Equation For Distribution Function Research Articles

Ballistic model of recycling of atomic and molecular hydrogen and its application to the ITER main chamber

A simple, computationally efficient 1D model is suggested for the recycling of atomic and molecular hydrogen between the main chamber first wall (FW) of a tokamak and the peripheral plasma. The model is based on iteratively solving the kinetic equation for the 1D velocity distribution function (VDF) of hydrogen neutral atoms in the projection of velocity on the direction normal to the FW. The model results in a fast-converging routine due to the domination of the contribution to the VDF from long-distance, ballistic flights of the neutral atoms produced by the charge exchange of colder atoms with hotter plasma ions. The proposed modification of the ballistic model (BM) enhances its original version along the following lines: the account of the inelastic reflections of the neutral atoms from the wall; parameterisation of the boundary condition for the VDF of the atoms produced by the reflection of ions from the wall with immediate recombination, without capture in the wall and thermalisation there; extension to a mixture of hydrogen isotopes; elaboration of algorithms for speeding up computation on graphical processing units. The complete set of equations of the BM is published for the first time. The model is verified by comparison with the EIRENE code simulations of the VDF of neutral atoms and molecules in the scrape-off-layer around mid-plane for typical conditions of ITER operation. The applicability of the BM to the synthetic diagnostics developed for the H-alpha high-resolution spectroscopy is demonstrated. That is, the simple 1D model reproduces reasonably well the 1D projection of the data computed with the 3D Monte-Carlo code on a 2D plasma background.

Analytical Equations and Numerical Simulation Describing the Pore and Solid Phase Distributions of Interfacial Transition Zones in Cement-Based Materials

AbstractBy combining the nearest surface distribution function and hydration kinetic equation, analytical equations are proposed to determine the pore phase and solid phase evolution in the interfa...

Modification of the singlet equation for a molecular system of solid spheres near a solid surface in the Percus-Yevick approximation

An analysis of theoretical methods for studying the molecular structure of liquids bordering a solid surface was carried out. It was established that the currently existing nonlinear integral equations for partial distribution functions do not have an analytical solution for spatially inhomogeneous liquids. A linear integral equation having an analytical solution for a system of solid spheres near a solid surface was proposed.

Simulating glow discharge positive column parameters in neon

The nonlocal self-consistent model for the low temperature plasma parameters in a glow discharge positive column is presented. The model implies relaxation method for the solving of the nonlocal Boltzmann equation for electron energy distribution function, continuity equation for ions distribution and Poisson equation for radial electric field. Calculations are provided for a cylindrical discharge tube used in the Plasma Kristall-4 experiments for neon as a plasma forming gas for the gas pressures of 30-70 Pa and discharge currents of 0.5-2.5 mA. Axial electric field strength, electron temperature and electron density are calculated as functions of the discharge current and the gas pressure. The results are in good agreement with the data provided by other authors.

A two-photon laser in a Kerr-like medium with cross-Kerr and intensity-dependent coupling

We have studied the photon statistics of a nondegenerate two-photon laser with a medium that has both self- and cross-Kerr interactions as well as intensity-dependent atom–field coupling in the Hamiltonian. The density matrix equation of the atoms and two-mode cavity fields was derived using the standard Scully–Lamb approach and numerically solved. Under certain conditions, the photon distribution function equation was derived and solved analytically. The first few moments of photon numbers were calculated in the region above the lasing threshold. Nonclassicality of the fields was investigated by means of the Mandel Q parameter, second-order correlation functions and the Cauchy–Schwarz inequality. The results are analyzed for a range of finite Kerr parameters. Intensity-dependent coupling suppresses the nonclassicality of the field. It pushes the statistics of the field towards super-Poissonian.The self- and cross-Kerr interactions have significant roles in the nonclassicality of the two-photon cavity field and its statistics. These results are interesting for the field of quantum nonlinear optics and useful in quantum communications.

On the evolution of classic charged particles

In this work, we obtain a closed system of equations for the non-probabilistic microscopic distribution functions of charged point particles moving in the electromagnetic field generated by them. The laws of change of both the total energy of a system of charged point particles and its momentum are also derived. All calculations have been performed in the covariant form. The resulting equations describe an irreversible evolution.

Helium and oxygen excited states densities in a He-air RF-driven atmospheric pressure plasma jet

The volumetric photon emission from a pulsed, RF-driven atmospheric pressure plasma jet was monitored from ultraviolet to the near infrared part of the spectrum. When a small admixture of dry air, as low as ∼0.01%, is added to the helium flow, the helium line intensities decrease, while those of oxygen increase dramatically. To understand the emission trends, a non-equilibrium, collisional-radiative model is developed to estimate He and O excited state densities as a function of relative air concentration. The model is based on a numerical solution of the electron Boltzmann equation for the electron energy distribution function self-consistently coupled to the balance equations for helium and air species. The spectral analyses reveal a good agreement between experiments and modeling, with the latter indicating that a complex interplay between electron kinetics and plasma chemistry is responsible for the substantial changes in emission profiles when He is diluted with a small amount of air.

The Impact of Cesium Vapor Bulk Condensation on the Transport of Aerosols in the Reactor Plant’s Primary Circuit during a Severe Accident at a Nuclear Power Plant with VVER

The article presents the results from numerical modeling of aerosol particles' deposition process during their transportation in the horizontal section of the reactor plant’s primary circuit model for the conditions typical for a severe accident at an NPP equipped with VVER reactors. The effect that the bulk condensation of cesium vapors has on the growth and deposition of aerosols during their transportation is analyzed. The influence of the initial size and concentration of aerosol particles on their growth rate and subsequent deposition is considered. Homogeneous condensation occurring in the absence of heterogeneous condensation nuclei (solid aerosol particles) and also heterogeneous condensation under the conditions in which nuclei of the new phase do not emerge according to the homogeneous mechanism are considered as the limit cases. Data on the average size of the produced droplets and aerosol particles along the channel axis in the condensation zone with the varied value of aerosol particles mass fraction in the flow have been obtained, which corresponded to different ratios of the condensate mass fraction in the form of droplets and liquid layer on the surface of solid aerosol particles. It has been established that the general deposition rate decreases if the flow contains droplets generated as a result of homogeneous bulk condensation and having a size smaller than the size of grown aerosol particles. The article describes the analysis scheme of the modeled section whose geometrical shape was specified based on the horizontal section of the experimental installation in the FALCON-ISP1 series of experiments on the deposition and transport of radionuclide aerosols. The cesium vapor homogeneous-heterogeneous condensation process is modeled by numerically solving—using the method of moments—the kinetic equation for the droplet size distribution function with involvement of the authors’ COND-KINET-1 computer program. The transport and deposition of aerosol particles in the primary circuit channel were modeled using the MAVR-TA computer code developed at the National Research Center Kurchatov Institute.

Unified mathematical model of the kinetics of nanoparticle phase condensation in magnetic fields

In this paper, we aim to present a unified mathematical modeling and description of the kinetics of magnetic nanoparticles phase condensation (conducting to the appearance of bulk elongated aggregates) under homogeneous permanent or alternating magnetic field. For such case, the aggregate growth rate usually takes the form dV/dt = G(V)∆(t), with V and t being the aggregate's volume and time, respectively, ∆(t)—the supersaturation of the nanoparticle suspension, and with the function G(V) depending on the precise configuration of the applied field. The Liouville equation for the aggregate size distribution function is solved by the method of characteristics. The solution is obtained in parametric form for an arbitrary function G(V), providing a general framework for any type of the applied magnetic field. In the particular case of low‐frequency rotating magnetic field (G(V)~V2/3), an explicit expression of the distribution function is obtained, while the dimensionless average aggregate volume 〈V〉 is found by the method of moments allowing a complete decoupling of the system of equations for the statistical moments 〈Vn〉 of the distribution function. Numerical examples are provided for the cases of permanent and low‐ or medium‐frequency rotating fields. It is shown that in all cases, the average volume 〈V〉 only slightly depends on the relative width of the initial size distribution. Nevertheless, at any times, t > 0, the size distribution shows a significant spreading around the average value 〈V〉, which increases progressively with time and achieves a final plateau at long times. This model can be helpful for several biomedical or environmental applications of magnetic nanoparticles in which the nanoparticle suspension undergoes a field‐induced phase condensation.

Characteristics and Parameters of Overstressed Nanosecond-Pulse Discharge Plasma between Chalcopyrite (CuInSe2) Electrodes in Argon

A bipolar nanosecond discharge with a voltage amplitude of one polarity of 15–40 kV and a current amplitude of 50–150 A in a pulse was ignited at an inter-electrode distance of 0.1 cm and the pressure of argon 101 or 202 kPa, at a frequency of the pulse voltage 40–100 Hz. The pulsed electric discharge power was in a range of 5–11 MW, with a plasma energy input of 0.40 and 0.44 J per pulse. Spectroscopic plasma diagnostics showed that the spectral lines of the copper atom in the range 200–230 nm and the spectral lines of the indium atom and Cu+ and In+ ions in a longer wavelength range of the spectrum are the most intense. The following lines from the spectral range of 300–460 nm can be used to diagnose the deposition of chalcopyrite films in real time: 307.38 nm Cu(I), 329.05 nm Cu(I), 410.17 nm In(I), 451.13 nm In(I). By solving the Boltzmann kinetic equation for the electron energy distribution function, the temperature and the density of electrons, the specific losses of the discharge power on the main electronic processes and the rate constants of the electronic processes depending on the value of the parameter of the ratio of the electric field strength E to the total concentration of Ar atoms and a small admixture of a Cu vapor N (E/N) are simulated. In a discharge in a mixture of copper vapor with argon, the electron temperature increased in a range of 300–110 000 K, with a change in the E/N parameter from 1 to 1800 Td. The electron concentration was in a range of 2.1 × 1020– 2.7 × 1020 m–3, at a current density (612–765) × 106 A/m2 on the electrode surface (at E/N = 1676 Td). Thin films of chalcopyrite were synthesized on substrates of transparent solid dielectrics, which, in a wide spectral range of 200–800 nm, absorb radiation incident of their surface quite effectively. This opens up prospects for their use in photovoltaic devices. It was shown that the smallest transmission of radiation is characteristic of chalcopyrite films synthesized at atmospheric pressure of argon and air.

Characteristics of the Nanosecond Overvoltage Discharge Between CuInSe2 Chalcopyrite Electrodes in Oxygen-Free Gas Media

The characteristics of the nanosecond overvoltage discharge ignited between semiconductor electrodes based on the CuInSe2 chalcopyrite compound in the argon and nitrogen atmospheres at gas pressures of 5.3–101 kPa are reported. Due to the electrode sputtering, chalcopyrite vapor enters the discharge plasma, so that some CuInSe2 molecules become destroyed, whereas the others become partially deposited in the form of thin films on solid dielectric substrates located near the plasma electrode system. The main products of the chalcopyrite molecule decomposition in the nanosecond overvoltage discharge are determined; these are atoms and singly charged ions of copper and indium in the excited and ionized states. Spectral lines emitted by copper and indium atoms and ions are proposed, which can be used to control the deposition of thin chalcopyrite films in the real-time mode. By numerically solving the Boltzmann kinetic equation for the electron energy distribution function, the electron temperature and density in the discharge, the specific losses of a discharge power for the main electronic processes, and the rate constants of electronic processes, as well as their dependences on the parameter E/N, are calculated for the plasma of vapor-gas mixtures on the basis of nitrogen and chalcopyrite. Thin chalcopyrite films that effectively absorb light in a wide spectral interval (200–800 nm) are synthesized on quartz substrates, by using the gas-discharge method, which opens new prospects for their application in photovoltaic devices.

A coupled hygro-thermo-mechanical model for the evolution of saturation in early-age concrete

Shrinkage can cause deformation of concrete and unexpected cracking in concrete structures under normal serviceable conditions, particularly at early age. The evolution of relative humidity and saturation in concrete at early age is the main reason for the shrinkage of concrete. In order to analyze the shrinkage and development of relative humidity for concrete at early age, a hygro-thermo-mechanical model based on hybrid mixture theory (HMT) is established by taking into consideration the evolution characteristics of saturation. Based on the Rayleigh-Ritz distribution function and Laplace equation, along with the saturation evolution characteristics, a two-stage relative humidity calculation model for early-age concrete is established. The model is verified using the experimental results from the literature. It is found that the hygro-thermo-mechanical coupling model accurately predicts the evolution of temperature, relative humidity, and deformation of concrete at early age. The modified saturation model better reflects the evolution of relative humidity and saturation in concrete at early age compared with others. Additionally, development and distribution of temperature, relative humidity and saturation in early-age concrete are discussed. The research provides a theoretical basis for the study of the deformation of and cracking in early-age concrete.

Method of accelerating convergence for gas kinetic algorithm based on digital constitutive relation of macroscopic equations

<sec>In the simulation of complex multi-scale flows covering various flow regimes, the computational efficiency of gas kinetic method by which the evolution equation of velocity distribution function is solved directly is the key to engineering applications. In order to accelerate simulation for steady flows, a gas kinetic algorithm accelerated by utilizing the macroscopic conservative equations with a digital constitutive relation is developed. In this algorithm, the contribution of the high-order terms of stress and heat flux in macroscopic conservative equations is determined by the gas kinetic solution. Meanwhile, the solution of the macroscopic conservative equations provides the macroscopic quantities for the equilibrium distribution function in the Boltzmann model equation, where a fully implicit scheme to solve the Boltzmann model equation is developed.</sec> <sec>Extensive validations are performed for the cavity flow, the supersonic flow around the cylinder, and the interactive rarefied flow around two side-by-side cylinders. The results from the above method are in good agreement with the results from the conventional gas kinetic unified algorithm and the direct simulation Monte Carlo method. It can be concluded that the nonlinear constitutive relation of rarefied flow can be well captured by the present method. And the ability of this method to simulate complex flows such as shock wave, strong wall shear and flow separation is demonstrated. Furthermore, the present method has shown to be much faster than the conventional gas kinetic unified algorithm, especially for the low-<i>Kn</i> flows. As the value of <i>Kn</i> increases, the acceleration rate decreases, because the effect of flow convection becomes weak. Meanwhile, more effort is needed to reduce inner loop iterations to improve its efficiency. </sec>

A Diffusion Limit for the Parabolic Kuramoto--Sakaguchi Equation with Inertia

In this paper, we study a macroscopic description on the ensemble of Kuramoto oscillators with finite inertia in a random media characterized by a white noise. In a mesoscopic regime, it is well known that the dynamics of a large Kuramoto ensemble in a random media is governed by the Kuramoto--Sakaguchi--Fokker--Planck (in short, parabolic Kuramoto--Sakaguchi) equation for one-oscillator distribution function. For this parabolic Kuramoto--Sakaguchi equation, we present a global existence of weak solutions in any finite-time interval. Furthermore, we rescale the kinetic equation using the diffusion scaling, and formally derive a drift-diffusion equation by using Hilbert-like expansion in a small parameter $\varepsilon$. For the rigorous justification of this asymptotic limit, we introduce a new free energy functional ${\mathcal E}$ consisting of total mass, kinetic energy, entropy functional, and interaction potential and show the uniform boundedness of this free energy with respect to the small parameter $\varepsilon$. This uniform boundedness of ${\mathcal E}$ combined with $L^1$-compactness argument enables us to derive the drift-diffusion equation. We also classified all ${\mathcal C}^2$-stationary solutions to the drift-diffusion equation in terms of synchronization parameters $\kappa$ and $\sigma$.

The initial high-energy phenomena of earthquake sources in fluid-saturated environments

Experimental data pertaining to weak faults requires construction of microscopic models of the earthquake sources. This study proposes a new high-energy mechanism of transformation of potential energy of strata, saturated with free or bounded fluids, into a kinetic energy of motion of rocks. The proposed approach relies within the framework of classic statistical theory. The high-energy phenomena at a nanometer level are generated by self-consistent molecular fields acting on individual particles. The analysis is carried out using the Bogolyubov-Born-Green-Kirkwood-Yvon equations for the particle group distribution functions. We have investigated the number of high-energy phenomena: (1) the emission of atoms and molecules with high energies from the abruptly opened surface of a condensed system; (2) the implosion of convergent streams of high-energy particles, accompanied by a phenomena of their shock dissociation and ionization, with a subsequent formation of partially ionized plasma; and (3) the recombination processes inside a plasma leading to a formation of molecules with high kinetic energies. The initiation of an earthquake occurs due to an abrupt opening of a cavity in the fluid-saturated medium. At certain thermodynamic conditions, the work function of atoms and molecules from the surface of the system may take negative values. As a result, the emission of molecules of fluids from the cavity walls will generate the high-speed streams of molecules. The emitted flux of molecules leads to the phenomena of implosion, impact dissociation, and ionization of the molecules. This plasma state of the medium is sufficient for an explosion. The explosion initiates a self-supporting chain of consequent explosions in a plane of the tectonic fault. As an example, we have considered the Bridgman explosion of serpentinite.

Simulations of the glow discharge positive column parameters in helium

The self-consistent model for the positive column of a glow discharge is presented. The model is based on the simultaneous solution of the nonlocal Boltzmann equation for electron energy distribution function, continuity equation for ions and Poisson equation for electric field. Calculations were performed for a discharge tube similar to the Plasma Kristall-4 experiments for plasma forming gas helium for the range of pressures of 20-70 Pa and the range of discharge currents of 0.5-2.5 mA. Axial electric field strength, electron temperature and electron density dependences on the discharge current and the gas pressure are obtained. The results show a good correlation with the data obtained by the other authors.

Isotopes of water molecules in a glow discharge in mixtures of helium with H2O and D2

The article presents the results of theoretical and experimental research on the concentrations of water isotopomers (H2O, HDO, and D2O) in plasma of a DC discharge in mixtures consisting of helium, molecular deuterium, and water vapor. Dependence of the isotopomers concentrations on the initial mixture composition is described. The measurements were performed using tunable diode laser absorption spectroscopy (TDLAS). Theoretical results were obtained with the use of a homogeneous kinetic model. The model included the balance equations for charged and neutral particles and the Boltzmann equation for the electron energy distribution function in the two-term approximation. The experimental and theoretical values of H2O and HDO concentrations were compared and showed good agreement.

Electric fields and concentrations of charged and neutral hydrogen isotopic particles in the plasma of low pressure DC discharge

This paper presents the results of experimental and theoretical studies of the electric discharge characteristics and the concentration dependences of water isotopomers (H2O, HDO, and D2O) in a glow discharge in mixtures of helium with water vapor and molecular deuterium depending on the initial concentrations of H2O and D2. Measurements of the H2O and HDO concentrations were carried out by absorption diode laser spectroscopy in parallel with measurements of the axial electric field. The calculations were performed using a homogeneous kinetic model including balance equations for neutral and charged particles, which are solved together with the Boltzmann equation for the electron energy distribution function in the two-term approximation and the equation for the external electric circuit. A comparison was made of the experimental and calculated values of the concentrations of H2O and HDO, the stationary values of the electric field strength, and the discharge current. All abovementioned experimental parameters showed satisfactory agreement with the calculations. It was found that in self-consistent calculations, as well as in the experiment, the ratio of the concentrations of H2O and HDO was the same for different proportions of the initial concentrations of H2O and D2, and this ratio remained the same during the discharge action.

Global Stability of Minkowski Space for the Einstein\u2013Vlasov System in the Harmonic Gauge

Minkowski space is shown to be globally stable as a solution to the massive Einstein–Vlasov system. The proof is based on a harmonic gauge in which the equations reduce to a system of quasilinear wave equations for the metric, satisfying the weak null condition, coupled to a transport equation for the Vlasov particle distribution function. Central to the proof is a collection of vector fields used to control the particle distribution function, a function of both spacetime and momentum variables. The vector fields are derived using a general procedure, are adapted to the geometry of the solution and reduce to the generators of the symmetries of Minkowski space when restricted to acting on spacetime functions. Moreover, when specialising to the case of vacuum, the proof provides a simplification of previous stability works.

Photothermal nanobubble nucleation on a plasmonic nanoparticle: A 3D lattice Boltzmann simulation

Nanobubble nucleation by plasmonic heating on a gold spherical nanoparticle under laser illumination is numerically investigated by a 3D liquid-vapor phase change lattice Boltzmann (LB) method. A volumetric heat source is incorporated to the evolution equation of temperature distribution function to simulate photothermal energy conversion effect in a plasmonic gold spherical nanoparticle induced by surface plasma resonance under laser illumination. A halfway bounce-back conjugate boundary condition on nanoparticle/liquid interface for LB simulation, with interfacial thermal resistance taken into consideration, is derived. It is confirmed that nanobubble nucleates on a laser heated hydrophilic gold spherical nanoparticle in a core-shell mode under normal gravity condition. To simulate nanobubble nucleation on a plasmonic nanoparticle in a spherical cap mode, an extraordinary (fictitious) hyper gravity environment would have to be imposed. Effects of nanoparticle surface’s wettability, nanoparticle-liquid interfacial thermal resistance, and nanoparticle’s diameter on nanobubble nucleation under normal gravity are investigated. The size of the nanoparticle is found to have non-monotonic effects on the threshold laser fluence of nanobubble nucleation, with the 60 nm gold nanoparticle having the lowest threshold laser fluence for nucleation. This finding is in agreement with existing experimental results.