The supersonic leading edge problem according to the ellipsoidal model.
Supersonic leading edge problem using nonlinear Boltzmann equation with ellipsoidal model, calculating molecular distribution functions for flow field
Huang, A. B.↗
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Acoustical theory of turbulence
Fluid functions are expanded in a series of functions constituting a complete and orthogonal system of wave solutions of the linearized Navier-Stokes system. The motion of characteristic waves is described by Hamiltonian equations of ray acoustics. Amplitude of the waves is described with the aid of nonlinear interaction terms. The distribution functions (squares of amplitudes) satisfy Boltzmann-type equations and completely describe the mean turbulence properties.
Kentzer, C. P.↗
Shock Radiation Tests for Saturn and Uranus Entry Probes
This paper describes a test series in the Electric Arc Shock Tube at NASA Ames Research Center with the objective of quantifying shock-layer radiative heating magnitudes for future probe entries into Saturn and Uranus atmospheres. Normal shock waves are measured in Hydrogen/Helium mixtures (89:11 by mole) at freestream pressures between 13-66 Pa (0.1-0.5 Torr) and velocities from 20-30 km/s. No shock layer radiation is detected below 25 km/s, a finding consistent with predictions for Uranus entries. Between 25-30 km/s, radiance is quantified from the Vacuum Ultraviolet through Near Infrared, with focus on the Lyman-alpha and Balmer series lines of Hydrogen. Shock profiles are analyzed for electron number density and electronic state distribution. The shocks do not equilibrate over several cm, and distributions are demonstrated to be non-Boltzmann. Radiation data are compared to simulations of Decadal survey entries for Saturn and shown to be significantly lower than predicted with the Boltzmann radiation model.
Saturn Probe↗
Approximation method for the kinetic Boltzmann equation
The further development of a method for approximating the Boltzmann equation is considered and a case of pseudo-Maxwellian molecules is treated in detail. A method of approximating the collision frequency is discussed along with a method for approximating the moments of the Boltzmann collision integral. Since the return collisions integral and the collision frequency are expressed through the distribution function moments, use of the proposed methods make it possible to reduce the Boltzmann equation to a series of approximating equations.
Shakhov, Y. M.↗
Anisotropic angular scattering models of elastic electron-neutral collisions for Monte Carlo plasma simulations
Many laboratory and industrial plasma applications require accurate modeling techniques to understand the interplay between microscopic and macroscopic processes. A prime example of this interplay is how particle and Monte Carlo (MC) simulation codes describe angular scattering of electrons following elastic scattering events. The forward peaked nature of high energy electron elastic scattering is relatively trivial to accurately describe in plasma simulations. However, for lower energy collisions, which produce near isotropic or backward peaked differential cross sections, there is not a strong consensus among the plasma modeling community on how to best describe these angular scattering trends. Here, in this study, we propose a systematic method to approximate the aforementioned non-trivial angular scattering behavior with a formula that can be readily implemented in particle-in-cell (PIC) and/or MC plasma simulation codes. The present approach is specifically applied to fusion relevant atomic hydrogen and helium, as well as for molecular hydrogen, and results are also applicable to the atomic isotopes and homonuclear molecular isotopologues of these species. Comparisons between the present angular distribution function and benchmark scattering data were used to validate the proposed models. In addition, two-term Boltzmann calculations and PIC direct simulation MC simulations revealed that the proposed angular distribution function is accurate, agreeing very well with benchmark convergent close-coupling scattering calculations, and electron transport measurements. These studies confirmed that the present angular distribution function model can be utilized without the need of renormalization to the momentum transfer cross section (as opposed to using the elastic scattering integrated cross section), which has been suggested by several studies in order to correct for deficient angular scattering models, and to agree with transport measurements. Hence, the present anisotropic angular scattering model can be utilized to accurately model the momentum transfer as well as the electron trajectories of elastic collisions.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Eight-moment approximation solar wind models
Heat conduction from the corona is important in the solar wind energy budget. Until now all hydrodynamic solar wind models have been using the collisionally dominated gas approximation for the heat conductive flux. Observations of the solar wind show particle distribution functions which deviate significantly from a Maxwellian, and it is clear that the solar wind plasma is far from collisionally dominated. We have developed a numerical model for the solar wind which solves the full equation for the heat conductive flux together with the conservation equations for mass, momentum, and energy. The equations are obtained by taking moments of the Boltzmann equation, using an 8-moment approximation for the distribution function. For low-density solar winds the 8-moment approximation models give results which differ significantly from the results obtained in models assuming the gas to be collisionally dominated. The two models give more or less the same results in high density solar winds.
Olsen, Espen Lyngdal↗
Anisotropic velocity distributions in M87 - Is a supermassive black hole necessary
A collisionless distribution of stars with an anisotropic velocity distribution is considered as a model for the cores of spherical galaxies. The first moment of the corresponding Boltzmann equation is integrated to give the resulting density and velocity dispersion profiles. These distributions plus a central point source of light give a good fit to the data from recent observations of M87 without requiring a supermassive central object. This model requires one less fitting parameter than published models incorporating a central black hole. The same model with no point source of light also provides a good fit to the normal E1 galaxy NGC 3379. The success of this model in describing the cores of both M87 and NGC 3379 suggests that a supermassive black hole is not necessary in either.
Duncan, M. J.↗
On the Maxwellian distribution, symmetric form, and entropy conservation for the Euler equations
The Euler equations of gas dynamics have some very interesting properties in that the flux vector is a homogeneous function of the unknowns and the equations can be cast in symmetric hyperbolic form and satisfy the entropy conservation. The Euler equations are the moments of the Boltzmann equation of the kinetic theory of gases when the velocity distribution function is a Maxwellian. The present paper shows the relationship between the symmetrizability and the Maxwellian velocity distribution. The entropy conservation is in terms of the H-function, which is a slight modification of the H-function first introduced by Boltzmann in his famous H-theorem. In view of the H-theorem, it is suggested that the development of total H-diminishing (THD) numerical methods may be more profitable than the usual total variation diminishing (TVD) methods for obtaining wiggle-free solutions.
Deshpande, S. M.↗
Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation
We report the Boltzmann equation with the Bhatnagar-Gross-Krook collision model (Boltzmann-BGK equation) has been employed to describe multiscale flows, i.e., from the hydrodynamic limit to free molecular flow. In this study we employ physics-informed neural networks (PINNs) to solve forward and inverse problems via the Boltzmann-BGK formulation (PINN-BGK), enabling PINNs to model flows in both the continuum and rarefied regimes. In particular, the PINN-BGK is composed of three sub-networks, i.e., the first for approximating the equilibrium distribution function, the second for approximating the non-equilibrium distribution function, and the third one for encoding the Boltzmann-BGK equation as well as the corresponding boundary/initial conditions. By minimizing the residuals of the governing equations and the mismatch between the predicted and provided boundary/initial conditions, we can approximate the Boltzmann-BGK equation for both continuous and rarefied flows. For forward problems, the PINN-BGK is utilized to solve various benchmark flows given boundary/initial conditions, e.g. Kovasznay flow, Taylor-Green flow, cavity flow, and micro Couette flow for Knudsen number up to 5. For inverse problems, we focus on rarefied flows in which accurate boundary conditions are difficult to obtain. We employ the PINN-BGK to infer the flow field in the entire computational domain given a limited number of interior scattered measurements on the velocity without using the (unknown) boundary conditions. Results for the two-dimensional micro Couette and micro cavity flows with Knudsen numbers ranging from 0.1 to 10 indicate that the PINN-BGK can infer the velocity field in the entire domain with good accuracy. Finally, we also present some results on using transfer learning to accelerate the training process. Specifically, we can obtain a three-fold speedup comparted to the standard training process (e.g., Adam plus L-BFG-B) for the two-dimensional flow problems considered in our work.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Excitation of whistler and slow-X waves by runaway electrons in a collisional plasma
Runaway electrons are known to provide robust ideal or collisionless kinetic drive for plasma wave instabilities in both the whistler and slow-X branches, via the anomalous Doppler-shifted cyclotron resonances. In a cold and dense post-thermal-quench plasma, collisional damping of the plasma waves can compete with the collisionless drive. Previous studies have found that, due to their higher wavelength and frequency, slow-X waves suffer stronger collisional damping than the whistlers, while the ideal growth rate of slow-X modes is higher. Here, we study runaway avalanche distributions that maintain the same eigen distribution and increase only in magnitude over time. The distributions are computed from the relativistic Fokker–Planck–Boltzmann solver, upon which a linear dispersion analysis is performed to search for the most unstable or least damped slow-X and whistler modes. Taking into account the effect of plasma density, plasma temperature, and effective charge number, we find that the slow-X modes tend to be excited before the whistlers in a runaway current ramp-up. Furthermore, even when the runaway current density is sufficiently high that both branches are excited, the most unstable slow-X mode has a much higher growth rate than the most unstable whistler mode. The qualitative and quantitative trends uncovered in the current study indicate that even though past experiments and modeling efforts have concentrated on whistler modes, there is a compelling case that slow-X modes should also be a key area of focus in the runaway self-mediation through wave instabilities.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Information divergences to parametrize astrophysical uncertainties in dark matter direct detection
Astrophysical uncertainties in dark matter direct detection experiments are typically addressed by parametrizing the velocity distribution in terms of a few uncertain parameters that vary around some central values. Here we propose a method to optimize over all velocity distributions lying within a given distance measure from a central distribution. We discretize the dark matter velocity distribution as a superposition of streams and use a variety of information divergences to parametrize its uncertainties. With this, we bracket the limits on the dark matter–nucleon and dark matter–electron scattering cross sections, when the true dark matter velocity distribution deviates from the commonly assumed Maxwell-Boltzmann form. The methodology pursued is general and could be applied to other physics scenarios where a given physical observable depends on a function that is uncertain.
particle astrophysics↗
N-body realizations of compound galaxies
A prescription for constructing N-body models of galaxies consisting of more than one component is described. Spatial density profiles are realized exactly, but the phase space distribution is approximated using moments of the collisionless Boltzmann equation. While this approach is not fully rigorous, empirical tests suggest that it is adequate for studies of, e.g., interacting galaxies and the forced response of galaxies to imposed perturbations such as bars.
Hernquist, Lars↗
Entropy Generation Across Earth's Bow Shock
Earth's bow shock is a transition layer that causes an irreversible change in the state of plasma that is stationary in time. Theories predict entropy increases across the bow shock but entropy has never been directly measured. Cluster and Double Star plasma experiments measure 3D plasma distributions upstream and downstream of the bow shock that allow calculation of Boltzmann's entropy function H and his famous H-theorem, dH/dt ~ O. We present the first direct measurements of entropy density changes across Earth's bow shock. We will show that this entropy generation may be part of the processes that produce the non-thermal plasma distributions is consistent with a kinetic entropy flux model derived from the collisionless Boltzmann equation, giving strong support that solar wind's total entropy across the bow shock remains unchanged. As far as we know, our results are not explained by any existing shock models and should be of interests to theorists.
Parks, George K.↗
Electron distribution functions in electric field environments
The amount of current carried by an electric discharge in its early stages of growth is strongly dependent on its geometrical shape. Discharges with a large number of branches, each funnelling current to a common stem, tend to carry more current than those with fewer branches. The fractal character of typical discharges was simulated using stochastic models based on solutions of the Laplace equation. Extension of these models requires the use of electron distribution functions to describe the behavior of electrons in the undisturbed medium ahead of the discharge. These electrons, interacting with the electric field, determine the propagation of branches in the discharge and the way in which further branching occurs. The first phase in the extension of the referenced models , the calculation of simple electron distribution functions in an air/electric field medium, is discussed. Two techniques are investigated: (1) the solution of the Boltzmann equation in homogeneous, steady state environments, and (2) the use of Monte Carlo simulations. Distribution functions calculated from both techniques are illustrated. Advantages and disadvantages of each technique are discussed.
Rudolph, Terence H.↗
Numerical investigation and experimental validation of water condensation in the gas diffusion layer with different properties
Liquid water in porous media hinders the transport of reactant to catalyst layer, where the electrochemical reactions occur, which in turn, reduces performance and lifetime of a proton exchange membrane fuel cell (PEMFC). In this study, two types of gas diffusion layer (GDL) materials are used to study the two-phase saturation distribution in a PEMFC. The correlations between the effective and anisotropic transport properties and porosity of the two GDLs are solved by the pore scale model. In addition, the anisotropic liquid water permeability and the relationship between saturation pressure and capillary pressure is determined by using the Lattice-Boltzmann method. Fuel cell performance as well as liquid water distribution in the GDL using neutron radiography are obtained for validation. Under wet conditions, the cell with Freudenberg GDL performs better than that using Toray GDL, especially at high current densities. The results of the two-phase model simulation show that the peak water saturation for Toray GDL occurs in the catalyst layer and can reach 40%, while the peak water saturation for Freudenberg GDL occurs inside the GDL directly under the ribs and only reaches 25% saturation. The combined results provide key insights to enable high power density operation of a PEMFC through GDL material optimization.
42 ENGINEERING↗
Temperature and density dependent pair potential for deuterium under shock
Large-scale classical molecular dynamics (CMD) simulations naturally include the microscopic physics necessary for atomistic modeling of shock release at the ablator-fuel interface in an inertial confinement fusion (ICF) capsule. Here, the multi-megabar shocks utilized in ICF experiments can drive the deuterium fuel from ambient to electron volt temperatures (T) and multi-fold compression. Modeling interatomic interactions over such an extreme range of conditions is challenging for empirical bond order potentials. We generate a pair potential for deuterium with explicit temperature and mass density dependence from ab initio density functional theory molecular dynamics using the iterative Boltzmann inversion method. This potential accurately reproduces the radial distribution functions and pressures from DFT in CMD equilibrium simulations across a wide range of thermodynamic conditions, yet fails to return the expected Hugoniot relations when used in direct CMD shock simulations.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Efficient quantum state tomography with mode-assisted training
Neural networks (NNs) representing quantum states are typically trained using Markov chain Monte Carlo based methods. However, unless specifically designed, such samplers only consist of local moves, making the slow-mixing problem prominent even for extremely simple quantum states. Here, we propose to use mode-assisted training that provides global information via the modes of the NN distribution. Further, applied to quantum state tomography using restricted Boltzmann machines, this method improves the quality of reconstructed quantum states by orders of magnitude. The method is applicable to other types of NNs and may efficiently tackle problems previously unmanageable.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Partially Ionized Plasma Physics and Technological Applications
Partially ionized plasma physics has attracted increased attention recently due to numerous technological applications made possible by the increased sophistication of computer modelling, the depth of the theoretical analysis, and the technological applications to a vast field of manufacturing for computer components. Partially ionized plasma is characterized by a significant presence of neutral particles in contrast to the fully ionized plasma. The theoretical analysis is based upon solutions of the kinetic Boltzmann equation, yielding the non-Maxwellian electron energy distribution function (EEDF), thereby emphasizing the difference with a fully ionized plasma. The impact of the effect on discharges in inert and molecular gases is described in detail, yielding the complex nonlinear phenomena resulting in plasma selforganization. A few examples of such phenomena are given, including the non-monotonic EEDFs in the discharge afterglow in a mixture of argon with the molecular gas NF3; the explosive generation of cold electron populations in capacitive discharges, hysteresis of EEDF in inductively coupled plasmas. Recently, highly advanced computer codes were developed in order to address the outstanding challenges in plasma technology. These developments are briefly described in general terms.
non-Maxwellian electron energy distribution functi↗