Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures

The linearized collision operator of the Boltzmann equation can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator. Compactness of the integral operator for monatomic single species is a classical result, while corresponding result for mixtures is more recently obtained. In this work the compactness of the operator for polyatomic single species, where the polyatomicity is modeled by a discrete internal energy variable, is studied. With a probabilistic formulation of the collision operator as a starting point, compactness is obtained by proving that the integral operator is a sum of Hilbert-Schmidt integral operators and approximately Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere model. Then it follows that the linearized collision operator is a Fredholm operator. The results can be extended to mixtures. For brevity, only the case of mixtures for monatomic species is accounted for.


Introduction
The Boltzmann equation is a fundamental equation of kinetic theory of gases. Considering deviations of an equilibrium, or Maxwellian, distribution, a linearized collision operator is obtained. The linearized collision operator can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator −K. Compact properties of the integral operator K (for angular cut-off kernels) are extensively studied for monatomic single species, see e.g. [8,5,4,13]. The integral operator can be written as the sum of a Hilbert-Schmidt integral operator and an approximately Hilbert-Schmidt integral operator (cf. Lemma 7 in Section 4.1.1) [7], and so compactness of the integral operator K can be obtained. More recently, compactness results were also obtained for monatomic multi-component mixtures [3]. In this work, we consider polyatomic single species, where the polyatomicity is modeled by a discrete internal energy variable [6,9]. We also consider the case of multi-component mixtures. For brevity -and clearness -we restrict ourselves to monatomic mixtures, even if compactness is already developed in that case in [3]. Our approach is different -maybe, also more direct and transparent -even if also similarities are obvious. The approach was crucial for us to understand the polyatomic case. In addition to trying to make the ideas and approach more clear, our intention by addressing the two cases separately -but in a corresponding way -, is to make the extension to multi-component mixtures of polyatomic species, at least at an formal level, clear. The case of polyatomic single species, where the polyatomicity is modeled by a continuous internal energy variable is considered in [1].
Motivated by an approach by Kogan in [12,Sect. 2.8] for the monatomic single species case, a probabilistic formulation of the collision operator is considered as the starting point. With this approach, it is shown, based on modified arguments from the monatomic case, that the integral operator K can be written as a sum of Hilbert-Schmidt integral operators and approximately Hilbert-Schmidt integral operators -and so compactness of the integral operator K follows -, also for the polyatomic model. The operator K is self-adjoint, as well as the collision frequency, why the linearized collision operator as the sum of two self-adjoint operators of which one is bounded, is also self-adjoint.
For hard sphere models, bounds on the collision frequency are obtained. Then the collision frequency is coercive and becomes a Fredholm operator. The set of Fredholm operators is closed under addition with compact operators, why also the linearized collision operator becomes a Fredholm operator by the compactness of the integral operator K.
The rest of the paper is organized as follows. In Section 2, the models considered are presented. The probabilistic formulation of the collision operators considered and its relations to more classical formulations [6,9] are accounted for in Section 2.1.1 for polyatomic molecules and in Section 2.2.1 for mixtures. Some classical results for the collision operators in Sections 2.1.2 and 2.2.2, respectively, and the linearized collision operators in Sections 2.1.3 and 2.2.3, respectively, are reviewed. Section 3 is devoted to the main results of this paper, while the main proofs are addressed in Section 4; proofs of compactness of the integral operators K are presented in Sections 4.1.1 and 4.2.1, respectively, while proof of the bounds on the collision frequency appears in Sections 4.1.2 and 4.2.2, respectively. Finally, the appendix concerns a new proof of a crucial -for the compactness in the mixture case -lemma by [3].

Models
This section concerns the models considered. Probabilistic formulations of the collision operators are considered, whose relations to more classical formulations are accounted for. Known properties of the models and corresponding linearized collision operators are also reviewed.

Polyatomic molecules modeled by a discrete internal energy variable
Consider a single species of polyatomic molecules with mass m, where the polyatomicity is modeled by r different internal energies I 1 , ..., I r . The inter-nal energies I i , i ∈ {1, ..., r}, are assumed to be nonnegative real numbers; {I 1 , ..., I r } ⊂ R + . The distribution functions are of the form f = (f 1 , ..., f r ), where f i = f i (t, x, ξ) = f (t, x, ξ, I i ), with t ∈ R + , x = (x, y, z) ∈ R 3 , and ξ = (ξ x , ξ y , ξ z ) ∈ R 3 . Moreover, consider the real Hilbert space h (r) := L 2 (dξ) r , with inner The evolution of the distribution functions is (in the absence of external forces) described by the (vector) Boltzmann equation where the (vector) collision operator Q = (Q 1 , ..., Q r ) is a quadratic bilinear operator that accounts for the change of velocities and internal energies of particles due to binary collisions (assuming that the gas is rarefied, such that other collisions are negligible).

Collision operator
The (vector) collision operator Q = (Q 1 , ..., Q r ) has components that can be written in the following form for some constant ϕ = (ϕ 1 , ..., ϕ r ) ∈ R r . Here and below the abbreviations are used. The transition probabilities are of the form, cf. [10], as well as [12,15,2] for the monatomic case, σ kl ij = σ kl ij (|g| , |cos θ|) > 0 and σ ij kl = σ ij kl (|g | , |cos θ|) > 0 a.e., with cos θ = g · g |g| |g | , g = ξ − ξ * , g = ξ − ξ * , and ∆I kl where δ 3 and δ 1 denote the Dirac's delta function in R 3 and R, respectively; taking the conservation of momentum and total energy into account. Here it is assumed that the scattering cross sections σ kl ij , {i, j, k, l} ⊆ {1, ..., r}, satisfy the microreversibility conditions Furthermore, to obtain invariance of change of particles in a collision, it is assumed that the scattering cross sections σ kl ij , {i, j, k, l} ⊆ {1, ..., r}, satisfy the symmetry relations The invariance under change of particles in a collision, which follows directly by the definition of the transition probability (3) and the symmetry relations (5) for the collision frequency, and the microreversibility of the collisions (4), implies that the transition probabilities (3) satisfy the relations Applying known properties of Dirac's delta function, the transition probabilities may be transformed to Remark 1 Note that, cf. [10], By a change of variables ξ , the observation that can be made, resulting in a more familiar form of the Boltzmann collision operator for polyatomic molecules modeled with a discrete energy variable, cf. e.g. [6,9].

Collision invariants and Maxwellian distributions
The following lemma follows directly by the relations (6).
The weak form of the collision operator Q(f, f ) reads for any function g = (g 1 , ..., g r ), such that the first integrals are defined for all {i, j, k, l} ⊆ {1, . . . , r}, while the following equalities are obtained by applying Lemma 1.
We have the following proposition.

Linearized collision operator
Considering a deviation of a Maxwellian distribution M = (M 1 , ..., M r ), where results, by insertion in the Boltzmann equation (1), in the system where the components of the linearized collision operator L = (L 1 , ..., L r ) are given by with M j * ϕ i ϕ j W (ξ, ξ * , I i , I j ξ , ξ * , I k , I l ) dξ * dξ dξ * , and while the components of the quadratic term Γ = (Γ 1 , ..., Γ r ) are given by The multiplication operator Λ defined by is a closed, densely defined, self-adjoint operator on L 2 (dξ) r . It is Fredholm, as well, if and only if Λ is coercive.
The following lemma follows immediately by Lemma 1.
The weak form of the linearized collision operator L reads (Lh, g) Proof. By Lemma 3, it is immediate that (Lh, g) = (h, Lg), and Furthermore, h ∈ ker L if and only if (Lh, h) = 0, which will be fulfilled if and only if for all {i, j, k, l} ⊆ {1, . . . , r} i.e. if and only if M −1/2 h is a collision invariant. The last part of the lemma now follows by Proposition 2.
Remark 3 Note also that the quadratic term is orthogonal to the kernel of L, i.e. Γ (h, h) ∈ (ker L) ⊥ h (r) .

Multicomponent mixtures
Consider a mixture of s monatomic species a 1 , ..., a s , with masses m α1 , ..., m αs , respectively (s = 1 corresponds to the case of a single species). The distribution functions are of the form f = (f 1 , ..., f s ), where f α = f α (t, x, ξ), with t ∈ R + , x = (x, y, z) ∈ R 3 , and ξ = (ξ x , ξ y , ξ z ) ∈ R 3 . Moreover, consider the real Hilbert space h (s) := L 2 (dξ) s , with inner The evolution of the distribution functions is (in the absence of external forces) described by the (vector) Boltzmann equation (1), where the (vector) collision operator Q = (Q 1 , ..., Q s ) is a quadratic bilinear operator that accounts for the change of velocities of particles due to binary collisions (assuming that the gas is rarefied, such that other collisions are negligible).

Collision operator
The (vector) collision operator Q = (Q 1 , ..., Q s ) has components that can be written in the following form -reminding the abbreviations (2), The transition probabilities are of the form cf. [4, p.65] where δ 3 and δ 1 denote the Dirac's delta function in R 3 and R, respectively; taking the conservation of momentum and kinetic energy into account. The scattering cross sections σ αβ , {α, β} ⊆ {1, ..., s}, satisfy the symmetry relation while Applying known properties of Dirac's delta function, the transition probabilities may be transformed to Due to invariance under change of particles in a collision and microreversibility of the collisions, which follows directly by the definition of the transition probability (16), (19), (18), and the symmetry relation (17) for the collision frequency, the transition probabilities (16) satisfy the relations By a change of variables ξ , the observation that can be made, resulting in a more familiar form of the Boltzmann collision operator for mixtures.

Collision invariants and Maxwellian distributions
The following lemma follows directly by the relations (20).

Lemma 4 The measures
The weak form of the collision operator Q(f, f ) reads for any function g = (g 1 , ..., g s ), such that the first integrals are defined for all {α, β} ⊆ {1, . . . , s}, while the following equalities are obtained by applying Lemma 4. We have the following proposition. Then It is clear that e 1 , ..., e s , mξ x , mξ y , mξ z , and m |ξ| 2 , where {e 1 , ..., e s } is the standard basis of R s and m = (m 1 , ..., m s ), are collision invariantscorresponding to conservation of mass(es), momentum, and kinetic energy.

Proposition 5
The vector space of collision invariants is generated by where m = (m 1 , ..., m s ) and {e 1 , ..., e s } is the standard basis of R s . Define

It follows by Proposition 4 that
with equality if and only if for all {α, β} ⊆ {1, . . . , s} or, equivalently, if and only if an H-theorem can be obtained.

Linearized collision operator
where while the components of the quadratic term Γ = (Γ 1 , ..., Γ s ) are of the form (15). The multiplication operator Λ defined by is a closed, densely defined, self-adjoint operator on L 2 (dξ) s . It is Fredholm, as well, if and only if Λ is coercive.
The following lemma follows immediately by Lemma 4.
The weak form of the linearized collision operator L reads for any function g = (g 1 , ..., g s ), such that the first integrals are defined for all {α, β} ⊆ {1, . . . , s}, while the following equalities are obtained by applying Lemma 5.
We have the following lemma.
Lemma 6 Let g = (g 1 , ..., g s ) be such that for any {α, β} ⊆ {1, . . . , s} Proposition 6 The linearized collision operator is symmetric and nonnegative, Proof. By Lemma 6, it is immediate that (Lh, g) = (h, Lg), and i.e. if and only if M −1/2 h is a collision invariant. The last part of the lemma now follows by Proposition 5.
Remark 5 Note also that the quadratic term is orthogonal to the kernel of L,

Main Results
This section is devoted to the main results, concerning compact properties in Theorems 1 and 3, respectively, and bounds of collision frequencies in Theorems 2 and 4, respectively.

Polyatomic molecules modeled by a discrete internal energy variable
Assume that for some positive number γ, such that 0 < γ < 1, there is a bound .., r}, satisfy the bound (28) for some positive number γ, such that 0 < γ < 1. Then the operator K = (K 1 , ..., K r ), with the components K i given by (14) is a selfadjoint compact operator on L 2 (dξ) r .
Theorem 1 will be proven in Section 4.1.1.

Corollary 1
The linearized collision operator L, with scattering cross sections satisfying (28), is a closed, densely defined, self-adjoint operator on L 2 (dξ) r .

Proof.
By Theorem 1, the linear operator L = Λ − K, where Λ(f ) = νf for ν = diag (ν 1 , ..., ν s ), is closed as the sum of a closed and a bounded operator, and densely defined, since the domains of the linear operators L and Λ are equal; D(L) = D(Λ). Furthermore, it is a self-adjoint operator, since the set of self-adjoint operators is closed under addition of bounded self-adjoint operators, see Theorem 4.3 of Chapter V in [11]. Now consider the scattering cross sections -cf. hard sphere models - for some positive constant C > 0 and all {i, j, k, l} ⊆ {1, ..., r}.
In fact, it would be enough with the bounds for some positive constants C ± > 0, on the scattering cross sections.

Corollary 2
The linearized collision operator L, with scattering cross sections (29) (or (30)), is a Fredholm operator, with domain Proof. By Theorem 2 the multiplication operator Λ is coercive and, hence, a Fredholm operator. The set of Fredholm operators is closed under addition of compact operators, see Theorem 5.26 of Chapter IV in [11] and its proof, so, by Theorem 2, L is a Fredholm operator.
Corollary 3 For the linearized collision operator L, with scattering cross sections (29) (or (30)), there exists a positive number λ, 0 < λ < 1, such that As a Fredholm operator, L is closed with a closed range, and as a compact operator, K is bounded, and so there are positive constants ν 0 > 0 and c K > 0, such that (h, Lh) ≥ ν 0 (h, h) and (h, Kh) ≤ c K (h, h).

Multicomponent mixtures
Assume that for some positive number γ, such that 0 < γ < 1, there is a bound, cf. [3], Theorem 3 will be proven in Section 4.2.1, but an alternative proof can be found in [3]. Essential ideas in the proof are inspired by [3], while the approach differs.

Corollary 4
The linearized collision operator L, with scattering cross sections satisfying (33), is a closed, densely defined, self-adjoint operator on L 2 (dξ) s .
In fact, it would be enough with the bounds for some positive constants C ± > 0 and all {α, β} ⊆ {1, ..., s}, on the scattering cross sections.

Corollary 5
The linearized collision operator L, for a hard-sphere model (34) (or (35)), is a Fredholm operator, with domain

Compactness and Bounds on the Collision Frequency
This section concerns the proofs of the main results presented in Theorem 1-4. Note that throughout this section C will denote a generic positive constant.
To show the compactness properties we will apply the following result. Denote, for any (non-zero) natural number N , Then we have the following lemma from [7], that will be of practical use for us to obtain compactness in this section .

Polyatomic molecules
This section is devoted to the proofs of the compactness properties in Theorem 1 and the bounds on the collision frequency in Theorem 2 of the linearized collision operator for polyatomic molecules modeled with a discrete number of internal energies.

Compactness
This section concerns the proof of Theorem 1. Note that in the proof the kernels are rewritten in such a way that ξ * -and not ξ and ξ * -always will be argument of the distribution functions. Then there will be essentially two different types of kernels; either ξ * is an argument in the loss term (as ξ) or in the gain term (opposite to ξ) of the collision operator. The kernels of the terms from the loss part of the collision operator will be shown to be Hilbert-Schmidt in a quite direct way, while the kernels of the terms from the gain parts of the collision operators will be shown to be approximately Hilbert-Schmidt in the sense of Lemma 7.

Bounds on the collision frequency
This section concerns the proof of Theorem 2.

Mixtures
This section is devoted to the proofs of the compactness properties in Theorem 3 and the bounds on the collision frequency in Theorem 4 of the linearized collision operator for (monatomic) multicomponent mixtures.

Compactness
This section concerns the proof of Theorem 3. Note that in the proof the kernels are rewritten in such a way that ξ * -and not ξ and ξ * -always will be argument of the distribution functions. As for single species, either ξ * is an argument in the loss term (as ξ) or in the gain term (opposite to ξ) of the collision operator. However, in the latter case, unlike for single species, for mixtures we have to differ between two different cases; either ξ * is associated to the same species as ξ, or not. The kernels of the terms from the loss part of the collision operator will be shown to be Hilbert-Schmidt in a quite direct way. The kernels of -some of -the terms -for which ξ * is associated to the same species as ξ -from the gain parts of the collision operators will be shown to be approximately Hilbert-Schmidt in the sense of Lemma 7. By applying the following lemma, Lemma 8, (for disparate masses) by Boudin et al in [3], it will be shown that the kernels of the remaining terms -i.e. for which ξ * is associated to the opposite species to ξ -from the gain parts of the collision operators, are Hilbert-Schmidt. and Then there exists a positive number ρ, 0 < ρ < 1, such that An alternative -maybe more basic, in the sense that only very basic calculations are used -to the proof of Lemma 8 in [3] is accounted for in the appendix. The proof is constructive, in the way that an explicit value of such a number ρ, namely is produced in the proof. Now we turn to the proof of Theorem 3.
Proof. By first renaming {ξ * } ξ and then {ξ * } ξ * Moreover, by renaming {ξ * } ξ , It follows that Note that, by applying the second relation in (20) and renaming ξ ξ * , Moreover, since, by applying the first relation in (20) and renaming ξ ξ * , while, by applying the two first relations in (20) and renaming ξ ξ * , We now continue by proving the compactness for the three different types of collision kernel separately. Note that, by applying the last relation in (20), k αβ (ξ, ξ * ) if α = β, and we will remain with only two casesthe first two below. Even if m α = m β , the kernels k (α) αβ (ξ, ξ * ) and k (β) αβ2 (ξ, ξ * ) are structurally equal, why we (in principle) remains with (first) two cases (the second one twice).