Vlasov Model Research Articles

Twenty-six moment equations for the Enskog–Vlasov equation

The Enskog–Vlasov equation is a phenomenological kinetic equation that extends the Enskog equation for the dense (non-ideal) hard-sphere fluid by adding an attractive soft potential tail to the purely repulsive hard-sphere contribution. Simplifying assumptions about pair correlations lead to a Vlasov-like self-consistent force field that adds to the Enskog non-local hard-sphere collision integral. Within the limitations imposed by the underlying assumptions, the extension gives the Enskog–Vlasov equation the ability to give a unified description of ideal and non-ideal fluid flows as well as of those fluid states in which liquid and vapour regions coexist, being separated by a resolved interface. Furthermore, the Enskog–Vlasov fluid can be arbitrarily far from equilibrium. Thus the Enskog–Vlasov model equation provides an excellent, although approximate, tool for modelling processes with liquid–vapour interfaces and adjacent Knudsen layers, and allows us to look at slip, jump and evaporation coefficients from a different perspective. Here, a set of 26 moment equations is derived from the Enskog–Vlasov equation by means of the Grad moment method. The equations provide a meaningful approximation to the underlying kinetic equation, and include the description of Knudsen layers. This work focuses on the – rather involved – derivation of the moment equations, with only a few applications shown.

Dynamic response analysis of Euler–Bernoulli beam on spatially random transversely isotropic viscoelastic soil

A novel dynamic soil-structure interaction model is developed for analysis for Euler–Bernoulli beam rests on a spatially random transversely isotropic viscoelastic foundation subjected to moving and oscillating loads. The dynamic equilibrium equation of beam-soil system is established using the extended Hamilton's principle, and the corresponding partial differential equations describing the displacement of beam and soil and boundary conditions are further obtained by the variational principles. These partial differential equations are discretized in spatial and time domains and solved by the finite difference (FD) method. After the differential equations of beam and soil are discretized in the spatial domain, the implicit iterative scheme is used to solve the equations in the time domain. The solving result shows the FD method is effective and convenient for solving the differential equations of beam-soil system. The spring foundation model adopted the modified Vlasov model, which is a two-parameter model considering the compression and shear of soil. The advantage of the present foundation model is avoided estimating input parameters of the modified Vlasov model using prior knowledge. The present solution is verified by publishing solution and equivalent three-dimensional FE analysis. The present model produced an accurate, faster, and effective displacement response. A few examples are carried out to analyze the parameter variation influence for beam on spatially random transversely isotropic viscoelastic soil under moving loads.

A fast convergence fourth-order Vlasov model for Hall thruster ionization oscillation analyses

A 1D1V hybrid Vlasov-fluid model was developed for this study to elucidate discharge current oscillations of Hall thrusters (HTs). The Vlasov equation for ions velocity distribution function with ionization source term is solved using a constrained interpolation profile conservative semi-Lagrangian method. The fourth-order weighted essentially non-oscillatory (4th WENO) limiter is applied to the first derivative value to minimize numerical oscillation in the discharge oscillation analyses. The fourth-order accuracy is verified through a 1D scalar test case. Nonoscillatory and high-resolution features of the Vlasov model are confirmed by simulating the test cases of the Vlasov–Poisson system and by comparing the results with a particle-in-cell (PIC) method. A 1D1V HT simulation is performed through the hybrid Vlasov model. The ionization oscillation is analyzed. The oscillation amplitude and plasma density are compared with those obtained from a hybrid PIC method. The comparison indicates that the hybrid Vlasov-fluid model yields noiseless results and that the steady-state waveform is calculable in a short time period.

Iterative technique for a beam lying on a transversely isotropic foundation

The effect of the foundation heterogeneity on the mechanical behaviour of a beam on Vlasov soils is discussed. According to a refined Vlasov soil model, the static problem of beams lying on transversely isotropic soils can be solved by an iterative method. In this paper, based on the energy variational principle, the differential equations for beams under an axial force on refined Vlasov foundations are derived. The methods for solving the internal forces and deformations of beams lying on refined elastic foundations are given. Additionally, an equation for the attenuation parameters is also established, and the characteristic parameters of the refined model are solved by iterative technique. Numerical results show that the foundation heterogeneity have a influence on the deformations and internal forces of the beam-soil system. Moreover, relatively accurate characteristic parameters can be obtained through the iterative process. The refined Vlasov model has broad application prospects.

Response Analysis of Second-Order Continuity Rectangular Plate Bending Element Resting on Elastic Foundation Using Modified Vlasov Model

Response Analysis of Second-Order Continuity Rectangular Plate Bending Element Resting on Elastic Foundation Using Modified Vlasov Model

Nonparametric estimation for interacting particle systems: McKean–Vlasov models

We consider a system of N interacting particles, governed by transport and diffusion, that converges in a mean-field limit to the solution of a McKean–Vlasov equation. From the observation of a trajectory of the system over a fixed time horizon, we investigate nonparametric estimation of the solution of the associated nonlinear Fokker–Planck equation, together with the drift term that controls the interactions, in a large population limit \(N \rightarrow \infty \). We build data-driven kernel estimators and establish oracle inequalities, following Lepski’s principle. Our results are based on a new Bernstein concentration inequality in McKean–Vlasov models for the empirical measure around its mean, possibly of independent interest. We obtain adaptive estimators over anisotropic Hölder smoothness classes built upon the solution map of the Fokker–Planck equation, and prove their optimality in a minimax sense. In the specific case of the Vlasov model, we derive an estimator of the interaction potential and establish its consistency.

Dynamic mechanical analysis with torsional rectangular geometry: A critical assessment of constrained warping models

Dynamic mechanical oscillatory shear measurements with torsional rectangular geometry are widely carried out in order to determine the mechanical properties of soft solid materials in a quick and practical way. This technique has the advantage of avoiding slip effects, thus being particularly attractive for testing stiff elastomers such as vulcanized rubber compounds. However, one of its drawbacks is the clamping system required to keep the specimen edges in place. Since it imposes a constraint to warping deformations (i.e., out-of-plane cross-section distortions about the torsional axis), a certain increase of dynamic moduli with respect to their values in simple shear is observed and considered as an experimental artifact (i.e., de Saint-Venant's assumption of primary torsion of the specimen is no longer valid). The increase of dynamic moduli in torsion depends on the specimen's cross-section geometry and relative dimensions. We test here the capability of different torsion models to describe the constrained warping effect for an industrial rubber with rectangular specimen cross section using a wide range of different geometric parameters (length-to-width p ratio and width-to-thickness u ratio). We compare two theoretical models (Vlasov's model and Timoshenko's approach) and a phenomenological model (based on finite element simulations by Diani and Gilormini) with our experimental data set. We propose a slight modification of Vlasov's model to obtain realistic predictions of torsion over a wider parameter range. It is shown to be particularly useful in soft rubber testing, and the limitations in the choice of specimen geometry to obtain sufficient torque signal are overcome.

Generalized non-ideal treatment and growth rates analysis of drift waves instabilities in a collisions-free magnetized dusty plasma

Dust acoustic wave propagation in weak magnetized homogeneous dusty plasmas has been extensively and successfully studied by different authors. Here, it is analyzed the behavior of electrostatic drift waves of very low frequency, appearing as due to the motion of plasma particles across the static magnetic field and the inhomogeneity introduced by the distribution of the particles in the dusty plasma generating the well-known diamagnetic effect. To study this situation, the kinetic Vlasov model in plasmas will be used. However, these models tend to behave unstably driven by the quasi-equilibrium of the diamagnetic current arising by the heavier species of dusty plasma particles. While other authors have used a standard state equation, such as the Van der Waals one, here, a more accurate model has been chosen as the equation of Ree and Hoover, a state equation that can be considered as a new form of the state equation. Together with the multipole approximation of the dispersion function, Z53 [P. Martin et al., J. Math. Phys. 21, 280–283 (1980)] allows a new derivation to obtain the simplest form of the dispersion relation in the form of a polynomial with complex coefficients. This mathematical approach was realized successfully and allowed straightforward calculus of the growth rate due to the drift unstable mode. This problem has been extensively analyzed in this work, including the non-ideal effect of dusty plasmas. These unstable modes are well discriminated and plotted as a function of the density of the grains in dusty plasma and their radius. Temperature gradients and charge fluctuation have been ignored.

High order semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators for nonlinear Vlasov dynamics

In this paper, we propose a semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinear Vlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators (EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlinear transport setting, the RKEI can be used to decompose the evolution of the nonlinear transport into a composition of a sequence of linearized dynamics. The resulting linearized transport equations can be solved by the semi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al. (JSC, 2017). The proposed method can achieve high order spatial accuracy via the SLDG framework, and high order temporal accuracy via the RK EI. Due to the SL nature, the proposed SLDG-RKEI method is not subject to the CFL condition, thus they have the potential in using larger time-stepping sizes than those in the Eulerian approach. Inheriting advantages from the SLDG method, the proposed SLDG-RKEI schemes are mass conservative, positivity-preserving, have no dimensional splitting error, perform well in resolving complex solution structures, and can be evolved with adaptive time stepping sizes. We show the performance of the SLDG-RKEI algorithm by classical test problems for the nonlinear Vlasov-Poisson system, as well as the Guiding center Vlasov model. Though that it is not our focus of this paper to explore the SLDG-RKEI scheme for nonlinear hyperbolic conservation laws that develop shocks, we show some preliminary results on schemes' performance on the Burgers' equation.

Complexation in germanium in accordance with Vlasov's model for solids

Using Vlasov's model for solids, the diffusion model of defect formation in germanium was verified. The possibility of applying Vlasov's model for solids to describe complexation in dislocation-free germanium single crystals is shown. Vlasov's model for solids allows one to interpret the processes of defect formation in a crystal from a single viewpoint both at the stage of its growth and at the stage of heat treatment.

Mean-field kinetic theory approach to Langmuir evaporation of polyatomic liquids

The evaporation of polyatomic liquids into near-vacuum conditions is investigated by using the Enskog–Vlasov model. Molecules are approximated as classical rigid rotators, and the collisional energy exchanges between the translational and rotational degrees of freedom are dealt with by the Borgnakke–Larsen method. The distribution function of evaporated molecules and the evaporation coefficient are evaluated in a wide range of liquid bulk temperatures and inelastic collision fractions. It is found that the translational velocity distribution function is well approximated by a drifted bi-Maxwellian, while the rotational energy follows the Boltzmann distribution at a temperature that varies between the separation and the parallel temperatures as the inelastic collision fraction increases. The evaporation coefficient based on the separation temperature turns out to be independent of the inelastic collision fraction and only mildly dependent on the liquid bulk temperature.

Bending Analysis of Circular Thin Plates Resting on Elastic Foundations Using Two Modified Vlasov Models

The influence of soil heterogeneity is studied on the bending of circular thin plates using two modified Vlasov foundation models. The model parameters are determined reasonably using an iterative technique. According to the principle of minimum potential energy and considering transversely isotropic soils and Gibson soils, the governing differential equations and boundary conditions for circular thin plates on two modified Vlasov foundations are derived using a variational approach, respectively. The determination of attenuation parameters is a difficult problem, which has hindered the further application of the Vlasov foundation model. The equation that must be satisfied by the attenuation parameter is determined, and an iterative method is used to solve the problem. A comparative analysis is conducted between two modified Vlasov models and the traditional Vlasov model. The results show that the governing equations and boundary conditions for circular thin plates resting on two modified foundations are consistent with those for a circular thin plate on traditional two-parameter foundation after degradation. The accuracy and reliability of the proposed solutions are demonstrated by comparing the obtained results with those reported in the literature. The heterogeneity of soils, including the transversely isotropic soils and Gibson soils, has a certain effect on characteristic parameters of the foundation models as well as the deformations and internal forces of circular thin plates. The present study could be employed as a reference for future engineering designs.

Solar coronal heating by Alfvén waves in bi-kappa distributed plasma

ABSTRACT In solar physics, there is a decades-old conundrum that is still unsolved. Why is the temperature of the corona so much larger than that of the surface of the Sun? To solve this, various approaches have been adopted so far, but they have certain limitations. In the present analysis, we invoke the standard Vlasov model and the steady-state Poynting theorem to unlock the mysterious coronal heating mechanism in terms of inertial and kinetic Alfvén waves whose electromagnetic energies turn into heat during wave–particle interaction. The coronal plasmas that support these waves are modelled by a non-thermal bi-kappa velocity distribution function. The non-thermal distribution function, which is assumed to pre-exist in the system, strongly influences the wave-heating process. Particularly, during heating by the waves in the inertial limit, the non-thermal features of the distribution function give rise to a unique competition (which is entirely absent in the usual Maxwellian plasmas) between waves of different perpendicular wavenumbers (kx). For small kx, when either the non-thermal parameter κ or the electron parallel temperature T||e increases, the inertial Alfvén waves can efficiently heat the plasma in their immediate vicinity. However, for relatively large kx, an increase in either κ or T||e enables the inertial Alfvén waves to effectively heat the plasma in remote regions in the corona. Although such competition is not seen in the kinetic limit, the non-thermal features still seem to control the heating process. The possible explanations behind the above-mentioned cases are provided by the bi-kappa velocity distribution function, which holds vital clues as to how the non-thermal features, together with kx, dictate the resonance conditions that play a crucial role in the heating process.

Iterative technique for circular thin plates on Gibson elastic foundation using modified Vlasov model

In this paper, to investigate the influence of soil inhomogeneity on the bending of circular thin plates on elastic foundations, the static problem of circular thin plates on Gibson elastic foundation is solved using an iterative method based on the modified Vlasov model. On the basis of the principle of minimum potential energy, the governing differential equations and boundary conditions for circular thin plates on modified Vlasov foundation considering the characteristics of Gibson soil are derived. The equations for the attenuation parameter in bending problem are also obtained, and the issue of unknown parameters being difficult to determine is solved using the iterative method. Numerical examples are analyzed and the results are in good agreement with those form other literatures. It proves that the method is practical and accurate. The inhomogeneity of modified Vlasov foundations has some influence on the deformation and internal force behavior of circular thin plates. The effects of various parameters on the bending of circular plates and characteristic parameters of the foundation are discussed. The modified model further enriches and develops the elastic foundations. Relevant conclusions that are meaningful to engineering practice are drawn.

High-Temperature Precipitation of Impurities within the Vlasov Model for Solids

It is shown that the Vlasov model for a solid describes the complexing processes when growing real crystals with allowance for the thermal growth conditions. It makes it possible (along with the classical theory of nucleation and growth of second-phase particles in solids) to calculate the defect crystal structure that was formed during the growth. It is established that the high-temperature impurity precipitation is directly related to the subsequent transformation of the defect structure when manufacturing of silicon devices. A qualitative model of the formation of electric centers is proposed, which directly relates their origin to the initial defect structure of silicon. It is shown that the concepts and principles of the Vlasov physics are absolutely applicable in solid-state physics.

Conservative Multi-dimensional Semi-Lagrangian Finite Difference Scheme: Stability and Applications to the Kinetic and Fluid Simulations

In this paper, we propose a mass conservative semi-Lagrangian finite difference scheme for multi-dimensional problems without dimensional splitting. The semi-Lagrangian scheme, based on tracing characteristics backward in time from grid points, does not necessarily conserve the total mass. To ensure mass conservation, we propose a conservative correction procedure based on a flux difference form. Such procedure guarantees local mass conservation, while introducing time step constraints for stability. We theoretically investigate such stability constraints from an ODE point of view by assuming exact evaluation of spatial differential operators and from the Fourier analysis for linear PDEs. The scheme is tested by classical two dimensional linear passive-transport problems, such as linear advection, rotation and swirling deformation. The scheme is applied to solve the nonlinear Vlasov-Poisson system using a a high order tracing mechanism proposed in [Qiu and Russo, 2016]. Such high order characteristics tracing scheme is generalized to the nonlinear guiding center Vlasov model and incompressible Euler system. The effectiveness of the proposed conservative semi-Lagrangian scheme is demonstrated numerically by our extensive numerical tests.

A High Order Semi-Lagrangian Discontinuous Galerkin Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model Without Operator Splitting

In this paper, we generalize a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for multi-dimensional linear transport equations without operator splitting developed in Cai et al. (J. Sci. Comput. 73: 514-542, 2017) to the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation and the guiding center Vlasov model. We adopt a local DG method for Poisson's equation of these models. For tracing the characteristics, we adopt a high order characteristics tracing mechanism based on a prediction-correction technique. The SLDG with large time-stepping size might be subject to extreme distortion of upstream cells. To avoid this problem, we propose a novel adaptive time-stepping strategy by controlling the relative deviation of areas of upstream cells.

Physics-Based-Adaptive Plasma Model for High-Fidelity Numerical Simulations

A physics-based-adaptive plasma model and an appropriate computational algorithm are developed to numerically simulate plasma phenomena in high fidelity. %The physics-based-adaptive plasma model is dynamically refined based on the local conditions to provide uniform model fidelity throughout the domain at all times of the simulation. The physics-based-adaptive plasma model can be dynamically refined based on the local plasma conditions to increase model fidelity uniformity throughout the domain at all times of the simulation. %The adaptive plasma model uses continuum representations of the plasma, which include a kinetic Boltzmann model for the highest fidelity, multi-fluid plasma models (13N-moment and 5$N$-moment), and single-fluid MHD models for the lowest fidelity. The adaptive plasma model uses continuum representations of the plasma, which include a kinetic Vlasov model for the highest fidelity, multi-fluid 5$N$-moment plasma model, and single-fluid MHD model for the lowest fidelity. The models include evolution equations for the electromagnetic fields, electron species, ion species, and neutral species. A nodal discontinuous Galerkin finite element method is implemented and is coupled with various implicit and explicit Runge-Kutta methods. Various model coupling techniques are investigated for a 5$N$-moment multi-fluid models with a Vlasov-Maxwell model, and a 5$N$-moment two-fluid model with an MHD model. Continuum plasma models using consistent normalizations and identical spatial representations provide straightforward and accurate coupling between the models. %The solution approach offers the potential for high-order accuracy and computational efficiency. The solution approach offers high-order accuracy and computational efficiency. Target compute platforms are heterogeneous computer architectures using a compute model that minimizes data movement.

Energy Conversion and Collisionless Plasma Dissipation Channels in the Turbulent Magnetosheath Observed by the Magnetospheric Multiscale Mission

Abstract Analysis of high-resolution Magnetospheric Multiscale Mission plasma and magnetic field data directly reveals the exchanges of energy between electromagnetic and flow energy and between microscopic flows and random kinetic energy in the inhomogeneous turbulent magnetosheath. The computed rates of exchange are based on exact results from the collisionless Vlasov model of plasma dynamics, without appeal to viscous or other closures. The description includes analyses of several structures observed in intervals of burst mode data in the magnetosheath, revealing pathways of energy exchange at sub-ion scales. Time-series of the work done by the electromagnetic field, and the pressure–stress interaction, enable description of the pathways to dissipation in this low-collisionality plasma. This method does not require any specific mechanism for its application, such as reconnection or a selected mode, although with increased experience it will be useful for distinguishing between proposed possibilities.

High-order discretization of a gyrokinetic Vlasov model in edge plasma geometry

We describe a new spatial discretization of a continuum gyrokinetic Vlasov model in axisymmetric tokamak edge plasma geometries. The geometries are represented using a multiblock decomposition in which logically distinct blocks are smoothly mapped from rectangular computational domains and are aligned with magnetic flux surfaces to accommodate strong anisotropy induced by the magnetic field. We employ a fourth-order, finite-volume discretization in mapped coordinates to mitigate the computational expense associated with discretization on 4D phase space grids. Applied to a conservative formulation of the gyrokinetic system, a finite-volume approach expresses local conservation discretely in a natural manner involving the calculation of normal fluxes at cell faces. In the approach presented here, the normal fluxes are computed in terms of face-averaged velocity normals in such a way that (i) the divergence-free property of the gyrokinetic velocity is preserved discretely to machine precision, (ii) the configuration space normal velocities are independent of mapping metrics, and (iii) the configuration space normal velocities are computed from exact pointwise evaluation of magnetic field data except for one term. The algorithms described in this paper form the foundation of a continuum gyrokinetic edge code named COGENT, which is used here to perform a convergence study verifying the accuracy of the spatial discretization.