue series, as opposed to those coming from stationary-phase points, correspond to wave trajectories containing a curved path in addition to rectilinear paths. For the sake of demonstration, approximate analytical formulas for computation are derived for a number of select cases. The method yields leading-order approximations for the reflected and refracted fields. The affinity with the reflection from a planar interface is analysed.

Generalized multirate models for conjugate transfer in heterogeneous materials

We propose a novel macroscopic model for conjugate heat and mass transfer between a \emph{mobile region}, where advective transport is significant, and a set of \emph{immobile regions} where diffusive transport is dominant. Applying a spatial averaging operator to the microscopic equations, we obtain a \emph{multi-continuum} model, where an equation for the average concentration in the mobile region is coupled with a set of equations for the average concentrations in the immobile regions. Subsequently, by mean of a spectral decomposition, we derive a set of equations that can be viewed as a generalisation of the multi-rate mass transfer (MRMT) model, originally introduced by Haggerty & Gorelick. This new formulation does not require any assumption on local equilibrium or geometry. We then show that the MRMT can be obtained as the leading order approximation, when the mobile concentration is in local equilibrium. The new Generalised Multi-Rate Mode (GMRM) has the advantage of providing a direct method for calculating the model coefficients for immobile regions of arbitrary shapes, through the solution of appropriate micro-scale cell problems. An important finding is that a simple re-scaling or re-parametrisation of the transfer rate coefficient (and thus, the memory function) is not sufficient to account for the flow field in the mobile region and the resulting non-uniformity of the concentration at the interfaces between mobile and immobile regions.

Asymptotic expansion of the velocity field within the front of viscoplastic surges: comparison with experiments

In this article, we investigate the internal dynamics (velocity profiles and shear rate) of free-surface surges made of viscoplastic fluids. Compared with fluids without a yield stress, additional complexity arises from the possible coexistence of sheared and unsheared (or pseudo-plug) zones in the flow. Expanding on the thin-layer approach of Fernandez-Nieto et al. (J. Non-Newtonian Fluid Mech., vol. 165, 2010, pp. 712–732), we derive formal asymptotic expansions of the velocity field and discharge up to with respect to flow aspect ratio . Detailed comparisons between these theoretical predictions and experimental data reveal that, although the leading-order approximation (equivalent to a lubrication model) satisfactorily accounts for the global dynamics of the flow, considering correction terms is required to capture the evolution of velocity and shear rate close to the tip. Notably, these correction terms are responsible for the vanishing of the unsheared layer in the tip region, a feature clearly observed in the experiments. Differences between the leading-order and models appear to be enhanced by the viscoplastic character of the fluid. In particular, correction terms related to the existence of plastic normal stresses in the pseudo-plug layer play a critical role. This study provides important insights for future development of consistent shallow-water models adapted to viscoplastic materials.

Third- and fourth-order virial coefficients of harmonically trapped fermions in a semiclassical approximation

Using a leading-order semiclassical approximation, we calculate the third- and fourth-order virial coefficients of nonrelativistic spin-1/2 fermions in a harmonic trapping potential in arbitrary spatial dimensions, and as functions of temperature, trapping frequency and coupling strength. Our simple, analytic results for the interaction-induced changes $\Delta b_3$ and $\Delta b_4$ agree qualitatively, and in some regimes quantitatively, with previous numerical calculations for the unitary limit of three-dimensional Fermi gases.

Electric potential generation of electrocytes: Modelling, analysis, and computation

In this paper, we developed a one-dimensional model for electric potential generation of electrocytes in electric eels. The model is based on the Poisson–Nernst–Planck system for ion transport coupled with membrane fluxes including the Hodgkin–Huxley type. Using asymptotic analysis, we derived a simplified zero-dimensional model, which we denote as the membrane model in this paper, as a leading order approximation. Our analysis provides justification for the assumption in membrane models that electric potential is constant in the intracellular space. This is essential to explain the superposition of two membrane potentials that leads to a significant transcellular potential. Numerical simulations are also carried out to support our analytical findings.