Equation For Density Research Articles

Symmetric teleparallel gravity with variable deceleration parameter

In this paper, our investigation focuses on the cosmic history in modified [Formula: see text] gravity, a different theory that describes the gravitational force using a non-metricity scalar. The equations of the theory are solved by assuming a parametric form for the deceleration parameter. The parameters of the model are estimated by using the Cosmic Chronometer dataset. A number of parameters, like the deceleration parameter, whose signature changes with redshift and shows typical behavior in late-time cosmology, are subject of our analysis. We also show the evolution of other parameters, such as the equation of state parameter and energy density. The parametrization approach used here in [Formula: see text] modified gravity provides a better description of the late-time cosmic evolution without dark energy.

Derivation of Power Density Equation of Hydro Turbine: Mathematical Basis of Osmosis Energy to Rotate a Reaction Turbine

Derivation of Power Density Equation of Hydro Turbine: Mathematical Basis of Osmosis Energy to Rotate a Reaction Turbine

The Time Response of a Uniformly Doped Transmission-Mode NEA AlGaN Photocathode Applied to a Solar-Blind Ultraviolet Detecting System

Due to the excellent quantum conversion and spectral response characteristics of the AlGaN photocathode, it has become the most promising III-V group semiconductor photocathode in solar-blind signal photoconversion devices in the ultraviolet band. Herein, the influence factors of the time-resolved characteristics of the AlGaN photocathode are researched by solving the photoelectron continuity equation and photoelectron flow density equation, such as the AlN/AlGaN interface recombination rate, AlGaN electron diffusion coefficient, and AlGaN activation layer thickness. The results show that the response time of the AlGaN photocathode decreases gradually with the increase in AlGaN photoelectron diffusion coefficient and AlN/AlGaN interface recombination rate, but the response time of the AlGaN photocathode gradually becomes saturated with the further increase in AlN/AlGaN interface recombination rate. When the thickness of the AlGaN photocathode is reduced from 250 nm to 50 nm, the response time of the AlGaN photocathode decreases from 63.28 ps to 9.91 ps, and the response time of AlGaN photocathode greatly improves. This study provides theoretical guidance for the development of a fast response UV detector.

Generalized Brenier Principle and the Closure Problem of Landgren–Monin–Novikov Hierarchy for Vorticity Field

Brenier’s concept – a representation of solutions to the equations of ideal incompressible fluids in terms of probability measures on the set of Lagrangian trajectories in the case of their stochasticity, is a generalization of Arnold’s principle of least action of finding smooth solutions of Euler’s equations. In this work, the variational generalized Brenier principle (Brenier, J. Am. Math. Soc. 1989) is used to close the infinite chain of Landgren–Monin–Novikov equations for the n-point probability density functions fn of the vortex field of two-dimensional turbulence. In addition, within the framework of the statistical approach, an approximation of the variational problem with conditions at the ends posed by Shnirelman (Mat. Sat. 1985) for the Euler equation is proposed.

Numerical solution of nonlinear equations of traffic flow density using spectral methods by filter

Numerical solution of nonlinear equations of traffic flow density using spectral methods by filter

Fokker-Planck equation for the crystal-size probability density in progressive nucleation and growth with application to lognormal, Gaussian and gamma distributions

The Fokker Planck (FP) equation for the probability density function (PDF) of crystal size in phase transformations ruled by progressive nucleation and growth, has been derived. Crystals are grouped in sub-sets, we refer to as τ-crystals, where τ is the birth time of the set. It is shown that the size PDF is the superposition of the PDF of the crystal sub-sets (τ-PDFs), with weight given by the nucleation rate. The growth and diffusion coefficients entering the FP equations are estimated as a function of both τ-PDFs and nucleation rate. The functional form of these coefficients is studied for solutions of the FP equation for τ-crystals given by the lognormal, Gaussian and gamma distributions. For the first two distributions, the effect of fluctuations, nucleation rate and growth rate, on the shape of the distribution has been investigated. It is shown that for an exponential decay of the fluctuation term, the shape of the PDF is mainly governed by both the time constant for nucleation and the strength of the fluctuation. It is found that τ-PDFs given by the one-parameter gamma distributions are suitable to deal with KJMA (Kolmogorov Johnson Mehl Avrami) compliant phase transformations, where the fluctuation term is proportional to crystal size. The connection between the FP equation for the size PDF and the evolution equation for the density of crystal populations is also discussed.

Predictive potentialities of the Quasi-Random Lattice model for electrolyte solutions, discussion and improvement strategies

This article focuses on the predictive potentialities of the Quasi-Random Lattice (QRL) model, developed for describing the activity behaviour of electrolytic solutions, and elaborates strategies for their improvement.First, the study critically discusses the computational-experimental procedure (previously published) for determining the QRL parameterization, whose convergence within few iterations is counterbalanced by known experimental issues concerning, in particular, the mean activity coefficient. An alternative procedure is proposed, that makes use of osmotic data at medium-high concentrations, so as to make QRL more interesting from a practical point of view.Second, the study explores the applicability of the model beyond the concentration ranges earlier considered. To this purpose, the solution density is evaluated in detail. Its thermodynamic relationship with the mean activity coefficient yields a parametric Abel Equation of the Second Kind valid in the medium-high range of concentrations. A further density equation is formulated, useful in the low-medium range, based on a classical power-series combined with appropriate analytical constraints to improve estimation and prediction methods.QRL theory, methods and procedures are applied to binary aqueous solutions at 25°C.

Correlation functions of weakly inhomogeneous plasma

This article focuses on the kinetic theory of inhomogeneous plasma and explores the interaction between a high-frequency electric field and weakly inhomogeneous plasma. Particularly, it examines the impact of an external variable field on the kinetic and high-frequency properties of the plasma, including kinetic equations, correlation functions, and distribution functions of charged particles. The study derives expressions for the pair (two-particle) correlation function and the corresponding distribution function, taking into ac- count the spatial inhomogeneity of the plasma and electric field, as well as the collisions between charged particles. The results were obtained using the kinetic equation for the spatial-temporal spectral density of fluctuations and the method of successive approximations (separation of slow motions and fast oscillations). The field amplitude is considered a slowly varying function of time and coordinates. The calculations neglect the contribution of the magnetic component of the electromagnetic field, which is applicable to longitudinal electric fields. The results obtained in this article are primarily of theoretical interest, they reveal the picture of the interaction of a weakly inhomogeneous plasma with a high-frequency electric field and can be used in the construction of a kinetic theory of an inhomogeneous plasma located in high-frequency electromagnetic fields. Note that for charged particles of the same sign, the correlation function is negative, and for particles of different signs it is positive. In addition, the correlation function is exponentially small when the distance between the particles is greater than the Debye radius. In all calculations, the contribution of the magnetic component of the electromagnetic field is neglected, which is quite true for the longitudinal electric field.

Stiffness properties of regular p-bars tensegrity prisms under the compressive loading

The regular tensegrity prisms are a kind of fundamental unit, which can be formed into different configurations by different splicing approaches to achieve specific structural functions. Taking regular prisms as the research object, this article presents an analytical method for structural deformations under compressive loading. The structural parameters after loading can be obtained directly by solving the nonlinear force density equation, which simplifies the form-finding process. This article also studies the structural stiffness under different connection modes. By defining the twist angle change rate and internal force utilization rate, we can compare the relative deformation of different tensegrity prisms. The results show that the relative deformation of the three-bar prism is minimal, and it has the best stiffness.

Constraining the parameters of generalized and viscous modified Chaplygin gas and black hole accretion in Einstein-Aether gravity

In this paper, we investigate the phenomenon of the accelerated cosmic expansion in the late universe and the mass accretion process for a four-dimensional Einstein-Aether black hole. Starting with the basics of Einstein-Aether gravity theory, we first consider the field equations and two well-known models of Chaplygin gas, i.e., the generalized cosmic Chaplygin gas model and viscous modified Chaplygin gas model. We then obtain the energy density and Hubble parameter equations for these models in terms of various dimensionless density parameters and some unknown parameters. After finding the required parameters, we proceed with the mass accretion process. For both models, we obtain the equation of mass in terms of the redshift function and graphically represent the change in the mass of the black hole with a redshift. At the same time, we perform a graphical comparison between these models and the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model of the universe. Finally, we conclude that the mass of a four-dimensional black hole will increase along the evolution of the universe within the framework of Einstein-Aether gravity.

Kinetic and hydrodynamic post-Newtonian equations in the Brans–Dicke theory

Abstract A kinetic theory for the post-Newtonian Brans–Dicke theory is developed. The Boltzmann equation and the equilibrium Maxwell-Jüttner distribution function are determined from the knowledge of the components of the metric tensor and Christoffel symbols in the post-Newtonian Brans–Dicke theory. The hydrodynamic equations for the mass density, momentum density and mass-energy density are obtained from a transfer equation derived from the Boltzmann equation. The problem of self-gravitating fluid instabilities in the post-Newtonian Brans–Dicke theory is investigated.

Barrow holographic dark energy models in Lyra and general relativity theories

This study investigates the Barrow holographic dark energy (BHDE) matter distribution in the Bianchi I universe model in Lyra and General Relativity Theories. To this end, it obtains exact solutions by Hubble parameter, conservation equation, and BHDE energy density equation and supports them with graphics. The results show that the solutions are in harmony with the functioning of the universe and the nature of dark energy. It finally discusses the need for further research.

Numerical modelling on dispersion behavior of particulate contamination induced by a moving operator in a semiconductor cleanroom: A eulerian-eulerian method

Contamination control in cleanrooms is crucial across various industries. In semiconductor manufacturing, operator-induced contamination presents a significant challenge, as it reduces the critical dimension. In this research, one comprehensive model describing contamination behaviour induced by operator motion was proposed, considering the moving operator as both a flow pattern changer and a contaminant source, combined with the number of particles accumulated (NPA) on the process instruments. To quantitatively characterise the transient contaminant dispersion behaviour caused by the operator, a Eulerian-Eulerian-based particle number density equation, integrated with overset mesh technology was employed. Our findings reveal that contamination dispersion behaviour is governed by convection and diffusion. The NPA on the side instruments at operator moving speeds at 0.5, 1, and 1.5 m/s were 4404.33, 8241.26, and 9384.41, respectively. And those of central instruments are 28463.75, 35848.60, and 43680.74 in corresponding cases. The largest reduction in influence on both the central and side instruments was found between cases with a maximum operator speed of 0.5 and 1.5 m/s, with NPA reaching reductions of 53.08 % and 34.84 % respectively. This indicates that the moving speed significantly affects contaminant dispersion.

Dark sink enhances the direct detection of freeze-in dark matter

We describe a simple dark sector structure which, if present, has implications for the direct detection of dark matter (DM); . A dark sink transports energy density from the DM into light dark-sector states that do not appreciably contribute to the DM density. As an example, we consider a light, neutral fermion ψ which interacts solely with DM χ via the exchange of a heavy scalar Φ. We illustrate the impact of a dark sink by adding one to a DM freeze-in model in which χ couples to a light dark photon γ′ which kinetically mixes with the Standard Model (SM) photon. This freeze-in model (absent the sink) is itself a benchmark for ongoing experiments. In some cases, the literature for this benchmark has contained errors; we correct the predictions and provide them as a public code. We then analyze how the dark sink modifies this benchmark, solving coupled Boltzmann equations for the dark-sector energy density and DM yield. We check the contribution of the dark sink ψ’s to dark radiation; consistency with existing data limits the maximum attainable cross section. For DM with a mass between MeV−O(10 GeV), adding the dark sink can increase predictions for the direct detection cross section all the way up to the current limits. Published by the American Physical Society 2024

Nonlinear Charge Transport and Excitable Phenomena in Semiconductor Superlattices.

Semiconductor superlattices are periodic nanostructures consisting of epitaxially grown quantum wells and barriers. For thick barriers, the quantum wells are weakly coupled and the main transport mechanism is a sequential resonant tunneling of electrons between wells. We review quantum transport in these materials, and the rate equations for electron densities, currents, and the self-consistent electric potential or field. Depending on superlattice configuration, doping density, temperature, voltage bias, and other parameters, superlattices behave as excitable systems, and can respond to abrupt dc bias changes by large transients involving charge density waves before arriving at a stable stationary state. For other parameters, the superlattices may have self-sustained oscillations of the current through them. These oscillations are due to repeated triggering and recycling of charge density waves, and can be periodic in time, quasiperiodic, and chaotic. Modifying the superlattice configuration, it is possible to attain robust chaos due to wave dynamics. External noise of appropriate strength can generate time-periodic current oscillations when the superlattice is in a stable stationary state without noise, which is called the coherence resonance. In turn, these oscillations can resonate with a periodic signal in the presence of sufficient noise, thereby displaying a stochastic resonance. These properties can be exploited to design and build many devices. Here, we describe detectors of weak signals by using coherence and stochastic resonance and fast generators of true random sequences useful for safe communications and storage.

Shortcut Approximation to the Coupled Equations of Heat and Electric Currents

A self-consistent approximation to the coupled equations of heat and electric current densities is developed to facilitate the integration of the relaxation time approximation of the Boltzmann transport equation in a realistic design, analysis, and optimization of energy conversion devices. By introducing two auxiliary energies in the empirical Boltzmann transport equations, the coupled equations can be simply expressed in these auxiliary energies. These auxiliary energies are then determined by optimizing both the heat current through charge neutrality equation without the presence of electric current and the carrier concentrations with the presence of electric current. The heat and the electric current densities as well as the transport properties can then be calculated. The calculated transport properties agreed reasonably well with existing experimental and computational data.

Experimental and simulation study of argon helicon discharge in multiple plasma simulation linear device (MPS-LD)

Abstract To obtain a high-parameter plasma in the target region of a multiple plasma simulation linear device and to realize the experimental simulation environment of tokamak divertor plasma, experimental and numerical simulations of argon helicon discharge are carried out. Langmuir probes are used to diagnose the electron density (ne ) in the source and target regions with different experimental parameters (magnetic field, radio frequency power, puffing flow rate). A three-dimensional discharge model is developed using drift-diffusion equations of electron density and electron energy with the aid of COMSOL. Helicon discharge with a long straight plasma beam and a bright blue core is experimentally achieved. The simulation and experimental results are compared, validating the model. The corresponding spatial ne distribution is obtained, and the dependence of ne on the main experimental parameters is confirmed. The energy conversion relationship between the helicon and plasma is found. Helicon waves prefer to transfer energy to the plasma in the source region, where ne is significantly increased. This results in a strong ne gradient, which acts as a barrier to prevent the propagation of helicon waves. Therefore, localized standing helicon waves are formed, which limits the increase in plasma density in the target region. By increasing the magnetic field strength (B < 1500 G) and RF power (P < 1500 W), ne in the source region can be increased, but they have little effect on ne in the target region.

A generalized Dean–Kawasaki equation for an interacting Brownian gas in a partially absorbing medium

The Dean–Kawasaki (DK) equation is a stochastic partial differential equation (SPDE) for the global density ρ = N − 1 ∑ j = 1 N δ ( x − X j ( t ) ) of a gas of N over-damped Brownian particles, where X j ( t ) is the position of the j th particle. In the thermodynamic limit N → ∞ with weak pairwise interactions, the expectation ⟨ ρ ⟩ with respect to the white noise processes converges in distribution to the solution of a McKean–Vlasov (MV) equation. In this article, we use an encounter-based approach to derive a generalized DK equation for an interacting Brownian gas on the half-line with a partially absorbing boundary at x = 0 . Each particle is independently absorbed when its local time L j ( t ) at x = 0 exceeds a random threshold ℓ ^ j . The global density is now summed over the set of particles that have not yet been absorbed, and expectations are taken with respect to the Gaussian noise and the random thresholds ℓ ^ j . Assuming the DK equation has a well-defined mean-field limit, we derive the corresponding MV equation on the half-line. We illustrate the theory by (i) analysing stationary solutions for a Curie–Weiss (quadratic) interaction potential and a totally reflecting boundary; and (ii) calculating the effective rate of particle loss in the weak absorption limit. Extensions to finite intervals and partially absorbing traps are also considered.

Yet anOther Dose Algorithm (YODA) for independent computations of dose and dose changes due to anatomical changes

Objective. To assess the viability of a physics-based, deterministic and adjoint-capable algorithm for performing treatment planning system independent dose calculations and for computing dosimetric differences caused by anatomical changes. Approach. A semi-numerical approach is employed to solve two partial differential equations for the proton phase-space density which determines the deposited dose. Lateral hetereogeneities are accounted for by an optimized (Gaussian) beam splitting scheme. Adjoint theory is applied to approximate the change in the deposited dose caused by a new underlying patient anatomy. Main results. The dose engine’s accuracy was benchmarked through three-dimensional gamma index comparisons against Monte Carlo simulations done in TOPAS. For a lung test case, the worst passing rate with (1 mm, 1%, 10% dose cut-off) criteria is 94.55%. The effect of delivering treatment plans on repeat CTs was also tested. For non-robustly optimized plans the adjoint component was accurate to 5.7% while for a robustly optimized plan it was accurate to 4.8%. Significance. Yet anOther Dose Algorithm is capable of accurate dose computations in both single and multi spot irradiations when compared to TOPAS. Moreover, it is able to compute dosimetric differences due to anatomical changes with small to moderate errors thereby facilitating its use for patient-specific quality assurance in online adaptive proton therapy.

Morphological human bone features and demography controlling damage accumulation and fracture: a finite element study

Prediction of bone fracture risk is clinically challenging. Computational modeling plays a vital role in understanding bone structure and diagnosing bone diseases, leading to novel therapies. The research objectives were to demonstrate the anisotropic structure of the bone at the micro-level taking into consideration the density and subject demography, such as age, gender, body mass index (BMI), height, weight, and their roles in damage accumulation. Out of 438 developed 3D bone models at the micro-level, 46.12% were female. The age distribution ranged from 23 to 95 years. The research unfolds in two phases: micro-morphological features examination and stress distribution investigation. Models were developed using Mimics 22.0 and SolidWorks. The anisotropic material properties were defined before importing into Ansys for simulation. Computational simulations further uncovered variations in maximum von-Misses stress, highlighting that young Black males experienced the highest stress at 127.852 ± 10.035 MPa, while elderly Caucasian females exhibited the least stress at 97.224 ± 14.504 MPa. Furthermore, age-related variations in stress levels for both normal and osteoporotic bone micro models were elucidated, emphasizing the intricate interplay of demographic factors in bone biomechanics. Additionally, a prediction equation for bone density incorporating demographic variables was proposed, offering a personalized modeling approach. In general, this study, which carefully examines the complexities of how bones behave at the micro-level, emphasizes the need for an enhanced approach in orthopedics. We suggest taking individual characteristics into account to make therapeutic interventions more precise and effective.