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# Self-consistent collisional closures for high-order fluid moment models with the electron inertial terms

A. Alvarez Laguna, Kentaro Hara

► To cite this version:

A. Alvarez Laguna, Kentaro Hara. Self-consistent collisional closures for high-order fluid moment models with the electron inertial terms. The 38th International Electric Propulsion Conference, Electric Rocket Propulsion Societ, Jun 2024, Toulouse, France. hal-05070326

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https://hal.science/hal-05070326v1

Submitted on 16 May 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Self-consistent collisional closures for high-order fluid
moment models with the electron inertial terms

IEPC-2024-685

Presented at the 38th International Electric Propulsion Conference, Toulouse, France
June 23-28, 2024

A. Alvarez Laguna*

Laboratoire de physique des plasmas, Centre National de la Recherche
Scientifique, Ecole polytechnique, Institut polytechnique de Paris

and

K. Hara†

Department of Aeronautics & Astronautics, Stanford University

Most of the fluid models for electrons in a partially-ionized plasma consider the electrons as having a small drift with respect to the other species. In this paper, we study a non-linear moment model based on a Hermitian expansion that considers 14 scalar fields (density, momentum, pressure tensor, heat-flux vector, and kurtosis). The main collisional processes in a partially-ionized plasma are taken into account (elastic collisions with electrons, ions and gas atoms, excitation and ionization collisions), by considering the electrons to have a velocity that is arbitrarily high with respect to the other species. The collisional exchanges are integrated analytically in a self-consistent manner with a non-equilibrium distribution function. We discuss about the validity of the drift-diffusion approximation by comparing two-term Boltzmann solutions to Monte-Carlo collision simulations. Finally, we compare our moment model to Monte-Carlo collision simulation under high electric fields, where the drift-diffusion approximation fails. The closure moment model can be applied to low-pressure discharges with high electric fields, where the electron inertial terms and the non-equilibrium effects may play an important role.

<footnote>*CNRS Researcher, Laboratoire de Physique des plasmas, alvarez@lpp.polytechnique.fr</footnote>

<footnote>†Assistant Professor, Department of Aeronautics & Astronautics, kenhara@stanford.edu</footnote>
Nomenclature

<table>
  <tr>
    <td>f<sub>e</sub></td>
    <td>= electron distribution function</td>
  </tr>
  <tr>
    <td>e, i, g</td>
    <td>= subindex corresponding to electrons, ions, and gas atoms (or molecules)</td>
  </tr>
  <tr>
    <td>el, inel, iz</td>
    <td>= superindex corresponding to elastic, inelastic, and ionization collisions</td>
  </tr>
  <tr>
    <td>v, x</td>
    <td>= velocity and physical space variables</td>
  </tr>
  <tr>
    <td>e</td>
    <td>= elementary charge</td>
  </tr>
  <tr>
    <td>m<sub>e</sub></td>
    <td>= electron mass</td>
  </tr>
  <tr>
    <td>E</td>
    <td>= electric field vector</td>
  </tr>
  <tr>
    <td>B</td>
    <td>= magnetic field vector</td>
  </tr>
  <tr>
    <td>C (f<sub>e</sub>)</td>
    <td>= electron collision source in kinetic equation</td>
  </tr>
  <tr>
    <td>&psi;</td>
    <td>= moment weight</td>
  </tr>
  <tr>
    <td>h<sup>(2n)</sup>, h<sub>r</sub><sup>(2n+1)</sup>, h<sub>rs</sub><sup>(2n)</sup></td>
    <td>= irreducible scalar, vectorial, and second rank tensorial Hermitian moments</td>
  </tr>
  <tr>
    <td>H<sup>(2n)</sup>, H<sub>r</sub><sup>(2n+1)</sup>, H<sub>rs</sub><sup>(2n)</sup></td>
    <td>= irreducible scalar, vectorial, and second rank tensorial Hermitian polynomials</td>
  </tr>
  <tr>
    <td>c<sub>e</sub></td>
    <td>= electron peculiar velocity</td>
  </tr>
  <tr>
    <td>&delta;<sub>ij</sub></td>
    <td>= Kronecker delta</td>
  </tr>
  <tr>
    <td>n<sub>e</sub></td>
    <td>= electron particle density</td>
  </tr>
  <tr>
    <td>u<sub>e</sub></td>
    <td>= electron mean velocity vector</td>
  </tr>
  <tr>
    <td>p<sub>e</sub></td>
    <td>= electron isotropic pressure</td>
  </tr>
  <tr>
    <td>P<sub>ej</sub></td>
    <td>= electron anisotropic pressure tensor</td>
  </tr>
  <tr>
    <td>&tau;<sub>ij</sub></td>
    <td>= electron pressure deviator</td>
  </tr>
  <tr>
    <td>T<sub>e</sub></td>
    <td>= electron isotropic temperature</td>
  </tr>
  <tr>
    <td>T<sub>ej</sub></td>
    <td>= electron anisotropic temperature tensor</td>
  </tr>
  <tr>
    <td>q<sub>e</sub></td>
    <td>= electron heat-flux vector</td>
  </tr>
  <tr>
    <td>p<sup>(4)</sup><sub>ijj</sub></td>
    <td>= electron contracted fourth-order moment</td>
  </tr>
  <tr>
    <td>&Delta;<sub>e</sub></td>
    <td>= electron normalized excess kurtosis</td>
  </tr>
  <tr>
    <td>q<sub>ijk</sub>, r<sub>ikll</sub>, r<sub>ijklj</sub></td>
    <td>= third, fourth-order and fifth-order moment closure fluxes</td>
  </tr>
  <tr>
    <td>&dot;n<sub>e</sub>, R<sub>i</sub>, Q<sub>ij</sub>, R<sub>i</sub><sup>hf</sup>, Q<sup>(4)</sup></td>
    <td>= moment collisional exchanges</td>
  </tr>
  <tr>
    <td>F<sup>0</sup>, F<sup>1</sup></td>
    <td>= zeroth and first spherical harmonics expansion distribution functions</td>
  </tr>
  <tr>
    <td>M<sub>e</sub></td>
    <td>= Mach number</td>
  </tr>
  <tr>
    <td>El</td>
    <td>= electric field parameter non-dimensional number</td>
  </tr>
</table>
# I. Introduction

UNDERSTANDING the transport processes in multicomponent plasmas is a fundamental problem that concerns all areas of plasma physics. In particular, in Hall thrusters, the interplay of the charged species with the electromagnetic field, together with the collisions between charged species and the gas atoms, results in complex transport phenomena, beyond the classical transport theory.¹

In this paper, we tackle the collisional transport of electrons in a partially-ionized plasma. In particular, we treat the collisional transport when the distribution function is out of equilibrium, i.e., non-Maxwellian, and the electron mean velocity is large as compared to the rest of species. Classical collisional transport models, e.g., the seminal works by Braginskii,² Balescu³ or Zhdanov,⁴ consider distribution function close to equilibrium and neglect the electron inertial terms in order to derive the transport coefficients for the transport of density, mass and momentum of electrons. This assumption (also done more recently in the works with non-Maxwellian energy distribution functions⁵,⁶) greatly simplifies the transport system as well as the integration of the collisional exchanges.

The development of analytical self-consistent transport models for multi-component plasmas is complex even when the plasma is close to thermodynamic equilibrium (we refer the reader to the classical works already mentioned above²⁻⁴,⁷). In particular, when the plasma is partially-ionized, the inelastic processes need to be taken into account, which increases the complexity of the underlying kinetic equation (see, e.g., the works by Zhdanov⁸ or Alvarez Laguna⁵,⁶).

Close to thermodynamic equilibrium, the most common method to derive analytical transport models is the Chapman-Enskog expansion.⁷,⁹ This method considers that the distribution function of the different species in the plasma is close to thermodynamic equilibrium. In particular, due to the disparity between the electron and the ion and atom masses, the equilibrium distribution function of electrons is at a different temperature than this of the heavy species. In order to include the transport processes (diffusion of particles, electrical conduction, thermal conduction, viscosity), the Chapman-Enskog expansion considers a small perturbation of the distribution function from the equilibrium. In low-pressure discharges, the distribution function of the different species are usually far from equilibrium and, for this reason, such a method might simplify some of the actual transport processes in low-pressure partially-ionized plasmas.

Far from thermodynamic equilibrium, partially-ionized plasmas have been traditionally studied with the two-term spherical harmonics expansion (SHE) Boltzmann equation.¹⁰,¹¹ With the spatially-homogenous solution of the two-term SHE kinetic equation, the electron transport coefficients and the collision rates are computed. After, this spatially-homogenous solution is used in the electron fluid equations in the drift-diffusion approximation. By doing this, there are two main hypothesis, that the electric field is locally in equilibrium with the collisional processes (so we can use a spatially-homogeneous solution for the transport) and that the electron inertial terms are negligible (so we can use the drift-diffusion approximation).

By considering high-order moment hierarchies, we aim at overcoming the above-mentioned limitations of Chapman-Enskog method and the local-field models (See Fig. 1). First, we will derive transport models that are not based on the spatially-homogeneous two-term Boltzmann. Instead, the transport properties are based on the resolution of high-order moment equations. This allows to consider non-equilibrium distribution functions without the local-field approximation. Second, we will include the inertial terms in the development of our transport model, as they can play an important role in low-pressure discharges.¹²⁻¹⁵ The transport models derived in this work can be relevant for full-fluid moment models for Hall thrusters.

<figure><img src="image_3.png" /><figcaption>Figure 1: Validity of the plasma models depending on the collisionality.</figcaption></figure>
The paper is organized as follows. First, we present the electron kinetic equation and the general moment equation. We discuss about the Hermitian moment closures and a Grad's 14-M closure. We present the moment equations as well as the closure fluxes. Second, we discuss about the validity of the drift-diffusion approximation. To do this we compare two-term Boltzmann simulations with MCC. We note that the two-term Boltzmann and hence the drift-diffusion approximation starts diverging from the kinetic solution for Mach numbers above ~ $10^{-1}$. Finally, we present the solution of the spatially-homogenous 14-M equations compared to the MCC solver for high reduced fields.

## II. Kinetic equation and high-order moment equations

We consider the electron kinetic equation in a partially-ionized plasma, which reads,

$$ \frac{\partial f_e}{\partial t} + \mathbf{v} \cdot \nabla f_e - e \frac{\mathbf{E} + \mathbf{v} \times \mathbf{B}}{m_e} \cdot \nabla \mathbf{v} f_e = C(f_e). \quad (1) $$

In partially-ionized plasmas, we consider electron-electron, electron-heavy species elastic collisions, electron-heavy species inelastic collisions and electron-gas ionization collisions, as follows,

$$ C(f_e) = \sum_{\alpha \in \{e, i, g\}} C(f_e, f_\alpha)|^{el} + C(f_e, f_g)|^{inel} + C(f_e, f_g)|^{iz}. \quad (2) $$

In this work, we consider the Boltzmann collision operator for the elastic collisions, the Wang Chang-Uhlenbeck operator for the inelastic collisions,¹⁶ and the Alexeev-Giovanigli operator for the ionization collisions,¹⁷,¹⁸ under the approximation of negligible recombination.⁵ We refer to these references for a detailed expression of the collisional source terms of the kinetic equation.

The velocity moment models, consider macroscopic equations that are obtained by integrating the kinetic equation in the full velocity space. The moment equations can be obtained with the general transfer equation, as follows,

$$ \frac{\partial}{\partial t} \int \psi f_e d\mathbf{v} + \frac{\partial}{\partial x_i} \int \psi v_i f_e d\mathbf{v} - \frac{F_i}{m} \int f_e \frac{\partial \psi}{\partial v_i} d\mathbf{v} = \int \psi C(f_e) d\mathbf{v}. \quad (3) $$

The weight of the moments $\psi(\mathbf{v})$ is a monomial of the velocity. Note, that we use the Einstein notation for the tensor notation.

Eq. (3) presents two main difficulties. First, the flux term, i.e., the second term in the left-hand-side of the equality, depends on the solution of the subsequent moment. This is the so-called the closure problem. Second, the right-hand-side term of the equality needs the integration of the collisional source terms of the kinetic equation. For both difficulties, we need to parametrize the non-equilibrium distribution function as a function of the moments. One way to do it is the so-called Grad's method.¹⁹

The Grad's method assumes the non-equilibrium distribution function to be an expansion in irreducible Hermite polynomials. The irreducible Hermite polynomials form a complete base in the velocity space. For this reason, without loss of generality, the distribution function is described as a Maxwellian distribution function that is perturbed by a polynomial expansion of Hermite polynomials, which in the case of irreducible Hermite polynomials reads

$$ f_e(\mathbf{x}, \mathbf{c}_e, t) = f_e^{(0)}(c_e, \mathbf{x}, t) \left( 1 + \sum_{n=0}^{\infty} h^{(2n)} H^{(2n)}(\mathbf{c}_e) + \sum_{n=0}^{\infty} h_r^{(2n+1)} H_r^{(2n+1)}(\mathbf{c}_e) \right. \\ \left. \qquad + \sum_{n=1}^{\infty} h_{rs}^{(2n)} H_{rs}^{(2n)}(\mathbf{c}_e) + \dots \right), \quad (4) $$

with the Maxwellian distribution function

$$ f_e^{(0)}(\mathbf{x}, \mathbf{c}_e, t) = n_e \left(\frac{\beta_e}{\pi}\right)^{3/2} e^{-\beta_e c_e^2} \quad \text{with} \quad \beta_e = \frac{m_e}{2eT_e}. \quad (5) $$

Here $n_e$ is the electron number density, $T_e$ is the electron temperature, $h^{(2n)}$, $h_r^{(2n+1)}$, and $h_{rs}^{(2n)}$ are the scalar, vectorial and tensorial moments and $H^{(2n)}$, $H_r^{(2n+1)}$, and $H_{rs}^{(2n)}$ are the scalar, vectorial and tensorial
irreducible Hermite polynomials. The peculiar velocity is defined as $c_e = v - u_e$, where $u_e$ is the electron mean velocity, computed from the distribution function as

$$ u_e = \frac{1}{n_e} \int v f_e dv \quad \text{with} \quad n_e = \int f_e dv. \qquad (6) $$

## A. The 14-Moment equations

The 14-M Grad non-equilibrium distribution function considers the following Hermitian expansion,

$$ f_{\epsilon}^{(14M)}(\mathbf{x}, c_{\epsilon}, t) = f_{\epsilon}^{(0)}(\mathbf{x}, c_{\epsilon}, t) \left[ 1 + \frac{2\beta_{\epsilon}^2}{\rho_{\epsilon}} \tau_{ij} c_{\epsilon_i} c_{\epsilon_j} + \frac{8\beta_{\epsilon}^2}{5\rho_{\epsilon}} q_{\epsilon_i} c_{\epsilon_i} \left( \beta_{\epsilon} c_{\epsilon}^2 - \frac{5}{2} \right) + \left( \frac{15}{8} - \frac{5\beta_{\epsilon}}{2} c_{\epsilon}^2 + \frac{\beta_{\epsilon}^2}{2} c_{\epsilon}^4 \right) \Delta_{\epsilon} \right]. \quad (7) $$

where the pressure deviator and anisotropic temperature are defined by the following relations

$$ p_{\epsilon} = \frac{1}{3} \int m_{\epsilon} c_{\epsilon}^{2} f_{\epsilon} dv, \quad \tau_{ij} = P_{\epsilon_{ij}} - p \delta_{ij}, \quad T_{\epsilon_{ij}} = \frac{P_{\epsilon_{ij}}}{n_{\epsilon} e}, \quad \text{and} \quad p_{\epsilon} = \frac{1}{3} P_{\epsilon_{ii}}, \qquad (8) $$

and the normalized excess kurtosis as

$$ \Delta_{\epsilon} = \frac{p_{ijj}^{(4)} - \frac{15 ne^{2}T_{\epsilon}^{2}}{2m}}{\frac{15 ne^{2}T_{\epsilon}^{2}}{2m}}. \qquad (9) $$

In Fig. 2, we present an example of the non-equilibrium distribution function corresponding to a 14-Moment (14-M) Hermitian model. In this figure, we see that when the distribution function has a non-zero heat-flux, the distribution function is skewed. Alternatively, the contracted fourth-order moment results in perturbations in the electron energy distribution function (EEDF). That is, the depletion of the distribution function in the tail is captured by the fourth-order moment. Finally, the pressure deviator, controls the anisotropy in the temperature of the distribution function.

<figure><img src="image_4.png" /><figcaption>Figure 2: Example of non-equilibrium distribution functions in the 14-Moment model.</figcaption></figure>
Below, we present the evolution equations of the 14-M model without magnetic field. In order to obtain the transport equations, we consider the following rates in the Eq. (3),

$$ \psi^{(14M)} = \left(1, m_e v_i, m_e v_i v_j, \frac{m_e}{2} v_i v^2, \frac{m_e}{2} v^4\right)^T. \quad (10) $$

The resulting system of 14-M equations that consider the balance laws for particle density, momentum, total energy tensor, total heat-flux, and total contracted fourth order moment, reads

$$ \frac{\partial n_e}{\partial t} + \frac{\partial}{\partial x_i} (n_e u_{\epsilon_i}) = \dot{n}_e \quad (11) $$

$$ \frac{\partial}{\partial t} (m_e n_e u_{\epsilon_i}) + \frac{\partial}{\partial x_j} (m_e n_e u_{\epsilon_i} u_{\epsilon_j} + P_{\epsilon_{ij}}) = -en_e E_i + R_i, \quad (12) $$

$$ \frac{\partial}{\partial t} (m_e n_e u_{\epsilon_i} u_{\epsilon_j} + P_{\epsilon_{ij}}) + \frac{\partial}{\partial x_k} (2q_{\epsilon_{ijk}} + P_{\epsilon_{ij}} u_{\epsilon_k} + P_{\epsilon_{jk}} u_{\epsilon_i} + P_{\epsilon_{ik}} u_{\epsilon_j} + m_e n_e u_{\epsilon_i} u_{\epsilon_j} u_{\epsilon_k}) = -en_e (E_i u_{\epsilon_j} + E_j u_{\epsilon_i}) + Q_{ij} \quad (13) $$

$$ \frac{\partial}{\partial t} \left( \frac{1}{2} m_e n_e u_{\epsilon}^2 u_{\epsilon_i} + P_{\epsilon_{ij}} u_{\epsilon_j} + \frac{1}{2} P_{\epsilon_{jj}} u_{\epsilon_i} + q_{\epsilon_i} \right) + \frac{\partial}{\partial x_k} \left( r_{\epsilon_{ikll}} + \frac{1}{2} P_{\epsilon_{ik}} u_{\epsilon}^2 + P_{\epsilon_{ij}} u_{\epsilon_j} u_{\epsilon_k} \right. \\ \left. + P_{\epsilon_{jk}} u_{\epsilon_i} u_{\epsilon_k} + \frac{1}{2} P_{\epsilon_{jj}} u_{\epsilon_i} u_{\epsilon_k} + q_{\epsilon_k} u_{\epsilon_i} + q_{\epsilon_i} u_{\epsilon_k} + 2q_{\epsilon_{ijk}} u_{\epsilon_j} + \frac{1}{2} m_e n_e u_{\epsilon}^2 u_{\epsilon_i} u_{\epsilon_k} \right) = \\ - en_e \left( E_j u_{\epsilon_i} u_{\epsilon_j} + \frac{1}{2} E_i u_{\epsilon}^2 + \frac{P_{\epsilon_{ij}}}{m_e n_e} E_j + \frac{P_{\epsilon_{jj}}}{2m_e n_e} E_i \right) + R_i^{hf} \quad (14) $$

$$ \frac{\partial}{\partial t} \left( \frac{1}{2} m_e n_e u_{\epsilon}^4 + P_{\epsilon_{ii}} u_{\epsilon}^2 + 2P_{\epsilon_{ij}} u_{\epsilon_i} u_{\epsilon_j} + 4q_{\epsilon_i} u_{\epsilon_i} + p_{\epsilon_{iijj}} \right) + \frac{\partial}{\partial x_k} \left( r_{\epsilon_{ijjk}} + 4u_{\epsilon_i} r_{\epsilon_{ikjj}} \right. \\ \left. + u_{\epsilon_k} p_{\epsilon_{ijjj}} + 4u_{\epsilon_j} u_{\epsilon_i} q_{\epsilon_i} + 4u_{\epsilon_i} u_{\epsilon_k} q_{\epsilon_k} + 2u_{\epsilon_i} u_{\epsilon_i} u_{\epsilon_k} P_{\epsilon_{ij}} \right. \\ \left. + 2u_{\epsilon_i} u_{\epsilon_i} u_{\epsilon_j} P_{\epsilon_{jk}} + u_{\epsilon_k} u_{\epsilon_i} u_{\epsilon_i} P_{\epsilon_{jj}} + \frac{1}{2} m_e n_e u_{\epsilon_k} u_{\epsilon}^4 \right) = \\ - en_e E_k \left( 2u_{\epsilon_j} u_{\epsilon_j} u_{\epsilon_k} + 2 \frac{P_{\epsilon_{jj}}}{m_e n_e} u_{\epsilon_k} + 4 \frac{P_{\epsilon_{jk}}}{m_e n_e} u_{\epsilon_j} + 4 \frac{q_{\epsilon_k}}{m_e} \right) + Q^{(4)} \quad (15) $$

With the non-equilibrium distribution function of Eq. (7), we can compute the closure fluxes as a function of the fluid variables, as follows,

$$ q_{\epsilon_{ijk}} = \frac{m_e}{2} \int c_i c_j c_k f_e dc = \frac{1}{5} (q_{\epsilon_i} \delta_{jk} + q_{\epsilon_j} \delta_{ik} + q_{\epsilon_k} \delta_{ij}), \quad (16) $$

$$ r_{\epsilon_{ikll}} = \frac{m_e}{2} \int c_i c_k c^2 f_e dc = \frac{1}{3} p_{\epsilon_{kkll}} \delta_{ik} + \frac{7}{2} \frac{p_e}{m_e n_e} \tau_{ik}, \quad (17) $$

$$ r_{\epsilon_{iijjk}} = \frac{m_e}{2} \int c_k c^4 f_e dc = 14 \frac{p_e}{m_e n_e} q_{\epsilon_k} \quad (18) $$

Note that the Hermitian closure provides different closure fluxes than those proposed by the Maximum-Entropy theory.²⁰

The integration of the collisional terms of the equations (11)-(15) (last term of the right-hand-side of the equations) is not straightforward (as shown in previous work⁵) and it will be detailed in a separate paper. The main novelty of the results presented in this paper, as compared to previous work,⁵ is that we resolve the equations of the anisotropic pressure tensor as well as we consider the inertial terms in the integration of the collisional exchanges.
### III. On the validity of the drift-diffusion approximation for electrons

In this section we discuss on the validity of the drift-diffusion approximation in electrons in spatially-homogeneous solutions. We consider the kinetic equation shown above of a partially-ionized plasma (i.e., we consider Coulomb collisions) with an ionization degree of $10^{-2}$ in an argon plasma. We compare the solution of a two-term Boltzmann solver¹¹ and a Monte-Carlo collision simulation with the null-collision method for the collisions between the electrons and the gas and a Nanbu collision method for the Coulomb collisions.²¹ In this example we neglect the electron-ion collisions and we consider the ionization collisions as inelastic collisions.

In order to analyze our results, we define two non-dimensional numbers. The Mach number and the ratio between the electric force and a characteristic electron-atom collision drag force (denoted in the following as electric field parameter), as follows,

$$ \text{Mach number} \quad M_e = \frac{u_e}{V_{th_e}} \qquad \text{Electron mean velocity/electron thermal velocity} \quad (19) $$

$$ \text{Electric parameter} \quad E_l = \frac{eE/m_e}{n_g V_{th_e}^2 \sigma_{eg}} \qquad \text{Electric field force/characteristic collision drag force} \quad (20) $$

<figure><img src="image_5.png" /><figcaption>Figure 3: Comparison between a two-term Boltzmann solver and a Monte-Carlo collision solver for electrons in a spatially-homogenous argon plasma. On the left, we present the electron temperature for different reduced fields, including the non-isotropic temperature tensor. On the center, we present the heat-flux vector for different reduced fields. On the right, we present the non-dimensional numbers for different reduced fields.</figcaption></figure>

In Fig. 3, we compare the solution of the two-term solver and the MCC solver. The two solutions consider an spatially-homogenous plasma with an imposed electric field. The MCC solution is run until convergence with $10^5$ particles and a $\Delta t < 0.05 \nu_{max}^{-1}$ where $\nu_{max}$ is the largest of the collision frequencies.

In this comparison, we see that the results of temperature and heat-flux are identical for reduced fields that are below $E/N < 100$ Td. However, for larger reduced fields, we find that there is a discrepancy in the representation of the macroscopic variables. In particular, the temperature becomes anisotropic, and, therefore, this effect cannot be captured by the two-term Boltzmann approximation. In addition, the representation of the heat-flux of the two-term Boltzmann is becomes incorrect at large electric fields.

In Fig. 3, we also present the non-dimensional numbers as a function of the reduced field. First, we note that both non-dimensional numbers are of the same order and that they evolve similarly with the electric field. In addition, we note that when the Mach number becomes larger than $M_e > 10^{-1}$ the two-term Boltzmann and the MCC solutions start to diverge.

In Fig. 4, we compare the EEDF and the first perturbation distribution function as resolved by the two-term Boltzmann and the MCC code for two large reduced fields, $E/N = 500$ Td and $E/N = 1000$ Td. In both cases, we note that the tail of the distribution function is underestimated by the two-term Boltzmann. This leads to the difference in the macroscopic variables that we have shown above.

The source of this error in the two-term Boltzmann solution can be explained as follows. The two-term Boltzmann approximation needs that $F^0/F^1 \ll 1$. That is, that the isotropic part of the distribution function needs to be much smaller than the anisotropic part. However, when the drift velocity becomes large, i.e., $M_e > 10^{-1}$, the anisotropic part become as large as the isotropic part for low energies. This is shown in the right panels of Fig. 4. We can note that the ratio between the isotropic and anisotropic part of the distribution function becomes of order one, i.e., $F^0/F^1 \sim 1$ even at low energies, i.e., $\sim 20$ eV.
<figure><img src="image_6.png" /><figcaption>Figure 4: Comparison between a two-term Boltzmann solver and a Monte-Carlo collision distribution functions for $E/N = 500$ Td and $E/N = 1000$ Td.</figcaption></figure>

With this comparison, we can conclude that for large electric fields, the drift velocity becomes large enough so the distribution function cannot be approximated as a large isotropic function with a small anisotropic perturbation. This can be characterized in spatially-homogenous regimes with two non-dimensional numbers, the electric field parameter and the Mach number. When these two parameters are of order $10^{-1}$, the two-term approximation starts diverging from the actual solution. Similarly, the fluid models based on the drift-diffusion approximation, will start diverging from the solution as the inertial terms will start playing an important role. We will present this in the next section.

## IV. Solution of the 14-M moment equations with the electron inertial terms

In this section we present one example of the resolution of the non-linear system of equation (11)-(15) in a spatially-homogenous plasma. As done in the previous section, we compare to the MCC solution, in the setup explained above.

In Fig. 5, we present the evolution of the first 5 moments (velocity, temperature tensor, heat-flux vector and normalized excess kurtosis) in time as solved by the moment hierarchy and the MCC solver. First, we note that the non-linear moment model captures remarkably well the temporal evolution of the moments, including the higher-order moments, i.e., the pressure anisotropy, heat-flux and excess kurtosis. The solution is not only accurate in the steady-state regime, but also captures the evolution of the transient.

In addition, with the moments and Eq. (7), we can reconstruct the distribution function in velocity space. In Fig. 5, we present a comparison with the distribution function of the MCC solution. We note that the moment method is able to capture the high-energy tail, as well as the small asymmetry in the $v_x$ space (which corresponds to the heat-flux). Similarly, the moment solution is able to capture that the distribution function has different temperatures depending on the direction (hotter in the $x$ direction than in the $y$). This is an effect of the large electric field in the $x$ direction.

Finally, we also present the results compared to a 5-M model that solves only for mass, momentum and energy. This model also includes the inertial terms in the collisional exchanges. The modified 5-M model is able to capture well the solved moments, despite not being able to capture the high-energy tail nor the transport of heat-flux as the distribution function of the 5-M model is a simple drifting Maxwellian.

This results show that a moment model that solves for the high-order moment equations and consider the inertial terms is able to represent the transport at high electric fields in spatially-homogeneous solutions.
In particular, we note that this solution also is used to validate the analytical integration of the collision
terms, which will be presented in a separate paper.

<figure><img src="image_7.png" /><figcaption>Figure 5: Comparison between a Monte-Carlo solution and the 14-M solution at $E/N = 1000$ Td.</figcaption></figure>

V. Conclusions

In this paper, we have presented a high-order moment model that represents electrons under conditions
that are far from thermodynamic equilibrium in partially-ionized plasmas. As discussed in the introduction,
analytical models far from the thermodynamic equilibrium are complex, in particular when the electrons
have a large drift and the distribution function cannot be considered as mostly isotropic in velocity space.
In order to study this problem, we consider a 14-M model that solves for moments up to the fourth-order
moment.

We have compare the spatially-homogeneous kinetic solution, obtained with an MCC solver, to a two-
term Boltzmann solver. We have shown that the two-term Boltzmann solution starts diverging when the
Mach number of electrons becomes of the order $M_e \sim 10^{-1}$. Similarly, we have defined a non-dimensional
number that characterizes the order of magnitude of the electric field as compared to the collisional term.
This non-dimensional number is of the order of the Mach at the studied conditions and also characterizes
the validity of the two-term Boltzmann solution.

Finally, we have presented a solution of the 14-M system of equations under spatially-homogeneous
solutions. The solution is shown at high fields, where the two-term Boltzmann was not able to correctly
represent the pressure-tensor or the heat-flux vector. In our model, we include the inertial terms and we
validate the collisional integrals. The moment model solution shows to be able to represent the pressure
anisotropy as well as the asymmetry in the distribution function and the high-energy tail of the EEDF. This
results can be useful for low-pressure discharges with strong electric fields.

References

<footnote n="1">¹K. Hara. An overview of discharge plasma modeling for hall effect thrusters. *Plasma Sources Sci. Techn.*., 28(4):044001, 2019.</footnote>

<footnote n="2">²S. I. Braginskii. Transport Processes in a Plasma. *Rev. Plasma Physics*, 1:205, 1965.</footnote>
³R. Balescu. The classical transport theory. In *Classical Transport*, Transport Processes in Plasmas, pages 211–276. North-Holland, 1988.

⁴V. M. Zhdanov. *Transport Processes in Multicomponent Plasma*, volume 44. 2002.

⁵A. Alvarez Laguna, B. Esteves, A. Bourdon, and P. Chabert. A regularized high-order moment model to capture non-maxwellian electron energy distribution function effects in partially ionized plasmas. *Phys. Plasmas*, 29(8):083507, 2022.

⁶A. Alvarez Laguna, B. Esteves, J.-L. Raimbault, A. Bourdon, and P. Chabert. Discussion on the transport processes in electrons with non-maxwellian energy distribution function in partially-ionized plasmas. *Plasma Phys. and Controlled Fusion*, 65(5):054002, 2023.

⁷J. H. Ferziger and H. G. Kaper. *Mathematical Theory of Transport Processes in Gases*. North-Holland Publishing Company, 1972.

⁸V. M. Zhdanov and A. A. Stepanenko. Kinetic theory of transport processes in partially ionized reactive plasma, ii: Electron transport properties. *Physica A*, 461:310–324, 2016.

⁹S. Chapman and T. G. Cowling. *The mathematical theory of non-uniform gases. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases*. 1970.

¹⁰W. P. Allis. Handbuch der physik. Springer-Verlag, Berlin, 21:383–444, 1956.

¹¹G. Hagelaar and L. Pitchford. Solving the boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models. *Plasma Sources Sci. Techn.*, 14:722–733, 2005.

¹²R. Sahu, A. R. Mansour, and K. Hara. Full fluid moment model for low temperature magnetized plasmas. *Phys. Plasmas*, 27(11):113505, 2020.

¹³A. Alvarez-Laguna, T. E. Magin, M. Massot, A. Bourdon, and P. Chabert. Plasma-sheath transition in multi-fluid models with inertial terms under low pressure conditions: comparison with the classical and kinetic theory. *Plasma Sources Sci. Tech.*, 29(2):025003, 2020.

¹⁴A. R. Mansour and K. Hara. Full fluid moment modeling of rotating spokes in penning-type configuration. *Plasma Sources Sci. Techn.*, 31(5):055012, 2022.

¹⁵Derek A. Kuldinow, Yusuke Yamashita, Adnan R. Mansour, and Kentaro Hara. Ten-moment fluid model with heat flux closure for gasdynamic flows. *Journal of Computational Physics*, 508:113030, 2024.

¹⁶C. S. Wang Chang and G. E. Uhlenbeck. Transport phenomena in polyatomic gases. *University of Michigan Research Report N CM-681*, 1951.

¹⁷B. V. Alexeev, A. Chikhaoui, and I. T. Grushin. Application of the generalized chapman-enskog method to the transport-coefficient calculation in a reacting gas mixture. *Phys. Rev. E*, 49:2809–2825, 1994.

¹⁸A. Ern and V. Giovangigli. The kinetic chemical equilibrium regime. *Physica A*, 260(1):49–72, 1998.

¹⁹H. Grad. On the kinetic theory of rarefied gases. *Commun. Pure Appl. Math.*, 2(4):331–407, 1949.

²⁰S. Boccelli, F. Giroux, T. E. Magin, C. P. T. Groth, and J. G. McDonald. A 14-moment maximum-entropy description of electrons in crossed electric and magnetic fields. *Phys. Plasmas*, 27(12):123506, 2020.

²¹K. Nanbu. Theory of cumulative small-angle collisions in plasmas. *Phys. Rev. E*, 55:4642–4652, Apr 1997.