Closed-form Analytical Expressions Research Articles

Semiclassical analytic theory of electronic energy transfer in 3D atomic collisions.

A previously developed semiclassical theory of nonadiabatic energy transfer is used to analyze electronic excitation and quenching in three-dimensional atomic collisions. The predicted transition probabilities, cross sections, and rate coefficients are compared with the quantum scattering calculations for O + O and N + N, for the same interaction potentials and nonadiabatic coupling, and with the experimental data where available. The theory predictions are in very good agreement with quantum scattering, at the conditions when the energy transfer is dominated by a single pair of adiabatic potentials. Closed-form analytic expressions for the cross sections and rate coefficients are obtained, for both the strongly and weakly coupled cases. The results quantify and illustrate the effect of the interaction potentials and their coupling on the energy transfer. The analytic cross sections and rate coefficients are in good agreement with the numerical predictions. The same approach has been used to predict the rate coefficients of electronic excitation and quenching in collisions of N + O atoms. The fidelity of these predictions may be improved considerably if accurate potentials for the excited electronic states of N + O and their coupling are available. The applicability of the semiclassical theory for the prediction of the rates of heavy particle impact excitation in atom-molecule collisions is discussed.

Three-body binary formation in clusters: analytical theory

ABSTRACT Binary formation in clusters through triple encounters between three unbound stars, ‘three-body’ binary formation, is one of the main dynamical formation processes of binary systems in dense environments. In this paper, we use an analytical probabilistic approach to study the process for the equal mass case and calculate a probability distribution for the orbital parameters of three-body-formed binaries, as well as their formation rate. For the first time, we give closed-form analytical expressions to the full orbital parameter distribution, accounting for both energy and angular momentum conservation. This calculation relies on the sensitive dependence of the outcomes of three-body scatterings on the initial conditions: here we compute the rate of three-body binaries from ergodic interactions, which allow for an analytical derivation of the distribution of orbital parameters of the binaries thus created. We find that soft binaries are highly favoured in this process and that these binaries have a superthermal eccentricity distribution, while the few hard three-body binaries have an eccentricity distribution much closer to thermal. The analytical results predict and reproduce simulation results of three-body scattering experiments in the literature well.

Strip electro-mechanical yield model for three equal straight collinear cracks in piezoelectric materials

This paper is devoted to the development of a model to study three straight collinear cracks in piezoelectric materials under the influence of combined electrical and mechanical loading conditions. Mathematical relationships for the size of yield zones with mechanical yielding conditions and the saturation zone sizes to applied electrical field has been established. To make this plausible, two different cases are considered: first when the electrical saturation zones are bigger than the mechanical yield zones while other case in which situation is just opposite. Closed-form analytic expressions are derived for various fracture parameters with the help of well known complex variable method. Numerical results are presented to study the multiple cracks in different piezoelectric materials. Results obtained are reported graphically, analyzed and concluded.

COMPREHENSIVE ANALYSIS OF ENERGY DISSIPATION IN SOIL-EMBEDDED PILES: AN ANALYTICAL APPROACH

The study delves into a comprehensive examination of the total energy dissipation associated with a pile entrenched in the soil, with the soil being conceptualized as viscoelastic springs. "Dissipation" can be utilized to gauge the decay rate of displacement amplitude over time. Understanding dissipation in piles is pivotal for discerning the pace at which a pile achieves its equilibrium state when exposed to impact or free vibrations. Elevated dissipation levels indicate a swifter attenuation of transient vibrations, a crucial aspect for effective vibration mitigation during significant occurrences such as earthquakes, severe weather conditions, explosions, and impacts. The dispersion relation, reliant on wavenumber and free wave frequency, is derived for the pile-soil system using the Bloch's Floquet theorem. This relation is approached via the free-wave method, where a real wavenumber produces a sequence of complex frequencies. We propose an analytical closed-form expression in this study to forecast the overall dissipation from these complex frequency values. Additionally, we validate the accuracy of the predicted dissipation against a finite element-based numerical model of the soil-pile system. Furthermore, we examine the influence of factors such as material damping, Young's modulus, and shear modulus of the soil on the dissipation characteristics of the pile-soil system.

COMPREHENSIVE ANALYSIS OF ENERGY DISSIPATION IN SOIL-EMBEDDED PILES: AN ANALYTICAL APPROACH

The study delves into a comprehensive examination of the total energy dissipation associated with a pile entrenched in the soil, with the soil being conceptualized as viscoelastic springs. "Dissipation" can be utilized to gauge the decay rate of displacement amplitude over time. Understanding dissipation in piles is pivotal for discerning the pace at which a pile achieves its equilibrium state when exposed to impact or free vibrations. Elevated dissipation levels indicate a swifter attenuation of transient vibrations, a crucial aspect for effective vibration mitigation during significant occurrences such as earthquakes, severe weather conditions, explosions, and impacts. The dispersion relation, reliant on wavenumber and free wave frequency, is derived for the pile-soil system using the Bloch's Floquet theorem. This relation is approached via the free-wave method, where a real wavenumber produces a sequence of complex frequencies. We propose an analytical closed-form expression in this study to forecast the overall dissipation from these complex frequency values. Additionally, we validate the accuracy of the predicted dissipation against a finite element-based numerical model of the soil-pile system. Furthermore, we examine the influence of factors such as material damping, Young's modulus, and shear modulus of the soil on the dissipation characteristics of the pile-soil system.

Damping analysis of a transversely isotropic piezothermoelastic nanobeam resonator based on the MGT thermoelasticity

The present work investigates the vibration of a piezothermoelastic nanobeam in the frame of Moore–Gibson–Thompson (MGT) thermoelasticity theory. Closed form analytical expressions for the thermoelastic damping in terms of quality factor and frequency shift for a homogeneous transversely isotropic piezothermoelastic (PTE) beam is derived by using Euler–Bernoulli beam theory and complex frequency approach. Detailed analysis of damping of vibration owing to thermal fluctuations and electric potential in the present context under three sets of boundary conditions is attempted to investigate the influence of modes of vibration, electric potential, and beam length on energy dissipation caused by thermoelastic damping in PTE beam resonators. Analytical results are illustrated with the help of graphical plots on numerical findings in the case of the nanobeam resonator of Lead Zirconate Titanate (PZT-5A) material. The investigation brings out some significant key findings and observations in view of MGT heat conduction model.

Photophysical image analysis: Unsupervised probabilistic thresholding for images from electron-multiplying charge-coupled devices.

We introduce the concept photophysical image analysis (PIA) and an associated pipeline for unsupervised probabilistic image thresholding for images recorded by electron-multiplying charge-coupled device (EMCCD) cameras. We base our approach on a closed-form analytic expression for the characteristic function (Fourier-transform of the probability mass function) for the image counts recorded in an EMCCD camera, which takes into account both stochasticity in the arrival of photons at the imaging camera and subsequent noise induced by the detection system of the camera. The only assumption in our method is that the background photon arrival to the imaging system is described by a stationary Poisson process (we make no assumption about the photon statistics for the signal). We estimate the background photon statistics parameter, λbg, from an image which contains both background and signal pixels by use of a novel truncated fit procedure with an automatically determined image count threshold. Prior to this, the camera noise model parameters are estimated using a calibration step. Utilizing the estimates for the camera parameters and λbg, we then introduce a probabilistic thresholding method, where, for the first time, the fraction of misclassified pixels can be determined a priori for a general image in an unsupervised way. We use synthetic images to validate our a priori estimates and to benchmark against the Otsu method, which is a popular unsupervised non-probabilistic image thresholding method (no a priori estimates for the error rates are provided). For completeness, we lastly present a simple heuristic general-purpose segmentation method based on the thresholding results, which we apply to segmentation of synthetic images and experimental images of fluorescent beads and lung cell nuclei. Our publicly available software opens up for fully automated, unsupervised, probabilistic photophysical image analysis.

Footprints of loop extrusion in statistics of intra-chromosomal distances: An analytically solvable model.

Active loop extrusion-the process of formation of dynamically growing chromatin loops due to the motor activity of DNA-binding protein complexes-is a firmly established mechanism responsible for chromatin spatial organization at different stages of a cell cycle in eukaryotes and bacteria. The theoretical insight into the effect of loop extrusion on the experimentally measured statistics of chromatin conformation can be gained with an appropriately chosen polymer model. Here, we consider the simplest analytically solvable model of an interphase chromosome, which is treated as an ideal chain with disorder of sufficiently sparse random loops whose conformations are sampled from the equilibrium ensemble. This framework allows us to arrive at the closed-form analytical expression for the mean-squared distance between pairs of genomic loci, which is valid beyond the one-loop approximation in diagrammatic representation. In addition, we analyze the loop-induced deviation of chain conformations from the Gaussian statistics by calculating kurtosis of probability density of the pairwise separation vector. The presented results suggest the possible ways of estimating the characteristics of the loop extrusion process based on the experimental data on the scale-dependent statistics of intra-chromosomal pair-wise distances.

Energy release rate in bimaterial specimens tested in pure modes I and II

New closed-form analytical expressions of the energy release rate of a bimaterial cracked specimen are derived, when it experiences pure mode I in an asymmetric double cantilever test, and pure mode II in an asymmetric end-notched flexure test. The analysis is based on the strain coenergy and on the laminated beam theory. The results of the analytical approach are compared with those obtained using the finite element method in two ways: by the virtual crack closure technique at a test point and by cohesive elements during a virtual delamination test. The agreement is excellent in both cases.

Incorporating the effect of frequency-independent attenuation within the on-axis spatial impulse response of a circular piston

The spatial impulse response is important for numerical simulations of diagnostic ultrasound. The spatial impulse response, which describes transient diffraction due to an impulsive input, yields closed form analytical expressions for various transducer geometries when the medium is lossless. These analytical expressions are advantageous for simulations that repeatedly evaluate these expressions at hundreds of thousands of points. However, spatial impulse responses evaluated for lossy materials typically require additional numerical calculations that substantially increase the computation time. Thus, analytical or rapidly converging numerical expressions for the lossy spatial impulse response are expected to greatly enhance present simulation methods. This motivates the derivation of on-axis spatial impulse responses for a circular piston that model frequency-independent attenuation. Closed-form analytical expressions for the on-axis spatial impulse response are introduced for two closely related frequency-independent attenuation models. The results show that, as the attenuation constant increases, the peak amplitudes of these lossy on-axis spatial impulse responses decrease. The lossy on-axis impulse response also decreases slightly as time increases beyond the initial arrival time, whereas the lossless on-axis spatial impulse response for a circular piston maintains a constant value after the initial arrival time.

Securing cooperative vehicular networks amid obstructing vehicles and mixed fading channels

Smart cities (SCs) were founded on the basis of the Internet of Vehicles paradigm, aiming to enhance the quality of life. Despite not having a unique composition, the intelligent transportation services (ITS) is a key component in most of the SCs. The ITS integrates the new communication systems along with the traditional transportation systems, forming a universal network of static and vehicular entities communicating over wireless broadcast channels. In addition to its vulnerability to co-channel interference (CCI), eavesdropping, and fading, the vehicle-to-vehicle communication channels could be subjected to shadowed fading. That is due to the probabilistic obstruction by big vehicles. Hence, degrading the signal-to-noise ratio at the receiving vehicle severely. This work evaluates the secrecy performance of a practical cooperative vehicular relaying network in terms of its secrecy outage probability (SOP). Due to the probabilistic existence of a vehicular obstacle, three different communication scenarios are considered and a novel closed-form analytical expression for the SOP is obtained over mixed Nakagami-m fading and Nakagami-N-Gamma shadowed fading channels. The system comprises legitimate source and destination operating in full-duplex mode, thus imposing CCI on both the relay and the passive eavesdropper. The result demonstrate the impacts of the secrecy data rate threshold, the strength of shadowing, the density of the obstructing vehicles, and several parameters on the SOP. High levels of shadowing strength and density of obstructing vehicles significantly degrades the secrecy performance by increasing the SOP. Accordingly, adopting the mixed fading channel model is of great importance for practical network modeling.

Intelligent Reflecting Surface Framework for ED Based Spectrum Sensing

According to a recent study, adding intelligent reflecting surfaces (IRS) to propagation environments improves the performance of wireless devices. The capabilities of cognitive radio’s spectrum sensing over environments with assisted wireless propagation via IRS are examined in this research. Taking channel fading effects into consideration, we derive closed-form analytical expressions for the average probability of detection (APD) for single user and cooperative users. Further, the derived APD expression is used to derive the average area under the receiver operating characteristics curve (AUC¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\overline{AUC}}$$\\end{document}) expression. Monte Carlo simulations are also carried out in order to validate the derived expressions. Results demonstarte the extent of detectability performance in cognitive radio on utilizing the IRS aided wireless propagation environment therefore witnessing its significance for upcoming wireless technological advancement by offering spectrum efficiency with good quality of service (QoS).

Effect of the Surface Charge-Dependent Boundary Slip on the Electrophoresis of a Hydrophobic Polarizable Rigid Colloid.

The electrophoresis of a hydrophobic charged rigid colloid is studied by considering the lateral movement of the adsorbed surface charge. The slip velocity condition at the hydrophobic surface is modified to take into account the impact of the frictional and electric forces created by the adsorbed laterally mobile surface charge. Though the dependency of the surface charge on the slip velocity in the context of electrophoresis has been addressed before, the effect of the laterally mobile adsorbed surface charge on the electrophoresis of hydrophobic colloids has not been studied. The dielectric colloid is considered to polarize and create an induced immobile surface charge when subjected to an imposed electric field. The impact of the mobile surface charge along with the immobile induced surface charge on electrophoresis of a hydrophobic colloid is elucidated by numerically solving the governing electrokinetic equations in their full form. We have also developed a simplified model under a weak applied field consideration, which can be further reduced to a closed-form analytic expression for the mobility under the Debye-Hückel approximation. This analytic model for mobility is in excellent agreement with the exact numerical solution for an entire range of the Debye length when the ζ-potential is in the order of the thermal potential. One of the notable features of this closed-form mobility expression is that it accounts for the mobile adsorbed surface charge on the hydrodynamic slip condition and the dielectric polarization of the particle. We find that the mobility of the surface charge decreases the electrophoretic mobility of the hydrophobic dielectric colloid. However, the mobile surface charge enhances the mobility of a conducting hydrophobic colloid.

A new analytical method for modeling a 2D electrostatic potential in MOS devices, applicable to compact modeling

This paper presents a new conformal mapping method to solve 2D Laplace and Poisson equations in MOS devices. More specifically, it consists of an analytical solution of the 2D Laplace equation in a rectangular domain with Dirichlet boundary conditions, with arbitrary values on the boundaries. The advantages of the new method are that all four edges of the rectangle are taken into account and the solution consists of closed-form analytical expressions, which make it fast and suitable for compact modeling. The new model was validated against other similar methods. It was found that the new model is much faster, easier to implement, and avoids many numerical issues, especially near the boundaries, at the cost of a very small loss in accuracy.

A coupled-mode-theory formulation for periodic multi-element metasurfaces in the presence of radiation losses

We derive a coupled-mode theory (CMT) formulation for the fast analysis of periodic multi-element metasurfaces in the presence of radiation losses. Full-wave simulations of periodic multi-element metasurfaces are very time- and memory-consuming, especially as the size and complexity of the metasurface increase. The CMT formulation provides a considerably faster and efficient alternative. It results in a small system of equations with size equal to the number of supported resonator modes in the frequency range of interest, allowing to calculate the resonator mode amplitudes and, consequently, the metasurface response. Subsequently, we systematically derive analytical closed-form expressions for the coupling coefficients between two weakly coupled resonators in the presence of radiation losses and incorporate them into the CMT model, which is found important for the accurate description of the metasurface, while also providing insight into the underlying physics of complex metasurfaces. We validate the proposed formulation on benchmark examples of both metal- and dielectric-based metasurface absorbers (MSAs) by comparing the CMT results to spectral FEM simulations of the composing supercell. To further demonstrate the potential of the proposed formulation, as a proof of concept, we use the CMT to synthesize a larger optimized periodic multi-element MSA. A comprehensive comparison to full-wave FEM simulations of the composing supercell is included in terms of time and computational requirements, which shows that our method provides a valuable and efficient alternative solver for synthesizing complex metasurfaces.

Cooperative Terrestrial–Underwater FSO System: Design and Performance Analysis

In this paper, we propose, design, and evaluate a new hybrid terrestrial–underwater optical communication link for providing high-speed connectivity between land and underwater systems. A device based on an amplify-and-forward strategy is considered and used for the hybrid optical link. A performance analysis of the proposed hybrid system is then carried out, taking into account both the atmospheric and underwater channels and their respective degradation sources. Different networking scenarios and conditions are evaluated. To this end, the channel model of the terrestrial free-space optical (FSO) link is modeled using the Gamma–Gamma distribution, while the underwater optical link is modeled using the Weibull distribution. The former takes into account atmospheric and turbulence attenuation, geometric spread and pointing errors, while the latter takes into account underwater and turbulence attenuation and geometric spread. Accordingly, a new analytical closed-form expression for the bit error rate (BER), which depends on the cumulative distribution function of the holistic hybrid system, is derived. Analytical results show that pointing errors as well as atmospheric and oceanic turbulence seriously degrade the performance of the hybrid system. In addition, ocean turbulence leads to the occurrence of a BER floor in some scenarios. This is the first time that such a network is proposed and modeled under the assumption of critical channel impairments.

Performance analysis of AUV-carried RISs-aided multihop UWOC convergent with RF MRC systems over WGG oceanic turbulence

To establish reliable connection between Internet of Underwater Things (IoUT) devices and terrestrial data centers, this work first proposes a reconfigurable intelligent surface (RIS) aided multihop underwater wireless optical communication (UWOC) convergent with radio frequency (RF) uplink system. Specifically, the RIS carried by an autonomous underwater vehicle (AUV) is introduced into UWOC link to relax the line-of-sight (LOS) requirement and the maximal-ratio combining (MRC) receiver is adopted at the terrestrial data center to mitigate the RF link fading. It is assumed that the underwater thermocline channel is subject to the newly proposed Weibull-generalized gamma (WGG) turbulence distribution and the RF link composite fading follows Fisher-Snedecor F distribution. Additionally, the optical link misalignment is characterized by the zero-boresight pointing errors model. With the decode-and-forward (DF) relaying scheme, the analytical closed-form expressions of the outage probability (OP) and average bit error rate (ABER) of this system are mathematically achieved, and then the impacts of air bubbles, thermohaline gradient, the number of RIS elements, pointing errors, system structure, and the number of receive antennas are further investigated. Meanwhile, the analytical results are verified by Monte Carlo (MC) simulations. Results reveal that this hybrid system performance would degrade with the increased air bubble levels and thermohaline gradients. Notably, RIS can effectively alleviate the impact of underwater turbulence and this effect would be more pronounced as the number of RIS elements increases. This work will benefit the design and research of hybrid UWOC-RF system.

Capturing non-equilibrium effects in cylindrical Couette flow problem using the Extended Onsager-13 moment equations

In the present work, cylindrical Couette flow is analyzed using the recently derived third-order accurate 13-moment transport equations by transforming them into cylindrical coordinates. Assuming the Mach number and normalized temperature difference between the cylinders to be relatively small, closed-form expressions for all relevant quantities, velocity, pressure, temperature, heat flux, and stresses, are obtained from the semi-linearized form of the equations. These closed-form expressions from the present study have been validated against the corresponding linearized Grad 13 moment (G13) equation solutions. It is further demonstrated that contrary to the G13 equations, the pressure in the cylinder annulus is not constant, while the temperature depends upon the magnitude of viscous heating apart from non-isothermal boundary conditions. The coupling among velocity, temperature, heat flux, and stress and its effect on the variation of various physical quantities across the annulus has been discussed. The obtained analytical solution shows that the equations correctly capture most known non-equilibrium effects, such as the presence of a Knudsen layer, non-Newtonian stresses, and non-Fourier heat flux for Knudsen numbers falling well into the transition regime through a quantitative agreement with direct simulation Monte Carlo data, G13, and regularized 13-moment equations. The presence of non-zero radial and tangential heat fluxes, even when both the cylinders are at the same temperature, has been observed. The analysis helps us to demonstrate the ability of the recently derived equations in accurately solving complex rarefied flow problems. Moreover, understanding of higher-order rarefaction effects should greatly improve with the availability of closed-form analytical expressions of all physical quantities obtained here.

Douglas–Rachford algorithm for control-constrained minimum-energy control problems

Splitting and projection-type algorithms have been applied to many optimization problems due to their simplicity and efficiency, but the application of these algorithms to optimal control is less common. In this paper we utilize the Douglas–Rachford (DR) algorithm to solve control-constrained minimum-energy optimal control problems. Instead of the traditional approach where one discretizes the problem and solves it using large-scale finite-dimensional numerical optimization techniques we split the problem in two subproblems and use the DR algorithm to find an optimal point in the intersection of the solution sets of these two subproblems hence giving a solution to the original problem. We derive general expressions for the projections and propose a numerical approach. We obtain analytic closed-form expressions for the projectors of pure, under-, critically- and over-damped harmonic oscillators. We illustrate the working of our approach to solving not only these example problems but also a challenging machine tool manipulator problem. Through numerical case studies, we explore and propose desirable ranges of values of an algorithmic parameter which yield smaller number of iterations.

Electrophoresis of hydrophobic and polarizable liquid droplets in hydrogel medium

The emulsion droplet-hydrogel composite is a new class of biomaterial which finds potential application in several fields, including chemical, biological, food and pharmaceutical industries, as well as to controlled release drug in targeted location, etc. Besides, the electric field response of polarizable emulsion droplets quenched in hydrogel medium offers several Lab-on-a-Chip applications. Note that the gel electrophoresis of such droplets is advantageous compared to free-flow electrophoresis. Given the importance, a general theory of electrophoresis applicable for any hydrophobic and polarizable liquid droplets quenched in hydrogel medium is developed. Such a theory is also applicable for electric field response of emulsion droplets quenched in a hydrogel medium. The mathematical model is based on conservation principles of mass and momentum considering the hydrogel skeleton as soft polymeric material saturated with electrolytic solution. Within the Debye-Hückel electrostatic framework, we have derived a closed form analytical expression for electrophoretic mobility of hydrophobic and polarizable liquid droplets embedded in hydrogel skeleton. The derived general expression is further approximated with negligible error and the corresponding approximate form is found to be very convenient and can be easily programmed on a personal computer. Additionally we further deduce several closed form analytical expressions derived under various limits. Thus, the newly derived analytical results may be useful to the experimentalists to analyze and interpret their observations.