Equilibrium Maxwellian Distribution Research Articles

A General Framework for the Kinetic Modelling of Polyatomic Gases

A general framework for the kinetic modelling of non-relativistic polyatomic gases is proposed, where each particle is characterized both by its velocity and by its internal state, and the Boltzmann collision operator involves suitably weighted integrals over the space of internal states. The description of the internal structure of a molecule is kept highly general, and this allows classical and semi-classical models, such as the monoatomic gas description, the continuous internal energy structure, and the description with discrete internal energy levels, to fit our framework. We prove the H-Theorem for the proposed kinetic equation of Boltzmann type in this general setting, and characterize the equilibrium Maxwellian distribution and the thermodynamic number of degrees of freedom. Euler equations are derived, as zero-order approximation in a suitable asymptotic expansion. In addition, within this general framework it is possible to build up new models, highly desirable for physical applications, where rotation and vibration are precisely described. Examples of models for the Hydrogen Fluoride gas are shown. A link between state-based and energy-based points of view is presented.

Large-eddy simulation of wall-bounded turbulent flow with high-order discrete unified gas-kinetic scheme

In this paper, we introduce the discrete Maxwellian equilibrium distribution function for incompressible flow and force term into the two-stage third-order Discrete Unified Gas-Kinetic Scheme (DUGKS) for simulating low-speed turbulent flows. The Wall-Adapting Local Eddy-viscosity (WALE) and Vreman sub-grid models for Large-Eddy Simulations (LES) of turbulent flows are coupled within the present framework. Meanwhile, the implicit LES are also presented to verify the effect of LES models. A parallel implementation strategy for the present framework is developed, and three canonical wall-bounded turbulent flow cases are investigated, including the fully developed turbulent channel flow at a friction Reynolds number (Re) about 180, the turbulent plane Couette flow at a friction Re number about 93 and lid-driven cubical cavity flow at a Re number of 12000. The turbulence statistics, including mean velocity, the r.m.s. fluctuations velocity, Reynolds stress, etc. are computed by the present approach. Their predictions match precisely with each other, and they are both in reasonable agreement with the benchmark data of DNS. Especially, the predicted flow physics of three-dimensional lid-driven cavity flow are consistent with the description from abundant literature. The present numerical results verify that the present two-stage third-order DUGKS-based LES method is capable for simulating inhomogeneous wall-bounded turbulent flows and getting reliable results with relatively coarse grids.

Numerical Modeling of Electron Transport in Solar Wind: Effects of Whistler Turbulence and Coulomb Collisions

The electron distribution function (eVDF) in the solar wind deviates significantly from an equilibrium Maxwellian distribution, and is comprised of a Maxwellian core, a suprathermal halo, a field-aligned component strahl, and a higher energy superhalo. Charged particle Coulomb collisions are ineffective in relaxing such a velocity distribution beyond a few solar radii. Therefore wave-particle interactions need to be considered. A wave-particle interaction term was introduced into the kinetic equation that describes the interaction of electrons with whistler waves, as well as particle collision terms. The kinetic equation has the form of an advection-diffusion-like equation in which the advection and diffusion coefficients describe the scattering and drag of electrons in whistler turbulence. A reliable numerical method has been developed to solve a full form of the advection-diffusion-like kinetic equation. Preliminary applications of the numerical method to the solar wind electron problem are presented. Comparison and analysis of the electron VDFs in the presence of Coulomb collisons and resonant wave-partcicle interactions are made.

Molecular dynamics investigation of energy transfer during gas-surface collisions

In this work, the energy transfer in gas-surface collisions is investigated using the molecular dynamics method. The numerical setup consists of a nitrogen molecule scattering from a graphite surface. The main focus is put on the energy redistribution between different molecular kinetic modes and the surface for the case of strong thermal non-equilibrium. The thermal non-equilibrium is defined as the state when either translational or rotational temperature of impinging molecules differs significantly from that of the surface. Accordingly, two different scenarios have been examined, including rotational and translational excitation of the initial molecular state. In contrast to the molecular beam method, the initial molecular velocities are sampled from the equilibrium Maxwellian distribution, ensuring isotropic incidence angles and energies. The obtained results are expressed in the form of energy transfer coefficients, which are used to quantify the normalized energy loss or gain in a specific mode. Furthermore, the velocity distributions of reflected molecules are analyzed and compared with some of the available wall kernels, providing further insight into the nature of the energy transfer dynamics and scattering process. Additionally, the numerical setup is tested against the available molecular beam experimental data and the obtained results were used to select a suitable numerical force field.

Convergence and Error Estimates for the Lagrangian-Based Conservative Spectral Method for Boltzmann Equations

We develop error estimates for the semidiscrete conservative spectral method for the approximation of the elastic and inelastic space homogeneous Boltzmann equation introduced by Gamba and Tharkabhushanam in [J. Comput. Phys., 228 (2009), pp. 2012--2036]. In addition we study the long time convergence of such semidiscrete solution to the equilibrium Maxwellian distribution that conserves the mass, momentum, and energy associated with the initial data. The numerical method is based on the Fourier transform of the collisional operator and a Lagrangian optimization correction that enforces the collision invariants, namely, conservation of mass, momentum, and energy in the elastic case, and just mass and momentum in the inelastic one. We present a detailed semidiscrete analysis on convergence of the proposed numerical method which includes the $L^{1}-L^{2}$ theory for the scheme. This analysis allows us to present, additionally, convergence in Sobolev spaces and convergence to equilibrium for the numerical approximation. The results of this work answer a long standing open problem posed by Cercignani, Illner, and Pulvirenti in [Mathematical Theory of Dilute Gases, Springer, New York, 1994, Chapter 12] about finding error estimates for a numerical scheme associated with the Boltzmann equation, as well as showing the semidiscrete numerical solution converges to the equilibrium Maxwellian distribution associated with the initial value problem.

A closed high-frequency Vlasov–Maxwell simulation model in toroidal geometry

A fully-kinetic ion and gyrokinetic electron Vlasov–Maxwell particle simulation model is derived through Lie transform perturbation theory for Hamiltonian systems in terms of four ordering parameters , , and . This model is closed by the field equations of Poisson’s equation and Ampere’ law. This scheme preserves the phase-space volume, and retains the ion cyclotron motion, while fine scale electron motion is ignored, so that the frequency falls in the range . Using a perturbative method (), and ignoring the high order terms, this model is then formulated in a magnetic flux coordinate system with equilibrium Maxwellian distribution of particles. Thus, this model is especially suitable for the global particle simulation of high-frequency physical processes such as lower hybrid waves and waves in the ion cyclotron range of frequencies.

Comparison of fluid neutral models for one-dimensional plasma edge modeling with a finite volume solution of the Boltzmann equation

We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assuming equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation.

Thermonuclear Fusion Reactivities for Drifting Tri-Maxwellian Ion Velocity Distributions

The presence of strong magnetic fields in magnetically confined thermonuclear fusion plasmas introduces a considerable anisotropy in the ion velocity distributions. Drifting or streaming of ionic species is another source of anisotropy in fusion plasmas. To account for these anisotropies, we consider in the present work a generalization of the equilibrium Maxwellian distribution in the form of a drifting tri-Maxwellian distribution function. It is shown that the calculation of thermonuclear reactivities for these distributions is best carried out in a transformed velocity space. Computational results for gyrotropic velocity distributions with drifts parallel and perpendicular to the magnetic field are presented.

A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation

We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect implying the ultra-analyticity of both the fluctuation and its Fourier transform for any positive time.

A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods

AbstractThe Boltzmann equation (BE) for gas flows is a time-dependent nonlinear differential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium distribution and to approximate the moment integrals by the discrete ordinate method (DOM) using a finite set of velocity quadrature points. Such simplification reduces the dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The main difficulty of the currently used (conventional) numerical procedures occurs when the mean velocity and the variation of temperature are large that requires an extremely large number of quadrature points. In this paper, a novel dynamic scheme that requires only a small number of quadrature points is proposed. This is achieved by a velocity-coordinate transformation consisting of Galilean translation and thermal normalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermite quadrature. The velocity quadrature points in the new velocity space are fixed while the correspondent quadrature points in the physical space change from time to time and from position to position. By this dynamic nature in the physical space, this new quadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS was implemented to the DOM and the lattice Boltzmann method (LBM). These new methods with DQS are therefore termed as the dynamic discrete ordinate method (DDOM) and the dynamic lattice Boltzmann method (DLBM), respectively. The new DDOM and DLBM have been tested and validated with several testing problems. Of the same accuracy in numerical results, the proposed schemes are much faster than the conventional schemes. Furthermore, the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM.

Adaptive and Recursive Time Relaxed Monte Carlo Methods for Rarefied Gas Dynamics

Recently a new class of Monte Carlo methods, called time relaxed Monte Carlo (TRMC), designed for the simulation of the Boltzmann equation close to fluid regimes has been introduced [L. Pareschi and G. Russo, SIAM J. Sci. Comput., 23 (2001), pp. 1253–1273]. A generalized Wild sum expansion of the solution is the basis of the simulation schemes. After a splitting of the equation, the time discretization of the collision step is obtained from the Wild sum expansion of the solution by replacing high order terms in the expansion with the equilibrium Maxwellian distribution; in this way speed-up of the methods close to fluid regimes is obtained by efficiently thermalizing particles close to the equilibrium state. In this work we present an improvement of such methods which allows us to obtain an effective uniform accuracy in time without any restriction on the time step and subsequent increase of the computational cost. The main ingredient of the new algorithms is recursivity [L. Pareschi and B. Wennberg, Monte Carlo Methods Appl., 7 (2001), pp. 349–358]. Several techniques can be used to truncate the recursive trees generated by the schemes without degrading the accuracy of the numerical solution. Techniques based on adaptive strategies are presented. Numerical results emphasize the gain of efficiency of the present simulation schemes with respect to standard DSMC (direct simulation Monte Carlo) methods.

Relaxation of Relativistic Plasmas Under the Effect of Wave‐Particle Interactions

We simulate the acceleration of electrons to relativistic energies due to the interaction of electrons with waves generated by longitudinal (i.e., electrostatic) streaming instabilities in plasmas. Two equal systems undergoing a streaming instability evolve, one according to the classical Newton's law and one according to the special relativity dynamics equation. The system that obeys Newton's law relaxes to a Maxwellian equilibrium distribution. In the case of the relativistic dynamics, the equilibrium distribution exhibits peaks in the phase space at high momenta and a larger number of particles at high energies. This steady electron population at higher energies could explain power-law energy distribution in many plasma physics and astrophysical systems.

Numerical solution of the Boltzmann equation by time relaxed Monte Carlo (TRMC) methods

AbstractA new family of Monte Carlo schemes has been recently introduced for the numerical solution of the Boltzmann equation of rarefied gas dynamics (SIAM J. Sci. Comput. 2001; 23:1253–1273). After a splitting of the equation the time discretization of the collision step is obtained from the Wild sum expansion of the solution by replacing high‐order terms in the expansion with the equilibrium Maxwellian distribution. The corresponding time relaxed Monte Carlo (TRMC) schemes allow the use of time steps larger than those required by direct simulation Monte Carlo (DSMC) and guarantee consistency in the fluid‐limit with the compressible Euler equations. Conservation of mass, momentum, and energy are also preserved by the schemes. Applications to a two‐dimensional gas dynamic flow around an obstacle are presented which show the improvement in terms of computational efficiency of TRMC schemes over standard DSMC for regimes close to the fluid‐limit. Copyright © 2005 John Wiley & Sons, Ltd.

Time Relaxed Monte Carlo Methods for the Boltzmann Equation

A new family of Monte Carlo schemes is introduced for the numerical solution of the Boltzmann equation of rarefied gas dynamics. The schemes are inspired by the Wild sum expansion of the solution of the Boltzmann equation for Maxwellian molecules and consist of a novel time discretization of the equation. In particular, high order terms in the expansion are replaced by the equilibrium Maxwellian distribution. The two main features of the schemes are high order accuracy in time and asymptotic preservation. The first property allows to recover accurate solutions with time steps larger than those required by direct simulation Monte Carlo (DSMC), while the latter guarantees that for the vanishing Knudsen number, the numerical solution relaxes to the local Maxwellian. Conservation of mass, momentum, and energy are preserved by the scheme. Numerical results on several space homogeneous problems show the improvement of the new schemes over standard DSMC. Applications to a one-dimensional shock wave problem are also presented.

Self-collimation in an atomic beam evaporated from a superfluid 4He film

Using a scaling Monte Carlo method, we have simulated the scattering in a pulsed atomic beam evaporated from a superfluid 4He film. The simulation assumes that the atoms leaving the surface have the equilibrium Maxwellian distribution at the temperature of the film T. This means that the initial particle flux varies as cos straight theta (Lambert's law). We find that the effect of atomic scattering just above the film is to bias the flux in favor of the forward (small straight theta) direction, in agreement with the experiment of Eckardt et al. The simulation predicts that the deviation from Lambert's law grows rapidly with increasing heat input Q and decreasing pulse length. At the same time, the average kinetic energy per particle in the forward direction is enhanced relative to the global average, 2k(B)T. The distribution with respect to speed is narrower than Maxwellian in the forward direction but broader at large angles. We find that a beam that has passed through a slit is slightly narrower than in a ballistic calculation with no collisions. This effect seems to saturate at values of Q that correspond to 30 or 40 collisions per atom.

Electron distribution function in a two-temperature plasma with cold ions

By modeling the dynamics of a large ensemble of particles, it is shown that slow electrons in a two-temperature plasma are in equilibrium with the electron component rather than with cold ions. The result of cooling by a cold ion component is that the number of the low-energy electrons only slightly exceeds that in the equilibrium Maxwellian distribution.

The magnetized plasma permittivity tensor

An alternative form of the nonrelativistic permittivity tensor for a homogeneous magnetized plasma in thermal equilibrium is derived in the linear approximation. The derivation follows the lines of Trubnikov’s derivation of the relativistic permittivity tensor and is restricted to a collisionless electron plasma. In the case of a Maxwellian equilibrium distribution, the final form is equivalent to Gordeev’s integral form of the permittivity tensor. The analysis is, with minor modification, applicable to plasma comprised of multiple particle species. Unlike the standard formulas, the alternative formula presented in this article yields the components of the permittivity tensor by means of partial derivatives of a scalar kernel integral. Besides the cyclotron harmonic resonance frequencies, where the permittivities in the nonrelativistic approximation become singular, the alternative formula is manifestly valid for all wave frequencies without any restrictions on the wave vector and the thermal energy.

Simulation of the scattering in an atomic beam evaporated from a [formula omitted] film

We have simulated the scattering in a pulsed atomic beam evaporated from a superfluid 4 He film. Assuming that the atoms leaving the surface of the film have an equilibrium Maxwellian distribution, we find that the experimentally observed deviations from Lambert's law are explained by scattering of the atoms after evaporation.

On convergence to equilibrium for Kac’s caricature of a Maxwell gas

The trend towards equilibrium of the solution to the one-dimensional Kac’s caricature of a Maxwell gas is investigated. When the initial value has finite energy and finite Linnik’s entropy, it is proven that the solution converges in L1∩L∞ to the equilibrium Maxwellian distribution.

Second-order oscillations of a Vlasov–Poisson plasma in Fourier-transformed velocity space

We study the Vlasov–Poisson system of equations in Fourier-transformed velocity space. First we reformulate some results of the linear theory: in the new representation the van Kampen–Case eigenmodes are found to be ordinary functions with convenient continuity properties. We give a transparent derivation of the free-streaming temporal echo in terms of the kinematics of wave packets in Fourier-transformed velocity space. We further extend this analysis to include Coulomb interactions, which allows us to establish a connection between the echo theory, the second-order oscillations of Best and the phenomenon of linear side bands. The calculation of the time evolution of the global second-order electric field is performed in detail in the case of a Maxwellian equilibrium distribution function. We conclude that the phenomenon of linear side bands may be properly explained in terms of the intrinsic features of the equilibrium distribution function.