Isotropic Maxwellian Research Articles

Ring momentum distributions as a general feature of Vlasov dynamics in the synchrotron dominated regime

We study how radiation reaction leads plasmas initially in kinetic equilibrium to develop features in momentum space, such as anisotropies and population inversion, resulting in a ring-shaped momentum distribution that can drive kinetic instabilities. We employ the Landau–Lifshiftz radiation reaction model for a plasma in a strong magnetic field, and we obtain the necessary condition for the development of population inversion; we show that isotropic Maxwellian and Maxwell–Jüttner plasmas, with thermal temperature T>mec2/3, will develop a ring-like momentum distribution. The timescales and features for forming ring-shaped momentum distributions, the effect of collisions, and non-uniform magnetic fields are discussed and compared with typical astrophysical and laboratory plasmas parameters. Our results show the pervasiveness of ring-like momentum distribution functions in synchrotron dominated plasma conditions.

Gyrokinetic modelling of the Alfvén mode and EGAM activity in ASDEX Upgrade

Energetic particles present in tokamak machines can drive through resonant wave-particle interaction different plasma instabilities, e.g Alfvén modes and energetic particle-driven geodesic acoustic modes (EGAMs). While the former are potentially detrimental as they can enhance the energetic particle transport and damage the machine wall, the latter are axisymmetric, possibly benign modes that can act to regulate turbulence. A unique scenario, the so-called NLED-AUG case, has been developed in ASDEX Upgrade by tuning the plasma parameters so that the energetic particle kinetic energy is 100 times higher than that of the background plasma, like in ITER. An intense energetic particle-driven activity is observed, most prominently various Alfvén mode bursts triggering chirping EGAMs. The present work reports studies on the Alfvén mode and EGAM dynamics showing, for the first time, many toroidal mode gyrokinetic simulations with ORB5 where the NLED-AUG case scenario is considered. We study the mode dynamics modelling the energetic particles with different equilibrium distribution functions, such as: isotropic slowing-down, double-bump-on-tail and equivalent Maxwellian. We retain, at the beginning, the nonlinearities only in the energetic particle dynamics. Later, also the background plasma species nonlinearities are taken into account.

Fusion yield of plasma with velocity-space anisotropy at constant energy

Velocity-space anisotropy can significantly modify fusion reactivity. The nature and magnitude of this modification depends on the plasma temperature, as well as the details of how the anisotropy is introduced. For plasmas that are sufficiently cold compared to the peak of the fusion cross section, anisotropic distributions tend to have higher yields than isotropic distributions with the same thermal energy. At higher temperatures, it is instead isotropic distributions that have the highest yields. However, the details of this behavior depend on exactly how the distribution differs from an isotropic Maxwellian. This paper describes the effects of anisotropy on fusion yield for the class of anisotropic distribution functions with the same energy distribution as a 3D isotropic Maxwellian and compares those results with the yields from bi-Maxwellian distributions. In many cases, especially for plasmas somewhat below reactor-regime temperatures, the effects of anisotropy can be substantial.

Velocity independent constraints on spin-dependent DM-nucleon interactions from IceCube and PICO

Adopting the Standard Halo Model (SHM) of an isotropic Maxwellian velocity distribution for dark matter (DM) particles in the Galaxy, the most stringent current constraints on their spin-dependent scattering cross-section with nucleons come from the IceCube neutrino observatory and the PICO-60 hbox {C}_3hbox {F}_8 superheated bubble chamber experiments. The former is sensitive to high energy neutrinos from the self-annihilation of DM particles captured in the Sun, while the latter looks for nuclear recoil events from DM scattering off nucleons. Although slower DM particles are more likely to be captured by the Sun, the faster ones are more likely to be detected by PICO. Recent N-body simulations suggest significant deviations from the SHM for the smooth halo component of the DM, while observations hint at a dominant fraction of the local DM being in substructures. We use the method of Ferrer et al. (JCAP 1509: 052, 2015) to exploit the complementarity between the two approaches and derive conservative constraints on DM-nucleon scattering. Our results constrain sigma _{mathrm{SD}} lesssim 3 times 10^{-39} mathrm {cm}^2 (6 times 10^{-38} mathrm {cm}^2) at gtrsim 90% C.L. for a DM particle of mass 1 TeV annihilating into tau ^+ tau ^- (bbar{b}) with a local density of rho _{mathrm{DM}} = 0.3~mathrm {GeV/cm}^3. The constraints scale inversely with rho _{mathrm{DM}} and are independent of the DM velocity distribution.

Spin distribution of asteroids - Statistical model revisited

The distribution of the rotational frequencies of asteroids is believed to carry important information on their formation and the subsequent evolutionary processes (Burns, 1975). In particular, it is commonly considered that during their formation stage the larger asteroids in the Main Belt have attained a statistical equilibrium (canonical ensemble) in the 3-dimensional isotropic velocity vector space. Subsequently, especially the smaller objects, suffered from various dynamical processes, such as collisions, fragmentation and YORP effect, for example, which modified their spin velocity and direction. In this work we re-examine the spin distribution of asteroids using more recent data and focusing on its statistical aspects, in particular, the dimensionality of the phase space. We find that the presently observed spin distribution of asteroids of any diameter bin is clearly consistent with a 2-dimensional phase space, or even less for the smaller objects. This is true also for those objects with diameter larger than 50 km, whose distribution is usually believed to be consistent with isotropic 3-dimensional Maxwellian. The present result casts open questions on the origin of the asteroids spin.

Collisional-radiative calculations for the J = 0−1 lasing line of neon-like germanium under anisotropic excitation conditions

A new asymmetry parameter characterizing the differences between the polarized π and σ gain components of the soft-x-ray J = 0–1 lasing line of neon-like ions is calculated in the case of Ge22+ assuming an electron distribution which is a weighted sum of an isotropic Maxwellian and a monoenergetic beam. Using a quasi steady-state collisional-radiative model, we determine in the weak amplification regime the relative populations of the upper M = 0 and lower magnetic sublevels of the lasing line as a function of electron density from 1020 to cm−3. This model includes inelastic and elastic collisional transitions, as well as spontaneous radiative decay between all the 337 M-sublevels arising from the 75 lowest-lying Ge22+ J-levels. The computations were performed for a temperature of the Maxwellian component between and K, a kinetic energy E0 and a fraction f of the beam component in the ranges and , respectively. The basic atomic data, such as level energies, radiative decay probabilities and inelastic collision strengths, were calculated with the flexible atomic code. However, some modifications of this code were made to get the collision strengths for transitions between M-sublevels due to impact with isotropic electrons as well as due to impact with an electron beam in the case of de-excitation. We find that the newly introduced asymmetry parameter may become significant under certain conditions of electron distribution corresponding to relatively low ( K) and E0 (3–6 keV). The results reported here may be useful in the evaluation of the polarization degree of the J = 0–1 x-ray laser output from a germanium plasma in the presence of fast directional electrons.

Self-diffusion in a stochastically heated two-dimensional dusty plasma

Diffusion in a two-dimensional dusty plasma liquid (i.e., a Yukawa liquid) is studied experimentally. The dusty plasma liquid is heated stochastically by a surrounding three-dimensional toroidal dusty plasma gas which acts as a thermal reservoir. The measured dust velocity distribution functions are isotropic Maxwellians, giving a well-defined kinetic temperature. The mean-square displacement for dust particles is found to increase linearly with time, indicating normal diffusion. The measured diffusion coefficients increase approximately linearly with temperature. The effective collision rate is dominated by collective dust–dust interactions rather than neutral gas drag, and is comparable to the dusty-plasma frequency.

Charge-exchange processes in collisions ofH+,H2+,H3+,He+, andHe2+ions with CO andCO2molecules at energies below 1000 eV

Absolute measurements of charge-exchange cross sections of ${\mathrm{H}}^{+},\phantom{\rule{0.16em}{0ex}}{\mathrm{H}}_{2}^{+},\phantom{\rule{0.16em}{0ex}}{\mathrm{H}}_{3}^{+},\phantom{\rule{0.16em}{0ex}}{\mathrm{He}}^{+}$, and ${\mathrm{He}}_{2}^{+}$ ions in CO and ${\mathrm{CO}}_{2}$ have been made for energies below 1000 eV, an equivalent of the energy of ionized particles at typical solar-wind conditions. An attenuation method for the case of complex ions of a molecule, taking into account the influence on the ion beam composition of the processes of disintegration of the primary ions into secondary ones with different charge-exchange cross sections, is described. Also the secondary effects, like three-body collisions and re-ionization processes that could emerge at higher pressures of the gas layer, are discussed. Dependence of the cross sections on the number of atomic centers in the projectile have been explained on the basis of the energy defect of the reactions and asymmetric near-resonant charge-exchange process between the ion and target molecule including the Doppler broadening in the interaction of the monoenergetic ion beam and target molecules having an isotropic Maxwellian velocity distribution corresponding to room temperature. Using the semiempirical approach based on the parametrized numerical coupled-channel two-state calculations, we have extrapolated the cross sections to a broader range of velocities.

Towards realistic parametrization of the kinetic anisotropy and the resulting instabilities in space plasmas. Electromagnetic electron–cyclotron instability in the solar wind

Measuredin situ, the particle velocity distributions in the solar wind plasma reveal two distinct components: a Maxwellian (thermal) core, and a less dense but hotter suprathermal halo with a power-law distribution described by Lorentzian/Kappa distribution function. Despite this evidence, the existing attempts to parametrize anisotropic distributions and the resulting wave instabilities are limited to idealized models, which either ignore the suprathermal populations, or minimize the core, assuming it is cold. Here, a more realistic approach is identified, combining an isotropic Maxwellian core and an anisotropic bi-Kappa halo. This model is relevant at large heliocentric distances and for the slow winds, when the field-aligned strahl is less pronounced and kinetic energy densities in the core and halo are comparable. A comparative study with the cold-core-based model is performed on the electron whistler–

ON THE COMPETITION BETWEEN RADIAL EXPANSION AND COULOMB COLLISIONS IN SHAPING THE ELECTRON VELOCITY DISTRIBUTION FUNCTION: KINETIC SIMULATIONS

We present numerical simulations of the solar wind using a fully kinetic model which takes into account the effects of particle's binary collisions in a quasi-neutral plasma in spherical expansion. Starting from an isotropic Maxwellian velocity distribution function for the electrons, we show that the combined effect of expansion and Coulomb collisions leads to the formation of two populations: a collision-dominated cold and dense population almost isotropic in velocity space and a weakly collisional, tenuous field-aligned and antisunward drifting population generated by mirror force focusing in the radially decreasing magnetic field. The relative weights and drift velocities for the two populations observed in our simulations are in excellent agreement with the relative weights and drift velocities for both core and strahl populations observed in the real solar wind. The radial evolution of the main moments of the electron velocity distribution function is in the range observed in the solar wind. The electron temperature anisotropy with respect to the magnetic field direction is found to be related to the ratio between the collisional time and the solar wind expansion time. Even though collisions are found to shape the electron velocity distributions and regulate the properties of the strahl, it is found that the heat flux is conveniently described by a collisionless model where a fraction of the electron thermal energy is advected at the solar wind speed. This reinforces the currently largely admitted fact that collisions in the solar wind are clearly insufficient to force the electron heat flux obey the classical Spitzer-Harm expression where heat flux and temperature gradient are proportional to each other. The presented results show that the electron dynamics in the solar wind cannot be understood without considering the role of collisions.

Numerical computation of Thomson scattering spectra for non-Maxwellian or anisotropic electron distribution functions

Monte Carlo techniques applied to Thomson scattering (TS) power spectrum computation have been extended so as to include non-Maxwellian and anisotropic electron distribution functions (EDFs). First, a simple model has been selected for the spatial (angular) anisotropy of electron velocities (uniformly distributed around an axis of angular symmetry on a cone of semiaperture θanis), while the energies are distributed according to a lower-hybrid-like model function. Spectra have been computed, and their dependence with the EDF model parameters has been given. The most noticeable changes in the spectrum with respect to the isotropic Maxwellian are the broadening and blue-shift of the spectrum due to suprathermal electrons, and the presence of satellite or additional maxima (that can be either red-shifted or blue-shifted with respect to the thermal Maxwellian maximum) coming from the anisotropy of the EDF. Also, extensive numerical computations have been carried out on angularly non-separable EDFs (meaning that the sampling of the distribution function cannot be done independently on angle and energy variables), like the relativistic bi-Maxwellian (with or without drift). The connection of these results with some recent TS measurements reported by Yamada et al (2010 Rev. Sci. Instrum. 81 10D522) and more generally, with the possibility of detecting non-Maxwellian or anisotropic EDF features with TS, has been discussed.

Molecular dynamics study of the processes in the vicinity of the n-dodecane vapour/liquid interface

Molecular dynamics (MD) simulation is used to study the evaporation and condensation of n-dodecane (C12H26), the closest approximation to Diesel fuel. The interactions in chain-like molecular structures are modelled using an optimised potential for liquid simulation (OPLS). The thickness of the transition layer between the liquid and vapour phases at equilibrium is estimated. It is shown that molecules at the liquid surface need to obtain relatively large translational energy to evaporate. The vapour molecules with large translational energy can easily penetrate deeply into the transition layer and condense in the liquid phase. The evaporation/condensation coefficient is estimated and the results are shown to be compatible with the previous estimates based on the MD analysis and the estimate based on the transition state theory. The velocity distribution functions of molecules at the liquid-vapour equilibrium state are found in the liquid phase, interface, and the vapour phase. These functions in the liquid phase and at the interface are shown to be close to isotropic Maxwellian for all velocity components. The velocity distribution function in the vapour phase is shown to be close to bi-Maxwellian with the temperature for the distribution normal to the interface being larger than the one for the distribution parallel to the interface.

A kinetic theory for particulate systems with bimodal and anisotropic velocity fluctuations

Observations of bubbles rising near a wall under conditions of large Reynolds and small Weber numbers have indicated that the velocity component of the bubbles parallel to the wall is significantly reduced upon collision with a wall. To understand the effect of such bubble-wall collisions on the flow of bubbly liquids bounded by walls, a model is developed and examined in detail by numerical simulations and theory. The model is a system of bubbles in which the velocity of the bubbles parallel to the wall is significantly reduced upon collision with the channel wall while the bubbles in the bulk are acted upon by gravity and linear drag forces. The inertial forces are accounted for by modeling the bubbles as rigid particles with mass equal to the virtual mass of the bubbles. The standard kinetic theory for granular materials modified to account for the viscous and gravity forces and supplemented with boundary conditions derived assuming an isotropic Maxwellian velocity distribution is inadequate for describing the behavior of the bubble-phase continuum near the walls since the velocity distribution of the bubbles near the walls is significantly bimodal and anisotropic. A kinetic theory that accounts for such a velocity distribution is described. The bimodal nature is captured by treating the system as consisting of two species with the bubbles (modeled as particles) whose most recent collision was with a channel wall treated as one species and those whose last collision was with another bubble as the other species. The theory is shown to be in very good agreement with the results of numerical simulations and provides closure relations that may be used in the analysis of bidisperse particulate systems as well as bounded bubbly flows.

Coordinated EISCAT Svalbard radar and Reimei satellite observations of ion upflows and suprathermal ions

The relationship between bulk ion upflows and suprathermal ions was investigated using data simultaneously obtained from the European Incoherent Scatter (EISCAT) Svalbard radar (ESR) and the Reimei satellite. Simultaneous observations were conducted in November 2005 and August 2006, and 14 conjunction data sets have been obtained at approximately 630 km in the dayside ionosphere. Suprathermal ions with energies of a few eV were present in the dayside cusp region, and the ion velocity distribution changed from an isotropic Maxwellian near the cusp region to tail heating at energies above a few eV in the cusp region. The velocity distribution of the suprathermal ions has a peak perpendicular or oblique to the geomagnetic field, and the temperature of the suprathermal ions was 0.9–1.4 eV. An increase in the phase space density (PSD) of the suprathermal ions, measured with the Reimei, was correlated with bulk ion upflow observed at the same altitude using EISCAT, and with the energy flux of precipitating electrons with energies of 50–500 eV. The PSD also has a good correlation with the electron temperature, which was increased by precipitation, but not with the ion temperature (0.1–0.3 eV) at the same altitude measured with EISCAT. These results suggest that plasma waves such as broadband extremely low frequency (BBELF) wavefields associated with precipitation are connected to the bulk ion upflows in the cusp and effectively cause the heating of suprathermal ions. The heating of suprathermal ions disagrees with anisotropic heating due to O+−O resonant charge exchange.

Current-voltage and kinetic energy flux relations for relativistic field-aligned acceleration of auroral electrons

Abstract. Recent spectroscopic observations of Jupiter's "main oval" auroras indicate that the primary auroral electron beam is routinely accelerated to energies of ~100 keV, and sometimes to several hundred keV, thus approaching the relativistic regime. This suggests the need to re-examine the classic non-relativistic theory of auroral electron acceleration by field-aligned electric fields first derived by Knight (1973), and to extend it to cover relativistic situations. In this paper we examine this problem for the case in which the source population is an isotropic Maxwellian, as also assumed by Knight, and derive exact analytic expressions for the field-aligned current density (number flux) and kinetic energy flux of the accelerated population, for arbitrary initial electron temperature, acceleration potential, and field strength beneath the acceleration region. We examine the limiting behaviours of these expressions, their regimes of validity, and their implications for auroral acceleration in planetary magnetospheres (and like astrophysical systems). In particular, we show that for relativistic accelerating potentials, the current density increases as the square of the minimum potential, rather than linearly as in the non-relativistic regime, while the kinetic energy flux then increases as the cube of the potential, rather than as the square.

Generating Equilibrium Dark Matter Halos: Inadequacies of the Local Maxwellian Approximation

We describe an algorithm for constructing N-body realizations of equilibrium spherical systems. A general form for the mass density ? is used, making it possible to represent most of the popular density profiles found in the literature, including the cuspy density profiles found in high-resolution cosmological simulations. We demonstrate explicitly that our models are in equilibrium. In contrast, many existing N-body realizations of isolated systems have been constructed under the assumption that the local velocity distribution is Maxwellian. We show that a Maxwellian halo with an initial ?(r) r-1 central density cusp immediately develops a constant-density core. Moreover, after just one crossing time the orbital anisotropy has changed over the entire system, and the initially isotropic model becomes radially anisotropic. These effects have important implications for many studies, including the survival of substructure in cold dark matter (CDM) models. Comparing the evolution and mass-loss rate of isotropic Maxwellian and self-consistent Navarro, Frenk, & White (NFW), satellites orbiting inside a static host CDM potential, we find that the former are unrealistically susceptible to tidal disruption. Thus, recent studies of the mass-loss rate and disruption timescales of substructure in CDM models may be compromised by using the Maxwellian approximation. We also demonstrate that a radially anisotropic, self-consistent NFW satellite loses mass at a rate several times higher than that of its isotropic counterpart on the same external tidal field and orbit.

Gas dynamic trap as high power 14 MeV neutron source

It is now widely recognized that further progress toward a commercial fusion reactor critically depends on the availability of low activated materials to be operated for many years in the fusion neutron environment without degradation of their properties. Therefore, high power neutron source (NS) for extensive material tests is in great demand. The realistic NS can be built on the basis of the gas dynamic trap (GDT) which is one of the simplest system for magnetic plasma confinement. GDT is an axisymmetric mirror machine of the Budker–Post type, but with a high mirror ratio ( R>10), and operated with warm and relatively dense plasma. Thus, due to frequent collisions, the plasma confined in a trap is very close to isotropic Maxwellian, and hence many instabilities are not excited and plasma behavior is similar to a classical one. It is proposed to use the oblique injection of neutral tritium and deuterium beams with an energy of the order of 100 keV into the GDT “warm” target plasma (electron temperature T e≥750 eV) to produce high intensity neutron flux. In this case, due to the ionization and charge-exchange collisions with plasma particles the energetic atoms will turn into ions confined in the trap. The axial density profile of these anisotropic energetic ions will be strongly inhomogeneous. Namely, the maximum of the fast ion density will be located in the vicinities of mirrors and the minimum—at the middle plane of the device. Thus, a strongly inhomogeneous fusion neutron flux, mostly generated in the fast deuterium–tritium (D–T) ion collisions will appear. GDT NS looks to be rather flexible system. It enables to construct the source stepwise, starting from rather moderate parameters. Increasing the testing zone length and neutral beam power one can increase both neutron flux and the flux density. Calculations show that even in the modest version of the NS with T e∼250 eV quite appreciable neutron flux density in the range of 0.2–0.3 MW/m 2 can be produced. At present, the maximum temperature about 130 eV is obtained in the experiments at the model GDT device at Budker INP. Further increase of the electron temperature in the device requires higher neutral beam power and now is considered to be straightforward.

Rheology of suspensions with high particle inertia and moderate fluid inertia

We consider the averaged flow properties of a suspension in which the Reynolds number based on the particle diameter is finite so that the inertia of the fluid phase is important. When the inertia of the particles is sufficiently large, their trajectories, between successive particle collisions, are only weakly affected by the interstitial fluid. If the particle collisions are nearly elastic the particle velocity distribution is close to an isotropic Maxwellian. The rheological properties of the suspension can then be determined using kinetic theory, provided that one knows the granular temperature (energy contained in the particle velocity fluctuations). This energy results from a balance of the shear work with the loss due to the viscous dissipation in the interstitial fluid and the dissipation due to inelastic collisions. We use lattice-Boltzmann simulations to calculate the viscous dissipation as a function of particle volume fraction and Reynolds number (based on the particle diameter and granular temperature). The Reynolds stress induced in the interstitial fluid by the random motion of the particles is also determined. We also consider the case where the interstitial fluid is moving relative to the particles, as would occur if the particles experienced an external body force. Owing to the nonlinearity of the equations of motion for the interstitial fluid, there is a coupling between the viscous dissipation caused by the fluctuating motion of the particles and the drag associated with a mean relative motion of the two phases, and this coupling is explored by computing the dissipation and mean drag for a range of values of the Reynolds numbers based on the mean relative velocity and the granular temperature.

A Coulomb collision model for PIC plasma simulation

A new approach to modeling partially collisional plasmas that provides a smooth transition from the fluid (Coulomb collision dominated) to the fully kinetic PIC (collisionless) limit is presented. In addition to the usual quantities of mass, charge, and velocity, each particle carries an isotropic Maxwellian velocity distribution. Higher resolution of velocity space is achieved by generating more particles using a procedure that preserves the first four velocity moments. This velocity space fragmentation is essential for capturing non-Maxwellian plasma behavior. The model developed here allows the efficient simulation of partially collisional plasmas by reducing both the number of particle pairings required per time step and the number of particles needed to retain non-Maxwellian plasma behavior. The method produces reasonable results when the time step is large relative to the collision frequencies and works in the limit of one particle per species per cell. Particle merging can be exploited to control the number of particles in a natural way. The collision process is fully three-dimensional and conserves energy and momentum exactly. Results from 3v and 1d3v simulations are presented and compared with previous multi-fluid and fully kinetic PIC simulations.

Determination of accurate solar wind electron parameters using particle detectors and radio wave receivers

We present a new, simple, and semiempirical method for determining accurate solar wind electron macroscopic parameters from the raw electron moments obtained from measured electron distribution functions. In the solar wind these measurements are affected by (1) photoelectrons produced by the spacecraft illumination, (2) spacecraft charging, and (3) the incomplete sampling of the electron distribution due to a nonzero low‐energy threshold of the energy sweeping in the electron spectrometer. Correcting fully for these effects is difficult, especially without the help of data from other experiments that can be taken as a reference. We take here advantage of the fact that high‐resolution solar wind electron parameters are obtained on board Wind using two different instruments: the electron electrostatic analyzer of the three‐dimensional Plasma experiment (3DP), which provides 3‐D electron velocity distribution functions every 99 s as well as 3‐s resolution computed onboard moments, and the thermal noise receiver (TNR), which yields unbiased electron density and temperature every 4.5 s from the spectroscopy of the quasi‐thermal noise around the electron plasma frequency. The present correction method is based on a simplified model evaluating the electron density and temperature as measured by the electron spectrometer, by taking into account both the spacecraft charging and the low‐energy cutoff effects: approximating the solar wind electron distributions by an isotropic Maxwellian, we derive simple analytical relations for the measured electron moments as functions of the real ones. These relations reproduce the qualitative behavior of the variation of the raw 3DP electron density and temperature as a function of the TNR ones. In order to set up a precise “scalar correction” of the raw 3DP electron moments, we use the TNR densities and temperatures as good estimates of the real ones; the coefficients appearing in the analytical relations are obtained by a best fit to the data from both instruments during a limited period of time, chosen as a reference. This set of coefficients is then used as long as the mode of operation of the electron spectrometer is unchanged. We show that this simple scalar correction of the electron density and temperature is reliable and can be applied routinely to the high‐resolution 3DP low‐order moments. As a by‐product, an estimate of the spacecraft potential is obtained. The odd‐order moments of the distribution function (electron bulk speed and heat flux) cannot be corrected by the model since the distribution is assumed to be an isotropic Maxwellian. We show, however, that a better estimate of the electron heat flux can be obtained by replacing the electron velocity by the proton velocity.