Nonequilibrium Velocity Distribution Research Articles

Analytical irreversible thermodynamics in conduction-radiative heat transfer of neutral gases through a lens of thermal radiation fields

An analytical investigation has been conducted on the steady-state conduction and a diluted gas exposed to a thermal radiation field experiences radiative heat transfer. Analytical solutions have been derived for the transport Boltzmann BGK partial differential equations system. The investigation delves into the irreversible thermodynamic behavior of the system through the utilization of the Liu-Lees model, which employs two-stream Maxwellian distribution functions. The moment approach is utilized to predict the behavior of macroscopic gas parameters like temperature, concentration, and Fourier heat flux. This process entails substituting these parameters into the two-stream Maxwellian distribution functions to analyze the system's non-equilibrium thermodynamic characteristics, which encompass gas particles and the heated plate. Through this analysis, kinetic coefficients, entropy, entropy flux, entropy production, and thermodynamic forces are determined. The investigation assesses the validation of Onsager's reciprocity relation, the Boltzmann H-theorem, the Le Chatelier principle, and the second law of thermodynamics within the system. Furthermore, the Gibbs formula is employed to estimate the ratios between various contributions to internal energy changes, which are determined by the total derivatives of extensive parameters. The non-equilibrium velocity distribution function was calculated for the first time, and its behavior was compared with that of the equilibrium one using 3D graphics. The behavior of the computed variables is predicted through graphical analysis and discussion of the results.

The Effects of Nonequilibrium Velocity Distributions on Alfvén Ion-cyclotron Waves in the Solar Wind

In this work, we investigate how the complex structure found in solar wind proton velocity distribution functions (VDFs), rather than the commonly assumed two-component bi-Maxwellian structure, affects the onset and evolution of parallel-propagating microinstabilities. We use the Arbitrary Linear Plasma Solver, a numerical dispersion solver, to find the real frequencies and growth/damping rates of the Alfvén modes calculated for proton VDFs extracted from Wind spacecraft observations of the solar wind. We compare this wave behavior to that obtained by applying the same procedure to core-and-beam bi-Maxwellian fits of the Wind proton VDFs. We find several significant differences in the plasma waves obtained for the extracted data and bi-Maxwellian fits, including a strong dependence of the growth/damping rate on the shape of the VDF. By applying the quasilinear diffusion operator to these VDFs, we pinpoint resonantly interacting regions in velocity space where differences in VDF structure significantly affect the wave growth and damping rates. This demonstration of the sensitive dependence of Alfvén mode behavior on VDF structure may explain why the Alfvén ion-cyclotron instability thresholds predicted by linear theory for bi-Maxwellian models of solar wind proton background VDFs do not entirely constrain spacecraft observations of solar wind proton VDFs, such as those made by the Wind spacecraft.

Instability of Langmuir-beam waves: Kappa-distributed electrons

In space plasmas, electron populations exhibit non-equilibrium velocity distributions with high-energy tails that are reproduced by the Kappa power-laws and contrast with the Maxwellian distributions often used in theoretical and numerical analyses. In this work, we investigate typical electron beam-plasma systems and show the influence of Kappa tails on the linear dispersion and stability spectra of Langmuir-beam waves. The most common scenarios invoke instabilities of Langmuir waves at the origin of radio emissions in solar flares and interplanetary shocks. However, the parametric domain of these instabilities is narrow (i.e., energetic beams but with very low density, nb/ne≲10−3), making their analytical and numerical characterization not straightforward, while the approximations used may lead to inconclusive results. Here, we provide exact numerical solutions of the Langmuir-beam mode, which distinguish from the classical ones (unaffected by the beam), and also from electron beam modes destabilized by more energetic and/or denser beams. Langmuir-beam solutions are only slightly modified by the Kappa distribution of the beam component, due to its very low density. However, if the main (core) population is Kappa distributed, the instability of the Langmuir-beam mode is strongly inhibited, if not suppressed. New analytical solutions are derived taking into account the more or less resonant involvement of the electron core and beam populations. As a result, the analytical solutions show an improved match with the exact solutions, making them applicable in advanced modeling of weak (weakly nonlinear) turbulence.

Quadrature-based moment methods for kinetic plasma simulations

Quadrature-based moment methods (QBMM) are applied to rarefied gas flow and plasma physics problems represented by the Boltzmann equation or the Vlasov-Poisson system of equations. QBMM are a computationally advantageous alternative to both direct Eulerian and Lagrangian particle solvers, able to provide a noise-free solution to the Boltzmann equation and capture non-equilibrium velocity distribution functions (VDF) without excessive computational cost. To provide a closure of the moment equations, the VDF is assumed to be represented by the sum of weighted kernel functions that are placed at given velocity abscissas. Three QBMM approaches, QMOM (Dirac delta kernels), HyQMOM (hyperbolic QMOM), and EQMOM (Gaussian kernels), are applied to canonical one-dimensional rarefied gas flow and plasma physics problems. A new algorithm for EQMOM that is able to treat pathological VDFs and provide control over large abscissas for more than two nodes is developed. QMOM and EQMOM are suitable for problems where the flow does not have shock-like features (for QMOM) or a discontinuity in the VDF (for EQMOM) and where the VDF is far from non-equilibrium and features strong phase-mixing. HyQMOM is generally applicable to all test problems and yields good representations of both the moments and the VDF. HyQMOM also provides significant wall-time improvement (factor of 10–100) compared to other direct Eulerian solvers.

The Mott-Smith solution to the regular shock reflection problem

The classical Mott-Smith solution for one-dimensional normal shock wave structure is extended to the two-dimensional regular shock reflection problem. The solution for the non-equilibrium molecular velocity distribution function along the symmetry-plane streamline is obtained as a weighted sum of four Maxwellians. An analysis of applicability of the solution has been performed using the results of direct simulation Monte Carlo calculations for a range of incident shock wave intensities. Accuracy of the solution improves with increasing$Ma_n$, the Mach number normal to the shock front, so that the solution becomes rather accurate for strong shocks with$Ma_n>8$.

Kinetic and thermodynamic examinations for the unsteady couette flow problem of a plasma using the BGK cylindrical model

We aim to investigate the kinetic and irreversible thermodynamic treatment of the Couette flow of gaseous plasma (GP) confined between two coaxial rotating circular rigid cylinders. The Bhatnagar-Gross-Krook (BGK) model of the Boltzmann kinetic equation (BKE) is solved by applying the perturbation method coupled with the moments' approach with a two-stream Maxwellian distribution function (MDF). Nonlinear partial differential equations are precisely solved. As an actual application, we are considering the laboratory Argon GP. An interesting comparison between the non-equilibrium electron velocity distribution function (EVDF) and the equilibrium EVDF is made carefully with 3-Dimensional graphics in various time values. We found that the system goes to an equilibrium state (ES) with time compatible with Le Chatelier's principles. Moreover, we determined with very high precision the value of the system equilibrium time, which is tequ.≅2.4. That great advantage cannot be reached by solving other macroscopic magnetohydrodynamic models. The relations between the various macroscopic variables of the GP are studied. The irreversible non-equilibrium thermodynamics (NT) properties of the system are presented. We present the extended Gibbs equation (EGE) for the internal energy variation (IEV) in cylindrical geometry for GP for the first time, to the best of our knowledge. We have shown that our model is compatible with the H-theorem of Boltzmann and the Onsager-Casimir relations. The various participations in IEV are introduced. The significance of this search is due to its vast implementations in various fields, such as in GP physics, electrical engineering, medicine, biological systems, and numerous commercial and industrial deployments.

2D particle-in-cell simulations of the electron temperature anisotropy driven whistler instability in plasmas having kappa velocity distributions

This study presents the results of 2D particle-in-cell (PIC) simulations of the electron temperature anisotropy driven whistler instability for plasmas in which the electron species is modeled by a bi-kappa velocity distribution. These simulations utilize our previously developed method to generate the initial multi-dimensional kappa velocity distributions. The use of multi-dimensional kappa loadings in PIC simulations provides insights into the non-linear regime of wave evolution in plasmas having non-equilibrium velocity distributions. Three cases are considered, corresponding to κe=2,3, and ∞ (Maxwellian case). Owing to the use of a large value of electron anisotropy required for reasonable simulation run times, the Maxwellian electron run has the fastest growth rate, reaching saturation earliest. The κe=2 case exhibits the slowest growth rate. Spectral analysis of the fluctuating fields reveals considerable wave intensity at frequencies and wavenumbers that satisfy the linear whistler wave dispersion relation. In the runs with kappa distributions, the regions of most intense fluctuations comprise frequencies and wavenumbers that agree only qualitatively with linear whistler wave theory. The results suggest that after saturation, there is some degree of Landau damping of the oblique whistler modes, which returns energy to the electron species. The rate of damping of the oblique modes is highest in the Maxwellian case and lowest for κe=2. Evidence of significant superthermal acceleration of electrons in the direction parallel to the ambient magnetic field is also observed. Thus, the power-law index of the electron distribution is reduced by the anisotropic whistler turbulence produced.

A measurement system of the nonequilibrium velocity distributions of evaporating molecules from liquid-vapor interfaces pinned at a nanoporous surface

A measurement system of the nonequilibrium velocity distributions of evaporating molecules from liquid-vapor interfaces pinned at a nanoporous surface

Hydrodynamic and Thermodynamic Nonequilibrium Effects around Shock Waves: Based on a Discrete Boltzmann Method.

A shock wave that is characterized by sharp physical gradients always draws the medium out of equilibrium. In this work, both hydrodynamic and thermodynamic nonequilibrium effects around the shock wave are investigated using a discrete Boltzmann model. Via Chapman–Enskog analysis, the local equilibrium and nonequilibrium velocity distribution functions in one-, two-, and three-dimensional velocity space are recovered across the shock wave. Besides, the absolute and relative deviation degrees are defined in order to describe the departure of the fluid system from the equilibrium state. The local and global nonequilibrium effects, nonorganized energy, and nonorganized energy flux are also investigated. Moreover, the impacts of the relaxation frequency, Mach number, thermal conductivity, viscosity, and the specific heat ratio on the nonequilibrium behaviours around shock waves are studied. This work is helpful for a deeper understanding of the fine structures of shock wave and nonequilibrium statistical mechanics.

A 14-moment maximum-entropy description of electrons in crossed electric and magnetic fields

A 14-moment maximum-entropy system of equations is applied to the description of non-equilibrium electrons in crossed electric and magnetic fields and in the presence of low collisionality, a characteristic of low-temperature plasma devices. The flexibility of this formulation is analyzed through comparison with analytical results for steady-state non-equilibrium velocity distribution functions and against particle-based solutions of the time-dependent kinetic equation. Electric and magnetic source terms are derived for the 14-moment equations, starting from kinetic theory. A simplified collision term based on the Bhatnagar–Gross–Krook operator is formulated to describe the collision of electrons with background neutrals, accounting for the large mass disparity and energy exchange. An approximated expression is proposed for the collision frequency, to include the effect of the electrons' drift velocity, showing good accuracy in the considered conditions. The capabilities of the proposed 14-moment closure to accurately capture the non-equilibrium behavior of electrons for space homogeneous problems under conditions representative of those found in Hall thrusters are demonstrated.

Study of regular reflection shock waves using a mesoscopic kinetic approach: Curvature pattern and effects of viscosity

The physical characteristics inside shock waves with nonequilibrium molecular motion are difficult to describe using conventional macroscopic methods. In this paper, nonequilibrium hydrodynamic and thermodynamic effects caused by the strong nonequilibrium molecular velocity distribution at a shock wave are studied using a mesoscopic kinetic approach. This approach is based on a lattice Boltzmann method and a kinetic nonequilibrium method. The former adopts a compressible double-distribution-function model with separated density and total energy distribution functions. The latter represents the nonequilibrium effects through nonequilibrium kinetic moments based on the nonequilibrium molecular velocity distribution. The nonequilibrium effects in the steady state and the process of the formation of a regular reflection shock wave are presented. Nonequilibrium effects inside the shock wave are further investigated. First, the curvature pattern during the formation of a regular reflection shock wave is addressed. The curvature characteristic leads to distinct features of nonequilibrium effects compared with the linear pattern. A vector-based approach for visualizing nonequilibrium effects is proposed to study the curvature pattern. Second, the influence of viscosity on nonequilibrium effects, which is related to the average collision time among molecules at the shock wave, is explored. The results obtained in this paper provide mesoscopic physical insight into the flow mechanisms occurring in shock waves.

Temperature relaxation in binary hard-sphere mixture system: Molecular dynamics and kinetic theory study.

Computational schemes to describe the temperature relaxation in the binary hard-sphere mixture system are given on the basis of molecular dynamics (MD) simulation and renormalized kinetic theory. Event-driven MD simulations are carried out for three model systems in which the initial temperatures and the ratios of diameter and mass of two components are different to study the temporal evolution of each component temperature in nanoscale molecular conditions mimicking those in living cells. On the other hand, the temperature changes of the two components are also described in terms of a mean-field kinetic theory with the correlation functions calculated in the Percus-Yevick approximation. The calculated results by both the computational approaches have shown fair agreement with each other, whereas slight deviations have been found in the temporal range of femto- to picoseconds when the initial temperatures of the two components are significantly different, such as 300 K vs 1000 K. This discrepancy can be ascribed to the fast intra-component temperature relaxation assumed in the kinetic theory, and its violation in the MD simulations can be evaluated in terms of the Kullback-Leibler divergence between the equilibrated Maxwell-Boltzmann distribution at each temperature and the actual non-equilibrium velocity distribution realized in the MD. Thus, the present analysis provides a quantitative basis for addressing the temperature inhomogeneities experimentally observed in nanoscale crowding conditions.

Limiting energy gain in deuterium plasma at powerful injection heating

The possibility of neutron production in deuterium plasma (with no external tritium breeding) is considered for the case of high ratio of fast particles. High amount of fast deuterons is sustained by a powerful injection of neutral beams. In this case, fusion reactivity is essentially higher that the reactivity in the Maxwellian plasma. The reactivity is calculated by taking into account non-equilibrium velocity distribution function of fast deuterons. Plasma power gain Q ∼ 0.5 can be achieved at electron temperature of approximately 100 keV and deuteron injection energy of approximately 2 MeV. To realize such a scheme for this type of fusion fuel, the plasma pressure should be approximately equal to the magnetic pressure.

Three-dimensional multidomain lattice Boltzmann grid refinement for passive scalar transport.

Simulations of passive scalar transport on a three-dimensional (3D) multiresolution grid are presented within the framework of single relaxation time lattice Boltzmann method. The combined modeling of the fluid flow and scalar transport is handled by a double distribution function approach in which the velocity in the fluid flow equation is solved first and then copied to the scalar equation to solve the corresponding scalar quantities. A 3D scaling technique, considering both external forces and scalar source terms, and two-dimensional bicubic interpolation scheme are developed for coupling nonequilibrium velocity and scalar distributions on the interfaces of different resolution grids. The proposed algorithm is validated for three benchmark cases, i.e., the forced convection in a 3D channel, natural convection in a cubical cavity, and turbulent channel flow with heat transfer. Good agreements are found between the present predictions and previous data, which confirms the capability of the proposed method for the computation of passive scalar transport in 3D domains.

Pulse-Periodic Regimes of Kinetic Instabilities in the Non-Equilibrium Plasma of an Electron Cyclotron Resonance Discharge Maintained by Continuous-Wave Radiation of a 24 GHz Gyrotron

We have experimentally discovered an instability, which manifests itself as precipitations of hot electrons occurring synchronously with generation of bursts of electromagnetic radiation, in the plasma of an electron cyclotron resonance discharge maintained by a high-power, continuous-wave radiation of a 24 GHz gyrotron, for the first time. The observed instability has the kinetic nature and is determined by the formation of the non-equilibrium velocity distribution of hot particles. Two possible explanations are proposed for the mechanism of wave excitation in a two-component plasma with a stationary source of non-equilibrium particles. The results of the studies performed are of interest for modeling of the dynamics of magnetospheric cyclotron masers.

A High-Order Accurate Gas-Kinetic Scheme for One- and Two-Dimensional Flow Simulation

AbstractThis paper develops a high-order accurate gas-kinetic scheme in the framework of the finite volume method for the one- and two-dimensional flow simulations, which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic polynomial reconstruction of the macroscopic variables and fourth-order accurate flux evolution. The first part reconstructs a piecewise cell-center based quartic polynomial and a cell-vertex based quartic polynomial according to the “initial” cell average approximation of macroscopic variables to recover locally the non-equilibrium and equilibrium single particle velocity distribution functions around the cell interface. It is in view of the fact that all macroscopic variables become moments of a single particle velocity distribution function in the gas-kinetic theory. The generalized moment limiter is employed there to suppress the possible numerical oscillation. In the second part, the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means of the simple particle transport mechanism in the microscopic level, i.e. free transport and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order flux evolution is based on the solution (i.e. the particle velocity distribution function) of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are numerically solved by using the proposed high-order accurate gas-kinetic scheme. By comparing with the exact solutions or the numerical solutions obtained the second-order or third-order accurate gas-kinetic scheme, the computations demonstrate that our scheme is effective and accurate for simulating invisid and viscous fluid flows, and the accuracy of the high-order GKS depends on the choice of the (numerical) collision time.

Ion acceleration by parallel propagating nonlinear Alfvén wave packets in a radially expanding plasma

Abstract. The numerical simulation of the nonlinear evolution of the parallel propagating Alfvén waves in a radially expanding plasma is performed by using a kinetic-fluid model (the Vlasov–MHD model). In our study, both the nonlinear evolution of the Alfvén waves and the radial evolution of the velocity distribution function (VDF) are treated simultaneously. On the other hand, important ion kinetic effects such as ion cyclotron damping and instabilities driven by the non-equilibrium ion velocity distributions are not included in the present model. The results indicate that the steepened Alfvén wave packets outwardly accelerate ions, which can be observed as the beam components in the interplanetary space. The energy of imposed Alfvén waves is converted into the longitudinal fluctuations by the nonlinear steepening and the nonlinear Landau damping. The wave shoaling due to the inhomogeneity of the phase velocity is also observed.

Anomalous velocity distributions in active Brownian suspensions

Large-scale simulations and analytical theory have been combined to obtain the nonequilibrium velocity distribution, f(v), of randomly accelerated particles in suspension. The simulations are based on an event-driven algorithm, generalized to include friction. They reveal strongly anomalous but largely universal distributions, which are independent of volume fraction and collision processes, which suggests a one-particle model should capture all the essential features. We have formulated this one-particle model and solved it analytically in the limit of strong damping, where we find that f(v) decays as 1/v for multiple decades, eventually crossing over to a Gaussian decay for the largest velocities. Many particle simulations and numerical solution of the one-particle model agree for all values of the damping.

Non-equilibrium pressure control of the height of a large-scale, ground-coupled, rotating fluid column

When a ground-coupled, rotating fluid column is modeled incorporating non-equilibrium pressure forces in the Navier-Stokes equations, a new exact solution results. The solution has been obtained in a similar manner to the classical equilibrium solution. Unlike the infinite-height, classical solution, the non-equilibrium pressure solution yields a ground-coupled rotating fluid column of finite height. A viscous, non-equilibrium Rankine vortex velocity distribution, developed previously, was used to demonstrate how the viscous and non-equilibrium pressure gradient forces, arising in the vicinity of the velocity gradient discontinuity that is present in the classical Rankine vortex model, effectively isolate the rotating central fluid column from the outer potential vortex region. Thus, the non-equilibrium region acts to confine and shield the central, rigid-body-like, rotating fluid core, justifying this examination of how such a rotating fluid column can interact with the ground. The resulting non-equilibrium ground-coupled, rotating fluid column solution was employed to estimate the central column heights of three well-documented dust devils, and the central column height predictions were consistent with published dust devil height statistics.

Space charge corrected electron emission from an aluminum surface under non-equilibrium conditions

A theoretical study has been conducted of ultrashort pulsed laser induced electron emission from an aluminum surface. Electron emission fluxes retrieved from the commonly employed Fowler-DuBridge theory were compared to fluxes based on a laser-induced non-equilibrium electron distribution. As a result, the two- and three-photon photoelectron emission parameters for the Fowler-DuBridge theory have been approximated. We observe that at regimes where photoemission is important, laser-induced electron emission evolves in a more smooth manner than predicted by the Fowler-DuBridge theory. The importance of the actual electron distribution decreases at higher laser fluences, whereas the contribution of thermionic emission increases. Furthermore, the influence of a space charge effect on electron emission was evaluated by a one dimensional particle-in-cell model. Depending on the fluences, the space charge reduces the electron emission by several orders of magnitude. The influence of the electron emission flux profiles on the effective electron emission was found to be negligible. However, a non-equilibrium electron velocity distribution increases the effective electron emission significantly. Our results show that it is essential to consider the non-equilibrium electron distribution as well as the space charge effect for the description of laser-induced photoemission.