Evolution Of Distribution Function Research Articles

Explaining the Evolution of Ion Velocity Distributions at a Low Activity Comet

AbstractAt a low activity comet the plasma is distributed in an asymmetric way. The hybrid simulation code Amitis is used to look at the spatial evolution of ion velocity distribution functions (VDFs), from the upstream solar wind (SW) to within the comet magnetosphere where the SW is heavily mass‐loaded by the cometary plasma. We find that the spatial structures of the ions and fields form a highly asymmetric induced magnetosphere. The VDFs of SW and cometary ions vary drastically for different locations in the comet magnetosphere. The shape of the VDFs differ for different species. The SW protons show high anisotropies that occasionally resemble partial rings, in particular at small cometocentric distances. A second, decoupled, proton population is also found. Solar wind alpha particles show similar anisotropies, although less pronounced and at different spatial scales. The VDFs of cometary ions are mostly determined by the structure of the electric field. We perform supplementary dynamic particle backtracing to understand the flow patterns of SW ions that lead to these anisotropic distributions. This tracing is needed to understand the origin of cometary ions in a given part of the comet magnetosphere. The particle tracing also aids in interpreting observed VDFs and relating them to spatial features in the electric and magnetic fields of the comet environment.

Complete quasilinear model for the acceleration-driven lower hybrid drift instability and a computational assessment of its validity.

A complete quasilinear model is derived for the electrostatic acceleration-driven lower hybrid drift instability in a uniform two-species low-beta plasma in which current is perpendicular to the background magnetic field. The model consists of coupled nonlinear velocity space diffusion equationsfor the volume-averaged ion and electron distribution functions. Each species' diffusion coefficient depends on a time-evolving spectral density of the electric-field energy per unit volume and a time-evolving dispersion relation. The dispersion relation is expressed analytically in integral form without the use of asymptotic limits and applies to arbitrary distribution functions, so long as they can be expressed as a function of one velocity coordinate, e.g., f(v_{y}) or f(v_{⊥}). The quasilinear model conserves energy and is complete in that it fully describes the evolution of the distribution functions, including resonant and nonresonant particle-wave interactions, while accounting for distribution-function-dependent mixed-complex frequencies. The quasilinear diffusion model is solved numerically and self-consistently using a Crank-Nicolson temporal discretization and a second-order finite-volume velocity-space discretization. Numerical solutions are compared to nonlinear fourth-order accurate continuum kinetic Vlasov-Poisson simulations. Evolution of electric-field energy, growth rates, distribution functions, and diffusion coefficients are shown to be in agreement with Vlasov simulations. The quasilinear model is shown to predict anomalous transport terms, like resistivity and heating, to within a factor of order unity. Discrepancies between the quasilinear model and Vlasov simulations are assessed and attributed primarily to lack of damping in the quasilinear description and to the use of unperturbed-orbit susceptibilities in the linear theory dispersion relation. The results illuminate the predictive accuracy of the quasilinear model, place approximate bounds on its validity, and provide much needed vetting of quasilinear theory's ability to predict the nonlinear state of a microturbulent plasma.

Linear operator theory of phase mixing

ABSTRACT We study solutions of the collisionless Boltzmann equation (CBE) in a functional Koopman representation. This facilitates the use of linear spectral techniques characteristic of the analysis of Schrödinger-type equations. For illustrative purposes, we consider the classical phase mixing of a non-interacting distribution function in a quartic potential. Solutions are determined perturbatively relative to a harmonic oscillator. We impose a form of coarse-graining by choosing a finite-dimensional basis to represent the distribution function and time evolution operators, which sets a minimum length-scale on phase space structure. We observe a relationship between the dimension of the representation and the multiplicity of the harmonic oscillator eigenvalues. System dynamics are understood in terms of degenerate subspaces of the linear operator spectra. Each subspace is associated with a mode of the harmonic oscillator, the first two being bending and breathing structures. The quartic potential splits the degenerate eigenvalues within each subspace. This facilitates the formation of spiral structure as deformations from the harmonic oscillator modes. We ultimately argue that this construction provides a promising avenue for study of self-interacting systems experiencing phase mixing, which is an outstanding problem in the context of the Gaia DR2 vertical phase space spirals.

Evolution of parton distribution functions in the short-distance factorization scheme

Lattice QCD offers the possibility of computing parton distributions from first principles, although not in the usual MS¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{MS} $$\\end{document} factorization scheme. Calculations are therefore matched to MS¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{MS} $$\\end{document} using a perturbative procedure which is the source of significant uncertainty within the currently accessible kinematics. We present the possibility of computing the z2 evolution of non-singlet pseudo-parton distribution functions within the short factorization scheme in a numerically improvable way. The goal is to have tools to evolve a calculation to a scale where perturbative uncertainties are less pronounced. We compare a numerical extraction of the evolution operator from lattice data to the computation of z2 dependence in perturbation theory. Finally, we discuss how this numerical work may be extended to address the two-scale problem that arises when the Ioffe time range must be made large to extend the reach of the calculation of the pseudo-PDF to smaller values of the momentum fraction.

Probabilistic Forecast of Multiphase Transport Under Viscous and Buoyancy Forces in Heterogeneous Porous Media

AbstractWe develop a probabilistic approach to map parametric uncertainty to output state uncertainty in first‐order hyperbolic conservation laws. We analyze this problem for nonlinear immiscible two‐phase transport in heterogeneous porous media in the presence of a stochastic velocity field. The uncertainty in the velocity field can arise from incomplete descriptions of either porosity field, injection flux, or both. This uncertainty leads to spatiotemporal uncertainty in the saturation field. Given information about spatial/temporal statistics of spatially correlated heterogeneity, we leverage the method of distributions to derive deterministic equations that govern the evolution of pointwise cumulative distribution functions (CDFs) of saturation for a vertical reservoir, while handling the manipulation of multiple shocks arising due to buoyancy forces. Unlike the Buckley‐Leverett equation, the equation describing the fine‐grained CDF is linear in space and time. Ensemble averaging of the fine‐grained CDF results in the CDF of saturation. Thus, we give routes to circumventing the computational cost of Monte Carlo simulations (MCS), while obtaining a pointwise description of the saturation field. We conduct a set of numerical experiments for one‐dimensional transport, and compare the obtained saturation CDFs, against those obtained using MCS as our reference solution, and the statistical moment equation method. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties, that is, standard deviation and correlation length of the underlying random fields, whereas the corresponding low‐order statistical moment equations significantly deviate from the MCS results, except for very small values of standard deviation and correlation length.

The Sivers asymmetry in charged Kaon and 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} hyperon produced SIDIS process at electron ion colliders

We investigate the single transverse-spin asymmetry with a sin(ϕh-ϕS)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sin (\\phi _h-\\phi _S)$$\\end{document} modulation in the charged Kaon produced and in 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} hyperon produced SIDIS process within the theoretical framework of transverse momentum dependent (TMD) factorization at the next-to-leading-logarithmic order. The asymmetry is contributed by the convolution of Sivers function and the unpolarized fragmentation function D1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_1$$\\end{document} for the produced hadron. The parametrization for the proton Qiu–Sterman function, which is closely related to the Sivers function, is adopted to numerically estimate the Sivers asymmetry at the kinematical region of Electron Ion Collider (EIC) and Electron Ion Collider in China (EicC). The TMD evolution of the TMD parton distribution functions are considered by employing the non-perturbative Sudakov form factor. It is found that the predicted Sivers asymmetries AUTsin(ϕh-ϕS)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{UT}^{\\sin (\\phi _h-\\phi _S)}$$\\end{document} as functions of x, z and PhT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{hT}$$\\end{document} are sizable at the kinematical configurations of both EIC and EicC. The strange constituent of the produced charged Kaon and 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} hyperon in the final state can be a promising probe of the sea quark Sivers function as well as the flavor dependence in the proton target. Therefore, it is important to utilize the future EIC facilities to constrain the sea quark distribution functions as well as the validity of the generalized universality of the Sivers function.

Nonlinear equilibria and transport processes in burning plasmas

In this work, we put forward a general phase space transport theory in axisymmetric tokamak plasmas based upon the concept of zonal state (ZS). Within this theoretical framework, the ZS corresponds to a renormalized plasma nonlinear equilibrium consisting of phase space zonal structures (PSZS) and zonal electromagnetic fields (ZFs) which evolve self-consistently with symmetry breaking fluctuations and sources/collisions. More specifically, our approach involves deriving governing equations for the evolution of particle distribution functions (i.e, PSZS), which can be used to compute the corresponding macro-/meso-scale evolving magnetized plasma equilibrium adopting the Chew Goldberger Low description, separating the spatiotemporal microscale structures. The nonlinear physics of ZFs and of geodesic acoustic modes (GAMs)/energetic particle driven GAMs is then analyzed to illustrate the applications of our theory.

Kinetic models for systems of interacting agents with multiple microscopic states

We propose and investigate general kinetic models with transition probabilities that can describe the simultaneous change of multiple microscopic states of the interacting agents. These models can be applied to many problems in socio-economic sciences, where individuals may change both their compartment and their characteristic microscopic variable, as for instance kinetic models for epidemic diffusion or for international trade with possible transfers of agents. Mathematical properties of the kinetic model are proved, as existence and uniqueness of a solution for the Cauchy problem in suitable Wasserstein spaces. The quasi-invariant asymptotic regime, leading to simpler kinetic Fokker–Planck-type equations, is investigated and commented on in comparison with other existing models. Some numerical tests are performed in order to show the time evolution of distribution functions and of meaningful macroscopic fields, even in case of non-constant interaction rates and transfer probabilities.

Kinetic Model of Vacuum Plasma Expansion in a Cylindrical Gap

Results of a theoretical description of collisionless kinetics of radial expansion of two-component (electron–ion) plasma in the one-dimensional cylindrical formulation of the problem are presented. The electric-field mechanism of supersonic expansion of the plasma flame due to the motion of the electron–ion ensemble and self-consistent electric field in the diode with the potential difference applied to it is demonstrated. The spatiotemporal evolution of the ion energy distribution function, electric potential, and rate of expansion of the emission boundary of the plasma flame is shown. The calculated rates of flame expansion at the copper cathode (~1.5 × 106 cm/s) well agree with the experimental data.

Kinetic Model of Vacuum Plasma Expansion in a Cylindrical Gap

Results of a theoretical description of collisionless kinetics of radial expansion of two-component (electron–ion) plasma in the one-dimensional cylindrical formulation of the problem are presented. The electric-field mechanism of supersonic expansion of the plasma flame due to the motion of the electron–ion ensemble and self-consistent electric field in the diode with the potential difference applied to it is demonstrated. The spatiotemporal evolution of the ion energy distribution function, electric potential, and rate of expansion of the emission boundary of the plasma flame is shown. The calculated rates of flame expansion at the copper cathode (~1.5 × 106 cm/s) well agree with the experimental data.

An improved calculation scheme of electron flow in a propagator method for solving the Boltzmann equation

In order to accurately evaluate the electron acceleration process in the calculation of the time evolution of the electron velocity distribution function (EVDF) based on the Boltzmann equation, an improved scheme blending upwind and central differences is introduced into the propagator method (PM). While the previous PM based on the upwind scheme needs fine cells to obtain an accurate EVDF at low electric fields, the improved PM is robust against coarse cells, which allows the reduction of cell resolution. Calculations of the EVDF in Ar under RF electric fields demonstrated that the blending scheme can provide satisfactorily accurate results even with cells about tenfold larger than the upwind case at low reduced electric fields below 1 Td, which leads to much shorter computational time because the reduction in the number of cells satisfactorily compensates for the complexity of the blending scheme. This technique has been built into a new user-friendly PM software named BOSPROM.

Renormalization of twist-two operators in covariant gauge to three loops in QCD

The leading short-distance contributions to hadronic hard-scattering cross sections in the operator product expansion are described by twist-two quark and gluon operators. The anomalous dimensions of these operators determine the splitting functions that govern the scale evolution of parton distribution functions. In massless QCD, these anomalous dimensions can be determined through the calculation of off-shell operator matrix elements, typically performed in a covariant gauge, where the physical operators mix with gauge-variant operators of the same quantum numbers. We derive a new method to systematically extract the counterterm Feynman rules resulting from these gauge-variant operators. As a first application of the new method, we rederive the unpolarized three-loop singlet anomalous dimensions, independently confirming previous results obtained with other methods. Employing a general covariant gauge, we observe the explicit cancellation of the gauge parameter dependence in these results.

Simulation of the spatio-temporal evolution of the electron energy distribution function in a pulsed hollow-cathode discharge

This article presents the 2D simulation results of a nanosecond pulsed hollow cathode discharge obtained through a combination of fluid and kinetic models. The spatio-temporal evolution of the electron energy distribution function (EEDF) of the plasma column and electrical characteristics of the nanosecond pulsed hollow cathode discharge at a gas pressure of 5 Torr are studied. The results show that the discharge development starts with the formation of an ionization front at the anode surface. The ionization front splits into two parts in the cathode cavity while propagating along its lateral surfaces. The ionization front formation leads to an increase in the fast isotropic EEDF component at its front, as well as in the anisotropic EEDF component. The accelerated electrons enter the cathode cavity, which significantly contributes to the formation of the high-energy EEDF component and EEDF anisotropy.

Evolution of the Soot-Particle Size Distribution Function in the Cylinder and Exhaust System of Piston Engines: Simulation

A computational tool for simulating the temporal evolution of the soot-particle size distribution function (SDF) in the internal combustion engine (ICE) and in the attached exhaust pipe is developed and tested against available experimental data on the soot-particle SDF at the outlet of the exhaust system. Firstly, a database of soot particle properties (particle mean diameter, dispersion, total particle number density vs. time for different fuels, fuel-to-air equivalence ratios, temperatures, pressures, and exhaust gas recirculation) is developed based on the thoroughly validated detailed model of soot formation under ICE conditions. The database is organized in the form of look-up tables. Secondly, the soot-particle SDF in the database is approximated using the log-normal SDF, which is directly used in the multidimensional calculations of the ICE operation process. Thirdly, the coagulation model of soot particles is developed, which includes three coagulation mechanisms: Brownian, turbulent–kinetic, and turbulent–diffusion. This model is applied for simulating the evolution of the soot-particle SDF in the exhaust pipe after opening the exhaust valve. Calculations show that the coagulation process of soot particles in the exhaust pipe has a significant effect on the mean size of particles at the outlet of the exhaust system (the mean particle diameter can increase by almost an order of magnitude), and the dominant mechanism of particle coagulation in the exhaust system of a diesel engine is the Brownian mechanism. The objective, approach, and obtained results are the novel features of the study.

A magnetic field coupling fractional step lattice Boltzmann model for the complex interfacial behavior in magnetic multiphase flows

In the present study, a magnetic field coupling fractional step lattice Boltzmann (FSLB) model is proposed to simulate the complex interfacial behaviors in magnetic multiphase flows with severe interface deformation and large density ratio, which can be recovered to the corresponding macroscopic governing equations with fully second-order accuracy. Since the equilibrium and the non-equilibrium terms are obtained by the Chapman-Enskog (C-E) expansion analysis, the basic idea of the present method is to predict the intermediate variables by updating the various moments of equilibrium distribution functions first, and then correct the physical variables by updating the various moments of non-equilibrium distribution functions. Besides, the present magnetic field coupling FSLB method does not need the evolution of distribution functions compared to the existing LB methods. The present method inherits the excellent performance from the conventional LB method and the relatively better numerical stability from the matured fractional-step method, which is more stable than the original LB method and more efficiency and easy-to-implement than the LB flux solver. Owing to those good properties, we successfully perform the first rigorous numerical simulation on the Rosensweig instability, which accurately captures the spikes forming at the surface of a ferrofluid under an applied magnetic field. Moreover, for the first time we numerically reproduce the falling ferrofluid droplet impacted on the solid surface. Intriguingly, our simulation results reveal that the falling velocity of the ferrofluid droplet is slightly accelerated by the elongation, although the direction of the magnetic field is opposite to that of the gravity. All the results for the interfacial behavior and the magnetic interaction highlights the comparative capability, stability, and accuracy of the present magnetic field coupling FSLB method for simulating the problems of complex fluid dynamics associated with a severe interface deformation and large density ratio in magnetic multiphase flows.

A Simplified Linearized Lattice Boltzmann Method for Acoustic Propagation Simulation.

A simplified linearized lattice Boltzmann method (SLLBM) suitable for the simulation of acoustic waves propagation in fluids was proposed herein. Through Chapman-Enskog expansion analysis, the linearized lattice Boltzmann equation (LLBE) was first recovered to linearized macroscopic equations. Then, using the fractional-step calculation technique, the solution of these linearized equations was divided into two steps: a predictor step and corrector step. Next, the evolution of the perturbation distribution function was transformed into the evolution of the perturbation equilibrium distribution function using second-order interpolation approximation of the latter at other positions and times to represent the nonequilibrium part of the former; additionally, the calculation formulas of SLLBM were deduced. SLLBM inherits the advantages of the linearized lattice Boltzmann method (LLBM), calculating acoustic disturbance and the mean flow separately so that macroscopic variables of the mean flow do not affect the calculation of acoustic disturbance. At the same time, it has other advantages: the calculation process is simpler, and the cost of computing memory is reduced. In addition, to simulate the acoustic scattering problem caused by the acoustic waves encountering objects, the immersed boundary method (IBM) and SLLBM were further combined so that the method can simulate the influence of complex geometries. Several cases were used to validate the feasibility of SLLBM for simulation of acoustic wave propagation under the mean flow.

Experimental investigation on the hysteresis in low-pressure inductively coupled neon discharge

A hysteresis phenomenon observed in neon inductive discharge at low gas pressure is investigated in terms of the evolution of the electron energy distribution function (EEDF). Generally, the hysteresis phenomenon has been reported at high-pressure Ramsauer gas discharges. However, in neon plasma, we found that the hysteresis phenomenon occurs even at low gas pressure (5 mTorr). Furthermore, the hysteresis vanishes with an increase in the gas pressure (10 and 25 mTorr). To analyze this hysteresis, the EEDF is measured depending on the radio frequency power. The EEDF at 10 mTorr sustains the bi-Maxwellian distribution during an E–H transition. On the other hand, the EEDF at 5 mTorr changes dramatically between discharge modes. At 5 mTorr, the measured EEDF for the E mode has the Maxwellian distribution due to high collisional heating in the bulk plasma. The EEDF for the H mode has the bi-Maxwellian distribution because collisionless heating in the skin depth is dominant. This apparent evolution of the EEDF causes a nonlinear energy loss due to collisions during the discharge mode transition. Therefore, the plasma can maintain the H mode discharge with high ionization efficiency, even at a lower applied power, which results in the hysteresis.

Quasilinear theory of general electromagnetic fluctuations including discrete particle effects for magnetized plasmas: General analysis

The general quasilinear Fokker–Planck kinetic equation for the gyrophase-averaged plasma particle distribution functions in magnetized plasmas is derived, making no restrictions on the energy of the particles and on the frequency of the electromagnetic fluctuations and avoiding the often made Coulomb approximation of the electromagnetic interactions. The inclusion of discrete particle effects breaks the dichotomy of nonlinear kinetic plasma theory divided into the test particle and the test fluctuation approximation because it provides expression of both the non-collective and collective electromagnetic fluctuation spectra in terms of the plasma particle distribution functions. Within the validity of the quasilinear approach, the resulting full quasilinear transport equation can be regarded as a determining nonlinear equation for the time evolution of the plasma particle distribution functions.

On the force scheme influence on pseudopotential method coexistence curve

Pseudopotential method has been drawing attention in simulating multiphase flows by adding an interparticle force to lattice Boltzmann method (LBM). However, implementing a known force field effect into evolution of particle distribution function is still an active field of interest, specially when higher order terms play significant role in resulting macroscopic equations. A considerable number of forcing schemes has already been proposed, and although many have shown their differences up to second order analysis, there is still some debate about high order terms. In this work, a study taking into account third order forcing terms is carried out, in order to better identify the reasons for existing distinct coexistence curves when the same interparticle force is employed through distinct forcing schemes. Third order terms related to spatial interparticle force gradients are taken into account, and a theoretical analysis is carried out in order to study gas–liquid coexistence curves. It is shown that this curve can be characterized by a single parameter, Γ, dependent on employed relaxation time and forcing scheme, which allows the current analysis to be used for other schemes as well. Flat interface simulations are carried out in order to study the behavior of gas–liquid coexistence curve. A good agreement between the numerical and theoretical results validates the presented approach. Static droplet simulations were carried out, where surface tension behavior also respect the expected trend of higher values for lower Γ. Maximum spurious velocities of renowned forcing schemes from literature were also studied, without a clear correlation with Γ parameter.

A one‐step simplified lattice Boltzmann method without evolution of distribution functions

AbstractRecently, a simplified lattice Boltzmann method without evolution of distribution functions was developed. This method adopts the predictor‐corrector scheme to evolve the macroscopic variables. Unlike the original simplified method, we develop a one‐step scheme based on the resultant governing equations given by the Chapman–Enskog expansion analysis of the lattice Boltzmann equation. The proposed method abandons the predictor‐corrector scheme and only needs one step to solve the governing equations without additional intermediate implementation. As a result, the computational efficiency can be improved. In addition, the theoretical analysis demonstrates that the present method has second‐order of accuracy. To treat the boundary, the second‐order approximations for equilibrium distribution functions are employed. To validate the proposed one‐step method, several cases including the Poiseuille flow, the Womersley flow, the Taylor‐Green vortex flow, and the lid‐driven cavity flow are simulated. The numerical results agree with the analytical solutions or the reference data. And the second‐order of accuracy is ensured. Moreover, simulations also suggest that our method has good performances on the numerical stability and the computational efficiency. And the improved efficiency is no less than in our tests.