A consistent BGK model with velocity-dependent collision frequency for gas mixtures
Jeff Haack, Cory Hauck, Christian Klingenberg, Marlies Pirner, Sandra Warnecke
aa r X i v : . [ m a t h . A P ] J a n A consistent BGK model withvelocity-dependent collision frequency for gasmixtures
J. Haack * C. Hauck † C. Klingenberg ‡ M. Pirner § S. Warnecke ¶ January 25, 2021
Abstract
We derive a multi-species BGK model with velocity-dependent colli-sion frequency for a non-reactive, multi-component gas mixture. The modelis derived by minimizing a weighted entropy under the constraint that thenumber of particles of each species, total momentum, and total energy areconserved. We prove that this minimization problem admits a unique so-lution for very general collision frequencies. Moreover, we prove that themodel satisfies an H-Theorem and characterize the form of equilibrium.
Keywords: multi-fluid mixture, kinetic model, BGK approximation plasmaphysics, velocity-dependent collision frequency, entropy minimization * Los Alamos National Laboratory, Los Alamos, NM 87545, USA, [email protected] † Oak Ridge National Laboratory, 1 Bethel Valley Road, Bldg. 5700, Oak Ridge, TN 37831-6164, USA, [email protected] ‡ [email protected], Universit¨at W¨urzburg, Emil-Fischer-Str. 40, 97074W¨urzburg, Germany § [email protected], Universit¨at W¨urzburg, Emil-Fischer-Str. 40,97074 W¨urzburg, Germany ¶ [email protected], Universit¨at W¨urzburg, Emil-Fischer-Str.40, 97074 W¨urzburg, Germany Introduction
In this paper, we present a BGK-type model for gas mixtures that, in the case oftwo species, takes the form ∂ t f + v · ∇ x f = ν ( M − f ) + ν ( M − f ) , ∂ t f + v · ∇ x f = ν ( M − f ) + ν ( M − f ) , (1)along with appropriate boundary and initial conditions. Here f = f ( x , v , t ) and f = f ( x , v , t ) are the number densities of species of mass m and m , respectively,with respect to the phase space measure dxdv ; x ∈ R is the position coordinate ofphase space; v ∈ R is the velocity coordinate; and t ≥ M k j = exp ( m k λ k j + m k λ k j · v + m k λ k j | v | ) , (2)which depend on parameters λ k j = ( λ k j , λ k j , λ k j ) ∈ R × R × R + , and (non-negative) collision frequencies ν k j . These parameters depend implicitly on f and f , and once specified, determine the BGK operator.The purpose of the relaxation operator in (1) is to provide an approximationof the multi-species Boltzmann collision operator that is more computationallytractable, but still maintains important structural properties. In the single-speciescase, the original BGK model [2] serves this purpose. In particular, it has thesame collision invariants as the Boltzmann operator (which lead to conservationof number, momentum, and energy) and it satisfies an H-Theorem. In the multi-species case, these requirements are not as straight-forward to satisfy, but it can bedone. There are many BGK models for gas mixtures proposed in the literature [14,16, 10, 12, 26, 21, 15, 5, 1], many of which satisfy these basic requirements and,in addition, are able to match some prescribed relaxation rates and/or transportcoefficients that come from more complicated physics models or from experiment.Many of these approaches have been extended to accommodate ellipsoid statistical(ES-BGK) models, polyatomic molecules, chemical reactions or quantum gases;see for example [22, 29, 13, 23, 24, 3, 4, 25].A common feature of all the models mentioned above is that they only allowfor collision frequencies which are independent of the microscopic velocity v ofthe particles [28]. However, the collision frequencies in principle should dependon the microscopic velocity, which is typically neglected for the reason of simplic-ity. In the case of neutral gases, velocity independent collision frequency leads totransport properties in the fluid regime that are inconsistent with the full kinetic2ollision operator, e.g., the Prandtl number. Models such as the ES-BGK modeland the Shakov model make changes to the target Maxwellian to provide extradegrees of freedom to the system, but still retain the constant collision frequencyassumption. Some attempts have been proposed to re-introduce velocity depen-dence in the case of variable hard spheres interactions for neutral gases [20], forwhich velocity-dependent collision frequencies are monotonically increasing andare well-defined. For particles interacting with long-ranged Coulomb interactions,i.e., a plasma, the canonical collision rate definition using the cross section is nolonger well defined due to a singularity at a zero relative velocity. A velocity-dependent collision frequency is instead defined by the momentum transfer crosssection without an integral, which results in a collision frequency that is decreas-ing in the limit of large relative velocities [19, 18].In this paper, we derive a model of the form (1) that allows for velocity-dependent collision frequencies. Our derivation includes as a by-product thesingle-species BGK model with velocity-dependent collision frequency that wasproposed in [27]. We identify target functions that are consistent with the conser-vation laws for (1) and satisfy an entropy minimization principle. In particular, intra-species collisions (between the same species) should preserve mass, mo-mentum, and energy within a species; that is, Z m k ν kk v | v | ( M kk − f k ) dv = , k ∈ { , } . (3)Meanwhile inter-species collisions (between different species) should preservethe mass of each species, but only the combined momentum and energy of both;that is, Z m ν ( M − f ) dv = , Z m ν ( M − f ) dv = Z m ν (cid:18) v | v | (cid:19) ( M − f ) dv + Z m ν (cid:18) v | v | (cid:19) ( M − f ) dv = . (4)When the collision frequencies are independent of v , the integrals in (3) and (4)can be computed explicitly, thereby providing relationships between the parame-ters λ k j and the moments of f and f with respect to { , v , | v | } . In the single-species case, this relationship defines the target function as the Maxwellian associ-ated to f , while in the multi-species case, additional constraints must be imposed.However, when the collision frequencies depend on v , the aforementioned inte-grals are not always computable in closed form and the relationship between the3arameters λ k j and the moments of f and f with respect to { , v , | v | } cannot bewritten down analytically.In spite of the difficulty of relating the target parameters to the moments of thekinetic distributions, the entropy minimization formulation can be still used to es-tablish a unique set of parameters, under the conditions λ = λ and λ = λ .We do so by adapting the strategy from [17] to fit the current setting. While a moreabstract approach based solely on convex optimization tools can also be used [6],we follow [17] because it provides a more concrete connection to the applicationat hand. Our proof provides a rigorous justification for the target function usedin [27] for the single species case. It also leads to an H-Theorem for the multi-species system (1).The remainder of the paper is organized as follows. In Section 2, we motivatethe choice of the target Maxwellians as solutions of minimization problems of theentropy under certain constraints. In Section 3, we prove existence and uniquenessof the minimization problems. In Section 4, we prove consistency of the modelmeaning that it satisfies the conservation properties, the H-Theorem and Maxwelldistributions with equal mean velocity and temperature in equilibrium. In Section5, we briefly summarize the straightforward extension to the case of N species,still with binary interactions. In this section, we motivate the form of the target functions in (2). It will beconvenient in what follows to define the strictly convex function h ( z ) = z ln z − z , z > , (5)and the vector-valued function a k ( v ) = a k ( v ) a k ( v ) a k ( v ) = m k m k vm k | v | . (6)Since h is convex and h ′ ( z ) = ln ( z ) , it follows that h ( x ) ≥ h ( y ) + ln ( y )( x − y ) , ∀ y , x ∈ R + . (7)4 .1 The one species target Maxwellians We seek a solution of the weighted entropy minimization problemmin g ∈ χ k Z ν kk h ( g ) dv , k ∈ { , } , (8)where χ k = (cid:26) g (cid:12)(cid:12)(cid:12) g ≥ , ν kk ( + | v | ) g ∈ L ( R ) , Z ν kk a k ( v )( g − f k ) dv = (cid:27) . (9)The choice of the set χ k ensures the conservation properties (3) for intra-speciescollisions. The motivation for weighting the usual objective by the collisionfrequencies in (8) is that the ansatz will take the form (2). Indeed, by stan-dard optimization theory, any critical point ( M kk , λ kk ) of the Lagrange functional L k : χ k × R → R , given by L k ( g , α ) = Z ν kk h ( g ) dv − α · Z ν kk a k ( v )( g − f k ) dv , (10)satisfies the first-order optimality condition δ L k δ g ( M kk , λ kk ) = ν kk ( ln M kk − λ kk · a k ( v )) = , (11)which implies then that M kk = exp ( λ kk · a k ) = exp ( m k λ kk + m k λ kk · v + m k λ kk | v | ) . (12)In Section 3.1, we prove in a rigorous way that there exists a unique function ofthe form (12) that satisfies these constraints. Theorem 2.1.
Suppose that there exists λ kk ∈ R × R × R such that the functionM kk given in (12) is an element of χ k . Then M kk is the unique minimizer of (8) .Proof. According to (7) h ( g ) ≥ h ( M kk ) + λ kk · a k ( g − M kk ) , (13)point-wise in v . Thus, because ν kk ≥
0, it follows that for all g ∈ χ k , Z ν kk h ( g ) dv ≥ Z ν kk h ( M kk ) dv + Z ν kk λ kk · a k ( g − M kk ) dv = Z ν kk h ( M kk ) dv (14)Hence M kk is a minimizer of (8), and uniqueness follows directly from the strictconvexity of h . 5 .2 The mixture target Maxwellians For interactions between species, we seek a solution of the weighted entropy min-imization problem min g , g ∈ χ Z ν h ( g ) dv + Z ν h ( g ) dv , (15)where χ = ( ( g , g ) (cid:12)(cid:12)(cid:12) g , g > , ν ( + | v | ) g , ν ( + | v | ) g ∈ L ( R ) , Z m ν g dv = Z m ν f dv , Z m ν g dv = Z m ν f dv , Z m ν (cid:18) v | v | (cid:19) ( g − f ) dv + Z m ν (cid:18) v | v | (cid:19) ( g − f ) dv = ) . (16)Here, χ is chosen such that the constraints (3) for inter-species collisions aresatisfied. Similar to the case of intra-species collisions, we consider the Lagrangefunctional L : χ × R × R × R × R → R L ( g , g , α , α , α , α ) = Z ν h ( g ) dv + Z ν h ( g ) dv − α Z m ν ( g − f ) dv − α Z m ν ( g − f ) dv − α · (cid:18) Z m ν v ( g − f ) dv + Z m ν v ( g − f ) dv (cid:19) − α (cid:18) Z m ν | v | ( g − f ) dv + Z m ν | v | ( g − f ) dv (cid:19) . (17)Any critical point ( M , M , λ , λ , λ , λ ) of L satisfies the first-order optimalityconditions δ L δ g ( M , M , λ , λ , λ , λ ) = ν ( ln M − λ · a ( v )) = , (18) δ L δ g ( M , M , λ , λ , λ , λ ) = ν ( ln M − λ · a ( v )) = , (19)6here λ = ( λ , λ , λ ) and λ = ( λ , λ , λ ) . Therefore M = exp ( λ · a ( v )) = exp (cid:0) m λ + m λ · v + m λ | v | (cid:1) (20) M = exp ( λ · a ( v )) = exp (cid:0) m λ + m λ · v + m λ | v | (cid:1) . (21)Since we only require conservation of the combined momentum and kinetic en-ergy, there is only one Lagrange multiplier for the momentum constraint andone Lagrange multiplier for the energy constraint. Therefore, λ = λ and λ = λ in (2). When the collision frequency is constant, this restriction isthe same as the one used in [15], but more restrictive than the model in [21].In the next section, we prove the existence of functions of the form (2) thatsatisfy the constraints in (3) and (4). As in the single species case, it follows thatthese functions are unique minimizer of the corresponding minimization problem. Theorem 2.2.
Assume that there exist λ ∈ R , λ ∈ R , λ = λ ∈ R , and λ = λ ∈ R such that the pair ( M , M ) , where M k j is defined in (2) , is anelement of χ . Then ( M , M ) is the unique minimizer of (15) .Proof. According to (7) h ( g ) ≥ h ( M k j ) + λ k j · a k ( g − M k j ) , (22)point-wise in v , for any measurable function g and k , j ∈ { , } . Therefore, since ν k j ≥
0, it follows that for any measureable functions g and g , Z ν h ( g ) dv + Z ν h ( g ) dv ≥ Z ν h ( M ) dv + Z ν h ( M ) dv + λ · Z ν a ( g − M ) dv + λ · Z ν a ( g − M ) dv . (23)Since λ = λ and λ = λ , λ · Z ν a ( g − M ) dv + λ · Z ν a ( g − M ) dv = λ Z ν m ( g − M ) dv + λ Z ν m ( g − M ) dv + λ · (cid:18) Z ν m v ( g − M ) dv + Z ν m v ( g − M ) dv (cid:19) + λ (cid:18) Z ν m | v | ( g − M ) dv + Z ν m | v | ( g − M ) dv (cid:19) . (24)7f ( g , g ) and ( M , M ) are elements of χ , then the constraints in (16) implythat each of the terms above is zero. In such cases, (23) reduces Z ν h ( g ) dv + Z ν h ( g ) dv ≥ Z ν h ( M ) dv + Z ν h ( M ) dv , (25)which shows that ( M , M ) solves (15). Since the collision frequencies ν and ν are non-negative and h is strictly convex, it follows that this solution is unique. In this section, we prove the existence of the multipliers λ , λ , λ and λ such that the single-species targets M and M satisfy (3) and the mixture targets M and M satisfy (4). We follow closey the strategy laid out in [17], althoughsome variations will be needed to account for the velocity-dependent collisionfrequencies and the mixture targets.Throughout the paper, we denote a distribution function of exponential formby exp k λ ( v ) : = exp ( λ · a k ( v )) , λ = ( λ , λ , λ ) ∈ R . (26)and let D k j = { g ≥ | ν k j ( + | v | ) g ∈ L ( R ) , g } , Λ k j = { λ ∈ R | exp k λ ∈ D k j } . (27)For any g ∈ D k j the moment map µ k j is given by µ k j ( g ) = µ k j µ k j µ k j ( g ) = Z ν k j a k ( v ) g ( v ) dv . (28)We make the following assumptions about the collision frequencies. Assumption 3.1.
Each frequency ν k j is strictly positive and defined such that Λ : = Λ k j = { λ | exp k λ ∈ L ( R ) } = { λ ∈ R | λ < } (29) is independent of k and j. Roughly speaking, these assumptions are used to ensure integrability proper-ties that are satisfied when the collision frequencies are independent of the veloc-ity. They are used in the technical details of the proofs below, but are in practicesatisfied by many realistic frequency models.8 .1 Target functions for intra-species collisions
We start the intra-species case; that is, for k ∈ { , } , we show the existence ofmultiplier λ kk such that M kk satisfies (3). The basic idea is to show that the dualfunction z ( λ ; ρ ) = µ kk ( exp k λ ) − λ · ρ (30)is differentiable and attains its minimum on Λ for any ρ ∈ µ kk ( D kk ) . Then thenecessary condition for an extremum in Λ yields0 = ∇ λ z ( λ kk ) = Z ν kk ( v ) a k ( v ) exp ( λ kk · a k ( v )) dv − ρ , (31)which gives ρ = µ kk ( exp k λ kk ) . Lemma 3.2.
The function z is strictly convex and twice Fr´echet differentiable on Λ .Proof. It is sufficient to prove that φ ( λ ) = µ kk ( exp k λ ) is strictly convex and twiceFr´echet differentiable, with first derivative D φ ( λ ) = µ kk ( exp k λ ) and Hessian H φ ( λ ) = R a k ( v ) ⊗ a k ( v ) exp k λ dv . Convexity following immediately from convexity of theexponential function and linearity of the integral. Specifically, given λ ( ) , λ ( ) andtwo positive scalars θ , θ such that θ + θ =
1, it follows that exp k θ λ ( ) + θ λ ( ) ≤ θ exp k λ ( ) + θ exp k λ ( ) . Hence φ ( θ λ ( ) + θ λ ( ) ) = µ kk ( exp k θ λ ( ) + θ λ ( ) ) ≤ µ kk ( θ exp k λ ( ) + θ exp k λ ( ) )= θ φ ( λ ( ) ) + θ φ ( λ ( ) ) . (32)For any nonzero δ ∈ R φ ( λ + δ ) − φ ( λ ) − D φ ( λ ) · δ | δ | = Z f δ ( v ) dv , (33)where f δ ( v ) = ν kk ( v ) exp k λ ( v ) exp k δ ( v ) − − a k ( v ) · δ | δ | ! . (34)9 Taylor series expansion shows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp k δ ( v ) − − δ · a k ( v ) | δ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ ∑ n = ( δ · a k ( v )) n n ! 1 | δ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | a k ( v ) | ∞ ∑ n = | δ · a k ( v ) | n n ! ≤ | a k ( v ) | exp ( | δ · a k ( v ) | ) (35)Therefore f δ ( v ) ≤ exp k λ / ( v ) g δ ( v ) , where g δ ( v ) : = ν kk ( v ) | a k ( v ) | exp k λ / ( v ) exp ( | δ · a k ( v ) | ) ≤ ν kk ( v ) | a k ( v ) | (cid:16) exp k λ / + δ ( v ) + exp k λ / − δ ( v ) (cid:17) . (36)Because Λ is open, for | δ | sufficiently small, exp λ / + δ ( v ) and exp λ / − δ ( v ) areelements of D kk , in which case g δ is integrable. Moreover, exp λ / is bounded.Hence f δ is bounded above by an integrable function and the dominated conver-gence theorem gives lim δ → Z f δ ( v ) dv = Z lim δ → f δ ( v ) dv = . (37)The existence of the Hessian can be proven in an analogous way. Lemma 3.3.
For fixed λ ∈ Λ , ξ ∈ S , and ρ ∈ µ kk ( D kk ) , the functionz ξ ( s ) = z ( λ + s ξ ; ρ ) (38) attains its unique minimum in the open intervalI ( ξ , λ ) : = ( − s b ( − ξ , λ ) , s b ( ξ , λ )) (39) where s b ( ξ , λ ) : = sup { s : λ + s ξ ∈ Λ } takes the value + ∞ if the boundary ∂Λ is not met in the direction ξ . Proof.
The fact that z is strictly convex and differentiable with respect to λ impliesthat z ξ is strictly convex and differentiable with respect to s . Hence it attains aunique minimum on the closure of I ( ξ , λ ) .10e now show that z ξ cannot attain its minimum on the boundary of I ( ξ , λ ) .Suppose first that s b ( ξ , λ ) < ∞ . According to Assumption 3.1, λ + s b ( ξ , λ ) ξ Λ .Hence by Fatou’s Lemma,lim s → s b ( ξ , λ ) Z ν kk exp k λ + s ξ dv ≥ Z ν kk exp k λ + s b ( ξ , λ ) ξ dv = ∞ (40)which implies that lim s → s b ( ξ , λ ) z ξ ( s ) = + ∞ .Suppose now that s b ( ξ , λ ) = ∞ . There are two cases: Case 1: ξ · a k ( v ) ≤ v ∈ R . Since ρ ∈ µ kk ( D kk ) , there exists g ∈ D kk such that ρ = µ kk ( g ) . By definition, g is not identically zero and byAssumption 3.1 ν kk >
0. Thus the set Ω : = { v ∈ R | ξ · a k ( v ) < } ∩ { v ∈ R | ν kk ( v ) g ( v ) > } (41)has positive measure. Hence ξ · ρ = ξ · µ kk ( g ) = Z ν kk ( v ) ξ · a k ( v ) g ( v ) dv < s → ∞ z ξ ( s ) = lim s → ∞ Z exp k λ + s ξ dv − ( λ + s ξ ) · ρ ≥ lim s → ∞ − ( λ + s ξ ) · ρ = ∞ . (43) Case 2: { v ∈ R : ξ · a k ( v ) > } has positive measure.Then there exists an ε > B = { v ∈ R : ξ · a k ( v ) ≥ ε } has positivemeasure. Hencelim s → ∞ z ξ ( s ) ≥ lim s → ∞ (cid:18)(cid:18) Z B ν kk ( v ) exp k λ dv (cid:19) exp ( s ε ) − ( λ + s ξ ) ρ (cid:19) = ∞ (44)due to exponential growth in s . Theorem 3.4.
For any ρ ∈ µ kk ( D kk ) , the function z ( · ; ρ ) has a unique minimizer λ ∗ ∈ Λ . roof. Let { λ ( ℓ ) } ∞ ℓ = be an infimizing sequence such that z ( λ ( ℓ ) ) → z ∗ , where z ∗ = inf λ ∈ Λ z ( λ ) . Let d ( ℓ ) = λ ( ℓ ) − λ ( ) ; ℓ ≥ ξ ( ℓ ) = d ( ℓ ) / || d ( ℓ ) || . Then ξ ( ℓ ) → ξ ∗ ∈ S possibly via a subsequence, because S is compact. For any ξ ∈ S , let s ∗ ( ξ ) = arg min s ∈ R z ( λ ( ) + s ξ ; ρ ) which, according to Lemma 3.3, is well-defined. Be-cause z is strictly convex and twice differentiable, ( i ) g ( ξ , s ) : = ∂ s z ( λ ( ) + s ξ ; ρ ) = s = s ∗ ( ξ )( ii ) ∂ s g ( ξ , s ) > s ∗ is a C function in a neighbour-hood N ( ξ ∗ ) ⊂ Λ that satisfies g ( ξ , s ∗ ( ξ )) = . (45)Let ℓ ∗ be large enough that ξ ( ℓ ) ∈ N ( ξ ∗ ) for all ℓ ≥ ℓ ∗ . Then z ( λ ( ℓ ) ; ρ ) = z ( λ ( ) + d ( ℓ ) ; ρ ) = z ( λ ( ) + || d ( ℓ ) || ξ ( ℓ ) ; ρ ) ≥ z ( λ ( ) + s ∗ ( ξ ( ℓ ) ) ξ ( ℓ ) ; ρ ) . (46)Because s ∗ is continuous on N ( ξ ∗ ) the sequence s ∗ ( ξ ( ℓ ) ) → s ∗ ( ξ ∗ ) with | s ∗ ( ξ ∗ ) | < ∞ . Moreover, since z is continuous z ∗ = lim ℓ → ∞ z ( λ ( ℓ ) ) ≥ lim ℓ → ∞ z ( λ ( ) + s ∗ ( ξ ( ℓ ) ) ξ ( ℓ ) ) = z ( λ ( ) + s ∗ ( ξ ∗ ) ξ ∗ ) ≥ z ∗ , (47)where first inequality follows from (46). Hence the infimum is attained at λ ∗ = λ ( ) + s ∗ ( ξ ∗ ) ξ ∗ ∈ Λ . Corollary 3.5.
Given any f k ∈ D kk , there exists a unique multiplier λ kk such thatM kk given by (2) solves (8) .Proof. Let ρ k = µ kk ( f k ) . According to Theorem 3.4, z ( · , ρ k ) has a unique mini-mizer in Λ , which we denote by λ kk . By Lemma 3.2, z ( · , ρ k ) is also differentiable,so the first-order optimality condition (31) implies that ρ k = µ kk ( exp λ kk ) . Theresult then follows from Theorem 2.1. 12 .2 Target functions for inter-species collisions In this section we show the existence of the multipliers λ = ( λ , λ , λ ) ∈ R × R × R and λ = ( λ , λ , λ ) ∈ R × R × R such that λ = λ , λ = λ , and M and M satisfy (4). Denote λ = ( λ , λ , λ , λ ) λ = ( λ , λ , λ ) λ = ( λ , λ , λ ) (48)and use this notation for other vectors when appropriate. Given g , g ∈ D , let¯ µ ( g , g ) = µ ( g ) µ ( g ) µ ( g ) + µ ( g ) µ ( g ) + µ ( g ) . (49)For any ¯ ρ ∈ ¯ µ ( D × D ) , introduce the dual function¯ z ( λ ; ¯ ρ ) = µ ( exp λ ) + µ ( exp λ ) − λ · ¯ ρ . (50)Similar to the intra-species case, our goal is to show that for any such ¯ ρ , z ( λ ; ¯ ρ ) attains its minimum on ¯ Λ = { λ ∈ R : λ , λ ∈ Λ } . (51)Then the necessary first-order condition for a minimum at λ = ∇ λ z ( λ ; ¯ ρ ) = ¯ µ ( exp λ ( v ) , exp λ ( v )) − ¯ ρ , (52)which recovers the required constraints in (4), if we set λ = λ and λ = λ . Lemma 3.6.
The function ¯ z defined in (50) is strictly convex and twice Fr´echetdifferentiable on ¯ Λ .Proof. Differentiability of the ¯ z can be deduced as in the intra-species case by sim-ply following the arguments of Lemma 3.2. We skip these details. Convexity alsofollows in a similar way. Let ¯ φ ( λ ) = µ ( exp λ ) + µ ( exp λ ) , then convexity ofthe exponential function implies that for any θ ∈ ( , ) , λ ∈ ¯ Λ , and β ∈ ¯ Λ ,¯ φ ( θλ ) + ¯ φ (( − θ ) β ) = µ ( exp θλ +( − θ ) β ) + µ ( exp θλ +( − θ ) β ) ≤ µ ( θ exp λ +( − θ ) exp β ) + µ ( θ exp λ +( − θ ) exp β )= θ ¯ φ ( λ ) + ( − θ ) ¯ φ ( β ) (53)Thus ¯ φ is strictly convex, as is ¯ z , since the two functions differ only by a linearterm. 13 emma 3.7. For λ ∈ ¯ Λ , ξ ∈ S , and ¯ ρ ∈ ¯ µ ( D × D ) , the function ¯ z ξ : s ¯ z ( λ + s ξ ; ¯ ρ ) (54) attains its unique minimum in the open interval ¯ I ( ξ , λ ) : = ( − ¯ s b ( − ξ , λ ) , ¯ s b ( ξ , λ )) , (55) where ¯ s b ( ξ , λ ) = sup { s : λ + s ξ , λ + s ξ ∈ Λ } . (56) Proof.
We follow the arguments of the proof of Lemma 3.3. The fact that ¯ z isstrictly convex and differentiable with respect to λ implies that ¯ z ξ is strictly con-vex and differentiable with respect to s . Hence ¯ z ξ attains a unique minimum on theclosure of ¯ I ( ξ , λ ) . We therefore need only show that ¯ z ξ cannot attain its minimumon the boundary of ¯ I ( ξ , λ ) .Suppose first that ¯ s b ( ξ , λ ) < ∞ . By Fatou’s Lemma,lim s → ¯ s b ( ξ , λ ) (cid:26) Z ν exp λ + s ξ dv + Z ν exp λ + s ξ dv (cid:27) ≥ (cid:26) Z ν exp λ + ¯ s b ( ξ , λ ) ξ dv + Z ν exp λ + s b ( ξ , λ ) ξ dv (cid:27) dv (57)Assumption 3.1 implies that λ + ¯ s b ( ξ , λ ) ξ Λ or λ + ¯ s b ( ξ , λ ) ξ Λ . Henceat least one of the integrals on the right-hand side above is ∞ , which implieslim s → ¯ s b ( ξ , λ ) z ξ ( s ) = µ ( exp λ ) + µ ( exp λ ) − λ · ¯ ρ = ∞ . (58)Now suppose instead that ¯ s b ( ξ , λ ) = ∞ . There are two cases: Case 1: ξ · a ( v ) ≤ ξ · a ( v ) ≤ v ∈ R .Since ¯ ρ ∈ ¯ µ ( D × D ) , there exist g , g ∈ D × D such that ¯ ρ = ¯ µ ( g , g ) ;that is ¯ ρ = ¯ µ ( g , g ) = µ ( g ) µ ( g ) µ ( g ) + µ ( g ) µ ( g ) + µ ( g ) . (59)14y definition, g and g are not identically zero, and by Assumption 3.1, ν k j > Ω : = { v ∈ R | ξ · a ( v ) < } ∩ { v ∈ R | ν ( v ) g ( v ) > } and (60) Ω : = { v ∈ R | ξ · a ( v ) < } ∩ { v ∈ R | ν ( v ) g ( v ) > } (61)both have positive measure. Hence ξ · ¯ ρ = ξ · µ ( g ) + ξ · µ ( g ) (62) = Z ν ξ · a ( v ) g ( v ) dv + Z ν ξ · a ( v ) g ( v ) dv < , (63)so thatlim s → ∞ ¯ z ξ ( s ) = lim s → ∞ n µ ( exp λ + s ξ ) + µ ( exp λ + s ξ ) − ( λ + s ξ ) · ¯ ρ o (64) > lim s → ∞ {− ( λ + s ξ ) · ¯ ρ } = ∞ . (65) Case 2:
The set { v ∈ R | ξ · a ( v ) > } or { v ∈ Ω | ξ · a ( v ) > } has positivemeasure.Without loss of generality, assume that { v ∈ R | ξ · a ( v ) > } has positivemeasure. Then, there exists some ε > B = { v ∈ R | ξ · a ( v ) > ε } also has positive measure. Hencelim s → ∞ ¯ z ξ ( s ) ≥ lim s → ∞ (cid:18)(cid:18) Z B ν exp λ dx (cid:19) exp ( s ε ) − ( λ + s ξ ) · ρ mix (cid:19) = ∞ . (66)due to exponential growth in s . Theorem 3.8.
For any ¯ ρ ∈ ¯ µ ( D × D ) , the function ¯ z ( · , ¯ ρ ) has a unique mini-mizer λ ∗ ∈ ¯ Λ . The proof of this theorem is analogous to the proof of Theorem 3.4 in theintra-species case.
Corollary 3.9.
Given any f ∈ D and f ∈ D , there exist multipliers λ and λ such that λ = λ , λ = λ , and the corresponding functions M andM given in (2) solve (4) . roof. Let ¯ ρ = ¯ µ ( f , f ) . According to Theorem 3.8, ¯ z ( · , ¯ ρ ) has a unique min-imizer, which we denote by λ ∗ = (( λ ∗ ) , ( λ ∗ ) , ( λ ∗ ) , ( λ ∗ ) ) . By Lemma 3.2,¯ z ( · , ¯ ρ ) is also differentiable, so the first-order optimality condition (52) impliesthat ¯ ρ = ¯ µ ( exp ( λ ∗ ) , exp ( λ ∗ ) ) . The result then follows from Theorem 2.2. Fi-nally, we set λ = (( λ ∗ ) , ( λ ∗ ) , ( λ ∗ ) ) and λ = (( λ ∗ ) , ( λ ∗ ) , ( λ ∗ ) ) (67)and define M and M according to (2). The conditions (3) and (4) lead to standard conservation laws and an entropy dis-sipation statement. We recall a few definitions:
Definition 4.1.
The mass density, momentum, and energy of an integrable distri-bution g = g ( v ) of particles with mass m are given by the moments ρ g = Z mg ( v ) dv , q g = Z mvg ( v ) dv , and E g = Z m | v | g ( v ) dv , (68) respectively. The associated mean velocity and temperature are given byu g = q g ρ g = R vg ( v ) dv R g ( v ) dv and T g = E g ρ g / m − | q g | ρ g = R m | v − u g | g ( v ) dv R g ( v ) dv . (69) An immediate consequence of (3) and (4) is the following.
Theorem 4.2 (Conservation of the number of each species, total momentum andtotal energy) . The space-homogeneous form of (1) satisfies ∂ t ρ f = ∂ t ρ f = , ∂ t (cid:0) q f + q f (cid:1) = , ∂ t (cid:0) E f + E f (cid:1) = .2 Entropy dissipation and the structure of equilibria Define the total entropy density H ( g , g ) = Z h ( g ) dv + Z h ( g ) dv (71)and the dissipation density S ( g , g ) = S ( g ) + S ( g , g ) + S ( g , g ) + S ( g ) (72) = Z ν ln g ( M − g ) dv + Z ν ln g ( M − g ) dv (73) + Z ν ln g ( M − f ) dv + Z ν ln g ( M − g ) dv (74) Theorem 4.3.
Assume g , g > . Then S ( g , g ) ≥ with equality if and only ifg and g are two Maxwellian distributions with equal mean velocity and temper-ature.Proof. In [27], it is shown that S kk ( g ) ≥ g is aMaxwellian. Thus it remains to show a similar result for the combined quantity S ( g , g ) + S ( g , g ) . We begin with the following claim: I ( g , g ) : = Z ν ln M ( M − g ) dv + Z ν ln M ( M − g ) dv = . (75)Indeed an explicit calculation givesln M = m λ + m λ · v + m λ | v | and ln M = m λ + m λ · v + m λ | v | , (76)which when substituted into (75) gives I ( g , g ) = Z ν ( m λ + m λ · v + m λ | v | )( M − g ) dv (77) + Z ν ( m λ + m λ · v + m λ | v | )( M − g ) dv = , (78)due to the constraints (4). From (75), it follows that S ( g , g ) + S ( g , g ) = S ( g , g ) + S ( g , g ) − I ( g , g )= Z ν ln (cid:18) g M (cid:19) ( M − g ) dv + Z ν ln (cid:18) g M (cid:19) ( M − g ) dv ≤ . (79)17ith equality if and only if g = M and g = M . Moreover, a direct calcu-lation shows that the functions M and M have the same mean velocity andtemperature: u M = u M = − λ λ and T M = T M = − λ (80) Corollary 4.4 (Entropy inequality for mixtures) . Assume that f , f > are asolution to (1) where the target Maxwellians have the shape (2) , then we have thefollowing entropy inequality ∂ t ( H ( f , f )) + ∇ x · (cid:18) Z v ( h ( f ) + h ( f )) dv (cid:19) ≤ with equality if and only if f and f are two Maxwellian distributions with equalmean velocity and temperature.Proof. A direct calculation with (1) gives ∂ t H ( f , f ) + ∇ x · Z ( h ( f ) + h ( f )) vdv = S ( f , f ) . (82)The result then follows immediately from the previous theorem. N -species case The two-species case can be extended to a system of N -species that undergo binarycollisions. We consider the N -species kinetic equation, ∂ t f i + v · ∇ x f i = N ∑ j = ν i j ( M i j − f i ) , i = , ..., N . (83)The quantity ν ii is the collision frequency of particles of species i with itselfwhereas ν i j is the collision frequency of particles of species i with species j , with i , j = , ..., N , i = j . We only have terms of this form and not terms containingindices of more than two species because we consider only binary interactions.For fixed i , j ∈ { , . . . , N } the target Maxwellians M ii , M j j , M i j and M ji aregiven by (2). The single species target Maxwellians M ii and M j j will be deter-mined such that they satisfy (3). The functions M i j and M ji will be determined18uch that we obtain conservation of mass of each species and conservation of totalmomentum and total energy in interactions between these two species, i.e., Z ν i j M i j dv = Z ν i j f i dv , Z ν ji M ji dv = Z ν ji f j dv Z ν i j (cid:18) m i vm i | v | (cid:19) ( M i j − f i ) dv = − Z ν ji (cid:18) m j vm j | v | (cid:19) ( M ji − f j ) dv . (84)as an obvious generalization of (4). All the proofs concerning existence anduniqueness of the target Maxwellians and the H-Theorem can be proven exactly inthe same way as for two species. For the total entropy H ( f , ..., f N ) = R ( h ( f ) + · · · + h ( f N )) dv we obtain ∂ t ( H ( f , ..., f N )) + ∇ x · (cid:18) Z v ( h ( f ) + · · · + h ( f N )) dv (cid:19) ≤ . (85) Conclusion
We have presented a multi-species BGK model in which the collision frequenciesdepend on the microscopic velocity. The model is formally derived based on anentropy minimization principle, which implies that the target functions take theform of Maxwellians. However, contrary to classical BGK models with velocity-independent frequencies, the relationship between the Maxwellian parameters andthe moments of the distribution function is not analytic. Thus some effort isrequired to establish rigorously the existence of parameters which satisfy first-order optimality conditions. We also show that the derived model satisfies anH-Theorem and that it can be extended to the case of arbitrarily many speciesundergoing binary collisions.In future work, we will develop numerical tools for discretizing the modeldeveloped here, including the numerical solution of the defining optimizationproblem. A numerical code will enable computational explorations about howto choose the collision frequencies and what benefit is providing by their flexibil-ity. Also, because the motivation for the model is the simulation of multi-speciesplasmas, we will extend it for use in such contexts by adding self-consistent fields.
Acknowledgements
Christian Klingenberg acknowledges a grant by the Bayrische Forschungsallianz.19arlies Pirner is supported from the Humboldt foundation and from the AustrianScience Fund (FWF) through grant number F65.The work of Jeff Haack was supported by the US Department of Energy throughthe Los Alamos National Laboratory. Los Alamos National Laboratory is oper-ated by Triad National Security, LLC, for the National Nuclear Security Adminis-tration of U.S. Department of Energy (Contract No. 89233218CNA000001). LosAlamos Report LA-UR-20-21464.The work of Cory Hauck is sponsored by the Office of Advanced Scientific Com-puting Research, U.S. Department of Energy, and performed at the Oak RidgeNational Laboratory, which is managed by UT-Battelle, LLC under Contract No.De-AC05-00OR22725 with the U.S. Department of Energy. The United StatesGovernment retains and the publisher, by accepting the article for publication,acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published formof this manuscript, or allow others to do so, for United States Government pur-poses. The Department of Energy will provide public access to these results offederally sponsored research in accordance with the DOE Public Access Plan(http://energy.gov/downloads/doe-public-access-plan).
References [1] P. Andries, K. Aoki and B. Perthame,
A consistent BGK-type model for gasmixtures,
Journal of Statistical Physics 106 (2002) 993-1018[2] PL Bhatnagar, EP Gross, M Krook
A model for collision processes in gases.I. Small amplitude processes in charged and neutral one-component systems,
Physical review, 94 (1954).[3] M.Bisi, M. C´aceres,
A BGK relaxation model for polyatomic gas mixtures,
Communication in Mathematical Sciences, 14 (2016) 297-325[4] Bisi, M., Groppi, M., Spiga, G. (2010). Kinetic Bhatnagar-Gross-Krookmodel for fast reactive mixtures and its hydrodynamic limit. Physical Re-view E, 81(3), 036327. 205] Bobylev, A. V., Bisi, M., Groppi, M., Spiga, G., Potapenko, I. F. (2018). Ageneral consistent BGK model for gas mixtures. Kinetic and Related Mod-els, 11(6).[6] Borwein, J. M., Lewis, A. S. (1991). Duality relationships for entropy-likeminimization problems. SIAM Journal on Control and Optimization, 29(2),325-338.[7] S. Brull, V. Pavan and J. Schneider, DerivationofaBGKmodelformixtures,European Journal of Mechanics B/Fluids, 33 (2012) 74-86[8] S. Brull, An ellipsoidal statistical model for gas mixtures, Communicationsin Mathematical Sciences, (2015), 1-13[9] A. Crestetto, C. Klingenberg, M. Pirner, Kinetic/fluid micro-macro numer-ical scheme for a two component gas mixture, SIAM Multiscale Modelingand Simulation 18.2, pp. 970-998 (2020)[10] J. Greene, Improved Bhatnagar-Gross-Krook model of electron-ion colli-sions. Phys. Fluids 16, 2022– 2023 (1973)[11] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kineticequationsand related problemswith stiffsources, Journal of ComputationalPhysics 20 (2010) 7625-7648[12] V. Garz´o, A. Santos and J. J. Brey, A kinetic model for a multicomponentgas Physics of Fluids, 1 (1989) 380-383[13] M. Groppi, S. Monica and G. Spiga, A kinetic ellipsoidal BGK model for abinarygasmixture, epljournal, (2011), 64002[14] E. P. Gross and M. Krook, Model for collision processes in gases: small-amplitudeoscillationsofcharged two-componentsystems, Physical Review3 (1956) 593[15] J. R. Haack, C.D. Haack, and M.S.Murillo . A conservative, entropic multi-species BGK model. Journal of Statistical Physics, 168 (2017), 826-856.[16] B. Hamel, Kineticmodelforbinarygasmixtures, Physics of Fluids 8 (1965)418-425 2117] M. Junk. Maximum entropy for reduced moment problems. MathematicalModels and Methods in Applied Sciences, 2000, 10. Jg., Nr. 07, S. 1001-1025.[18] N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, New York, 1973.[19] Y. T. Lee, and R. M. More, An electron conductivity model for dense plas-mas , Physics of Fluids 27 (1984), 1273–1286.[20] L. Mieussens, H. Struchtrup,
Numerical comparison of Bhatna-gar–Gross–Krook models with proper Prandtl number , Physics of Fluids16 (2004), 2797–2813.[21] C. Klingenberg, M.Pirner, G.Puppo,