A consistent kinetic model for a two-component mixture with an application to plasma

We consider a non reactive multi component gas mixture.We propose a class of models, which can be easily generalized to multiple species. The two species mixture is modelled by a system of kinetic BGK equations featuring two interaction terms to account for momentum and energy transfer between the species. We prove consistency of our model: conservation properties, positivity of the solutions for the space homogeneous case, positivity of all temperatures, H-theorem and convergence to a global equilibrium in the space homogeneous case in the form of a global Maxwell distribution. Thus, we are able to derive the usual macroscopic conservation laws. In particular, by considering a mixture composed of ions and electrons, we derive the macroscopic equations of ideal MHD from our model.


Introduction
In this paper we shall concern ourselves with a kinetic description of gases. This is traditionally done via the Boltzmann equation for the density distributions f 1 and f 2 . Under certain assumptions the complicated interaction terms of the Boltzmann equation can be simplified by a so called BGK approximation, consisting of a collision frequency multiplied by the deviation of the equilibrium distribution from f 1 respective f 2 . This approximation should be constructed in a way such that it has the same main properties of the Boltzmann equation namely conservation of mass, momentum and energy, further it should have an H-theorem with its entropy inequality and the equilibrium must still be Maxwellian.
Here we shall focus on gas mixtures modelled via a BGK approach. In the literature one can find two types of models. Just like the Boltzmann equation for gas mixtures contains a sum of collision terms on the right-hand side, one type of BKG models also has a sum of BGK-type interaction terms in the relaxation operator. Examples are the models of Gross and Krook [17], Hamel [18], Asinari [2], Garzo Santos Brey [15], Sofena [22], see also Cercignani [10]. The other type of models contains only one collision term on the right-hand side. Examples for this are Andries, Aoki and Perthame [1] (giving rise to an indifferentiability principle for the species), the models in [8,16] or the model by Brull [7] with an extension leading to a correct Prandtl number in the Navier Stokes equation, adapting the ES-BGK model for mixtures.
In this paper we are interested in the first type of models. We want to describe a gas mixture with a BGK approach containing a sum of collision terms on the right-hand side. Our interest in this kind of models comes from the fact that it is still used by engineers, chemists and physicists and in numerical applications, see for example [19,21]. Moreover BGK and ES-BGK models give rise to efficient numerical computations, which are asymptotic preserving, that is they remain efficient even approaching the hydrodynamic regime [20,14,13,4,12,5,11]. We are motivated by the example of a mixture of electrons and ions without chemical reactions, so there is no transfer of mass from one species to the other. The particles of the two species are so different that it is desirable to maintain their contribution separately.
Here the collision frequencies differ vastly, a characteristic we want to preserve in our model. Our model is so general that the related models of Gross and Krook [17] and Hamel [18] can be derived as special cases. For our model we are able to show conservation properties, the H-theorem, positivity of solutions with positive initial data in the space homogeneous case and positivity of all temperatures.
The outline of the paper is as follows: in section 2 we will present the model for two species and prove the conservation properties and the H-theorem. We discuss the relationship of the collision frequencies in the case of plasmas and show the positivity of solutions with positive initial data in the space homogeneous case and the positivity of all temperatures. In section 3, we compare our model with other models present in the literature. First we consider related models and next we compare our model with the model of Andries, Aoki and Perthame [1]. In section 4 we consider the asymptotic limit of a mixture given by the positive and negative particles of an ionized gas, and show that we obtain the macroscopic equations of classical MHD

A two species kinetic model
In this section we will present the model for two species and prove the conservation properties and the H-theorem. Further, we especially discuss the relationship between the collision frequencies and analyse the positivity of the temperatures.

The general form of the model
For simplicity in the following we consider a mixture composed of two different species. Thus, our kinetic model has two distribution functions f 1 (x, v, t) > 0 and f 2 (x, v, t) > 0 where x ∈ R 3 and v ∈ R 3 are the phase space variables and t ≥ 0 the time. They are determined by two equations to describe their time evolution. Furthermore we consider binary interactions. So the particles of one species can interact with either themselves or with particles of the other species. In the model this is accounted for introducing two interaction terms in both equations. These considerations allow us to write formally the system of equations for the evolution of the mixture. The following structure containing a sum of the collision operator is also given in [9,10]. We describe the time evolution of the number distribution functions f 1 and f 2 by the Boltzmann equation with binary interactions for two species of particles as in [9], chapter 6.2 , where F1 m1 respective F2 m2 are the acceleration of the respective species due to forces F k on particles of species k with mass m k for k = 1, 2 and Q kl , k, l = 1, 2 are the collision operators for interactions of species k with species l.
Furthermore we relate the distribution functions to macroscopic quantities by where n k is the number density, u k the mean velocity and T k the temperature which is related to the pressure p k by p k = n k T k . Note that in this paper we shall write

Conservation properties of the collision operators
A model for the evolution of a mixture should satisfy the following conservation properties: Conservation of mass, momentum and energy of the individual species in interaction with the species itself: Conservation of total mass, momentum and energy

The BGK approximation
We are interested in a BGK approximation of the interaction terms. This leads us to define equilibrium distributions not only for each species itself but also for the two interspecies equilibrium distributions. Choose the collision terms Q 11 , Q 12 , Q 21 and Q 22 in section 2.2 as BGK operators. Then the model can be written as: , with the Maxwell distributions , where ν 11 and ν 22 are the collision frequencies of the particles of each species with itself, while ν 12 and ν 21 are related to interspecies collisions. The structure of the collision terms ensures that if one collision frequency ν kl → ∞ the corresponding distribution function becomes Maxwell distribution. In addition at global equilibrium, the distribution functions become Maxwell distributions with the same velocity and temperature (see section 2.8). The Maxwell distributions M 1 and M 2 in (3) have the same moments as f 1 respective f 2 . With this choice, we guarantee the conservation of mass, momentum and energy in interactions of one species with itself (see section 2.2). The remaining parameters u 12 , u 21 , T 12 and T 21 will be determined using conservation of total momentum and energy, together with some symmetry considerations.

Relationship between the collision frequencies
The goal of this subsection is to derive an estimate for the ratio of all the relaxation parameters ν 11 , ν 12 , ν 22 and ν 21 in the case of a plasma. The parameters ν 12 and ν 21 are linked to the interspecies collision frequency. In plasmas, the mass ratio of the two kinds of particles is m2 m1 << 1, where 1 denotes ions and 2 denotes electrons. In this case a common relationship found in literature [3] is A motivation for this relationship in the case of a plasma can be found in [3], chapter 1.9, which we want to mention here shortly. The collision frequency is proportional to the differential cross section and the relative velocity. For the typical velocity of ions and electrons close to equilibrium one can take the thermal velocity v Ti = ( 2Ti mi ) 1 2 , i = 1, 2 and assume that the temperatures are of the same order, T 1 ≈ T 2 . The cross sections are considered equal, because they depend on the . Therefore For an estimate of ν 12 and ν 21 we consider a collision of an electron head-on with an ion. The velocities after a collision of an ion with an electron are given by where ω is a unit vector in S 2 . Since we consider a head-on collision this simplifies Since m 2 is small compared to m 1 , we get , which reflects the physical fact that collisions of a heavy particle with a light one have a bigger influence on the lighter one than on the heavy one. Hence ν 12 = m2 m1 ν 22 .
To summarize, in the case of ions and electrons, the collision frequencies can be ordered as follows: See also [23]. To be flexible in choosing the relationship between the collision frequencies, we now assume the relationship If ε > 1, exchange the notation 1 and 2 and choose 1 ε .

Conservation properties
This section shows how the macroscopic quantities in the interspecies Maxwellians have to be chosen in order to ensure the macroscopic conservation properties. then Proof. Conservation of mass implies that in the homogeneous case ∂ t f 1 dv = 0. Therefore ν 11 n 1 (M 1 − f 1 )dv + ν 12 n 2 (M 12 − f 1 )dv = 0.
Theorem 2.2 (Conservation of total momentum). Assume the relationships (5) and (6) hold and assume further that u 12 is a linear combination of u 1 and u 2 Then we have conservation of total momentum Proof. The flux of momentum of species 1 is given by The flux of momentum of species 2 is given by In order to get conservation of momentum we therefore need which holds provided u 21 satisfies (8) under the assumption that ν 12 and ν 21 satisfy (5).
we obtain a similar structure for u 21 as for u 12 (5), conditions (6), (7) and (8) and assume that T 12 is of the following form

Theorem 2.3 (Conservation of total energy). Assume
Then we have conservation of total energy provided that Proof. Using the energy flux of species 1 where we used (7) and (11). Analogously the energy flux of species 2 towards 1 is Substitute u 21 with (8) and T 21 from (12). This permits to rewrite the energy fluxes as Adding these two terms, we see that the total energy is conserved.
the two terms with the temperatures are also a convex combination of T 1 and T 2 .
Remark 2.3. The remaining free parameters can be fixed for specific situations.
For example, if we see the parameters α, δ, γ and ε from the model presented in this paper as functions of the masses m 1 and m 2 , we can get more restrictions on these parameters by physical considerations.
• In the limit m1 m1+m2 → 0, we expect that u 12 = u 2 and T 12 = T 2 , since we expect that light particles are driven by the flow of the heavy particles, so they adapt the velocity and the fluctuations to the mean velocity of the heavy particles. If we look at (7), (8), (11) and (12), the definitions of u 12 , u 21 , T 12 and T 21 , we see in order to realize this, we need δ → 0, α → 0 and γ → 0.
• In the limit m1 m1+m2 → 1 2 , when the mass of the particles become indistinguishable, we expect T 12 = T 21 and • In the limit m1 m1+m2 → 1, the heavy particles don't feel the other particles, so we expect that we have no change in the mean velocity and in the temperature, e.g u 12 = u 1 and T 12 = T 1 . Here we need δ → 1, α → 1 and γ → 0.

Positivity of the distribution function
We want to show that in the space homogeneous case positive initial values of the distribution functions stay non-negative when their time evolution is described by the two species BGK model described in this paper.
Theorem 2.4 (Non-negative solutions of the BGK equation for two species). As- Proof. In the space-homogeneous case we get from the conservation properties that n 1 and n 2 are constant in time. Rewrite (2) in the space homogeneous case as Define By using (15) we get The initial value of α is chosen such that f 1 and g 1 have the same initial values. Then g 1 solves Since we assumed ν 11 (t), ν 12 (t), ν 21 (t), ν 22 (t) ≥ 0 for every t ≤ t 0 and positive initial values, all terms on the right-hand side are positive. Hence f 1 is positive. Similar for f 2 .

Positivity of the temperatures
Then all temperatures T 1 , T 2 , T 12 given by (11) and T 21 given by (12) are positive provided that Proof. T 1 and T 2 are positive as integrals of positive functions. T 12 is positive because by construction it is a convex combination of T 1 and T 2 . For T 21 we consider the coefficients in front of |u 1 − u 2 | 2 , T 1 and T 2 . The term in front of T 1 is positive by definition. The positivity of the term in front of T 2 is equivalent to the condition α ≥ 1 − 1 ε , which is satisfied since ε ≤ 1, the positivity of the term in front of |u 1 − u 2 | 2 is equivalent to the condition (16).

Remark 2.4.
According to the definition of γ, γ is a non-negative number, so the right-hand side of the inequality in (16) must be non-negative. This condition is equivalent to If the collision frequencies are linked as in (5), ε = m2 m1 , then the right-hand side of (16) is always positive.

H-theorem for mixtures
Remark 2.5. From the case of one species BGK model we know that for k = 1, 2, see for example problem 1.7.1 in [9]. Lemma 2.6. Assuming (11) and (12) and the positivity of the temperatures (16), we have the following inequality Proof. We start with the left-hand side of (2.6). First we insert the definition of T 12 and T 21 from (11) and (12). Since γ and the term in front of |u 1 − u 2 | 2 in (12) are positive, we can use the monotonicity of the logarithm and get ε ln T 12 + ln T 21 If we now use the concavity of the logarithm and the assumptions 0 ≤ α ≤ 1, ε < 1, the expression above can be bounded from below by which gives the inequality stated in lemma 2.6.
which is equivalent to the condition (18) proven in Lemma 2.6. With this inequality we get The last inequality follows from remark (2.5). Here we also have equality if and only if f 1 = M 1 and f 2 = M 2 , but since we already noticed that equality also implies f 1 = M 12 and f 2 = M 21 , we also have T 21 = T 2 = T 1 = T 12 and u 1 = u 2 = u 12 = u 21 .
Define the total entropy H(f 1 , f 2 ) = (f 1 ln f 1 + f 2 ln f 2 )dv. We can compute by multiplying the BGK equation for the species 1 by ln f 1 , the BGK equation for the species 2 by ln f 2 and integrating the sum with respect to v.

2.9
The structure of the equilibrium

Special cases of this model in the literature
In this section, we review models that have been previously introduced, [17] and [18], and that can be considered as special cases of the class described here. Thanks to this, all of them enjoy an H-theorem, conservation properties and positivity of the interspecies temperatures.

Model of Gross and Krook
The model of Gross and Krook [17] is obtained by choosing ε = 1, while δ, α and γ are free parameters. In the case of a plasma they suggest δ = m1 m1+m2 . They also assume (6) for conservation of mass. They assume one of the mixture velocities to be a linear combination of u 1 and u 2 , similar to (7) and deduce (8) from conservation of momentum. They further choose T 12 of the form and deduce with conservation of energy that T 21 is given by where five of the variables A, B, C, D, E ∈ R are determined in order to get conservation of energy. From the present work the constants must be chosen in order to satisfy (16). In this case the model satisfies the H-Theorem.

Comparison with the model of Andries, Aoki and Perthame
The next model also describes a gas mixture of Maxwellian molecules, but it contains only one term on the right-hand side [1]. , The Maxwell distributions are given by 2T (1) , with the interspecies velocities u (1) = u 1 + 2 m 2 m 1 + m 2 χ 12 ν 11 n 1 + ν 12 n 2 n 2 (u 2 − u 1 ), and the interspecies temperatures , where χ 12 , χ 21 , ν 12 and ν 21 are parameters which are related to the differential cross section. For the detailed expressions see [1]. The model also satisfies the conservation properties and the H-theorem with equality if and only if the distribution functions are Maxwell distributions with equal mean velocity and temperature.
The flux of the energy of species 1 is given by So the model discussed here reproduces the same momentum and energy fluxes between the species (9), (10), (13) and (14) choosing the parameters δ, α and γ as: For χ 12 ≤ ν 12 the parameter γ is non-negative.
The model of Andries, Aoki and Perthame has another property, proposition 3.2 in [1], which the models described above do not have. It is called the indifferentiability principle. It denotes the following property: See also [7] for another model which also has the indifferentiability principle. The model in this paper does not satisfy the indifferentiability principle. The indifferentiability principle in our model holds only in the global equilibrium. On physical grounds it is reasonable to assume that two species of identical particles become really indifferentiable when they have the same macroscopic speeds and temperatures.

Deriving macroscopic MHD equations
In this section we want to illustrate the model in the case of ions and electrons. Finally, we derive the typical macroscopic equations for a mixture composed of ions and electrons, the equations of ideal Magnetohydrodynamics, from our model. You can also find a similar derivation in [6] but for an isothermal flow.

The BGK model for ions and electrons
We consider the case of ions and electrons and set ε = m2 m1 as it is motivated in section 2.4 or [3]. For simplicity we take δ = 0, α = m2 m1+m2 and γ = 0, although the MHD equations can also derived from the general model. We replace the index 1 by i for the ions and 2 by e for the electrons. Then the particles are subjected to the Lorentz force F i = e(E + v × B) and F e = −e(E + v × B), where e is the elementary charge and E and B the mean electric and magnetic fields given by the Maxwell equations. In this case the model (2) rewrites as (24)

Macroscopic equations for ions and electrons
In order to derive macroscopic equations, we multiply the first equation of (2) with (1, m i v, mi 2 |v| 2 ), and the second with (1, m e v, me 2 |v| 2 ). Then we integrate them with respect to the velocity.The obtained macroscopic system is not closed since we obtain terms of the form v ⊗ vf i dv, v ⊗ vf e dv, |v| 2 vf i dv and |v| 2 vf e dv. All the other terms are functions of known quantities given in (1). There are plasmas where the two species first relax to its own equilibrium and then to a global one. According to Chapter 1.9 in [3], we expect that plasmas are typically not in thermodynamic equilibrium, although the components may be in a partial equilibrium. This means, the electrons are in thermal equilibrium with itself but not with the ions, and the other way round. So in our considerations we assume that each species is in equilibrium with itself, e.g. setting f i = M i and f e = M e . In this way, we obtain a closed system of equations for the conservation of mass, momentum and energy.
∂ t (m e n e u e ) + ∇ x (n e T e ) + ∇ x · (m e u e ⊗ u e n e ) + en e (E + u e × B) = ν ei m e n e n i (u i − u e ), In order to determine the time evolution of the electric and magnetic field, we couple the system with the Maxwell equations.
where c 2 = 1 µ0ε0 is the speed of light and µ 0 , ε 0 the magnetic and electric vacuum permittivity.

Dimensionless equations
First we define dimensionless variables of the time t, the length x, the velocities u e , u i , the number densities n e , n i , the temperatures T e , T i , the magnetic field B, the electric field E, the electron-ion collision frequency ν ei , the ion-electron collision frequency ν ie and the current density j, for example t ′ = t /t for a typical time scalet. In particular, the order of magnitudes of some quantities are assumed to be linked: We assume that both species have densities, mean velocities and temperatures of the same order of magnitude, e.g.n i =n e =n,ū i =ū e =ū =x/t and T i =T e =T . With the last two assumptions we assume that we are close to a thermodynamic equilibrium in which the two mean velocities and temperatures would be equal. Further, we assume thatĒ =Bū. From non-dimensionalizing the first two Maxwell equations we see that this means that the electric field induced by a change of the magnetic field in time dominates over the fields which arise from charges and currents. Further we assume thatB = µ 0xj , which means that the magnetic field induced by currents dominates over the magnetic field due to changes of the electric field in time. Finally, we assume thatν ie = me miν ei This leads to the following equations, where now the variables are non-dimensional together with the Maxwell equations coming from non-dimensionalizing. The physical meaning is the following: C 1 describes the ratio over the typical scale of thermal energynT and of the kinetic energy m inū 2 of ions. If we consider an ion travelling with a speed perpendicular to a magnetic field at distance r, the force due to the magnetic field eBū on the particle acts as a centripetal force mū 2 r , so the norm of the forces is equal which is equivalent to ω :=ū r = eBū m which describes a frequency called cyclotron frequency. So

The limits to the MHD equations
Now we consider the formal limit of the mass ratio C 4 → 0 and the non-relativistic limit M → 0.
Theorem 4.1. The formal limit of the mass ratio C 4 → 0 and the non-relativistic limit M → 0 of the system non-dimensionalized system with the remaining parameters remain finite is the system For the interested reader the proof is given in the Appendix. Next, we consider the formal limit C 5 → 0 and C3 C2 → 0, such that C 2 C 5 and remain bounded away from zero. Physically the first limit means that the current from moving particles enū dominates over the current due to electric forcesj. The second limit means that the cyclotron frequency eB mi , dominates over the collision frequencyν ien , while the current due to electric fieldsj per cyclotron time 1/ eB mi over the current induced by the flow enū in a typical time scalet, remains bounded away from zero. Moreover, the ratio of the collision frequency and the cyclotron frequency is assumed to be of the same order of the electric current per collision time 1 νien over the current induced by the flow per typical time scale. All in all, we get the following theorem Theorem 4.2. As C 5 → 0 and C3 C2 → 0, such that C 2 C 5 and C2 remain bounded away from zero, formally the solution of the system in Theorem 4.1 tends to the solution of This is a direct consequence of Theorem 4.1. Last we consider the formal limit C 3 → 0 which means that interactions of ions and electrons can be neglected. In addition, we choose the special regime where C 1 = 1, that isnT = m inū 2 and C 2 C 5 = 1 in order to obtain the well-known conservation form for ideal MHD. Theorem 4.3. As C 3 → 0 and in the special regime C 1 = 1 and C 2 C 5 = 1, formally, we obtain the system of ideal MHD equations ∂ t n + ∇ x · (nu) = 0, Again, for the interested reader the proof is given in the appendix.

Conclusion and perspectives
We derived a BGK equation for mixtures that replaces the Boltzmann collision operator satisfying the conservation properties, the H-theorem, positivity of solutions of positive initial data and positivity of all temperatures. The BGK collision operator contains a sum of relaxation terms corresponding to each type of interaction, interaction with each species with itself and interaction with the other species. It has the advantage to single out the influence of a certain type of collision directly.
For example, if one species has already reached a Maxwell distribution and the other one not.
First, we expect to extend the micro-macro decomposition [11] to gas mixtures. Further work will be concentrated on the problem of deriving the Navier-Stokes equations for mixtures from kinetic equations as in [16,8] in order to estimate accurate values for Fick's diffusion coefficient, the viscosity coefficient, the thermal conductivity and the thermal diffusion parameter. This model offers a possibility of matching experimental data because of the remaining free parameters α, δ and γ.

Appendix
Proof of Theorem 4.1. We start with the non-dimensionalized system from section 4.3. In the limit M → 0, we get from the first Maxwell equation that n i and n e converge formally to the same limit n. The third Maxwell equation simplifies to ∇ x × B = j.
We denote the limit of u i by u. Then we get from conservation of the number of ions ∂ t n + ∇ x · (nu) = 0.
The momentum equation of the electrons turns into C 1 ∇ x (nT e ) + C 2 n(E + u e × B) = C 3 ν ie nn(u − u e ).
The limit of the sum of the momentum equations with T := T i + T e gives ∂ t (nu) + C 1 ∇ x (nT ) + ∇ x · (u ⊗ un) + C 2 n(u e − u) × B = 0.
From the sum of the energy equations we get (nT e u e + nT i u)) + ∇ x · ( 1 2 n|u| 2 u) = C 2 En(u − u e ).