General Analytical Expression Research Articles

Anomalous exciton Rydberg series in two-dimensional semiconductors on high- κ dielectric substrates

Engineering of the dielectric environment represents a powerful strategy to control the electronic and optical properties of two-dimensional (2D) materials without compromising their structural integrity. Here we show that the recent development of high-$\kappa$ 2D materials present new opportunities for dielectric engineering. By solving a 2D Mott-Wannier exciton model for WSe$_2$ on different substrates using a screened electron-hole interaction obtained from first principles, we demonstrate that the exciton Rydberg series changes qualitatively when the dielectric screening within the 2D semiconductor becomes dominated by the substrate. In this regime, the distance dependence of the screening is reversed and the effective screening increases with exciton radius, which is opposite to the conventional 2D screening regime. Consequently, higher excitonic states become underbound rather than overbound as compared to the Hydrogenic Rydberg series. Finally, we derive a general analytical expression for the exciton binding energy of the entire 2D Rydberg series

Application of bipolar coordinates to the analysis of the structure of viscous fluid flow between two rotating cylinders

The motion of a viscous fluid between two circular rotating cylinders is studied. The transition to the bipolar coordinates is applied to determine the general analytical expressions for stream function. The structure of the stream lines is investigated in the cases when one of the cylinders is motionless and when the both cylinders rotate.

Retarded hydrodynamic fluctuations effects on the light scattering spectrum and elastic moduli of a linear viscoelastic liquid

Fluctuating hydrodynamics (FH) is one of the theories that describes the dynamics of the fluctuations for fluids at mesoscopic scales. However, it is only valid for Markovian processes. Here we extend this approach for a viscoelastic liquid by using a generalized Langevin equation (GLE). We obyain general analytic expressions for the density fluctuations correlation function, the dynamic structure factor and the light scattering spectrum. In particular, we calculate the intermediate scattering function when the time memory and the noise correlation function are power-laws. We find that the difference in values of this quantity as functions of the viscoelasticity may be ∼62% and might be measurable.

Propagation properties of partially coherent modified Bessel-Gauss beams through the gradient-index medium.

Based on the ABCD matrix method and Collins diffraction integral formula, the general analytical expression for the partially coherent modified Bessel-Gauss beam propagating in a gradient-index medium is derived. The propagation trajectory, intensity, and phase distribution of such a beam are numerically investigated. The effects of the topological charge, the coherence parameter, and the coefficient of the gradient refractive index on propagation properties are considered. Results show that the propagation trajectory of such beam focuses and diverges periodically, which is different from free-space propagation. The period of intensity distribution is consistent with that of phase distribution under different cases. As propagation distance increases, the dark core always exists and the phase singularities remain stable and do not split. The dark core can be modulated by topological charge and coherence parameter, and the periodical distance can be modulated by the coefficient of the gradient refractive index. These results will help to explore such beams and find applications in optical communication and optical trapping.

High-eccentricity migration of planetesimals around polluted white dwarfs

ABSTRACT Several white dwarfs (WDs) with atmospheric metal pollution have been found to host small planetary bodies (planetesimals) orbiting near the tidal disruption radius. We study the physical properties and dynamical origin of these bodies under the hypothesis that they underwent high-eccentricity migration from initial distances of several astronomical units. We examine two plausible mechanisms for orbital migration and circularization: tidal friction and ram-pressure drag in a compact disc. For each mechanism, we derive general analytical expressions for the evolution of the orbit that can be rescaled for various situations. We identify the physical parameters that determine whether a planetesimal’s orbit can circularize within the appropriate time-scale and constrain these parameters based on the properties of the observed systems. For tidal migration to work, an internal viscosity similar to that of molten rock is required, and this may be naturally produced by tidal heating. For disc migration to operate, a minimal column density of the disc is implied; the inferred total disc mass is consistent with estimates of the total mass of metals accreted by polluted WDs.

Mechanical Squeezing via Fast Continuous Measurement.

We revisit quantum state preparation of an oscillator by continuous linear position measurement. Quite general analytical expressions are derived for the conditioned state of the oscillator. Remarkably, we predict that quantum squeezing is possible outside of both the backaction dominated and quantum coherent oscillation regimes, relaxing experimental requirements even compared to ground-state cooling. This provides a new way to generate nonclassical states of macroscopic mechanical oscillators, and opens the door to quantum sensing and tests of quantum macroscopicity at room temperature.

The κ-cookbook: a novel generalizing approach to unify κ-like distributions for plasma particle modelling

ABSTRACT In the literature different so-called κ-distribution functions are discussed to fit and model the velocity (or energy) distributions of solar wind species, pickup ions, or magnetospheric particles. Here, we introduce a generalized (isotropic) κ-distribution as a ‘cookbook’, which admits as special cases, or ‘recipes’, all the other known versions of κ-models. A detailed analysis of the generalized distribution function is performed, providing general analytical expressions for the velocity moments, Debye length, and entropy, and pointing out a series of general requirements that plasma distribution functions should satisfy. From a contrasting analysis of the recipes found in the literature, we show that all of them lead to almost the same macroscopic parameters with a small standard deviation between them. However, one of these recipes called the regularized κ-distribution provides a functional alternative for macroscopic parametrization without any constraint for the power-law exponent κ.

RKKY interaction in a doped pseudospin-1 fermion system at finite temperature

We study the RKKY interaction of magnetic impurities in the $\alpha-\mathcal{T}_3$ model which hosts pseudospin-1 fermions with two dispersive and one flat bands. By using the effective low-energy Hamiltonian we calculate the RKKY coupling for impurities placed on the same or different sublattices. We find that there are three types of interaction, which depend on the model parameter defining the relative strength of hoppings between sublattices, two of them can be reduced to graphene case while the third one is new and is due to the presence of a flat zero-energy band. We derive general analytical expressions for the RKKY interaction in terms of Mellin-Barnes type integrals and analyze different limiting cases. The cases of finite chemical potential and temperature, as well as asymptotic at large distances are considered. We show that the interaction between impurities located at different rim sites displays a very strong temperature dependence at small doping being a direct consequence of the flat band. The subtleties of the theorem for signs of the RKKY interaction at zero doping, as applied to the $\mathcal{T}_3$ lattice, related to the existence of a dispersionless flat band are discussed.

Quantum work statistics close to equilibrium

We study the statistics of work, dissipation, and entropy production of a quantum quasi-isothermal process, where the system remains close to the thermal equilibrium along the transformation. We derive a general analytic expression for the work distribution and the cumulant generating function. All work cumulants split into a classical (non-coherent) and quantum (coherent) term, implying that close to equilibrium there are two independent channels of dissipation at all levels of the statistics. For non-coherent or commuting protocols, only the first two cumulants survive, leading to a Gaussian distribution with its first two moments related through the classical fluctuation-dissipation relation. On the other hand, quantum coherence leads to positive skewness and excess kurtosis in the distribution, and we demonstrate that these non-Gaussian effects are a manifestation of asymmetry in relation to the resource theory of thermodynamics. Furthermore, we also show that the non-coherent and coherent contributions satisfy independently the Evans-Searles fluctuation theorem, which sets strong bounds on the statistics, with negative values of the dissipation being exponentially suppressed. Our findings are illustrated in a driven two-level system and an Ising chain, where quantum signatures of the work distribution in the macroscopic limit are discussed.

Is There Still Something Left That Gravity Probe B Can Measure?

We perform a full analytical and numerical treatment, to the first post-Newtonian (1pN) order, of the general relativistic long-term spin precession of an orbiting gyroscope due to the mass quadrupole moment J 2 of its primary without any restriction on either the gyro’s orbital configuration and the orientation in space of the symmetry axis k ^ of the central body. We apply our results to the past spaceborne Gravity Probe B (GP-B) mission by finding a secular rate of its spin’s declination δ which may be as large as ≲30–40 milliarcseconds per year mas yr − 1 , depending on the initial orbital phase f 0 . Both our analytical calculation and our simultaneous integration of the equations for the parallel transport of the spin 4-vector S and of the geodesic equations of motion of the gyroscope confirm such a finding. For GP-B, the reported mean error in measuring the spin’s declination rate amounts to σ δ ˙ GP − B = 18.3 mas yr − 1 . We also calculate the general analytical expressions of the gravitomagnetic spin precession induced by the primary’s angular momentum J . In view of their generality, our results can be extended also to other astronomical and astrophysical scenarios of interest like, e.g., stars orbiting galactic supermassive black holes, exoplanets close to their parent stars, tight binaries hosting compact stellar corpses.

First passage time moments of asymmetric Lévy flights

We investigate the first-passage dynamics of symmetric and asymmetric Lévy flights in semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Lévy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Lévy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.

Spiraling interaction of multi-mode solitons in self-induced harmonic potentials

We investigate the propagation and spiraling interaction of multi-mode solitons in self-induced harmonic potentials. The general analytical propagation expression for the interaction of the multi-mode solitons is derived. The interaction characteristics, such as the evolution of the intensity of the combined optical field, the spacing of each constituent soliton to the longitudinal axis, the spiraling angle and velocity, the propagation trajectory, and the phase shift are discussed with detail. We show that it is possible to control the spiraling interacting trajectories of the solitons by carefully designing the input parameters. It is proved that the motion state of each soliton satisfies the conservation of mechanical energy. The results may have potential applications in all-optical information processing and particle control.

Spatiotemporal antiresonance in coupled reaction-diffusion systems.

We present a theoretical study of the spatiotemporal antiresonance in a system of two diffusively coupled chemical reactions, one of which is driven by an external periodic forcing. Although antiresonance is well known in various physical systems, the phenomenon in coupled chemical reactions has largely been overlooked. Based on the linearized dynamics around the steady state of the two-component coupled reaction-diffusion systems we have derived the general analytical expressions for the amplitude-frequency response functions of the driven and undriven components of the system. Our theoretical analysis is well corroborated by detailed numerical simulations on coupled Gray-Scott reaction-diffusion systems exhibiting antiresonance dip in the amplitude-frequency response curve as a result of destructive interference between the coupling and the periodic external forcing imparting differential stability of the two subsystems. This leads to the emergence of spatiotemporal patterns in an undriven subsystem, while the driven one settles down to a homogeneously stable steady state.

Analytical evaluation of geometric dilution of precision for three-dimensional angle-of-arrival target localization in wireless sensor networks

This article focuses on the evaluation of geometric dilution of precision for three-dimensional angle-of-arrival target localization in wireless sensor networks. We calculate a general analytical expression for the geometric dilution of precision for three-dimensional angle-of-arrival target localization. Unlike the existing works in the literature, in this article, no assumptions are made regarding the observation ranges, noise variances, or the number of sensors in the derivation of the geometric dilution of precision. Necessary and sufficient conditions regarding the existence of geometric dilution of precision are also derived, which can be readily used to evaluate the observability of three-dimensional angle-of-arrival target localization in wireless sensor networks. Moreover, a concise procedure is also presented to calculate the geometric dilution of precision when it exists. Finally, several examples are used to illustrate our results, and it is shown that the performance of the proposed regular deployment configurations of angle-of-arrival sensors is better than the one with random deployment patterns.

Tortuosity in the Brick and Mortar Model Based on Chemical Conduction

Diffusion is a reoccurring phenomena in many fields and is affected by the geometry in which it takes place. Here we investigate the effects of geometry on diffusion in a Brick and Mortar model system. The tortuous effects are evaluated based on generalized Fick’s law, i.e. diffusion driven by differences in chemical potential. The presented formalism gives a general (semi-) exact analytic expression for the tortuosity using impermeable bricks, which is successfully validated against standard techniques and finite element method results. The approach allows for anisotropic properties of the mortar, which we show can be significant and is not captured with known analytic techniques. Based on the introduced concept of chemical conductivity we also find generalized Fick’s law consistent with Ohm’s and Fourier’s law in terms of their constituent parts, which further makes the main results for Brick and Mortar structures directly applicable to diffusion of either charge, heat, or mass.

Propagation properties of partially coherent anomalous hollow beams in quadratic-index media

Based on the generalized diffraction integral formula, analytical propagation expressions for a partially coherent anomalous hollow beam (AHB) propagating through a quadratic-index medium are derived. The distributions of intensity and Poynting vector for a partially coherent AHB through the quadratic-index medium are numerically and theoretically studied. It is found that the intensity and Poynting vector of the partially coherent AHB take on a periodical change. The evolution properties of the partially coherent AHB in a quadratic-index medium are closely relevant to its transverse coherence width and the quadratic-index parameter of media.

Electric dipole image forces in three-layer systems: The classical electrostatic model.

General exact analytical expressions have been derived for the image force energy Wi(Z, φ) of a point dipole in a classical three-layer system composed of dispersionless media with arbitrary constant dielectric permittivities εi. Here, i = 1-3 is the layer number, and Z and φ are the dipole coordinate and orientation angle, respectively. It was found that the long-range asymptotics Wi(Z→∞,φ) in both covers (i = 1, 3) are reached unexpectedly far from the interlayer (i = 2). Another specific feature of the solution consists in that the interference of the fields created by polarization charges emerging at both interfaces leads to the appearance of a constant contribution inside the interlayer with a non-standard dependence on the dipole orientation angle φ. It was shown that by changing the dielectric constants of the structure components, one can realize two peculiar regimes of the Wi(Z, φ) behavior in the covers; namely, there arises either a potential barrier preventing adsorption or a well far from the interface, both being of a totally electrostatic origin, i.e., without involving the Pauli exchange repulsion, which is taken into account in the conventional theories of physical adsorption. The results obtained provide a fresh insight into the physics of adsorption in physical electronics, chemical physics, and electrochemistry.

Study of the Parameters of the Substitution Scheme and Calculation of Mechanical Characteristics of Asynchronous Motors of the AIR Series Taking into Account the Skin Effect in the Rotor Winding

Studies of catalog data and parameters of substitution schemes for asynchronous motors of the air series with a short-circuited rotor have been performed in order to obtain general laws and analytical expressions depending on the engine power. In the Mathcad computer-aided design system, 32 engines were studied, and machines with a synchronous speed of 1500 rpm were taken as the main ones. Catalog data in the reference literature, according to manufacturers and other sources, differ significantly from each other, and so analytical expressions that reflect their average values are needed. General regularities and analytical expressions for determining the necessary parameters are obtained. An approximate method for calculating mechanical characteristics is proposed that takes into account the skin effect and current displacement in the rotor winding, which allows obtaining mechanical characteristics with an increased starting torque.

What Would Happen if We Were About 1 pc Away from a Supermassive Black Hole?

Abstract We consider a hypothetical planet with the same mass m, radius R, angular momentum S, oblateness J 2, semimajor axis a, eccentricity e, inclination I, and obliquity ε of the Earth orbiting a main-sequence star with the same mass and radius of the Sun at a distance from a supermassive black hole in the center of the hosting galaxy with the same mass of, say, M87*. We preliminarily investigate some dynamical consequences of its presence in the neighborhood of such a stellar system on the planet’s possibility of sustaining complex life over time. In particular, we obtain general analytic expressions for the long-term rates of change, doubly averaged over both the planetary and the galactocentric orbital periods and , of e, I, ε, which are the main quantities directly linked to stellar insolation. We find that, for certain orbital configurations, the planet’s perihelion distance may greatly shrink and even lead to, in some cases, an impact with the star. I may also notably change, with variations even of the order of tens of degrees. On the other hand, ε does not seem to be particularly affected, being shifted, at most, by over 1 Myr. Our results strongly depend on the eccentricity of the galactocentric motion.

Beam displacement and blur caused by fast electron beam deflection

Electrostatic beam blankers are an alternative to photo-emission sources for generating pulsed electron beams for Time-resolved Cathodoluminescence and Ultrafast Electron Microscopy. While the properties of beam blankers have been extensively investigated in the past for applications in lithography, characteristics such as the influence of blanking on imaging resolution have not been fully addressed. We derive general analytical expressions for the spot displacement and loss in resolution induced by deflecting the electron beam in a blanker. In particular, we analyze the sensitivity of both measures to how precise the conjugate focus is aligned in between the deflector plates. We then work out the specific case of a beam blanker driven by a linear voltage ramp as was used in recent studies by others and by us. The result shows that the spot displacement and focus blur can be reduced to the same order as the electron beam probe size, even when using a beam blanker of millimeter or larger scale dimensions. An interesting result is that, by the right choice of the focus position in the deflector, either the spot displacement from the stationary position can be minimized, or the blur can be made zero but not both at the same time. Our results can be used both to characterize existing beam blanker setups and to design novel blankers. This can further develop the field of time-resolved electron microscopy by making it easier to generate pulses with a typical duration of tens of picoseconds in a regular scanning electron microscope at high spatial resolution.