Linearized Boltzmann collision operator for a mixture of monatomic and polyatomic chemically reacting species

Abstract

At higher altitudes near space shuttles moving at hypersonic speed the air is excited to high temperatures. Then not only mechanical collisions are affecting the gas flow, but also chemical reactions have an impact on such hypersonic flows. In this work we insert chemical reactions, in form of dissociations and associations, in a model for a mixture of mono- and polyatomic (non-reacting) species. More general chemical reactions, e.g., bimolecular ones, can be obtained by instant combinations of the considered reactions. Polyatomicity is here modelled by a continuous internal energy variable and the evolution of the gas is described by a Boltzmann equation. In the Chapman-Enskog process—and related half-space problems—the linearized Boltzmann collision operator plays a central role. Here we extend some important properties of the linearized operator to the considered model with chemical reactions. A compactness result, that the linearized operator can be decomposed into a sum of a positive multiplication operator—the collision frequency—and a compact integral operator, is obtained. The terms of the integral operator are shown to be (at least) uniform limits of Hilbert-Schmidt integral operators and, thereby, compact operators. Self-adjointness of the linearized operator follows as a direct consequence. Also, bounds on—including coercivity of—the collision frequency is obtained for hard sphere, as well as hard potentials with cutoff, like models. As consequence, Fredholmness as well as the domain of the linearized operator are obtained.

1 Introduction

At atmospheric reentry of space shuttles, or, higher altitudes flights at hypersonic speed the vehicles excite the air around them to high temperatures. At high altitudes the pressure is lower and therefore the air is excited at lower temperatures than at the sea level [20]. In high temperature gases, not only mechanical collisions affect the flow, but also chemical reactions have an impact on such flows [14, 20]. Typical chemical reactions in air at high temperatures are dissociation of oxygen, $\textrm{O}_{2}\rightleftarrows 2\textrm{O}$, and nitrogen, $\textrm{ N}_{2}\rightleftarrows 2\textrm{N,}$ but at higher temperatures even ionization of oxygen and nitrogen atoms, $\textrm{O}\rightleftarrows \textrm{ O}^{+}+\textrm{e}^{-}$and $\textrm{N}\rightleftarrows \textrm{N}^{+}+\textrm{ e}^{-}$, respectively [20].

At high altitude the gas is rarefied and the Boltzmann equation is used to describe the evolution of the gas flow, e.g., around a space shuttle in the upper atmosphere during reentry [2, 14]. Studies of the main properties of the linearized Boltzmann collision operator are of great importance in gaining increased knowledge about related problems, see, e.g., [13, 14] and references therein.

The linearized collision operator, obtained by considering deviations of an equilibrium, or Maxwellian, distribution, 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., [13, 17,18,19, 23], and more recently for monatomic multi-component mixtures [4, 10]. Extensions to polyatomic gases, where the polyatomicity is modeled by either a discrete, or, a continuous internal energy variable for single species [4, 5, 12], or, mixtures [6, 7] have very recently been conducted. Compactness results are also recently obtained for models of polyatomic single gases, with a continuous internal energy variable, where the molecules undergo resonant collisions (for which internal energy and kinetic energy, respectively, are conserved) [9]. In this work,¹ we extend the results for mixtures of mono- and polyatomic (non-reacting) species in [6], where the polyatomicity is modeled by a continuous internal energy variable, cf., [1, 2, 15], to include chemical reactions, in form of dissociations and recombinations (associations) [14]. More general chemical reactions, e.g., bimolecular ones [15], can be obtained by instant combinations of associations and dissociations.

Following the lines of a series of papers [4,5,6,7], motivated by an approach by Kogan in [22, 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 that the integral operator K can be written as a sum of Hilbert-Schmidt integral operators and operators, which are uniform limits of Hilbert-Schmidt integral operators (cf. Lemma 3 in Sect. 4)—and so compactness of the integral operator K follows. Self-adjointness of the operator K and the collision frequency, imply that the linearized collision operator, as the sum of two self-adjoint operators whereof (at least) one is bounded, is also self-adjoint.

For models corresponding to hard sphere models, as well as hard potentials with cut off models, in the monatomic case, bounds on the collision frequency are obtained. Then the collision frequency is coercive and, hence, a Fredholm operator. The resultant Fredholmness—vital in the Chapman-Enskog process—of the linearized operator, is due to that the set of Fredholm operators is closed under addition with compact operators. Unlike for hard potential models, the linearized operator is not Fredholm for soft potential models, even in the monatomic case. The domain of collision frequency—and, hence, of the linearized collision operator as well—follows directly by the obtained bounds.

For hard sphere like models the linearized collision operator satisfies all the properties of the general linear operator in the abstract half-space problem considered in [3], and, hence, the general existence results in [3] apply.

The rest of the paper is organized as follows. In Sect. 2, the model considered is presented. The probabilistic formulation of the collision operators considered and its relations to a more classical parameterized expression are accounted for in Sect. 2.1. Some results for the collision operator in Sect. 2.2 and the linearized collision operator in Sect. 2.3 are presented. Section 3 is devoted to the main results of this paper, while the main proofs are addressed in Sects. 4–5; a proof of compactness of the integral operators K is presented in Sect. 4, while a proof of the bounds on the collision frequency appears in Sect. 5.

2 Model

Consider a multicomponent mixture of s species $a_{1},\ldots ,a_{s}$, with $s_{0}$ monatomic and $s_{1}:=s-s_{0}$ polyatomic species, and masses $m_{1},\ldots ,m_{s}$, respectively, and introduce the index sets

$$\begin{aligned} {\mathcal {I}}= & {} \left\{ 1,\ldots ,s\right\} \text {, }{\mathcal {I}}_{mono}=\left\{ \alpha ;\text { }a_{\alpha }\text { is monatomic}\right\} =\left\{ 1,\ldots ,s_{0}\right\} \text {, and } \\ {\mathcal {I}}_{poly}= & {} \left\{ \alpha ;\text { }a_{\alpha }\text { is polyatomic }\right\} =\left\{ s_{0}+1,\ldots ,s\right\} \text {.} \end{aligned}$$

Here the polyatomicity is modeled by a continuous internal energy variable $I\in$ ${\mathbb {R}}_{+}$ [8]. Denote by $\varepsilon _{\alpha 0}$ the potential energy due to configuration of species $a_{\alpha }$ for $\alpha \in {\mathcal {I}}$. Moreover, $\delta ^{\left( 1\right) }=...=\delta ^{\left( s_{0}\right) }=2$, while $\delta ^{\left( \alpha \right) }$, with $\delta ^{\left( \alpha \right) }\ge 2$, denote the number of internal degrees of freedom of the species for $\alpha \in {\mathcal {I}}_{poly}$.

The distribution functions are of the form $f=\left( f_{1},\ldots ,f_{s}\right)$, where any component $f_{\alpha }=f_{\alpha }\left( t,{\textbf{x}},{\textbf{Z}}\right)$, with

$$\begin{aligned} \mathbf {Z=Z}_{\alpha }=\left\{ \begin{array}{l} \quad \varvec{\xi } \qquad \, \text {for }\alpha \in {\mathcal {I}} _{mono}\quad \\ \left( \varvec{\xi },I\right) \quad \text {for }\alpha \in {\mathcal {I}} _{poly} \end{array} \right. \text {,} \end{aligned}$$

$\left\{ t,I\right\} \subset {\mathbb {R}}_{+}$, ${\textbf{x}}=\left( x,y,z\right) \in {\mathbb {R}}^{3}$, and $\varvec{\xi }=\left( \xi _{x},\xi _{y},\xi _{z}\right) \in {\mathbb {R}}^{3}$, is the distribution function for species $a_{\alpha }$.

Moreover, consider the real Hilbert space

$$\begin{aligned} {{\mathfrak {h}}}=\left( L^{2}\left( d\varvec{\xi \,}\right) \right) ^{s_{0}}\times \left( L^{2}\left( d\varvec{\xi \,}dI\right) \right) ^{s_{1}}, \end{aligned}$$

with inner product

$$\begin{aligned} \left( f,g\right) =\sum _{\alpha =1}^{s_{0}}\int _{{\mathbb {R}}^{3}}f_{\alpha }g_{\alpha }\,d\varvec{\xi \,}+\sum _{\alpha =s_{0}+1}^{s}\int _{{\mathbb {R}} ^{3}\times {\mathbb {R}}_{+}}f_{\alpha }g_{\alpha }\,d\varvec{\xi \,}dI \text { for }f,g\in {{\mathfrak {h}}}\text {.} \end{aligned}$$

(1)

The evolution of the distribution functions is (in the absence of external forces) described by the (vector) Boltzmann equation

$$\begin{aligned} \frac{\partial f}{\partial t}+\left( \varvec{\xi }\cdot \nabla _{{\textbf{x}}}\right) f=Q\left( f\right) \text {, with }Q\left( f\right) =Q_{mech}\left( f,f\right) +Q_{chem}\left( f\right) \text {,} \end{aligned}$$

(2)

where the (vector) collision operator $Q_{mech}=\left( Q_{m1},\ldots ,Q_{ms}\right)$ is a quadratic bilinear operator, see [1, 6], 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) and the transition operator $Q_{chem}=\left( Q_{c1},\ldots ,Q_{cs}\right)$ accounts for the change of velocities and internal energies under possible chemical reactions—processes where two particles associate into one, or, vice versa, one particle dissociates into two.

A chemical process (dissociation or association) $\left\{ a_{\gamma },a_{\varsigma }\right\} \longleftrightarrow \left\{ a_{\beta }\right\}$; one given particle of species $a_{\beta }$ for $\beta \in {\mathcal {I}}_{poly}$ dissociating in two particles of species $a_{\gamma }$ and $a_{\varsigma }$ for $\left( \gamma ,\varsigma \right) \in {\mathcal {I}} ^{2}$, or, vice versa, two given particles of species $a_{\gamma }$ and $a_{\varsigma }$ for $\left( \gamma ,\varsigma \right) \in {\mathcal {I}}^{2}$ associating in a particle of species $a_{\beta }$ for $\beta \in {\mathcal {I}} _{poly}$, can be represented by two post-/preprocess elements, each element consisting of a microscopic velocity and possibly also an internal energy, ${\textbf{Z}}^{\prime }$ and ${\textbf{Z}}_{*}^{\prime }$, and one corresponding post-/pre-process element, ${\textbf{Z}}_{*}$, consisting of a microscopic velocity and an internal energy. Denote the set of all indices $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {I}}_{poly}\times {\mathcal {I}}^{2}$ of valid considered chemical processes $\left\{ a_{\gamma },a_{\varsigma }\right\} \longleftrightarrow \left\{ a_{\beta }\right\}$ by ${\mathcal {C}}$. Assume below that $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$.

Due to mass, momentum, and total energy conservation, the following relations have to be satisfied by the elements

$$\begin{aligned}{} & {} m_{\gamma }+m_{\zeta }=m_{\beta }{ };\qquad m_{\gamma } \varvec{\xi }^{\prime }+m_{\zeta }\varvec{\xi }_{*}^{\prime }=m_{\beta }\varvec{\xi }_{*}; \\{} & {} m_{\gamma }\frac{\left| \varvec{\xi }^{\prime }\right| ^{2}}{2} +I^{\prime }{\textbf{1}}_{\gamma \in {\mathcal {I}}_{poly}}+\varepsilon _{\gamma 0}+m_{\zeta }\frac{\left| \varvec{\xi }_{*}^{\prime }\right| ^{2}}{2}+I_{*}^{\prime }{\textbf{1}}_{\zeta \in {\mathcal {I}}_{poly}}+\varepsilon _{\zeta 0}=m_{\beta }\frac{\left| \varvec{\xi } _{*}\right| ^{2}}{2}+I_{*}+\varepsilon _{\beta 0}\text {.} \end{aligned}$$

We denote the internal energy gap, and a multiple of it, by

$$\begin{aligned} \Delta I=I_{*}-I^{\prime }{\textbf{1}}_{\gamma \in {\mathcal {I}} _{poly}}-I_{*}^{\prime }{\textbf{1}}_{\zeta \in {\mathcal {I}}_{poly}}\text { and }{\widetilde{\Delta }}_{\alpha \beta }I=\frac{m_{\alpha }+m_{\beta }}{ m_{\alpha }m_{\beta }}\Delta I\text {,} \end{aligned}$$

the potential energy gap by

$$\begin{aligned} \Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}=\varepsilon _{\gamma 0}+\varepsilon _{\zeta 0}-\varepsilon _{\beta 0}>0\text {,} \end{aligned}$$

and the relative velocities by

$$\begin{aligned}{} & {} {\textbf{g}}^{\prime }=\varvec{\xi }^{\prime }-\varvec{\xi }_{*}^{\prime }\text {, }{\textbf{g}}_{*}=\varvec{\xi }_{*}-\varvec{ \xi }_{*}^{\prime }=\frac{m_{\gamma }}{m_{\beta }}{\textbf{g}}^{\prime }=\left( 1-\frac{m_{\zeta }}{m_{\beta }}\right) {\textbf{g}}^{\prime }\text {, and } \\{} & {} \widetilde{{\textbf{g}}}=\varvec{\xi }_{*}-\varvec{\xi }^{\prime }=-\frac{m_{\zeta }}{m_{\beta }}{\textbf{g}}^{\prime }=\left( \frac{m_{\gamma } }{m_{\beta }}-1\right) {\textbf{g}}^{\prime }\text {.} \end{aligned}$$

It follows that the center of mass velocity

$$\begin{aligned} \varvec{\xi }_{*}=\frac{m_{\gamma }\varvec{\xi }^{\prime }+m_{\zeta }\varvec{\xi }_{*}^{\prime }}{m_{\beta }}=\frac{m_{\gamma }\varvec{\xi }^{\prime }+m_{\zeta }\varvec{\xi }_{*}^{\prime }}{ m_{\gamma }+m_{\zeta }}={\textbf{G}}_{\gamma \zeta }^{\prime }\text {,} \end{aligned}$$

is conserved, as well as the total energy in the center of mass frame

$$\begin{aligned} E_{\gamma \zeta }^{\prime }=\dfrac{m_{\gamma }m_{\zeta }}{2\left( m_{\gamma }+m_{\zeta }\right) }\left| {\textbf{g}}^{\prime }\right| ^{2}+I^{\prime }{\textbf{1}}_{\gamma \in {\mathcal {I}}_{poly}}+I_{*}^{\prime }{\textbf{1}}_{\zeta \in {\mathcal {I}}_{poly}}+\varepsilon _{\gamma 0}+\varepsilon _{\zeta 0}=I_{*}+\varepsilon _{\beta 0}=E_{\beta } \text {,} \end{aligned}$$

while, by straightforward calculations,

$$\begin{aligned}{} & {} \Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}=\Delta I-\Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}=\frac{m_{\gamma }}{2}\left| \varvec{\xi } ^{\prime }\right| ^{2}+\frac{m_{\zeta }}{2}\left| \varvec{\xi } _{*}^{\prime }\right| ^{2}-\frac{\left| m_{\gamma }\varvec{ \xi }^{\prime }+m_{\zeta }\varvec{\xi }_{*}^{\prime }\right| ^{2} }{2\left( m_{\gamma }+m_{\zeta }\right) }\\{} & {} \quad =\frac{m_{\gamma }m_{\zeta }}{ 2\left( m_{\gamma }+m_{\zeta }\right) }\left| {\textbf{g}}^{\prime }\right| ^{2}\text {,} \\{} & {} {\widetilde{\Delta }}_{\gamma \zeta }^{\beta }{\mathcal {E}}=\frac{2m_{\beta }}{ m_{\gamma }m_{\zeta }}\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}=\left| {\textbf{g}}^{\prime }\right| ^{2}\text { and }{\widetilde{\Delta }}_{\beta }^{\gamma \zeta }{\mathcal {E}}=\frac{2m_{\gamma }}{m_{\beta }m_{\zeta }}\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}=\left| {\textbf{g}}_{*}\right| ^{2}\text {.} \end{aligned}$$

Moreover, for any possible chemical process $\left\{ a_{\gamma },a_{\varsigma }\right\} \longleftrightarrow \left\{ a_{\beta }\right\}$, the following relation on the numbers of internal degrees of freedom is assumed:

$$\begin{aligned} \delta ^{\left( \gamma \right) }+\delta ^{\left( \zeta \right) }\ge \delta ^{\left( \beta \right) }+1\text {.} \end{aligned}$$

(3)

On the other hand, a mechanical collision can, given two particles of species $a_{\alpha }$ and $a_{\beta }$, $\left\{ \alpha ,\beta \right\} \subset \left\{ 1,\ldots ,s\right\}$, respectively, be represented by two pre-collisional elements ${\textbf{Z}}$ and ${\textbf{Z}}_{*}$, and two corresponding post-collisional elements ${\textbf{Z}}^{\prime }$ and ${\textbf{Z}}_{*}^{\prime }$ [6]. The notation for pre- and post-collisional pairs may be interchanged as well. Due to momentum and total energy conservation, the following relations have to be satisfied by the elements

$$\begin{aligned}{} & {} m_{\alpha }\varvec{\xi }+m_{\beta }\varvec{\xi }_{*}=m_{\alpha }\varvec{\xi }^{\prime }+m_{\beta }\varvec{\xi }_{*}^{\prime } \text {;} \\{} & {} m_{\alpha }\left| \varvec{\xi }\right| ^{2}+m_{\beta }\left| \varvec{\xi }_{*}\right| ^{2}+I{\textbf{1}}_{\alpha \in {\mathcal {I}}_{poly}}+I_{*}{\textbf{1}}_{\beta \in {\mathcal {I}}_{poly}} \\{} & {} \quad =m_{\alpha }\left| \varvec{\xi }^{\prime }\right| ^{2}+m_{\beta }\left| \varvec{\xi }_{*}^{\prime }\right| ^{2}+I^{\prime }{\textbf{1}}_{\alpha \in {\mathcal {I}}_{poly}}+I_{*}^{\prime }{\textbf{1}}_{\beta \in {\mathcal {I}}_{poly}}\text {.} \end{aligned}$$

Here the internal energy gap is

$$\begin{aligned} \Delta I=\left( I^{\prime }-I\right) {\textbf{1}}_{\alpha \in {\mathcal {I}} _{poly}}+\left( I_{*}^{\prime }-I_{*}\right) {\textbf{1}}_{\beta \in {\mathcal {I}}_{poly}}\text {,} \end{aligned}$$

and we have an additional relative velocity denoted by

$$\begin{aligned} {\textbf{g}}=\varvec{\xi }-\varvec{\xi }_{*}\text {.} \end{aligned}$$

Conservation of the total energy in the center of mass frame is now given by

$$\begin{aligned} E_{\alpha \beta }= & {} \dfrac{m_{\alpha }m_{\beta }}{2\left( m_{\alpha }+m_{\beta }\right) }\left| {\textbf{g}}\right| ^{2}+I{\textbf{1}} _{\alpha \in {\mathcal {I}}_{poly}}+I_{*}{\textbf{1}}_{\beta \in {\mathcal {I}} _{poly}} \\= & {} \dfrac{m_{\alpha }m_{\beta }}{2\left( m_{\alpha }+m_{\beta }\right) } \left| {\textbf{g}}^{\prime }\right| ^{2}+I^{\prime }{\textbf{1}} _{\alpha \in {\mathcal {I}}_{poly}}+I_{*}^{\prime }{\textbf{1}}_{\beta \in {\mathcal {I}}_{poly}}=E_{\alpha \beta }^{\prime }\text {.} \end{aligned}$$

2.1 Collision operator

The mechanical collision operator $Q_{mech}=\left( Q_{m1},\ldots ,Q_{ms}\right)$ [6] has components that can be written in the following form

$$\begin{aligned}{} & {} Q_{m\alpha }(f,f)=\sum _{\beta =1}^{s}Q_{\alpha \beta }(f,f)=\sum _{\beta =1}^{s}\int _{{\mathcal {Z}}_{\alpha }\times {\mathcal {Z}}_{\beta }^{2}}W_{\alpha \beta }\Lambda _{\alpha \beta }(f)\,d{\textbf{Z}}_{*}d {\textbf{Z}}^{\prime }d{\textbf{Z}}_{*}^{\prime }\text {, where} \\{} & {} \Lambda _{\alpha \beta }(f)=\frac{f_{\alpha }^{\prime }f_{\beta *}^{\prime }}{\varphi _{\alpha }\left( I^{\prime }\right) \varphi _{\beta }\left( I_{*}^{\prime }\right) }-\frac{f_{\alpha }f_{\beta *}}{ \varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) } \text { and }{\mathcal {Z}}_{\gamma }=\left\{ \begin{array}{l} {\mathbb {R}}^{3}\text { if }\gamma \in {\mathcal {I}}_{mono} \\ {\mathbb {R}}^{3}\times {\mathbb {R}}_{+}\text { if }\gamma \in {\mathcal {I}}_{poly} \end{array} \right. \text {.} \end{aligned}$$

Here and below the abbreviations

$$\begin{aligned} f_{\alpha *}=f_{\alpha }\left( t,{\textbf{x}},{\textbf{Z}}_{*}\right) \text {, }f_{\alpha }^{\prime }=f_{\alpha }\left( t,{\textbf{x}},{\textbf{Z}} ^{\prime }\right) \text {, and }f_{\alpha *}^{\prime }=f_{\alpha }\left( t,{\textbf{x}},{\textbf{Z}}_{*}^{\prime }\right) \end{aligned}$$

where ${\textbf{Z}}_{*}$, ${\textbf{Z}}^{\prime }$, and ${\textbf{Z}}_{*}^{\prime }$, are defined as the natural extensions of definition (1), i.e. $\mathbf {Z_{*}=Z_{*\alpha }}=\left\{ \begin{array}{l} \quad \varvec{\xi }_{*} \qquad \quad \text {for }\alpha \in {\mathcal {I}}_{mono}\quad \\ \left( \varvec{\xi }_{*},I_{*}\right) \quad \text {for }\alpha \in {\mathcal {I}}_{poly} \end{array} \right.$ etc., are used for $\alpha \in {\mathcal {I}}$. Moreover, the degeneracies $\varphi _{\alpha }=\varphi _{\alpha }\left( I\right)$, $\alpha \in {\mathcal {I}}$, with $\varphi _{\alpha }=1$ for $\alpha \in {\mathcal {I}}_{mono}$, are positive functions for $I>0$. A typical choice of the degeneracies is [1, 6, 16]

$$\begin{aligned} \varphi _{\alpha }\left( I\right) =I^{\delta ^{\left( \alpha \right) }/2-1} \text {, }\alpha \in {\mathcal {I}}\text {,} \end{aligned}$$

where $\delta ^{\left( 1\right) }=...=\delta ^{\left( s_{0}\right) }=2$, while $\delta ^{\left( \alpha \right) }$, with $\delta ^{\left( \alpha \right) }\ge 2$, denote the number of internal degrees of freedom of the species for $\alpha \in {\mathcal {I}}_{poly}$. Our main results below will be proven for this particular choice of degeneracies.

Note that in the literature it is usual to use a slightly different setting [2, 11, 15], where already renormalized distribution functions are considered, opting to consider a weighted measure—where the renormalization weights appear as weights—with respect to I. However, this is merely due to a different scaling of the distribution functions considered.

The transition probabilities $W_{\alpha \beta }$ are of the form [6]

$$\begin{aligned}{} & {} W_{\alpha \beta }=W_{\alpha \beta }(\textbf{Z,Z}_{*}\left| {\textbf{Z}}^{\prime },{\textbf{Z}}_{*}^{\prime }\right. )= \left( m_{\alpha }+m_{\beta }\right) ^{2}m_{\alpha }m_{\beta }\varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) \sigma _{\alpha \beta }\frac{\left| {\textbf{g}}\right| }{\left| {\textbf{g}}^{\prime }\right| }\widehat{\varvec{\delta }}_{1}\widehat{ \varvec{\delta }}_{3}\text {,} \\{} & {} \text {where }\sigma _{\alpha \beta }=\sigma _{\alpha \beta }\left( \left| {\textbf{g}}\right| ,\cos \theta ,I,I_{*},I^{\prime },I_{*}^{\prime }\right) >0\text { a.e., with} \\{} & {} \widehat{\varvec{\delta }}_{1}=\varvec{\delta }_{1}\left( \frac{1}{ 2}\left( m_{\alpha }\left| \varvec{\xi }\right| ^{2}+m_{\beta }\left| \varvec{\xi }_{*}\right| ^{2}-m_{\alpha }\left| \varvec{\xi }^{\prime }\right| ^{2}-m_{\beta }\left| \varvec{ \xi }_{*}^{\prime }\right| ^{2}\right) -\Delta I\right) \text {,} \\{} & {} \widehat{\varvec{\delta }}_{3}=\varvec{\delta }_{3}\left( m_{\alpha }\varvec{\xi }+m_{\beta }\varvec{\xi }_{*}-m_{\alpha } \varvec{\xi }^{\prime }-m_{\beta }\varvec{\xi }_{*}^{\prime }\right) \text { }\text { and }\cos \theta =\frac{{\textbf{g}}\cdot {\textbf{g}}^{\prime }}{\left| {\textbf{g}}\right| \left| {\textbf{g}} ^{\prime }\right| }\text {.} \end{aligned}$$

Here and below $\varvec{\delta }_{3}$ and $\varvec{\delta }_{1}$ denote the Dirac’s delta function in ${\mathbb {R}} ^{3}$ and ${\mathbb {R}}$, respectively — $\widehat{\varvec{\delta }}_{1}$ and $\widehat{\varvec{\delta }}_{3}$ taking the conservation of momentum and total energy into account. Note that for $\alpha \in {\mathcal {I}}_{mono}$ the scattering cross sections $\sigma _{\alpha \beta }$ are independent of $\ I$ and $I^{\prime }$, while for $\beta \in {\mathcal {I}} _{mono}$ they are independent of $I_{*}$ and $I_{*}^{\prime }$.

Furthermore, it is assumed that for $\left( \alpha ,\beta \right) \in {\mathcal {I}}^{2}$ the scattering cross sections $\sigma _{\alpha \beta }$ satisfy the microreversibility and symmetry relations

$$\begin{aligned}{} & {} \varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) \left| {\textbf{g}}\right| ^{2}\sigma _{\alpha \beta }\left( \left| {\textbf{g}}\right| ,\left| \cos \theta \right| ,I,I_{*},I^{\prime },I_{*}^{\prime }\right) \\{} & {} \quad =\varphi _{\alpha }\left( I^{\prime }\right) \varphi _{\beta }\left( I_{*}^{\prime }\right) \left| {\textbf{g}}^{\prime }\right| ^{2}\sigma _{\alpha \beta }\left( \left| {\textbf{g}}^{\prime }\right| ,\left| \cos \theta \right| ,I^{\prime },I_{*}^{\prime },I,I_{*}\right) \text {;} \\{} & {} \sigma _{\alpha \beta }\left( \left| {\textbf{g}}\right| ,\cos \theta ,I,I_{*},I^{\prime },I_{*}^{\prime }\right) =\sigma _{\beta \alpha }\left( \left| {\textbf{g}}\right| ,\cos \theta ,I_{*},I,I_{*}^{\prime },I^{\prime }\right) \text {;} \\{} & {} \sigma _{\alpha \alpha }\left( \left| {\textbf{g}} \right| ,\left| \cos \theta \right| ,I,I_{*},I^{\prime },I_{*}^{\prime }\right) =\sigma _{\alpha \alpha }\left( \left| {\textbf{g}}\right| ,\left| \cos \theta \right| ,I_{*},I,I^{\prime },I_{*}^{\prime }\right) \text {.} \end{aligned}$$

Applying known properties of Dirac’s delta function, the transition probabilities may be transformed to [6]

$$\begin{aligned} W_{\alpha \beta }=\varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) \sigma _{\alpha \beta }\frac{\left| {\textbf{g}}\right| }{\left| {\textbf{g}}^{\prime }\right| ^{2}} {\textbf{1}}_{\left| {\textbf{g}}\right| ^{2}>2{\widetilde{\Delta }} _{\alpha \beta }I}\varvec{\delta }_{1}\left( \sqrt{\left| {\textbf{g}} \right| ^{2}-2{\widetilde{\Delta }}_{\alpha \beta }I}-\left| {\textbf{g}} ^{\prime }\right| \right) \varvec{\delta }_{3}\left( {\textbf{G}} _{\alpha \beta }-{\textbf{G}}_{\alpha \beta }^{\prime }\right) \text {.} \end{aligned}$$

For more details about the mechanical collision operator, including more familiar formulations, we refer to [6].

The chemical process operator $Q_{chem}=\left( Q_{c1},\ldots ,Q_{cs}\right)$ has components that can be written in the following form

$$\begin{aligned}{} & {} Q_{c\alpha }(f)=\sum _{\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}}Q_{\gamma \zeta }^{\beta ,\alpha }(f)=\sum _{\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}}\int _{{\mathcal {Z}}}W_{\gamma \zeta }^{\beta }\Delta _{\gamma \zeta }^{\beta }\left( \alpha ,{\textbf{Z}}\right) \Lambda _{\gamma \zeta }^{\beta }(f)\,d{\textbf{Z}}_{*}d{\textbf{Z}}^{\prime }d{\textbf{Z}}_{*}^{\prime }\text {,} \\{} & {} \quad \text {where }\Lambda _{\gamma \zeta }^{\beta }=\frac{f_{\gamma }^{\prime }f_{\zeta *}^{\prime }}{\varphi _{\gamma }\left( I^{\prime }\right) \varphi _{\zeta }\left( I_{*}^{\prime }\right) }-\frac{f_{\beta *}}{\varphi _{\beta }\left( I_{*}\right) }\text {, } {\mathcal {Z}}={\mathcal {Z}}_{\beta }\times {\mathcal {Z}}_{\gamma }\times \mathcal {Z }_{\zeta }\text {, and } \\{} & {} \Delta _{\gamma \zeta }^{\beta }\left( \alpha ,{\textbf{Z}}\right) =\delta _{\alpha \beta }\varvec{\delta }_{c}\left( {\textbf{Z}}-\mathbf {Z_{*}} \right) -\delta _{\alpha \gamma }\varvec{\delta }_{c}\left( {\textbf{Z}}- {\textbf{Z}}^{\prime }\right) -\delta _{\alpha \zeta }\varvec{\delta } _{c}\left( {\textbf{Z}}-\mathbf {Z_{*}^{\prime }}\right) \text {.} \end{aligned}$$

Here $\varvec{\delta }_{c}$ denotes the Dirac’s delta function in ${\mathbb {R}}^{3}$ or ${\mathbb {R}}^{4}$ if $\alpha \in {\mathcal {I}}_{mono}$ or $\alpha \in {\mathcal {I}}_{poly}$, respectively.

For $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$ introduce a positive number—the energy of the transition state of the process—$\kappa _{\gamma \zeta }^{\beta }$, such that

$$\begin{aligned} \kappa _{\gamma \zeta }^{\beta }=\kappa _{\zeta \gamma }^{\beta }\ge \varepsilon _{\gamma 0}+\varepsilon _{\zeta 0}>\varepsilon _{\beta 0}>0\text {.} \end{aligned}$$

That is, for a chemical process $\left\{ a_{\gamma },a_{\varsigma }\right\} \longleftrightarrow \left\{ a_{\beta }\right\}$ to take place we have to have

$$\begin{aligned} E_{\gamma \zeta }^{\prime }=E_{\beta }=I_{*}+\varepsilon _{\beta 0}\ge \kappa _{\gamma \zeta }^{\beta }>0\text {, or, equivalently, }I_{*}\ge \kappa _{\gamma \zeta }^{\beta }-\varepsilon _{\beta 0}>0\text {.} \end{aligned}$$

The transition probabilities $W_{\gamma \zeta }^{\beta }$ are for $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$ of the form

$$\begin{aligned} W_{\gamma \zeta }^{\beta }=W_{\gamma \zeta }^{\beta }({\textbf{Z}}_{*}, {\textbf{Z}}^{\prime },{\textbf{Z}}_{*}^{\prime })= & {} m_{\beta }^{2}m_{\gamma }m_{\zeta }\varphi _{\beta }\left( I_{*}\right) \sigma _{\beta }^{\gamma \zeta }\widehat{\varvec{\delta }}_{1}\widehat{ \varvec{\delta }}_{3}{\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }} \\= & {} m_{\beta }^{2}m_{\gamma }m_{\zeta }\varphi _{\gamma }\left( I^{\prime }\right) \varphi _{\zeta }\left( I_{*}^{\prime }\right) \sigma _{\gamma \zeta }^{\beta }\widehat{\varvec{\delta }}_{1}\widehat{\varvec{ \delta }}_{3}{\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }} \text {,} \end{aligned}$$

where

$$\begin{aligned}{} & {} \sigma _{\beta }^{\gamma \zeta }=\sigma _{\beta }^{\gamma \zeta }\left( \left| {\textbf{g}}^{\prime }\right| ,\cos \theta ,I^{\prime },I_{*}^{\prime }\right)>0\text { and }\sigma _{\gamma \zeta }^{\beta }=\sigma _{\gamma \zeta }^{\beta }\left( \left| {\textbf{g}}^{\prime }\right| ,\cos \theta ,I_{*}\right) >0\text { a.e., } \\{} & {} \quad \text {with }\widehat{\varvec{\delta }}_{1}=\varvec{\delta } _{1}\left( \frac{1}{2}\left( m_{\gamma }\frac{\left| \varvec{\xi } ^{\prime }\right| ^{2}}{2}+m_{\zeta }\frac{\left| \varvec{\xi } _{*}^{\prime }\right| ^{2}}{2}-m_{\beta }\frac{\left| \varvec{\xi }_{*}\right| ^{2}}{2}\right) -\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}\right) \text {,} \\{} & {} \quad \widehat{\varvec{\delta }}_{3}=\varvec{\delta }_{3}\left( m_{\gamma }\varvec{\xi }^{\prime }+m_{\zeta }\varvec{\xi }_{*}^{\prime }-m_{\beta }\varvec{\xi }_{*}\right) \text {, } \text { and }\cos \theta =\frac{\varvec{\xi }\cdot {\textbf{g}}^{\prime }}{ \left| \varvec{\xi }\right| \left| {\textbf{g}}^{\prime }\right| }\text {.} \end{aligned}$$

In general, we, for the chemcial process case, use the convention that index $\alpha$ relates to argument ${\textbf{Z}}$, $\beta$ to ${\textbf{Z}}_{*}$, $\gamma$ to ${\textbf{Z}}^{\prime }$, and $\zeta$ to ${\textbf{Z}} _{*}^{\prime }$. Note that for $\gamma \in {\mathcal {I}} _{mono}$ the scattering cross sections $\sigma _{\beta }^{\gamma \zeta }$ are independent of $I^{\prime }$, while for $\zeta \in {\mathcal {I}} _{mono}$ they are independent of $I_{*}^{\prime }$.

Furthermore, it is assumed that the scattering cross sections $\sigma _{\beta }^{\zeta \gamma }$ and $\sigma _{\gamma \zeta }^{\beta }$ for $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$ satisfy the microreversibility and symmetry (due to invariance of particles associating or resulting from dissociation) relations

$$\begin{aligned} \varphi _{\beta }\left( I_{*}\right) \sigma _{\beta }^{\gamma \zeta }\left( \left| {\textbf{g}}^{\prime }\right| ,\left| \cos \theta \right| ,I^{\prime },I_{*}^{\prime }\right)= & {} \varphi _{\gamma }\left( I^{\prime }\right) \varphi _{\zeta }\left( I_{*}^{\prime }\right) \sigma _{\gamma \zeta }^{\beta }\left( \left| {\textbf{g}} ^{\prime }\right| ,\left| \cos \theta \right| ,I_{*}\right) ; \\ \sigma _{\beta }^{\gamma \zeta }\left( \left| {\textbf{g}}^{\prime }\right| ,\left| \cos \theta \right| ,I^{\prime },I_{*}^{\prime }\right)= & {} \sigma _{\beta }^{\zeta \gamma }\left( \left| {\textbf{g}}^{\prime }\right| ,\left| \cos \theta \right| ,I_{*}^{\prime },I^{\prime }\right) \text {.} \end{aligned}$$

Applying known properties of Dirac’s delta function, the transition probabilities may for $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$ be transformed to

$$\begin{aligned}{} & {} W_{\gamma \zeta }^{\beta }({\textbf{Z}}_{*},{\textbf{Z}} ^{\prime },{\textbf{Z}}_{*}^{\prime })=W_{\zeta \gamma }^{\beta }(\textbf{Z }_{*},{\textbf{Z}}_{*}^{\prime },{\textbf{Z}}^{\prime }) \\{} & {} \quad =\frac{m_{\gamma }m_{\zeta }}{m_{\beta }}\varphi _{\beta }\left( I_{*}\right) \sigma _{\beta }^{\gamma \zeta }\varvec{\delta }_{3}\left( \varvec{\xi }_{*}-{\textbf{G}}_{\gamma \zeta }^{\prime }\right) \varvec{\delta }_{1}\left( \frac{m_{\gamma }m_{\zeta }}{2m_{\beta }} \left| {\textbf{g}}^{\prime }\right| ^{2}-\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}\right) {\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }} \\{} & {} \quad =\varphi _{\beta }\left( I_{*}\right) \frac{\sigma _{\beta }^{\gamma \zeta }}{\left| {\textbf{g}}^{\prime }\right| }{\textbf{1}}_{\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}>0}\varvec{\delta }_{1}\left( \left| {\textbf{g}}^{\prime }\right| -\sqrt{{\widetilde{\Delta }} _{\gamma \zeta }^{\beta }{\mathcal {E}}}\right) \varvec{\delta }_{3}\left( \varvec{\xi }_{*}-{\textbf{G}}_{\gamma \zeta }^{\prime }\right) {\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }}\text {,} \end{aligned}$$

or, with $\widetilde{{\textbf{G}}}_{\beta \zeta }=\dfrac{m_{\beta } \varvec{\xi }_{*}-m_{\zeta }\varvec{\xi }_{*}^{\prime }}{ m_{\gamma }}$, to

$$\begin{aligned} W_{\gamma \zeta }^{\beta }= & {} \frac{m_{\beta }^{2}m_{\zeta }}{m_{\gamma }^{2}}\varphi _{\beta }\left( I_{*}\right) \sigma _{\beta }^{\gamma \zeta }{\textbf{1}}_{\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}>0}\varvec{\delta }_{1}\left( \frac{m_{\beta }m_{\zeta }}{2m_{\gamma }}\left| {\textbf{g}}_{*}\right| ^{2}-\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}\right) \varvec{\delta } _{3}\left( \varvec{\xi }^{\prime }-\widetilde{{\textbf{G}}}_{\beta \zeta }\right) {\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }} \\= & {} \frac{m_{\beta }}{m_{\gamma }\left| {\textbf{g}}_{*}\right| }\varphi _{\beta }\left( I_{*}\right) \sigma _{\beta }^{\gamma \zeta }\varvec{\delta }_{1}\left( \sqrt{ {\widetilde{\Delta }}_{\beta }^{\gamma \zeta }{\mathcal {E}}}-\left| {\textbf{g}}_{*}\right| \right) \varvec{\delta }_{3}\left( \varvec{\xi } ^{\prime }-\widetilde{{\textbf{G}}}_{\beta \zeta }\right) {\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }} \\= & {} \varphi _{\beta }\left( I_{*}\right) \sigma _{\beta }^{\gamma \zeta } \frac{\left| {\textbf{g}}^{\prime }\right| }{\left| {\textbf{g}} _{*}\right| ^{2}}\varvec{\delta }_{1}\left( \sqrt{{\widetilde{\Delta }}_{\beta }^{\gamma \zeta }{\mathcal {E}}}-\left| {\textbf{g}}_{*}\right| \right) \varvec{\delta }_{3}\left( \varvec{\xi } ^{\prime }-\widetilde{{\textbf{G}}}_{\beta \zeta }\right) {\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }}\text {.} \end{aligned}$$

Remark 1

Note that for $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$, denoting ${\widetilde{E}}_{\gamma \zeta }^{\prime }=E_{\gamma \zeta }^{\prime }-\varepsilon _{\beta 0}$,

$$\begin{aligned} \varvec{\delta }_{1}\left( \frac{m_{\gamma }m_{\zeta }}{2\left( m_{\gamma }+m_{\zeta }\right) }\left| {\textbf{g}}^{\prime }\right| ^{2}-\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}\right) =\varvec{\delta } _{1}\left( E_{\beta }-E_{\gamma \zeta }^{\prime }\right) =\varvec{\delta }_{1}\left( I_{*}-{\widetilde{E}}_{\gamma \zeta }^{\prime }\right) . \end{aligned}$$

By a series of change of variables:$\left\{ \varvec{\xi }^{\prime },\varvec{\xi }_{*}^{\prime }\right\} \rightarrow \!\left\{ {\textbf{g}}^{\prime }=\varvec{\xi } ^{\prime }-\varvec{\xi }_{*}^{\prime }\text {,}{\textbf{G}}_{\gamma \zeta }^{\prime }=\dfrac{m_{\gamma }\varvec{\xi }^{\prime }+m_{\zeta } \varvec{\xi }_{*}^{\prime }}{m_{\gamma }+m_{\zeta }}\!\right\}$, followed by a change to spherical coordinates $\left\{ {\textbf{g}}^{\prime }\right\} \rightarrow \left\{ \left| {\textbf{g}}^{\prime }\right| , \varvec{\sigma \,}=\dfrac{{\textbf{g}}^{\prime }}{\left| {\textbf{g}} ^{\prime }\right| }\right\}$, observe that

$$\begin{aligned} d\varvec{\xi }^{\prime }d\varvec{\xi }_{*}^{\prime }=d{\textbf{G}} _{\gamma \zeta }^{\prime }d{\textbf{g}}^{\prime }=\left| {\textbf{g}} ^{\prime }\right| ^{2}d{\textbf{G}}_{\gamma \zeta }^{\prime }d\left| {\textbf{g}}^{\prime }\right| d\varvec{\sigma \,}\text {,} \end{aligned}$$

then—assuming that $\zeta \in {\mathcal {I}}_{poly}$—$\left\{ \left| {\textbf{g}}^{\prime }\right| ,I_{*}^{\prime }\right\} \rightarrow \left\{ R=\dfrac{m_{\gamma }m_{\zeta }}{2m_{\beta }} \dfrac{\left| {\textbf{g}}^{\prime }\right| ^{2}}{{\widetilde{E}}_{\beta }},{\widetilde{E}}_{\gamma \zeta }^{\prime }=\dfrac{m_{\gamma }m_{\zeta }}{ 2m_{\beta }}\left| {\textbf{g}}^{\prime }\right| ^{2}+I^{\prime } {\textbf{1}}_{\gamma \in {\mathcal {I}}_{poly}}+I_{*}^{\prime }+\Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}\right\}$, with ${\widetilde{E}} _{\beta }=I_{*}-\Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}=\left( 1-\dfrac{\Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}}{I_{*}}\right) I_{*}$, follows that

$$\begin{aligned} d\varvec{\xi }^{\prime }d\varvec{\xi }_{*}^{\prime }dI_{*}^{\prime }=\sqrt{2}\left( \frac{m_{\beta }}{m_{\gamma }m_{\zeta }}\right) ^{3/2}{\widetilde{E}}_{\beta }^{3/2}R^{1/2}dRd\varvec{\sigma \,}d{\textbf{G}} _{\gamma \zeta }^{\prime }d{\widetilde{E}}_{\gamma \zeta }^{\prime }\text {,} \end{aligned}$$

and, finally, if also $\gamma \in {\mathcal {I}}_{poly}$, $\left\{ I^{\prime }\right\} \rightarrow \left\{ r=\dfrac{I^{\prime }}{\left( 1-R\right) {\widetilde{E}}_{\beta }}\right\}$, then

$$\begin{aligned} d\varvec{\xi }^{\prime }d\varvec{\xi }_{*}^{\prime }dI^{\prime }dI_{*}^{\prime }=\sqrt{2}\left( \frac{m_{\beta }}{m_{\gamma }m_{\zeta }} \right) ^{3/2}{\widetilde{E}}_{\beta }^{5/2}(1-R)R^{1/2}drdRd\varvec{ \sigma \,}d{\textbf{G}}_{\gamma \zeta }^{\prime }d{\widetilde{E}}_{\gamma \zeta }^{\prime }\text {.} \end{aligned}$$

Similarly, with $\widetilde{{\textbf{G}}}_{\beta \zeta }=\dfrac{ m_{\beta }\varvec{\xi }_{*}-m_{\zeta }\varvec{\xi }_{*}^{\prime }}{m_{\gamma }}$,

$$\begin{aligned} d\varvec{\xi }_{*}d\varvec{\xi }_{*}^{\prime }= & {} d \widetilde{{\textbf{G}}}_{\beta \zeta }d{\textbf{g}}_{*}=\left| {\textbf{g}}_{*}\right| ^{2}d\widetilde{{\textbf{G}}}_{\beta \zeta }d\left| {\textbf{g}}_{*}\right| d\varvec{\sigma \,}\text {,} \\ d\varvec{\xi }_{*}d\varvec{\xi }_{*}^{\prime }dI_{*}= & {} \sqrt{2}\left( \frac{m_{\gamma }}{m_{\beta }m_{\zeta }}\right) ^{3/2} {\widetilde{E}}_{\beta }^{3/2}R^{1/2}dRd\varvec{\sigma \,}d\widetilde{ {\textbf{G}}}_{\beta \zeta }d{\widetilde{E}}_{\beta }\text { if }\left( \gamma ,\zeta \right) \notin {\mathcal {I}}_{mono}^{2}\text {, and} \\ d\varvec{\xi }_{*}d\varvec{\xi }_{*}^{\prime }dI_{*}dI_{*}^{\prime }= & {} \sqrt{2}\left( \frac{m_{\gamma }}{m_{\beta }m_{\zeta }}\right) ^{3/2}{\widetilde{E}}_{\beta }^{5/2}(1-R)R^{1/2}drdRd \varvec{\sigma \,}d\widetilde{{\textbf{G}}}_{\beta \zeta }d{\widetilde{E}} _{\beta }\text { if }\left\{ \gamma ,\zeta \right\} \subset {\mathcal {I}}_{poly} \text {,} \end{aligned}$$

valid also for $\left( \zeta ,\varvec{\xi }_{*}^{\prime }\right)$ and $\left( \gamma ,\varvec{\xi }^{\prime }\right)$ interchanged.

Then, with $\varphi _{\alpha }\left( I\right) =I^{\delta ^{\left( \alpha \right) }/2-1}$ for $\alpha \in {\mathcal {I}}$, for two monatomic constituents, i.e., with $\left( \gamma ,\zeta \right) \in {\mathcal {I}} _{mono}^{2}$, (mono/mono-case)

or, for one monatomic and one polyatomic constituent, respectively, i.e., either with $\left( \gamma ,\zeta \right) \in {\mathcal {I}}_{mono}\times {\mathcal {I}}_{poly}$ (mono/poly-case),

$$\begin{aligned} Q_{\gamma \zeta }^{\beta ,\alpha }(f)= & {} \int _{[0,1]^{2}\mathbb {\times S} ^{2}}B_{1\gamma \zeta }^{\beta }\Delta _{1\gamma \zeta }^{\beta }\left( \frac{f_{\gamma }^{\prime }f_{\zeta *}^{\prime }}{\left( I_{*}^{\prime }\right) ^{\delta ^{\left( \zeta \right) }/2-1}}-\frac{f_{\beta *}}{I_{*}^{\delta ^{\left( \beta \right) }/2-1}}\right) I_{*}^{\delta ^{\left( \beta \right) }/2-1} \\{} & {} \times \left( 1-R\right) ^{\delta ^{\left( \zeta \right) }/2-1}\left( 1- \dfrac{\Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}}{I_{*}}\right) ^{\delta ^{\left( \zeta \right) }/2}drdRd\varvec{\sigma }\text {, } \end{aligned}$$

with

$$\begin{aligned} B_{1\gamma \zeta }^{\beta }= & {} \sqrt{\frac{2m_{\beta }}{m_{\gamma }m_{\zeta } }}\frac{{\widetilde{E}}_{\beta }^{1/2}I_{*}^{\delta ^{\left( \zeta \right) }/2}}{\left( I_{*}^{\prime }\right) ^{\delta ^{\left( \zeta \right) }/2-1}}R^{1/2}\sigma _{\beta }^{\gamma \zeta }{\textbf{1}}_{\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}>0}{\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }} \\= & {} \left| {\textbf{g}}^{\prime }\right| \frac{I_{*}^{\delta ^{\left( \zeta \right) }/2}}{\left( I_{*}^{\prime }\right) ^{\delta ^{\left( \zeta \right) }/2-1}}\sigma _{\beta }^{\gamma \zeta }{\textbf{1}} _{\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}>0}{\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }} \\= & {} \left| {\textbf{g}}^{\prime }\right| I_{*}^{\left( \delta ^{\left( \zeta \right) }-\delta ^{\left( \beta \right) }\right) /2+1}\sigma _{\gamma \zeta }^{\beta }{\textbf{1}}_{\Delta _{\gamma \zeta }^{\beta } {\mathcal {E}}>0}{\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }} \text { and} \\ \Delta _{1\gamma \zeta }^{\beta }\left( \alpha ,{\textbf{Z}}\right)= & {} \varvec{\delta }_{1}\left( 1-r\right) {\textbf{1}}_{\left( \alpha ,{\textbf{Z}}\right) =\left( \beta ,\left( \varvec{\xi }\varvec{_{*}},I_{*}\right) \right) }-\varvec{\delta }_{1}\left( 1-r\right) {\textbf{1}} _{\left( \alpha ,{\textbf{Z}}\right) =\left( \gamma ,\varvec{\xi }^{\prime }\right) }-{\textbf{1}}_{\left( \alpha ,{\textbf{Z}}\right) =\left( \zeta ,\left( \varvec{\xi }\varvec{_{*}^{\prime },}I_{*}^{\prime }\right) \right) }\text {,} \end{aligned}$$

or, correspondingly, with $\left( \gamma ,\zeta \right) \in {\mathcal {I}} _{poly}\times {\mathcal {I}}_{mono}$ (poly/mono-case),

$$\begin{aligned} Q_{\gamma \zeta }^{\beta ,\alpha }(f)= & {} \int _{[0,1]^{2}\mathbb {\times S} ^{2}}B_{1\zeta \gamma }^{\beta }\Delta _{1\zeta \gamma }^{\beta }\left( \frac{f_{\gamma }^{\prime }f_{\zeta *}^{\prime }}{\left( I^{\prime }\right) ^{\delta ^{\left( \gamma \right) }/2-1}}-\frac{f_{\beta *}}{ I_{*}^{\delta ^{\left( \beta \right) }/2-1}}\right) I_{*}^{\delta ^{\left( \beta \right) }/2-1} \\{} & {} \times \left( 1-R\right) ^{\delta ^{\left( \gamma \right) }/2-1}\left( 1- \dfrac{\Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}}{I_{*}}\right) ^{\delta ^{\left( \gamma \right) }/2}drdRd\varvec{\sigma }\text {,} \end{aligned}$$

and, finally, for two polyatomic constituents, i.e., with $\left( \gamma ,\zeta \right) \in {\mathcal {I}}_{poly}^{2}$, (poly/poly-case)

$$\begin{aligned}{} & {} Q_{\gamma \zeta }^{\beta ,\alpha }(f) \\{} & {} \quad =\int _{[0,1]^{2}\mathbb {\times S}^{2}}B_{2\gamma \zeta }^{\beta }\Delta _{2\gamma \zeta }^{\beta }\left( \frac{f_{\gamma }^{\prime }f_{\zeta *}^{\prime }}{\left( I^{\prime }\right) ^{\delta ^{\left( \gamma \right) }/2-1}\left( I_{*}^{\prime }\right) ^{\delta ^{\left( \zeta \right) }/2-1}}-\frac{f_{\beta *}}{I_{*}^{\delta ^{\left( \beta \right) }/2-1}}\right) I_{*}^{\delta ^{\left( \beta \right) }/2-1} \\{} & {} \qquad \times r^{\delta ^{\left( \gamma \right) }/2-1}\left( 1-r\right) ^{\delta ^{\left( \zeta \right) }/2-1}(1-R)^{\left( \delta ^{\left( \gamma \right) }+\delta ^{\left( \zeta \right) }\right) /2-1}\left( 1-\dfrac{\Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}}{I_{*}}\right) ^{\left( \delta ^{\left( \gamma \right) }+\delta ^{\left( \zeta \right) }\right) /2}drdRd \varvec{\sigma }\text {,} \end{aligned}$$

with

$$\begin{aligned} B_{2\gamma \zeta }^{\beta }= & {} \sqrt{\frac{2m_{\beta }}{m_{\gamma }m_{\zeta } }}\frac{R^{1/2}{\widetilde{E}}_{\beta }^{1/2}I_{*}^{\left( \delta ^{\left( \gamma \right) }+\delta ^{\left( \zeta \right) }\right) /2}}{\left( I^{\prime }\right) ^{\delta ^{\left( \gamma \right) }/2-1}\left( I_{*}^{\prime }\right) ^{\delta ^{\left( \zeta \right) }/2-1}}\sigma _{\beta }^{\gamma \zeta }{\textbf{1}}_{\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}>0} {\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }} \\= & {} \left| {\textbf{g}}^{\prime }\right| \frac{I_{*}^{\left( \delta ^{\left( \gamma \right) }+\delta ^{\left( \zeta \right) }\right) /2}}{\left( I^{\prime }\right) ^{\delta ^{\left( \gamma \right) }/2-1}\left( I_{*}^{\prime }\right) ^{\delta ^{\left( \zeta \right) }/2-1}}\sigma _{\beta }^{\gamma \zeta }{\textbf{1}}_{\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}>0} {\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }} \\= & {} \left| {\textbf{g}}^{\prime }\right| I_{*}^{\left( \delta ^{\left( \gamma \right) }+\delta ^{\left( \zeta \right) }-\delta ^{\left( \beta \right) }\right) /2+1}\sigma _{\gamma \zeta }^{\beta }{\textbf{1}} _{\Delta _{\gamma \zeta }^{\beta }{\mathcal {E}}>0}{\textbf{1}}_{E_{\beta }\ge \kappa _{\gamma \zeta }^{\beta }}\text {, and} \\ \Delta _{2\gamma \zeta }^{\beta }\left( \alpha ,{\textbf{Z}}\right)= & {} {\textbf{1}}_{\left( \alpha ,{\textbf{Z}}\right) =\left( \beta ,\left( \varvec{\xi } \varvec{_{*}},I_{*}\right) \right) }-{\textbf{1}}_{\left( \alpha , {\textbf{Z}}\right) =\left( \gamma ,\left( \varvec{\xi }^{\prime },I^{\prime }\right) \right) }-{\textbf{1}}_{\left( \alpha ,{\textbf{Z}}\right) =\left( \zeta ,\left( \varvec{\xi }\varvec{_{*}^{\prime },}I_{*}^{\prime }\right) \right) }\text {,} \end{aligned}$$

where

or, correspondingly,

resulting in more explicit forms of the chemical process operators. For the mono/mono-case, note that

$$\begin{aligned} \frac{m_{\beta }}{m_{\gamma }m_{\zeta }}\left| {\textbf{g}}^{\prime }\right| \,dI_{*}d\varvec{\sigma }=\left| {\textbf{g}}^{\prime }\right| ^{2}d\left| {\textbf{g}}^{\prime }\right| d\varvec{ \sigma }=d{\textbf{g}}^{\prime }=\left\{ \begin{array}{c} d\varvec{\xi }^{\prime }\text { for fixed }\varvec{\xi }_{*}^{\prime } \\ d\varvec{\xi }_{*}^{\prime }\text { for fixed }\varvec{\xi } ^{\prime } \end{array} \right. \text {.} \end{aligned}$$

Explicitly, the internal energy gaps are given by $\Delta I=I_{*}$ in the mono/mono-case, $\Delta I=I_{*}-I_{*}^{\prime }$ in the mono/poly-case, $\Delta I=I_{*}-I^{\prime }$ in the poly/mono-case, while $\Delta I=I_{*}-I^{\prime }-I_{*}^{\prime }$ in the poly/poly-case.

2.2 Collision invariants and Maxwellian distributions

Denote for $\left( \beta ,\gamma ,\varsigma ,\alpha \right) \in \mathcal {C\times I}$

$$\begin{aligned} dA_{\gamma \zeta }^{\beta }= & {} W_{\gamma \zeta }^{\beta }({\textbf{Z}}_{*}, {\textbf{Z}}^{\prime },{\textbf{Z}}_{*}^{\prime })\,d{\textbf{Z}}_{*}d {\textbf{Z}}^{\prime }d{\textbf{Z}}_{*}^{\prime }\text { }\text { and} \\ \widetilde{{\mathcal {Z}}}= & {} \mathcal {Z\times Z}_{\alpha }={\mathcal {Z}}_{\beta }\times {\mathcal {Z}}_{\gamma }\times {\mathcal {Z}}_{\zeta }\mathcal {\times Z} _{\alpha }\text {.} \end{aligned}$$

Furthermore, denote

$$\begin{aligned} m=\left( m_{1},\ldots ,m_{s}\right) \text { and }{\mathbb {I}}=(\varepsilon _{10},\ldots ,\varepsilon _{s_{0}0},I+\varepsilon _{s_{0}+10},\ldots ,I+\varepsilon _{s0})\text {,} \end{aligned}$$

while, denote by $\left\{ e_{1},\ldots ,e_{s}\right\}$ the standard basis of ${\mathbb {R}}^{s}$.

The weak form of the collision operator $Q_{chem}(f)$ reads

$$\begin{aligned} \left( Q_{chem}(f),g\right) =\sum _{\left( \beta ,\gamma ,\varsigma ,\alpha \right) \in \mathcal {C\times I}}\int _{\widetilde{{\mathcal {Z}}}}\Delta _{\gamma \zeta }^{\beta }\left( \alpha \text {,}{\textbf{Z}}\right) \Lambda _{\gamma \zeta }^{\beta }(f)g_{\alpha }\,d{\textbf{Z}}dA_{\gamma \zeta }^{\beta } \end{aligned}$$

for any function $g=\left( g_{1},\ldots ,g_{s}\right)$, with $g_{\alpha }=g_{\alpha }(\varvec{\xi },I)$, such that the integrals are defined for all $\left( \beta ,\gamma ,\varsigma ,\alpha \right) \in \mathcal {C\times I}$, and we have the following proposition.

Proposition 1

Let $g=\left( g_{1},\ldots ,g_{s}\right)$, with $g_{\alpha }=g_{\alpha }({\textbf{Z}})$, be such that

is defined for all $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$. Then

$$\begin{aligned} \left( Q_{chem}(f),g\right) =\sum _{\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}}\int _{{\mathcal {Z}}}\Lambda _{\gamma \zeta }^{\beta }(f)\left( g_{{\beta *}}-g_{\gamma }^{\prime }-g_{\zeta *}^{\prime }\right) \,dA_{\gamma \zeta }^{\beta }. \end{aligned}$$

Definition 1

A function $g=\left( g_{1},\ldots ,g_{s}\right)$, with $g_{\alpha }=g_{\alpha }({\textbf{Z}})$,is

• (i) a chemical process invariant if for all $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$

  

  (4)

• (ii) a mechanical collision invariant if for all $\left\{ \alpha ,\beta \right\} \subseteq \left\{ 1,\ldots ,s\right\}$

  $$\begin{aligned} \left( g_{\alpha }+g_{{\beta *}}-g_{\alpha }^{\prime }-g_{\beta *}^{\prime }\right) W_{\alpha \beta }({\textbf{Z}},{\textbf{Z}}_{*}\left| {\textbf{Z}}^{\prime },{\textbf{Z}}_{*}^{\prime }\right. )=0 \text { a.e.;} \end{aligned}$$

• (iii) a collision invariant if it is a chemical process as well as a mechanical collision invariant.

Remind that $\left\{ e_{1},\ldots ,e_{s},m\xi _{x},m\xi _{y},m\xi _{z},m\left| \varvec{\xi }\right| ^{2}+2{\mathbb {I}}\right\}$ is a basis for the vector space of mechanical collision invariants [2, 6, 15] and introduce the vector space

$$\begin{aligned} {\mathcal {U}}=\left\{ u\in {\mathbb {R}}^{s};\text { }u\cdot \left( e_{\beta }-e_{\gamma }-e_{\varsigma }\right) =0\text { for all }\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}\right\} \text {.} \end{aligned}$$

(5)

We have the following proposition.

Proposition 2

For any basis ${\mathcal {U}}_{0}=\left\{ u_{1},\ldots ,u_{{\widetilde{s}} }\right\}$ of ${\mathcal {U}}$ the vector space of collision invariants is generated by

$$\begin{aligned} \left\{ u_{1},\ldots ,u_{{\widetilde{s}}},m\xi _{x},m\xi _{y},m\xi _{z},m\left| \varvec{\xi }\right| ^{2}+2{\mathbb {I}}\right\} . \end{aligned}$$

For example, if $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$, such that the species $a_{\beta }$, $a_{\gamma }$, and $a_{\varsigma }$ do not take part in any other chemical process, then if $\gamma \ne \varsigma$ one can replace $\left\{ e_{\beta },e_{\gamma },e_{\varsigma }\right\}$ in the set of generating mechanical collision invariants by $\left\{ e_{\beta }+e_{\gamma },e_{\beta }+e_{\varsigma }\right\}$ to describe generators for the collision invariants, while if $\gamma =\varsigma$ one can correspondingly replace $\left\{ e_{\beta },e_{\gamma }\right\}$ in the set of generating mechanical collision invariants by $\left\{ 2e_{\beta }+e_{\gamma }\right\}$.

It is known that [6]

$$\begin{aligned} {\mathcal {W}}_{mech}\left[ f\right] =\left( Q_{mech}(f),\log \left( \varphi ^{-1}f\right) \right) \le 0 \end{aligned}$$

where $\varphi =\textrm{diag}\left( \varphi _{1}\left( I\right) ,\ldots ,\varphi _{s}\left( I\right) \right)$, with equality if and only if $Q_{mech}(f)\equiv 0$. Furthermore, the mechanical equilibrium distributions $M_{mech}=(M_{m1},\ldots ,M_{ms})$ are of the form [6]

$$\begin{aligned} M_{\alpha }=\left\{ \begin{array}{l} \dfrac{n_{\alpha }m_{\alpha }^{3/2}}{\left( 2\pi k_{B}T\right) ^{3/2}} e^{-m_{\alpha }\left| \varvec{\xi }-{\textbf{u}}\right| ^{2}/\left( 2k_{B}T\right) }\text { if }\alpha \in {\mathcal {I}}_{mono}\text { \ \ } \\ \dfrac{n_{\alpha }\varphi _{\alpha }\left( I\right) m_{\alpha }^{3/2}e^{-\left( m_{\alpha }\left| \varvec{\xi }-{\textbf{u}} \right| ^{2}+2I\right) /\left( 2k_{B}T\right) }}{\left( 2\pi k_{B}T\right) ^{3/2}q_{\alpha }}\text { if }\alpha \in {\mathcal {I}}_{poly} \end{array} \right. , \end{aligned}$$

(6)

where $q_{\alpha }=\int _{0}^{\infty }\varphi _{\alpha }\left( I\right) e^{-I/\left( k_{B}T\right) }\,dI$ is an additional normalization factor for any $\alpha \in {\mathcal {I}}_{poly}$, $n_{\alpha }=\left( M,e_{{\alpha }}\right)$, ${\textbf{u}}=\dfrac{1}{\rho }\left( M,m\varvec{\xi }\right)$, and $T= \dfrac{1}{3nk_{B}}\left( M,m\left| \varvec{\xi }-{\textbf{u}} \right| ^{2}\right)$, with $n=\sum \nolimits _{\alpha =1}^{s}n_{\alpha }$ and $\rho =\sum \nolimits _{\alpha =1}^{s}m_{\alpha }n_{\alpha }$, while $k_{B}$ denote the Boltzmann constant.

For the particular case when $\varphi _{\alpha }\left( I\right) =I^{\delta ^{\left( \alpha \right) }/2-1}$ for all $\alpha \in {\mathcal {I}}$

$$\begin{aligned} q_{\alpha }=\left( k_{B}T\right) ^{\delta ^{\left( \alpha \right) }/2}\Gamma \left( \delta ^{\left( \alpha \right) }/2\right) \text {,} \end{aligned}$$

where $\Gamma =\Gamma (n)$ denote the Gamma function $\Gamma (n)=\int _{0}^{\infty }x^{n-1}e^{-x}\,dx$.

Correspondingly, define

$$\begin{aligned} {\mathcal {W}}_{chem}\left[ f\right] =\left( Q_{chem}(f),\log \left( \varphi ^{-1}f\right) \right) , \end{aligned}$$

It follows by Proposition 1 that

$$\begin{aligned}{} & {} {\mathcal {W}}_{chem}\left[ f\right] \\{} & {} \quad =-\sum _{\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}}\int _{ {\mathcal {Z}}}\left( \frac{f_{\gamma }^{\prime }f_{\zeta *}^{\prime }\varphi _{\beta }\left( I_{*}\right) }{f_{\beta *}\varphi _{\gamma }\left( I^{\prime }\right) \varphi _{\zeta }\left( I_{*}^{\prime }\right) }-1\right) \log \left( \frac{\varphi _{\beta }\left( I_{*}\right) f_{\gamma }^{\prime }f_{\zeta *}^{\prime }}{f_{\beta *}\varphi _{\gamma }\left( I^{\prime }\right) \varphi _{\zeta }\left( I_{*}^{\prime }\right) }\right) \frac{f_{\beta *}}{\varphi _{\beta }\left( I_{*}\right) }\,dA_{\gamma \zeta }^{\beta }\text {.} \end{aligned}$$

Reminding that $\left( x-1\right) \textrm{log} \left( x\right) \ge 0$ for $x>0$, with equality if and only if $x=1$,

$$\begin{aligned} {\mathcal {W}}_{chem}\left[ f\right] \le 0\text {,} \end{aligned}$$

with equality if and only if for all $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$

$$\begin{aligned} \Lambda _{\gamma \zeta }^{\beta }(f)W_{\gamma \zeta }^{\beta }\left( {\textbf{Z}}_{*},{\textbf{Z}}^{\prime },{\textbf{Z}}_{*}^{\prime }\right) =0\text { a.e.,} \end{aligned}$$

(7)

or, equivalently, if and only if $\ Q_{chem}(f)\equiv 0$.

Note that $Q(M)\equiv 0$ if and only if $Q_{chem}(f)\equiv 0$ and $Q_{mech}(f)\equiv 0$. Hence, any equilibrium, or, Maxwellian, distribution $M=(M_{1},\ldots ,M_{s})$, i.e., such that $Q(M)\equiv 0$, has to be of the form (6), but also, it follows by relation (7), that

$$\begin{aligned} \left( \log \frac{M_{\beta *}}{\varphi _{\beta }\left( I_{*}\right) } -\log \frac{M_{\gamma }^{\prime }}{\varphi _{\gamma }\left( I^{\prime }\right) }-\log \frac{M_{\zeta *}^{\prime }}{\varphi _{\zeta }\left( I_{*}^{\prime }\right) }\right) W_{\gamma \zeta }^{\beta }\left( {\textbf{Z}}_{*},{\textbf{Z}}^{\prime },{\textbf{Z}}_{*}^{\prime }\right) =0\text { a.e..} \end{aligned}$$

Hence, $\log \left( \varphi ^{-1}M\right) =\left( \log \dfrac{M_{1}}{\varphi _{1}\left( I\right) },\ldots ,\log \dfrac{M_{s}}{\varphi _{s}\left( I\right) } \right)$ is a chemical process invariant, and the components of the Maxwellian distributions $M=(M_{1},\ldots ,M_{s})$ are, of the form (6), with

$$\begin{aligned} \frac{n_{\gamma }n_{\varsigma }}{n_{\beta }}=\left( \frac{2\pi k_{B}Tm_{\beta }}{m_{\gamma }m_{\varsigma }}\right) ^{3/2}e^{-\Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}/\left( k_{B}T\right) }\dfrac{ q_{\gamma }q_{\varsigma }}{q_{\beta }} \end{aligned}$$

(8)

for all $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$, where $q_{\alpha }=\left\{ \begin{array}{l} 1\text { if }\alpha \in {\mathcal {I}}_{mono}\text { \ \ } \\ \int _{0}^{\infty }\varphi _{\alpha }\left( I\right) e^{-I/\left( k_{B}T\right) }\,dI\text { if }\alpha \in {\mathcal {I}}_{poly} \end{array} \right.$. In particular, for the particular case when $\varphi _{\alpha }\left( I\right) =I^{\delta ^{\left( \alpha \right) }/2-1}$ for all $\alpha \in {\mathcal {I}}$

$$\begin{aligned} \frac{n_{\gamma }n_{\varsigma }}{n_{\beta }}= & {} \left( \frac{2\pi m_{\beta } }{m_{\gamma }m_{\varsigma }}\right) ^{3/2}e^{-\Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}/\left( k_{B}T\right) }\dfrac{\Gamma \left( \delta ^{\left( \gamma \right) }/2\right) \Gamma \left( \delta ^{\left( \varsigma \right) }/2\right) }{\Gamma \left( \delta ^{\left( \beta \right) }/2\right) } \\{} & {} \times \left( k_{B}T\right) ^{\left( \delta ^{\left( \gamma \right) } {\textbf{1}}_{\gamma \in {\mathcal {I}}_{poly}}+\delta ^{\left( \varsigma \right) }{\textbf{1}}_{\varsigma \in {\mathcal {I}}_{poly}}+3-\delta ^{\left( \beta \right) }\right) /2} \end{aligned}$$

for all $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$. Note that any Maxwellian distribution M satisfies the relation (7) for any $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$.

Remark 2

An ${\mathcal {H}}$-theorem can be obtained, by introducing the ${\mathcal {H}}$-functional

$$\begin{aligned} {\mathcal {H}}\left[ f\right] =\left( f,\log \left( \varphi ^{-1}f\right) \right) \text {.} \end{aligned}$$

2.3 Linearized operator

Consider (without loss of generality) a deviation of a non-drifting, i.e., with $\mathbf {u=0}$ in expression (6), Maxwellian distribution M (6), (8) of the form

$$\begin{aligned} f=M+{\mathcal {M}}^{1/2}h\text {, with }M=(M_{1},\ldots ,M_{s})\text { and }{\mathcal {M}}=\textrm{diag}\left( M_{1},\ldots ,M_{s}\right) \text {.} \end{aligned}$$

Insertion in the Boltzmann equation (2) results in the system

$$\begin{aligned} \frac{\partial h}{\partial t}+\left( \varvec{\xi }\cdot \nabla _{{\textbf{x}}}\right) h+{\mathcal {L}}h=S\left( h,h\right) \text {,} \end{aligned}$$

with the linearized operator ${\mathcal {L}}={\mathcal {L}}_{mech}+{\mathcal {L}} _{chem}$, where the components of the linearized reactive operator $\mathcal { L}_{chem}=\left( {\mathcal {L}}_{c1},\ldots ,{\mathcal {L}}_{cs}\right)$ are given by

$$\begin{aligned} {\mathcal {L}}_{c\alpha }h= & {} -M_{\alpha }^{-1/2}\sum _{\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}}\int _{{\mathcal {Z}}}W_{\gamma \zeta }^{\beta }\Delta _{\gamma \zeta }^{\beta }\left( \alpha ,{\textbf{Z}}\right) {\widetilde{\Lambda }}_{\gamma \zeta }^{\beta }(f)\,d{\textbf{Z}}_{*}d {\textbf{Z}}^{\prime }d{\textbf{Z}}_{*}^{\prime }\text {, where} \\ {\widetilde{\Lambda }}_{\gamma \zeta }^{\beta }= & {} \frac{M_{\gamma }^{\prime }\left( M_{\zeta *}^{\prime }\right) ^{1/2}h_{\zeta *}^{\prime }+M_{\zeta *}^{\prime }\left( M_{\gamma }^{\prime }\right) ^{1/2}h_{\gamma }^{\prime }}{\varphi _{\gamma }\left( I^{\prime }\right) \varphi _{\zeta }\left( I_{*}^{\prime }\right) }-\frac{M_{\beta *}^{1/2}h_{\beta *}}{\varphi _{\beta }\left( I_{*}\right) } \\= & {} \left( \frac{M_{\beta *}M_{\gamma }^{\prime }M_{\zeta *}^{\prime } }{\varphi _{\beta }\left( I_{*}\right) \varphi _{\gamma }\left( I^{\prime }\right) \varphi _{\zeta }\left( I_{*}^{\prime }\right) } \right) ^{1/2}\left( \frac{h_{\gamma }^{\prime }}{\left( M_{\gamma }^{\prime }\right) ^{1/2}}+\frac{h_{\zeta *}^{\prime }}{\left( M_{\zeta *}^{\prime }\right) ^{1/2}}-\frac{h_{\beta *}}{M_{\beta *}^{1/2}} \right) \text {,} \end{aligned}$$

while $S=S_{mech}+S_{chem}$, where the components of the non-linear reactive operator $S_{chem}=\left( S_{c1},\ldots ,S_{cs}\right)$ are given by

$$\begin{aligned} S_{c\alpha }\left( h\right)= & {} M_{\alpha }^{-1/2}\sum _{\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}}\int _{{\mathcal {Z}}}W_{\gamma \zeta }^{\beta }\Delta _{\gamma \zeta }^{\beta }\left( \alpha ,{\textbf{Z}}\right) {\widehat{\Lambda }}_{\gamma \zeta }^{\beta }(h)\,d{\textbf{Z}}_{*}d\textbf{Z }^{\prime }d{\textbf{Z}}_{*}^{\prime }\text {, where} \\ {\widehat{\Lambda }}_{\gamma \zeta }^{\beta }= & {} \frac{\left( M_{\gamma }^{\prime }\right) ^{1/2}\left( M_{\zeta *}^{\prime }\right) ^{1/2}}{ \varphi _{\gamma }\left( I^{\prime }\right) \varphi _{\zeta }\left( I_{*}^{\prime }\right) }h_{\gamma }^{\prime }h_{\zeta *}^{\prime }=\left( \frac{M_{\beta *}}{\varphi _{\beta }\left( I_{*}\right) \varphi _{\gamma }\left( I^{\prime }\right) \varphi _{\zeta }\left( I_{*}^{\prime }\right) }\right) ^{1/2}h_{\gamma }^{\prime }h_{\zeta *}^{\prime }\text {.} \end{aligned}$$

Here, for $\alpha \in {\mathcal {I}}$

$$\begin{aligned} {\mathcal {L}}_{c\alpha }h=\nu _{c\alpha }h_{\alpha }-K_{c\alpha }\left( h\right) \text {,} \end{aligned}$$

(9)

with (in $W_{**}^{*}$ index $\alpha$ relates to argument ${\textbf{Z}}$, $\beta$ to ${\textbf{Z}}_{*}$, $\gamma$ to ${\textbf{Z}}^{\prime }$, and $\zeta$ to ${\textbf{Z}}_{*}^{\prime }$)

$$\begin{aligned} \nu _{c\alpha }= & {} \frac{1}{\varphi _{\alpha }\left( I\right) }\sum _{\beta ,\gamma =1}^{s}\int \limits _{{\mathcal {Z}}_{\beta }\mathcal {\times Z}_{\gamma }} W_{\beta \gamma }^{\alpha }{\textbf{1}}_{\left( \alpha ,\beta ,\gamma \right) \in {\mathcal {C}}}+\frac{2M_{\gamma }^{\prime }}{ \varphi _{\gamma }\left( I^{\prime }\right) }W_{\alpha \gamma }^{\beta }{\textbf{1}}_{\left( \beta ,\alpha ,\gamma \right) \in {\mathcal {C}} }\,d{\textbf{Z}}_{*}d{\textbf{Z}}^{\prime }, \nonumber \\ K_{c\alpha }= & {} -\frac{2M_{\alpha }^{1/2}}{\varphi _{\alpha }\left( I\right) }\sum _{\beta ,\gamma =1}^{s}\int \limits _{{\mathcal {Z}}_{\beta }\mathcal {\times Z}_{\gamma }}\frac{M_{\gamma }^{\prime }}{\varphi _{\gamma }\left( I^{\prime }\right) }W_{\alpha \gamma }^{\beta }\left( \frac{h_{\beta *} }{M_{\beta *}^{1/2}}-\frac{h_{\gamma }^{\prime }}{\left( M_{\gamma }^{\prime }\right) ^{1/2}}\right) {\textbf{1}}_{\left( \beta ,\alpha ,\gamma \right) \in {\mathcal {C}}} \nonumber \\{} & {} -\frac{h_{\beta }^{\prime }}{\left( M_{\beta }^{\prime }\right) ^{1/2}}W_{\beta \gamma }^{\alpha }{\textbf{1}}_{\left( \alpha ,\beta ,\gamma \right) \in {\mathcal {C}}}\,d{\textbf{Z}}_{*}d{\textbf{Z}}^{\prime }=2\sum _{\beta ,\gamma =1}^{s}\int _{{\mathcal {Z}}_{\beta }}k_{\alpha \beta \gamma }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right) h_{\beta *}d\textbf{Z }_{*}\text {,} \nonumber \\ \end{aligned}$$

(10)

with

$$\begin{aligned} k_{\alpha \beta \gamma }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right) =\int _{ {\mathcal {Z}}_{\gamma }}\frac{W_{\beta \gamma }^{\alpha }{\textbf{1}}_{\left( \alpha ,\beta ,\gamma \right) \in {\mathcal {C}}}+W_{\alpha \beta }^{\gamma } {\textbf{1}}_{\left( \gamma ,\alpha ,\beta \right) \in {\mathcal {C}}}-W_{\alpha \gamma }^{\beta }{\textbf{1}}_{\left( \beta ,\alpha ,\gamma \right) \in {\mathcal {C}}}}{\left( \varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) \varphi _{\gamma }\left( I^{\prime }\right) \right) ^{1/2}}\left( M_{\gamma }^{\prime }\right) ^{1/2}d{\textbf{Z}}^{\prime }\text {. } \nonumber \\ \end{aligned}$$

(11)

Moreover, the components of the linear operator ${\mathcal {L}}_{mech}=\left( {\mathcal {L}}_{m1},\ldots ,{\mathcal {L}}_{ms}\right)$ [6] are given by

$$\begin{aligned} {\mathcal {L}}_{m\alpha }h=-M_{\alpha }^{-1/2}\left( Q_{\alpha }(M, {\mathcal {M}}^{1/2}h)+Q_{\alpha }({\mathcal {M}}^{1/2}h,M)\right) =\nu _{m\alpha }h_{\alpha }-K_{m\alpha }\left( h\right) \text {,} \nonumber \\ \end{aligned}$$

(12)

with

$$\begin{aligned} \nu _{m\alpha }={} & {} \sum \limits _{\beta =1}^{s}\int _{\mathcal {Z}_{\alpha }\times \mathcal {Z}_{\beta }^{2}}\frac{M_{\beta *}}{\varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) } W_{\alpha \beta }d\textbf{Z}_{*}d\textbf{Z}^{\prime }d\textbf{Z}_{*}^{\prime }\text{, } \\ K_{m\alpha }={} & {} \sum \limits _{\beta =1}^{s}\int _{\mathcal {Z}_{\alpha }\times \mathcal {Z}_{\beta }^{2}}W_{\alpha \beta }\left( \frac{h_{\alpha }^{\prime }}{\left( M_{\alpha }^{\prime }\right) ^{1/2}}+\frac{h_{\beta *}^{\prime }}{\left( M_{\beta *}^{\prime }\right) ^{1/2}}-\frac{h_{\beta *}}{ M_{\beta *}^{1/2}}\right) \\ {}{} & {} \times \left( \frac{M_{\beta *}M_{\alpha }^{\prime }M_{\beta *}^{\prime }}{\varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) \varphi _{\alpha }\left( I^{\prime }\right) \varphi _{\beta }\left( I_{*}^{\prime }\right) }\right) ^{1/2}d\textbf{Z}_{*}d\textbf{Z}^{\prime }d\textbf{Z}_{*}^{\prime }\text{. } \end{aligned}$$

The components of the quadratic term $S_{mech}=\left( S_{m1},\ldots ,S_{ms}\right)$ [6] are given by

$$\begin{aligned} S_{m\alpha }\left( h,h\right) =M_{\alpha }^{-1/2}Q_{m\alpha }( {\mathcal {M}}^{1/2}h,{\mathcal {M}}^{1/2}h)\text { for }\alpha \in {\mathcal {I}} \text {.} \end{aligned}$$

We remind the following properties of ${\mathcal {L}} _{mech}$ [6].

Proposition 3

The linearized collision operator ${\mathcal {L}}_{mech}$ is symmetric and nonnegative, i.e., $\left( {\mathcal {L}} _{mech}h,g\right) =\left( h,{\mathcal {L}}_{mech}g\right)$ and $\left( {\mathcal {L}}_{mech}h,h\right) \ge 0$, and the kernel of ${\mathcal {L}}_{mech}$, $\ker {\mathcal {L}}_{mech}$, is generated by

$$\begin{aligned} \left\{ {\mathcal {M}}_{1}^{1/2}e_{1},\ldots ,{\mathcal {M}}^{1/2}e_{s},{\mathcal {M}} ^{1/2}m\xi _{x},{\mathcal {M}}^{1/2}m\xi _{y},{\mathcal {M}}^{1/2}m\xi _{z}, {\mathcal {M}}^{1/2}\left( m\left| \varvec{\xi }\right| ^{2}+2 {\mathbb {I}}\right) \right\} \text {.} \end{aligned}$$

The multiplication operator $\Lambda$ defined by

$$\begin{aligned} \Lambda (f)=\nu f\text {, where }\nu =\textrm{diag}\left( \nu _{1},\ldots ,\nu _{s}\right) \text {, with }\nu _{\alpha }=\nu _{m\alpha }+\nu _{c\alpha } \text {,} \end{aligned}$$

is a closed, densely defined, self-adjoint operator on ${{\mathfrak {h}} }$. It is Fredholm as well if and only if $\Lambda$ is coercive.

Denote for $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$

$$\begin{aligned} d{\widetilde{A}}_{\gamma \zeta }^{\beta }=\left( \frac{M_{\beta *}M_{\gamma }^{\prime }M_{\zeta *}^{\prime }}{\varphi _{\beta }\left( I_{*}\right) \varphi _{\gamma }\left( I^{\prime }\right) \varphi _{\zeta }\left( I_{*}^{\prime }\right) }\right) ^{1/2}dA_{\gamma \zeta }^{\beta } \text {.} \end{aligned}$$

The weak form of the linearized operator ${\mathcal {L}}_{chem}$ reads

$$\begin{aligned}{} & {} \left( {\mathcal {L}}_{chem}h,g\right) \nonumber \\{} & {} \quad =\sum _{\left( \beta ,\gamma ,\varsigma ,\alpha \right) \in \mathcal { C\times I}}\int _{\widetilde{{\mathcal {Z}}}}\Delta _{\gamma \zeta }^{\beta }\left( \alpha \text {,}{\textbf{Z}}\right) \left( \frac{h_{\gamma }^{\prime }}{ \left( M_{\gamma }^{\prime }\right) ^{1/2}}+\frac{h_{\zeta *}^{\prime }}{ \left( M_{\zeta *}^{\prime }\right) ^{1/2}}-\frac{h_{\beta *}}{ M_{\beta *}^{1/2}}\right) \frac{g_{\alpha }}{M_{\alpha }^{1/2}}\,d {\textbf{Z}}d{\widetilde{A}}_{\gamma \zeta }^{\beta } \nonumber \\ \end{aligned}$$

(13)

for any function $g=\left( g_{1},\ldots ,g_{s}\right)$, with $g_{\alpha }=g_{\alpha }(\varvec{\xi },I)$, such that the integrals are defined for all $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$.

We have the following lemma and concluding proposition.

Lemma 1

Let $g=\left( g_{1},\ldots ,g_{s}\right)$, with $g_{\alpha }=g_{\alpha }(\varvec{\xi },I)$, be such that the integrals in the right hand side of the weak form (13) are defined for all $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$. Then

$$\begin{aligned} \left( {\mathcal {L}}_{chem}h,g\right)= & {} \sum _{\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}}\int \limits _{{\mathcal {Z}}}\left( \frac{h_{\gamma }^{\prime }}{\left( M_{\gamma }^{\prime }\right) ^{1/2}}+\frac{h_{\zeta *}^{\prime }}{\left( M_{\zeta *}^{\prime }\right) ^{1/2}}-\frac{h_{\beta *}}{M_{\beta *}^{1/2}}\right) \\{} & {} \times \left( \frac{g_{\gamma }^{\prime }}{\left( M_{\gamma }^{\prime }\right) ^{1/2}}+\frac{g_{\zeta *}^{\prime }}{\left( M_{\zeta *}^{\prime }\right) ^{1/2}}-\frac{g_{\beta *}}{M_{\beta *}^{1/2}} \right) \,d{\widetilde{A}}_{\gamma \zeta }^{\beta }. \end{aligned}$$

Proposition 4

The linearized operator ${\mathcal {L}}$ is symmetric and nonnegative,

$$\begin{aligned} \left( {\mathcal {L}}h,g\right) =\left( h,{\mathcal {L}}g\right) \text { and } \left( {\mathcal {L}}h,h\right) \ge 0\text {,} \end{aligned}$$

and the kernel of ${\mathcal {L}}$, $\ker {\mathcal {L}}$, is for any basis ${\mathcal {U}}_{0}=\left\{ u_{1},\ldots ,u_{{\widetilde{s}}}\right\}$ of the set ${\mathcal {U}}$ (5) generated by

$$\begin{aligned} \left\{ {\mathcal {M}}^{1/2}u_{1},\ldots ,{\mathcal {M}}^{1/2}u_{{\widetilde{s}}}, {\mathcal {M}}^{1/2}m\xi _{x},{\mathcal {M}}^{1/2}m\xi _{y},{\mathcal {M}}^{1/2}m\xi _{z},{\mathcal {M}}^{1/2}\left( m\left| \varvec{\xi }\right| ^{2}+2 {\mathbb {I}}\right) \right\} \text {.} \end{aligned}$$

Proof

By Lemma 1 and Proposition 3, $\left( {\mathcal {L}} h,g\right) =\left( h,{\mathcal {L}}g\right)$ is immediate, as well as $\left( {\mathcal {L}}h,h\right) \ge 0$. Furthermore, $h\in \ker {\mathcal {L}}$ if and only if $\left( {\mathcal {L}}h,h\right) =0$, which will be fulfilled if and only if $\left( {\mathcal {L}}_{chem}h,h\right) =\left( {\mathcal {L}}_{mech}h,h\right) =0$. However, $\left( {\mathcal {L}} _{chem}h,h\right) =0$ if and only if ${\mathcal {M}}^{-1/2}h$ satisfies relations (4), i.e., if and only if ${\mathcal {M}}^{-1/2}h$ is a chemical process invariant. The last part of the lemma follows by Propositions 2 and 3. $\square$

Remark 3

Note also that it trivially follows that the quadratic term is orthogonal to the kernel of ${\mathcal {L}}$, i.e., $S\left( h\right) \in \left( \ker {\mathcal {L}}\right) ^{\perp _{{{\mathfrak {h}}}}}$.

3 Main results

This section is devoted to the main results, concerning compact properties in Theorem 2 and bounds of collision frequencies in Theorem 3. Below we consider the particular case $\varphi _{\alpha }\left( I\right) =I^{\delta ^{\left( \alpha \right) }/2-1}$ for $\alpha \in {\mathcal {I}}$.

Assume that for some positive number $\tau$, such that $0<\tau <1$, there is for all $\left( \alpha ,\beta \right) \in {\mathcal {I}}^{2}$ a bound

$$\begin{aligned}{} & {} 0\le \sigma _{\alpha \beta }\left( \left| {\textbf{g}}\right| ,\cos \theta ,I,I_{*},I^{\prime },I_{*}^{\prime }\right) \le C\frac{\Psi _{\alpha \beta }+\left( \Psi _{\alpha \beta }\right) ^{\tau /2}}{\left| {\textbf{g}}\right| ^{2}}\Upsilon _{\alpha \beta }\text {, where} \nonumber \\{} & {} \Upsilon _{\alpha \beta }=\frac{\varphi _{\alpha }\left( I^{\prime }\right) \varphi _{\beta }\left( I_{*}^{\prime }\right) }{\widetilde{ {\mathcal {E}}}_{\alpha \beta }^{\delta ^{\left( \alpha \right) }/2}\left( \widetilde{{\mathcal {E}}}_{\alpha \beta }^{*}\right) ^{\delta ^{\left( \beta \right) }/2}}\text { and }\Psi _{\alpha \beta }=\left| {\textbf{g}} \right| \sqrt{\left| {\textbf{g}}\right| ^{2}-2{\widetilde{\Delta }} _{\alpha \beta }I}\text {,} \end{aligned}$$

(14)

for $\left| {\textbf{g}}\right| ^{2}>2\widetilde{\Delta }_{\alpha \beta }I$, on the scattering cross sections. Here and below $\widetilde{{\mathcal {E}}}_{\alpha \beta }=1$ if $\alpha \in {\mathcal {I}}_{mono}$ and $\widetilde{{\mathcal {E}}}_{\alpha \beta }^{*}=1$ if $\beta \in {\mathcal {I}}_{mono}$, while, otherwise,

$$\begin{aligned} \widetilde{{\mathcal {E}}}_{\alpha \beta }=\widetilde{{\mathcal {E}}} _{\alpha \beta }^{*}= & {} \frac{m_{\alpha }m_{\beta }}{ 2\left( m_{\alpha }+m_{\beta }\right) }\left| {\textbf{g}}\right| ^{2}+I{\textbf{1}}_{\alpha \in {\mathcal {I}}_{poly}}+I_{*}{\textbf{1}}_{\beta \in {\mathcal {I}}_{poly}} \\= & {} \frac{m_{\alpha }m_{\beta }}{2\left( m_{\alpha }+m_{\beta }\right) } \left| {\textbf{g}}^{\prime }\right| ^{2}+I^{\prime }{\textbf{1}} _{\alpha \in {\mathcal {I}}_{poly}}+I_{*}^{\prime }{\textbf{1}}_{\beta \in {\mathcal {I}}_{poly}}\text {.} \end{aligned}$$

We remind the following compactness result for $K_{mech}$ in [6].

Theorem 1

Assume that for all $\left( \alpha ,\beta \right) \in {\mathcal {I}}^{2}$ the scattering cross sections $\sigma _{\alpha \beta }$ satisfy the bound (14) for some positive number $\tau$, such that $0<\tau <1$. Then the operator $K_{mech}=\left( K_{m1},\ldots ,K_{ms}\right)$ is a self-adjoint compact operator on ${{\mathfrak {h}}}$.

Assume now also that for some positive number $\chi$, such that $0<\chi <1$, there is for all $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$ a bound

$$\begin{aligned} 0\le & {} \sigma _{\gamma \zeta }^{\beta }\left( \left| {\textbf{g}} ^{\prime }\right| ,\left| \cos \theta \right| ,I_{*}\right) \le C\left( 1+\left| {\textbf{g}}^{\prime }\right| ^{\chi -1}\right) \frac{\varphi _{\beta }\left( I_{*}\right) }{{\mathcal {E}}_{\gamma \zeta }^{\delta ^{\left( \gamma \right) }/2}\left( {\mathcal {E}}_{\gamma \zeta }^{*}\right) ^{\delta ^{\left( \zeta \right) }/2}}\text {, or,} \nonumber \\ 0\le & {} B_{i\gamma \zeta }^{\beta }\left( \left| {\textbf{g}}^{\prime }\right| ,\cos \theta ,I_{*},I^{\prime },I_{*}^{\prime }\right) \le C\left| {\textbf{g}}^{\prime }\right| \left( 1+\left| {\textbf{g}}^{\prime }\right| ^{\chi -1}\right) \end{aligned}$$

(15)

for $E_{\beta }\ge K_{\gamma \zeta }^{\beta }$, on the scattering cross sections, or, the collision kernels. Here and below ${\mathcal {E}}_{\gamma \zeta }=1$ if $\gamma \in {\mathcal {I}}_{mono}$ and ${\mathcal {E}}_{\gamma \zeta }^{*}=1$ if $\zeta \in {\mathcal {I}}_{mono}$, while, otherwise,

$$\begin{aligned} {\mathcal {E}}_{\gamma \zeta }={\mathcal {E}}_{\gamma \zeta }^{*}=\frac{ m_{\gamma }m_{\zeta }}{2m_{\beta }}\left| {\textbf{g}}^{\prime }\right| ^{2}+I^{\prime }{\textbf{1}}_{\gamma \in {\mathcal {I}} _{poly}}+I_{*}^{\prime }{\textbf{1}}_{\zeta \in {\mathcal {I}}_{poly}}+\Delta _{\gamma \zeta }^{\beta }\varepsilon _{0}=I_{*}. \end{aligned}$$

Then the following result may be obtained.

Lemma 2

Assume that for all $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$ the scattering cross sections $\sigma _{\gamma \zeta }^{\beta }$ satisfy the bound (15) for some positive number $\chi$, such that $0<\chi <1$. Then the operator $K_{chem}=\left( K_{c1},\ldots ,K_{cs}\right)$ is a self-adjoint compact operator on ${{\mathfrak {h}}}$.

The proof of Lemma 2 will be addressed in Sect. 4. The following theorem follows by Theorem 1 and Lemma 2.

Theorem 2

Assume that for all $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$ the scattering cross sections $\sigma _{\gamma \zeta }^{\beta }$ satisfy the bound (15) for some positive number $\chi$, such that $0<\chi <1$, and that for all $\left( \alpha ,\beta \right) \in {\mathcal {I}}_{poly}^{2}$ the scattering cross sections $\sigma _{\alpha \beta }$ satisfy the bound (14) for some positive number $\tau$, such that $0<\tau <1$. Then the operator

$$\begin{aligned} K=K_{mech}+K_{chem} \end{aligned}$$

is a self-adjoint compact operator on ${{\mathfrak {h}}}$.

By Theorem 2, the linear operator ${\mathcal {L}}=\Lambda -K$ is closed as the sum of a closed and a bounded operator, and densely defined, since the domains of the linear operators ${\mathcal {L}}$ and $\Lambda$ are equal; $D({\mathcal {L}})=D(\Lambda )$. 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 [21].

Corollary 1

The linearized collision operator ${\mathcal {L}}$, with scattering cross sections satisfying (15), is a closed, densely defined, self-adjoint operator on ${{\mathfrak {h}}}$.

Now consider, for some nonnegative number $\eta$ less than 1, $0\le \eta <1,$ - cf. hard sphere models for $\eta =0$ - the scattering cross sections

$$\begin{aligned} \sigma _{\gamma \zeta }^{\beta }= & {} C_{\gamma \zeta }^{\beta }\frac{1}{E_{\beta }^{\eta /2}}\frac{\varphi _{\beta }\left( I_{*}\right) }{{\mathcal {E}}_{\gamma \zeta }^{\delta ^{\left( \gamma \right) }/2}\left( {\mathcal {E}}_{\gamma \zeta }^{*}\right) ^{\delta ^{\left( \zeta \right) }/2}}\text {, if }E_{\beta }\ge K_{\gamma \zeta }^{\beta }\text {, and} \nonumber \\ \sigma _{\alpha \beta }= & {} C_{\alpha \beta }\dfrac{\sqrt{\left| {\textbf{g}}\right| ^{2}-2{\widetilde{\Delta }}_{\alpha \beta }I}}{\left| {\textbf{g}}\right| E_{\alpha \beta }^{\eta /2}}\frac{\varphi _{\alpha }\left( I^{\prime }\right) \varphi _{\beta }\left( I_{*}^{\prime }\right) }{ {\mathcal {E}}_{\alpha \beta }^{\delta ^{\left( \alpha \right) }/2}\left( {\mathcal {E}}_{\alpha \beta }^{*}\right) ^{\delta ^{\left( \beta \right) }/2}}\text {, if }\left| {\textbf{g}}\right| ^{2}>2{\widetilde{\Delta }} _{\alpha \beta }I\text {,} \end{aligned}$$

(16)

for some positive constants $C_{\gamma \zeta }^{\beta }>0$ for $\left( \beta ,\gamma ,\varsigma \right) \in {\mathcal {C}}$ and $C_{\alpha \beta }>0$ for $\left( \alpha ,\beta \right) \in {\mathcal {I}}$.

In fact, it would be enough with the bounds

$$\begin{aligned} C_{-}\frac{1}{E_{\beta }^{\eta /2}}\frac{\varphi _{\beta }\left( I_{*}\right) }{{\mathcal {E}}_{\gamma \zeta }^{\delta ^{\left( \gamma \right) }/2}\left( {\mathcal {E}}_{\gamma \zeta }^{*}\right) ^{\delta ^{\left( \zeta \right) }/2}}\le & {} \sigma _{\gamma \zeta }^{\beta }\le C_{+}\frac{1 }{E_{\beta }^{\eta /2}}\frac{\varphi _{\beta }\left( I_{*}\right) }{ {\mathcal {E}}_{\gamma \zeta }^{\delta ^{\left( \gamma \right) }/2}\left( {\mathcal {E}}_{\gamma \zeta }^{*}\right) ^{\delta ^{\left( \zeta \right) }/2}}\text {, and} \nonumber \\ C_{-}\dfrac{\sqrt{\left| {\textbf{g}}\right| ^{2}-2{\widetilde{\Delta }} _{\alpha \beta }I}}{\left| {\textbf{g}}\right| E_{\alpha \beta }^{\eta /2}}\Upsilon _{\alpha \beta }\le & {} \sigma _{\alpha \beta }\le C_{+}\dfrac{ \sqrt{\left| {\textbf{g}}\right| ^{2}-2\widetilde{\Delta }_{\alpha \beta }I}}{\left| {\textbf{g}}\right| E_{\alpha \beta }^{\eta /2}} \Upsilon _{\alpha \beta } \end{aligned}$$

(17)

if $E_{\beta }\ge K_{\gamma \zeta }^{\beta }$, and $\left| {\textbf{g}}\right| ^{2}>2{\widetilde{\Delta }}_{\alpha \beta }I$, for some nonnegative number $\eta$ less than 1, $0\le \eta <1$, and some positive constants $C_{\pm }>0$, on the scattering cross sections—cf. hard potential with cut-off models.

Theorem 3

The linearized collision operator ${\mathcal {L}}$, with scattering cross sections (16) (or (17)), can be split into a positive multiplication operator $\Lambda$, where $\Lambda \left( f\right) =\nu f$, with $\nu =\nu _{mech}(\left| \varvec{\xi }\right| )+\nu _{chem}(\left| \varvec{\xi }\right| )$, minus a compact operator K on ${{\mathfrak {h}}}$, such that there exist positive numbers $\nu _{-}$ and $\nu _{+}$, with $0<\nu _{-}<\nu _{+}$, such that for any $\alpha \in {\mathcal {I}}$

$$\begin{aligned} \nu _{-}\left( 1+\left| \varvec{\xi }\right| +{\textbf{1}}_{\alpha \in {\mathcal {I}}_{poly}}\sqrt{I}\right) ^{1-\eta }\le \nu _{\alpha }\le \nu _{+}\left( 1+\left| \varvec{\xi }\right| +{\textbf{1}}_{\alpha \in {\mathcal {I}}_{poly}}\sqrt{I}\right) ^{1-\eta }\text {.} \end{aligned}$$

(18)

The decomposition follows by decomposition (9), (12) and Theorem 2, while the bounds (18) on the collision frequency will be proven in Sect. 5.

By Theorem 3 the multiplication operator $\Lambda$ is coercive, and thus it is a Fredholm operator. Furthermore, the set of Fredholm operators is closed under addition of compact operators, see Theorem 5.26 of Chapter IV in [21] and its proof, so, by Theorem 3, ${\mathcal {L}}$ is a Fredholm operator.

Corollary 2

The linearized collision operator ${\mathcal {L}}$, with scattering cross sections (16) (or (17)), is a Fredholm operator with domain

$$\begin{aligned} D({\mathcal {L}})=\left( L^{2}\left( \left( 1+\left| \varvec{\xi } \right| \right) ^{1-\eta }d\varvec{\xi \,}\right) \right) ^{s_{0}}\times \left( L^{2}\left( \left( 1+\left| \varvec{\xi } \right| +\sqrt{I}\right) ^{1-\eta }d\varvec{\xi \,}dI\right) \right) ^{s_{1}}\text {.} \end{aligned}$$

For hard sphere like models we obtain the following result.

Corollary 3

For the linearized collision operator ${\mathcal {L}}$, with scattering cross sections (16) (or (17)), where $\eta =0$, there exists a positive number $\lambda$, $0<\lambda <1$, such that

$$\begin{aligned} \left( h,{\mathcal {L}}h\right) \ge \lambda \left( h,\nu (\left| \varvec{\xi }\right| ,I)h\right) \ge \lambda \nu _{-}\left( h,\left( 1+\left| \varvec{\xi }\right| \right) h\right) \end{aligned}$$

for all $h\in D\left( {\mathcal {L}}\right) \cap \textrm{Im}{\mathcal {L}}$.

The proof is the same as in the non-reactive case, cf., e.g., [4,5,6,7].

Remark 4

By Proposition 4 and Corollary 1–3 the linearized operator ${\mathcal {L}}$ fulfills the properties assumed on the linear operators in [3], and hence, the results for half-space problems with general boundary conditions therein can be applied to hard sphere like models.

4 Compactness

This section concerns the proof of Lemma 2. Note that in Sect. 2.3 the kernels are written in such a way that ${\textbf{Z}}_{*}$ always will be an argument of the distribution function. Either ${\textbf{Z}}$ and ${\textbf{Z}}_{*}$ are both arguments in the dissociated term of the collision operator, or, either of ${\textbf{Z}}$ and ${\textbf{Z}}_{*}$ is the argument in the associated term. In the former case, the kernel will be shown to be Hilbert-Schmidt, while, in the latter case, the terms will be shown to be uniform limits of Hilbert-Schmidt integral operators, i.e., approximately Hilbert-Schmidt in the sense of Lemma 3.

To obtain the compactness properties we will apply the following result. Denote, for any (nonzero) natural number N,

$$\begin{aligned} {\mathfrak {h}}_{N}&:=\left\{ ({\textbf{Z}},{\textbf{Z}}_{*})\in \mathbb { Y\times Y}_{*}:\left| \varvec{\xi }-\varvec{\xi }_{*}\right| \ge \frac{1}{N}\text {; }\left| \varvec{\xi } \right| \le N\right\} \text {, and} \nonumber \\ b^{(N)}&=b^{(N)}({\textbf{Z}},{\textbf{Z}}_{*}):=b({\textbf{Z}},{\textbf{Z}} _{*}){\textbf{1}}_{{\mathfrak {h}}_{N}}\text {.} \end{aligned}$$

(19)

Here, either ${\textbf{Z}}=\varvec{\xi }$ and $\mathbb {Y=R}^{3}$, or, ${\textbf{Z}}=\left( \varvec{\xi },I\right)$ and $\mathbb {Y=R}^{3}\times {\mathbb {R}}_{+}$, and correspondingly, either ${\textbf{Z}}_{*}=\varvec{ \xi }_{*}$ and $\mathbb {Y_{*}=R}^{3}$, or, ${\textbf{Z}}_{*}=\left( \varvec{\xi }_{*},I_{*}\right)$ and $\mathbb {Y_{*}=R}^{3}\times {\mathbb {R}}_{+}$. Then we have the following lemma, cf. Glassey [18, Lemma 3.5.1] and Drange [17].

Lemma 3

Assume that $Tf\left( {\textbf{Z}}\right) =\int _{{\mathbb {Y}}_{*}}b({\textbf{Z}},{\textbf{Z}}_{*})f\left( {\textbf{Z}}_{*}\right) \,d {\textbf{Z}}_{*}$, with $b({\textbf{Z}},{\textbf{Z}}_{*})\ge 0$. Then T is compact on $L^{2}\left( d{\textbf{Z}}\right)$ if

• (i) $\int _{{\mathbb {Y}}}b({\textbf{Z}},{\textbf{Z}}_{*})\,d{\textbf{Z}}$ is bounded in ${\textbf{Z}}_{*}$;

• (ii) $b^{(N)}\in L^{2}\left( d{\textbf{Z}}\varvec{\,}d{\textbf{Z}}_{*}\right)$ for any (nonzero) natural number N;

• (iii) $\underset{{\textbf{Z}}\in {\mathbb {Y}}}{\sup }\int _{{\mathbb {Y}}_{*}}b( {\textbf{Z}},{\textbf{Z}}_{*})-b^{(N)}({\textbf{Z}},{\textbf{Z}}_{*})\,d {\textbf{Z}}_{*}\rightarrow 0$ as $N\rightarrow \infty$.

Then T [18, Lemma 3.5.1] is the uniform limit of Hilbert-Schmidt integral operators and we say that the kernel $b({\textbf{Z}},{\textbf{Z}}_{*})$ is approximately Hilbert-Schmidt, while T is an approximately Hilbert-Schmidt integral operator. The reader is referred to Glassey [18, Lemma 3.5.1] for a proof of Lemma 3.

Now we turn to the proof of Theorem 2. Note that throughout the proof C will denote a generic positive constant and $\varphi _{\alpha }\left( I\right) =I^{\delta ^{\left( \alpha \right) }/2-1}$ for $\alpha \in {\mathcal {I}}$.

Proof

For $\alpha \in {\mathcal {I}}$ write expression (10)–(11) as

$$\begin{aligned} K_{c\alpha }=2\sum _{\beta ,\gamma =1}^{s}\int _{ {\mathcal {Z}}_{\beta }}k_{\alpha \beta \gamma }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right) h_{\beta *}d{\textbf{Z}}_{*}\text {, where }k_{\alpha \beta \gamma }\left( {\textbf{Z}}, {\textbf{Z}}_{*}\right) =k_{\beta \gamma }^{1\alpha }+k_{\alpha \beta }^{2\gamma }-k_{\alpha \gamma }^{3\beta }\text {,} \end{aligned}$$

with

$$\begin{aligned} k_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right)= & {} \int _{{\mathcal {Z}}_{\gamma }}W_{\beta \gamma }^{\alpha }\left( {\textbf{Z}}, {\textbf{Z}}_{*},{\textbf{Z}}^{\prime }\right) \frac{M_{\alpha }^{1/2}}{ \varphi _{\alpha }\left( I\right) M_{\beta *}^{1/2}}d{\textbf{Z}}^{\prime } \\= & {} \int _{{\mathcal {Z}}_{\gamma }}W_{\beta \gamma }^{\alpha }\left( {\textbf{Z}}, {\textbf{Z}}_{*},{\textbf{Z}}^{\prime }\right) \left( \frac{M_{\alpha }M_{\gamma }^{\prime }}{\left( \varphi _{\alpha }\left( I\right) \right) ^{3}\varphi _{\beta }\left( I_{*}\right) \varphi _{\gamma }\left( I^{\prime }\right) M_{\beta *}}\right) ^{1/4}d{\textbf{Z}}^{\prime }\text {, } \\ k_{\alpha \beta }^{2\gamma }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right)= & {} \int _{{\mathcal {Z}}_{\gamma }}W_{\alpha \beta }^{\gamma }\left( {\textbf{Z}} ^{\prime },{\textbf{Z}},{\textbf{Z}}_{*}\right) \frac{M_{\alpha }^{1/2}M_{\beta *}^{1/2}}{\varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) }d{\textbf{Z}}^{\prime }\text {, and} \\ k_{\alpha \gamma }^{3\beta }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right)= & {} \int _{{\mathcal {Z}}_{\gamma }}W_{\alpha \gamma }^{\beta }\left( {\textbf{Z}} _{*},{\textbf{Z}},{\textbf{Z}}^{\prime }\right) \frac{M_{\beta *}^{1/2}}{ \varphi _{\beta }\left( I_{*}\right) M_{\alpha }^{1/2}}d{\textbf{Z}} ^{\prime }=k_{\alpha \gamma }^{1\beta }\left( {\textbf{Z}}_{*},{\textbf{Z}} \right) \text {.} \end{aligned}$$

I. Compactness of $K_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}} \right) =\int _{{\mathcal {Z}}_{\beta }}k_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right) h_{\beta *}d{\textbf{Z}}_{*}$ for $\left( \alpha ,\beta ,\gamma \right) \in {\mathcal {C}}$.

The following bound for the transition probabilities $W_{\beta \gamma }^{\alpha }=W_{\beta \gamma }^{\alpha }\left( {\textbf{Z}},{\textbf{Z}}_{*}, {\textbf{Z}}^{\prime }\right)$, see also Remark 1, may be obtained under assumption (15)

$$\begin{aligned} W_{\beta \gamma }^{\alpha }\le & {} C\frac{\varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) \varphi _{\gamma }\left( I^{\prime }\right) }{{\mathcal {E}}_{\beta \gamma }^{\delta ^{\left( \beta \right) }/2}\left( {\mathcal {E}}_{\beta \gamma }^{*}\right) ^{\delta ^{\left( \gamma \right) }/2}}\left( 1+\left| \varvec{\xi } ^{\prime }-\varvec{\xi }_{*}\right| ^{\chi -1}\right) {\textbf{1}} _{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }}\widetilde{ \varvec{\delta }} \\\le & {} C\frac{\varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) }{{\mathcal {E}}_{\beta \gamma }^{\delta ^{\left( \beta \right) }/2}\left( {\mathcal {E}}_{\beta \gamma }^{*}\right) ^{\delta ^{\left( \gamma \right) }/2}}\left( \frac{\varphi _{\gamma }\left( I^{\prime }\right) }{M_{\gamma }^{\prime }}\right) ^{1/4} \left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) {\textbf{1}}_{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }} \widetilde{\varvec{\delta }}\text {, where} \\{} & {} \text { }\widetilde{\varvec{\delta }}=\varvec{\delta } _{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) \varvec{ \delta }_{3}\left( \varvec{\xi }^{\prime }-{\textbf{g}}_{\alpha \beta }\right) \text {,}\text { }{\textbf{g}}_{\alpha \beta }=\frac{ m_{\alpha }\varvec{\xi }-m_{\beta }\varvec{\xi }_{*}}{m_{\alpha }-m_{\beta }}\text { and }{\textbf{g}}=\varvec{\xi }-\varvec{\xi } _{*}\text {.} \end{aligned}$$

(i) Firstly, assume that $\gamma \in {\mathcal {I}} _{poly}$. By relation (3), since $\delta ^{\left( \gamma \right) }\ge 2$,

$$\begin{aligned} \frac{\delta ^{\left( \alpha \right) }}{2}-1\le \frac{\delta ^{\left( \beta \right) }}{2}+\frac{\delta ^{\left( \gamma \right) }}{2}-\frac{3}{2}\le \frac{\delta ^{\left( \beta \right) }}{2}+\delta ^{\left( \gamma \right) }- \frac{5}{2}\text {.} \end{aligned}$$

Then, since $I\ge \Delta _{\beta \gamma }^{\alpha }\varepsilon _{0} >0$ and $\varphi _{\beta }\left( I_{*}\right) \le C{\mathcal {E}}_{\beta \gamma }^{\delta ^{\left( \beta \right) }}$,

$$\begin{aligned} k_{\beta \gamma }^{1\alpha }\le & {} C\left( \frac{\varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) }{{\mathcal {E}}_{\beta \gamma }^{\delta ^{\left( \beta \right) }}\left( {\mathcal {E}}_{\beta \gamma }^{*}\right) ^{\delta ^{\left( \gamma \right) }}}\right) ^{1/2}\left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) \left( \frac{M_{\alpha }\varphi _{\beta }\left( I_{*}\right) }{M_{\beta *}\varphi _{\alpha }\left( I\right) }\right) ^{1/4}{\textbf{1}}_{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }} \\\le & {} \frac{C}{I^{3/4}}\left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) \left( \frac{M_{\alpha }\varphi _{\beta }\left( I_{*}\right) }{M_{\beta *}\varphi _{\alpha }\left( I\right) } \right) ^{1/4}{\textbf{1}}_{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }} \text {, with }\frac{C}{I^{3/4}}\le C\text {.} \end{aligned}$$

Note that, denoting ${\widetilde{I}}=I-I_{*}{\textbf{1}}_{\beta \in {\mathcal {I}}_{poly}}-\dfrac{m_{\alpha }m_{\beta }}{2\left( m_{\alpha }-m_{\beta }\right) }\left| {\textbf{g}}\right| ^{2}>0$,

$$\begin{aligned} m_{\alpha }\frac{\left| \varvec{\xi }\right| ^{2}}{2}-m_{\beta } \frac{\left| \varvec{\xi }_{*}\right| ^{2}}{2}+I-I_{*} {\textbf{1}}_{\beta \in {\mathcal {I}}_{poly}}=\frac{m_{\alpha }-m_{\beta }}{2}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}+ {\widetilde{I}}\text {.} \end{aligned}$$

Also, for $0\le a_{1}<3$, $a_{2}>0$, and $b=b(t)\in L^{1}(\left[ -1,1\right] )$

$$\begin{aligned}&\int _{{\mathbb {R}}^{3}}\left| {\textbf{g}}\right| ^{-a_{1}}\exp \left( -a_{2}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) b\left( \cos \phi \right) d\varvec{\xi } \\&\quad \le \int _{{\mathbb {R}}^{3}}\left| {\textbf{g}}\right| ^{-a_{1}}\exp \left( -a_{2}\left| {\textbf{g}}\right| ^{2}\right) {\textbf{1}} _{\left| {\textbf{g}}\right| \le \left| {\textbf{g}}_{\alpha \beta }\right| }b\left( \cos \phi \right) d{\textbf{g}} \\&\qquad +\int _{{\mathbb {R}}^{3}}\left| {\textbf{g}}_{\alpha \beta }\right| ^{-a_{1}}\exp \left( -a_{2}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) {\textbf{1}}_{\left| {\textbf{g}}_{\alpha \beta }\right| \le \left| {\textbf{g}}\right| }b\left( \cos \phi \right) d{\textbf{g}} _{\alpha \beta } \\&\quad \le 4\pi \int _{0}^{\infty }R^{2-a_{1}}\exp \left( -a_{2}R^{2}\right) dR\int _{0}^{\pi }b\left( \cos \phi \right) \sin \phi d\phi =C\text {.} \end{aligned}$$

Then

$$\begin{aligned}{} & {} \int _{{\mathcal {Z}}_{\alpha }}k_{\beta \gamma }^{1\alpha }d {\textbf{Z}}=\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}_{+}}k_{\beta \gamma }^{1\alpha }d\varvec{\xi }dI \\{} & {} \quad \le C\int _{{\mathbb {R}}^{3}}\left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) \exp \left( -\frac{m_{\alpha }-m_{\beta }}{8k_{B}T} \left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) d\varvec{ \xi }\int _{0}^{\infty }\exp \left( -\frac{{\widetilde{I}}}{4k_{B}T}\right) d {\widetilde{I}}=C. \end{aligned}$$

But, $k_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}}, {\textbf{Z}}_{*}\right) {\textbf{1}}_{{\mathfrak {h}}_{N}}\in L^{2}\left( d {\textbf{Z}}\varvec{\,}d{\textbf{Z}}_{*}\right)$ for any (nonzero) natural number N, by

$$\begin{aligned}{} & {} \int _{{\mathcal {Z}}_{\alpha }\times {\mathcal {Z}}_{\beta }}\left( k_{\beta \gamma }^{1\alpha }\right) ^{2}{\textbf{1}}_{{\mathfrak {h}} _{N}}d{\textbf{Z}}d{\textbf{Z}}_{*}\le CN^{2}\int \limits _{{\mathbb {R}}^{3}}\exp \left( -\frac{m_{\alpha }-m_{\beta }}{ 4k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) d {\textbf{g}}_{\alpha \beta } \\{} & {} \quad \times \int \limits _{\left| \varvec{\xi }\right| \le N}d \varvec{\xi }\int \limits _{0}^{{\mathcal {E}}_{\beta \gamma }}\frac{I_{*}^{\delta ^{\left( \beta \right) }/2-1}}{{\mathcal {E}}_{\beta \gamma }^{\delta ^{\left( \beta \right) }/2}}dI_{*}\int \limits _{\Delta _{\beta \gamma }^{\alpha }\varepsilon _{0}}^{\infty }\frac{1}{I^{3/2}}dI=CN^{5}\text {.} \end{aligned}$$

Note that, with $\cos \phi =\dfrac{\varvec{\xi }}{\left| \varvec{\xi }\right| }\cdot \dfrac{{\textbf{g}}}{\left| {\textbf{g}} \right| }$,

$$\begin{aligned} \frac{\left| m_{\alpha }\varvec{\xi }-m_{\beta }\varvec{\xi } _{*}\right| ^{2}}{2\left( m_{\alpha }-m_{\beta }\right) }\ge 8k_{B}T\Upsilon _{\phi }\text {, where }\Upsilon _{\phi }=\dfrac{ m_{\beta }}{8k_{B}T}\left| \varvec{\xi }\right| \left| {\textbf{g}}\right| \left( 1+\cos \phi \right) \text {,} \end{aligned}$$

since for any $\left( {\textbf{x}},{\textbf{y}}\right) \in \left( {\mathbb {R}} ^{3}\right) ^{2}$, with $\cos \theta =\dfrac{{\textbf{x}}}{ \left| {\textbf{x}}\right| }\cdot \dfrac{{\textbf{y}}}{\left| {\textbf{y}}\right| }$,

$$\begin{aligned} \left| {\textbf{x}}\pm {\textbf{y}}\right| ^{2}= \left| \left| {\textbf{x}}\right| -\left| {\textbf{y}} \right| \right| ^{2}+2\left| {\textbf{x}}\right| \left| {\textbf{y}}\right| \left( 1\pm \cos \theta \right) \ge 2\left| {\textbf{x}}\right| \left| {\textbf{y}} \right| \left( 1\pm \cos \theta \right) \text {.} \end{aligned}$$

Hence, it follows that

$$\begin{aligned}{} & {} \sup \int _{{\mathcal {Z}}_{\beta }}k_{\beta \gamma }^{1\alpha }\left( 1-{\textbf{1}}_{{\mathfrak {h}}_{N}}\right) d{\textbf{Z}}_{*}\le C\int _{0}^{{\mathcal {E}}_{\beta \gamma }}\frac{I_{*}^{\delta ^{\left( \beta \right) }/4-1/2}}{{\mathcal {E}}_{\beta \gamma }^{\delta ^{\left( \beta \right) }/4+1/2}}dI_{*}\sup \left( \int _{\left| {\textbf{g}}\right| \le \frac{1}{N}}1+\left| {\textbf{g}}\right| ^{\chi -1}d{\textbf{g}}\right. \\{} & {} \qquad \left. +\int _{\left| {\textbf{g}}\right| \ge \frac{1}{N} \text {, }\left| \varvec{\xi }\right| \ge N\ge 1} \left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) \exp \left( -\Upsilon _{\phi }\right) \exp \left( -\frac{m_{\alpha }-m_{\beta }}{16k_{B}T }\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) d\varvec{ \xi }_{*}\right) \\{} & {} \quad \le C\int _{0}^{1/N}R^{\chi +1}dR+\frac{C}{N}\int _{{\mathbb {R}} ^{3}}\left( \left| {\textbf{g}}\right| ^{-1}+\left| {\textbf{g}} \right| ^{\chi -2}\right) \exp \left( -\frac{m_{\alpha }-m_{\beta }}{ 16k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) \\{} & {} \qquad \left. \times \left| \varvec{\xi }\right| \left| {\textbf{g}} \right| \exp \left( -\Upsilon _{\phi }\right) d\varvec{\xi }_{*}\right. \le \frac{C}{N}\rightarrow 0\text { as } N\rightarrow \infty . \end{aligned}$$

By Lemma 3, $K_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}}\right) =\int _{{\mathcal {Z}}_{\beta }}k_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}}, {\textbf{Z}}_{*}\right) h_{\beta *}d{\textbf{Z}}_{*}$ is compact on $L^{2}\left( d{\textbf{Z}}\right)$ for any $\left( \alpha ,\beta ,\gamma \right) \in {\mathcal {C}}$ such that $\gamma \in {\mathcal {I}}_{poly}$.

(ii) Now, assume that $\left( \beta ,\gamma \right) \in {\mathcal {I}}_{mono}^{2}$. Then

$$\begin{aligned} k_{\beta \gamma }^{1\alpha }\le & {} CI^{1/4}\left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) \left( \frac{M_{\alpha } }{M_{\beta *}\varphi _{\alpha }\left( I\right) }\right) ^{1/2} \varvec{\delta }_{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) {\textbf{1}}_{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }} \\\le & {} C\left( 1+\left| {\textbf{g}}\right| ^{1/2}\right) \left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) \left( \frac{M_{\alpha } }{M_{\beta *}\varphi _{\alpha }\left( I\right) }\right) ^{1/2} \varvec{\delta }_{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) {\textbf{1}}_{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }} \\\le & {} C\left( \left| {\textbf{g}}\right| ^{1/2}+\left| {\textbf{g}} \right| ^{\chi -1}\right) \exp \left( -\frac{m_{\alpha }-m_{\beta }}{ 4k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) \varvec{\delta }_{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) {\textbf{1}}_{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }}\text {,} \end{aligned}$$

implies, since $\left| {\textbf{g}}\right| \le \left| \varvec{\xi }\right| +\left| {\textbf{g}}_{\alpha \beta }\right|$, that

$$\begin{aligned}{} & {} \int _{{\mathcal {Z}}_{\alpha }}k_{\beta \gamma }^{1\alpha }d {\textbf{Z}}=\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}_{+}}k_{\beta \gamma }^{1\alpha }d\varvec{\xi }dI\le C\int _{0}^{\infty } \varvec{\delta }_{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) dI \\{} & {} \qquad \times \int _{{\mathbb {R}}^{3}}\left( \left| \varvec{\xi } \right| ^{1/2}+\left| {\textbf{g}}_{\alpha \beta }\right| ^{1/2}+\left| {\textbf{g}}\right| ^{\chi -1}\right) \exp \left( -\frac{m_{\alpha }-m_{\beta }}{4k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) d\varvec{\xi } \\{} & {} \quad \le C\left( 1+\int _{{\mathbb {R}}^{3}}\left| \varvec{\xi } \right| \left| {\textbf{g}}\right| \exp \left( -2\Upsilon _{\phi }\right) \frac{1}{\left| {\textbf{g}}\right| }\exp \left( -\frac{ m_{\alpha }-m_{\beta }}{8k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) {\textbf{1}}_{1\le \left| \varvec{\xi } \right| }d\varvec{\xi }\right) =C\text {.} \end{aligned}$$

Furthermore, it follows, for ${\mathfrak {h}}_{N}$ given by notation (19), that

$$\begin{aligned} \left( k_{\beta \gamma }^{1\alpha }\right) ^{2} {\textbf{1}}_{{\mathfrak {h}}_{N}}\le & {} C(1+N)\left( 1+N^{2-2\chi }\right) \frac{\left| {\textbf{g}}\right| M_{\alpha }}{ M_{\beta *}\varphi _{\alpha }\left( I\right) }\varvec{\delta } _{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) {\textbf{1}}_{ {\mathfrak {h}}_{N}} \\\le & {} CN^{3}\left( \left| \varvec{\xi }\right| +\left| {\textbf{g}}_{\alpha \beta }\right| \right) \exp \left( -\frac{m_{\alpha }-m_{\beta }}{2k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) \varvec{\delta }_{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) {\textbf{1}}_{{\mathfrak {h}}_{N}} \\\le & {} CN^{4}\left( 1+\left| {\textbf{g}}_{\alpha \beta }\right| \right) \exp \left( -\frac{m_{\alpha }-m_{\beta }}{2k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) \varvec{\delta } _{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) {\textbf{1}}_{ {\mathfrak {h}}_{N}}\text {.} \end{aligned}$$

Then $k_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}}, {\textbf{Z}}_{*}\right) {\textbf{1}}_{{\mathfrak {h}}_{N}}\in L^{2}\left( d {\textbf{Z}}\varvec{\,}d{\textbf{Z}}_{*}\right)$ for any (nonzero) natural number N, by

$$\begin{aligned}{} & {} \int _{\left( {\mathbb {R}}^{3}\right) ^{2}\times {\mathbb {R}}_{+}}\left( k_{\beta \gamma }^{1\alpha }\right) ^{2}{\textbf{1}}_{ {\mathfrak {h}}_{N}}d\varvec{\xi }d\varvec{\xi }_{*}dI\le CN^{4} \int \limits _{\left| \varvec{\xi }\right| \le N}d \varvec{\xi }\int _{{\mathbb {R}}^{3}}\left( 1+\left| {\textbf{g}}_{\alpha \beta }\right| \right) \\{} & {} \quad \times \exp \left( -\frac{m_{\alpha }-m_{\beta }}{ 2k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) d {\textbf{g}}_{\alpha \beta }\int _{0}^{\infty }\varvec{\delta }_{1}\left( I- {\widetilde{E}}_{\beta \gamma }^{\prime }\right) dI\le CN^{7}\text {.} \end{aligned}$$

Moreover,

$$\begin{aligned}{} & {} \sup \int _{{\mathbb {R}}^{3}}k_{\beta \gamma }^{1\alpha }\left( 1- {\textbf{1}}_{{\mathfrak {h}}_{N}}\right) d\varvec{\xi }_{*} \\{} & {} \quad \le C\sup \left( \int _{\left| {\textbf{g}}\right| \le \frac{1}{N}}\left( \left| {\textbf{g}}\right| ^{1/2}+\left| {\textbf{g}}\right| ^{\chi -1}\right) \varvec{\delta }_{1}\left( I- {\widetilde{E}}_{\beta \gamma }^{\prime }\right) d{\textbf{g}}\right. \\{} & {} \qquad +\int _{\left| {\textbf{g}}\right| \ge \frac{1}{N}\text {, }\left| \varvec{\xi }\right| \ge N\ge 1}\left( \left| \varvec{\xi }\right| ^{1/2}+\left| {\textbf{g}}_{\alpha \beta }\right| ^{1/2}\exp \left( -\frac{m_{\alpha }-m_{\beta } }{8k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) +\left| {\textbf{g}}\right| ^{\chi -1}\right) \\{} & {} \qquad \left. \times \exp \left( -2\Upsilon _{\phi }\right) \varvec{\delta }_{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) d{\textbf{g}}\right) \\{} & {} \quad \le C\sup \left( \int _{\left| {\textbf{g}}\right| \le \frac{1}{N} }\left( \left| {\textbf{g}}\right| ^{3/2}+\left| {\textbf{g}} \right| ^{\chi }\right) \varvec{\delta }_{1}\left( I-{\widetilde{E}} _{\beta \gamma }^{\prime }\right) d\left| {\textbf{g}}\right| ^{2}+\left( \frac{1}{N^{1/2}}+\frac{1}{N}+\frac{1}{ N^{\chi }}\right) \right. \\{} & {} \qquad \left. \times \int _{0}^{\infty }\int _{0}^{\pi }\left| \varvec{\xi } \right| \left| {\textbf{g}}\right| \exp \left( -2\Upsilon _{\phi }\right) \sin \phi d\phi \varvec{\delta } _{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) d\left| {\textbf{g}}\right| ^{2}\right) \\{} & {} \quad \le C\left( \frac{1}{N^{1/2}}+\frac{1}{N^{\chi }} \right) \int _{0}^{\infty }\varvec{\delta }_{1}\left( I- {\widetilde{E}}_{\beta \gamma }^{\prime }\right) d\left| {\textbf{g}} \right| ^{2}\\{} & {} \quad \le C\left( \frac{1}{N^{1/2}}+\frac{1}{N^{\chi }}\right) \rightarrow 0\text { as }N\rightarrow \infty . \end{aligned}$$

Now by Lemma 3, $K_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}} \right) =\sum _{\beta ,\gamma =1}^{s}\int _{{\mathcal {Z}}_{\beta }}k_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right) h_{\beta *}d{\textbf{Z}}_{*}$ is compact on $L^{2}\left( d{\textbf{Z}}\right)$ for any $\left( \alpha ,\beta ,\gamma \right) \in {\mathcal {C}}$ such that $\left( \beta ,\gamma \right) \in {\mathcal {I}}_{mono}^{2}$.

(iii) Finally, assume that $\left( \gamma ,\beta \right) \in {\mathcal {I}}_{mono}\times {\mathcal {I}}_{poly}$. Then

$$\begin{aligned} k_{\beta \gamma }^{1\alpha }\le C\frac{\left( \varphi _{\beta }\left( I_{*}\right) \right) ^{1/2}}{{\mathcal {E}}_{\beta \gamma }^{\delta ^{\left( \beta \right) }/4}}\left( 1+\left| {\textbf{g}} \right| ^{\chi -1}\right) \exp \left( -\frac{m_{\alpha }-m_{\beta }}{ 4k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) \varvec{\delta }_{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) {\textbf{1}}_{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }}\text {,} \end{aligned}$$

implies that

$$\begin{aligned}{} & {} \int _{{\mathcal {Z}}_{\alpha }}k_{\beta \gamma }^{1\alpha }d \varvec{\xi }dI=\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}_{+}}k_{\beta \gamma }^{1\alpha }d\varvec{\xi }dI \\{} & {} \quad \le C\int _{{\mathbb {R}}^{3}}\left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) \exp \left( -\frac{m_{\alpha }-m_{\beta }}{4k_{B}T} \left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) d\varvec{ \xi }\int _{0}^{\infty }\varvec{\delta }_{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) dI=C \end{aligned}$$

Also, $k_{\beta \gamma }^{1\alpha }{\textbf{1}}_{ {\mathfrak {h}}_{N}}\in L^{2}\left( d{\textbf{Z}}\varvec{\,}d{\textbf{Z}}_{*}\right)$ for any (nonzero) natural number N, since

$$\begin{aligned} \int _{{\mathcal {Z}}_{\alpha }\times {\mathcal {Z}}_{\beta }}\left( k_{\beta \gamma }^{1\alpha }\right) ^{2}{\textbf{1}}_{{\mathfrak {h}} _{N}}d{\textbf{Z}}d{\textbf{Z}}_{*}\le&CN^{2}\int \limits _{{\mathbb {R}}^{3}} \exp \left( -\frac{m_{\alpha }-m_{\beta }}{2k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) d{\textbf{g}} _{\alpha \beta }\int \limits _{\left| \varvec{\xi }\right| \le N}d \varvec{\xi } \\ \times&\int \limits _{0}^{{\mathcal {E}}_{\beta \gamma }}\frac{I_{*}^{\delta ^{\left( \beta \right) }/2-1}}{{\mathcal {E}}_{\beta \gamma }^{\delta ^{\left( \beta \right) }/2}}dI_{*}\int \limits _{0}^{\mathcal {\infty }} \varvec{\delta }_{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) dI=CN^{5}\text {.} \end{aligned}$$

Furthermore,

$$\begin{aligned}{} & {} \sup \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}_{+}}k_{\beta \gamma }^{1\alpha }\left( 1-{\textbf{1}}_{{\mathfrak {h}}_{N}}\right) d\varvec{\xi } _{*}dI_{*}\le C\int _{0}^{\mathcal {\infty }}\varvec{ \delta }_{1}\left( I-{\widetilde{E}}_{\beta \gamma }^{\prime }\right) dI_{*} \\{} & {} \quad \times \sup \left( \int _{\left| {\textbf{g}}\right| \le \frac{1}{N} }1+\left| {\textbf{g}}\right| ^{\chi -1}d{\textbf{g}}+ \int _{\left| {\textbf{g}}\right| \ge \frac{1}{N}\text {, }\left| \varvec{\xi }\right| \ge N\ge 1}\left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) \right. \\{} & {} \quad \left. \times \exp \left( -2\Upsilon _{\phi }\right) \exp \left( -\frac{m_{\alpha }-m_{\beta }}{8k_{B}T}\left| {\textbf{g}}_{\alpha \beta }\right| ^{2}\right) d\varvec{\xi }_{*}\right) \le \frac{C }{N}\rightarrow 0\text { as }N\rightarrow \infty . \end{aligned}$$

Then by Lemma 3, $K_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}} \right) =\int _{{\mathcal {Z}}_{\beta }}k_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right) h_{\beta *}d{\textbf{Z}}_{*}$ is compact on $L^{2}\left( d{\textbf{Z}}\right)$ for any $\left( \alpha ,\beta ,\gamma \right) \in {\mathcal {C}}$ such that $\left( \gamma ,\beta \right) \in {\mathcal {I}}_{mono}\times {\mathcal {I}}_{poly}$.

Concluding, $K_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}}\right) =\int _{ {\mathcal {Z}}_{\beta }}k_{\beta \gamma }^{1\alpha }\left( {\textbf{Z}},{\textbf{Z}} _{*}\right) h_{\beta *}d{\textbf{Z}}_{*}$ is compact on $L^{2}\left( d{\textbf{Z}}\right)$ for any $\left( \alpha ,\beta ,\gamma \right) \in {\mathcal {C}}$.

II. Compactness of $K_{\alpha \beta }^{2\gamma }\left( {\textbf{Z}} \right) =\int _{{\mathcal {Z}}_{\beta }}k_{\alpha \beta }^{2\gamma }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right) h_{\beta *}d{\textbf{Z}}_{*}$ for $\left( \gamma ,\alpha ,\beta \right) \in {\mathcal {C}}$.

The following bound for the transition probabilities $W_{\alpha \beta }^{\gamma }=W_{\alpha \beta }^{\gamma }\left( {\textbf{Z}}^{\prime },{\textbf{Z}},{\textbf{Z}}_{*}\right)$, see also Remark 1, may be obtained under assumption (15)

$$\begin{aligned} W_{\alpha \beta }^{\gamma }\le & {} C\left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) \frac{\varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) \varphi _{\gamma }\left( I^{\prime }\right) }{{\mathcal {E}}_{\alpha \beta }^{\delta ^{\left( \alpha \right) }/2}\left( {\mathcal {E}}_{\alpha \beta }^{*}\right) ^{\delta ^{\left( \beta \right) }/2}}\varvec{\delta }_{1}\left( I^{\prime }-{\widetilde{E}}_{\alpha \beta }^{\prime }\right) \varvec{ \delta }_{3}\left( \varvec{\xi }^{\prime }-{\textbf{G}}_{\alpha \beta }\right) {\textbf{1}}_{E_{\gamma }\ge \kappa _{\alpha \beta }^{\gamma }} \\\le & {} C\left( 1+\left| {\textbf{g}}\right| ^{\chi -1}\right) \varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) \left( {\widetilde{E}}_{\alpha \beta }^{\prime }\right) ^{1/2}\varvec{\delta } _{1}\left( I^{\prime }-{\widetilde{E}}_{\alpha \beta }^{\prime }\right) \varvec{\delta }_{3}\left( \varvec{\xi }^{\prime }-{\textbf{G}} _{\alpha \beta }\right) {\textbf{1}}_{E_{\gamma }\ge \kappa _{\alpha \beta }^{\gamma }}\text {.} \end{aligned}$$

Here the first of the following bounds was applied to obtain the last inequality,

$$\begin{aligned} \frac{\left( {\widetilde{E}}_{\alpha \beta }^{\prime }\right) ^{\delta ^{\left( \gamma \right) }/2-1}}{{\mathcal {E}}_{\alpha \beta }^{\delta ^{\left( \alpha \right) }/2}\left( {\mathcal {E}}_{\alpha \beta }^{*}\right) ^{\delta ^{\left( \beta \right) }/2}}\le C\left( {\widetilde{E}}_{\alpha \beta }^{\prime }\right) ^{1/2}\le C\left( 1+\left| {\textbf{g}} \right| \right) \left( 1+{\textbf{1}}_{\alpha \in {\mathcal {I}} _{poly}}I\right) \left( 1+{\textbf{1}}_{\beta \in {\mathcal {I}}_{poly}}I_{*}\right) \end{aligned}$$

while the second bound implies that

$$\begin{aligned} \left( k_{\alpha \beta }^{2\gamma }\right) ^{2}\le & {} C\left( 1+\left| {\textbf{g}}\right| ^{2\chi -2}\right) \left( 1+\left| {\textbf{g}}\right| ^{2}\right) \left( 1+{\textbf{1}}_{\alpha \in {\mathcal {I}} _{poly}}I^{2}\right) \left( 1+{\textbf{1}}_{\beta \in {\mathcal {I}} _{poly}}I_{*}^{2}\right) M_{\alpha }M_{\beta *} \\\le & {} C\left( \left| {\textbf{g}}\right| ^{2}+\left| {\textbf{g}} \right| ^{2\chi -2}\right) M_{\alpha }^{1/2}M_{\beta *}^{1/2}\text {.} \end{aligned}$$

Noting that $m_{\alpha }\dfrac{\left| \varvec{\xi }\right| ^{2}}{ 2}+m_{\beta }\dfrac{\left| \varvec{\xi }_{*}\right| ^{2}}{2}= \dfrac{m_{\alpha }m_{\beta }}{2\left( m_{\alpha }+m_{\beta }\right) } \left| {\textbf{g}}\right| ^{2}+\dfrac{m_{\alpha }+m_{\beta }}{2} \left| {\textbf{G}}_{\alpha \beta }\right| ^{2}$, we obtain that $k_{\alpha \beta }^{2\gamma }\left( {\textbf{Z}}, {\textbf{Z}}_{*}\right) \in L^{2}\left( d{\textbf{Z}}d{\textbf{Z}}_{*}\right)$, since

$$\begin{aligned}{} & {} \int _{{\mathcal {Z}}_{\alpha }\times {\mathcal {Z}}_{\beta }}\left( k_{\alpha \beta }^{2\gamma }\right) ^{2}d{\textbf{Z}}d{\textbf{Z}} _{*}\le C\int _{{\mathbb {R}}^{3}}\exp \left( - \frac{m_{\alpha }+m_{\beta }}{4k_{B}T}\left| {\textbf{G}}_{\alpha \beta }\right| ^{2}\right) d{\textbf{G}}_{\alpha \beta } \\{} & {} \quad \times \int _{{\mathbb {R}}^{3}}\left( \left| {\textbf{g}}\right| ^{2}+\left| {\textbf{g}}\right| ^{2\chi -2}\right) \exp \left( -\frac{m_{\alpha }m_{\beta }}{4\left( m_{\alpha }+m_{\beta }\right) k_{B}T}\left| {\textbf{g}}\right| ^{2}\right) d {\textbf{g}}\left( \int _{0}^{\infty }e^{-I/\left( 2k_{B}T\right) }dI\right) ^{2}=C\text {.} \end{aligned}$$

Therefore, $K_{\alpha \beta }^{2\gamma }\left( {\textbf{Z}}\right) =\int _{ {\mathcal {Z}}_{\beta }}k_{\alpha \beta }^{2\gamma } h_{\beta *}d{\textbf{Z}}_{*}$ are Hilbert-Schmidt integral operators and as such compact on $L^{2}\left( d{\textbf{Z}}\right)$, see, e.g., Theorem 7.83 in [24], for $\left( \gamma ,\alpha ,\beta \right) \in {\mathcal {C}}$.

III. Compactness of $K_{\alpha \gamma }^{3\beta }\left( {\textbf{Z}} \right) =\int _{{\mathcal {Z}}_{\beta }}k_{\alpha \gamma }^{3\beta }\left( {\textbf{Z}},{\textbf{Z}}_{*}\right) h_{\beta *}d{\textbf{Z}}_{*}$ for $\left( \beta ,\alpha ,\gamma \right) \in {\mathcal {C}}$ follows directly by I., since $k_{\alpha \gamma }^{3\beta }\left( {\textbf{Z}},{\textbf{Z}} _{*}\right) =k_{\alpha \gamma }^{1\beta }\left( {\textbf{Z}}_{*}, {\textbf{Z}}\right)$ for all $\left( \beta ,\alpha ,\gamma \right) \in {\mathcal {C}}$.

Concluding, the operator

$$\begin{aligned} K_{c}= & {} 2(K_{c1},\ldots ,K_{cs}) \\= & {} 2\sum \limits _{\beta .\gamma =1}^{s}(K_{\beta \gamma }^{11}+K_{1\beta }^{2\gamma }-K_{1\gamma }^{3\beta },K_{\beta \gamma }^{12}+K_{2\beta }^{2\gamma }-K_{2\gamma }^{3\beta },\ldots ,K_{\beta \gamma }^{1s}+K_{s\beta }^{2\gamma }-K_{s\gamma }^{3\beta })\text {,} \end{aligned}$$

is a compact self-adjoint operator on ${{\mathfrak {h}}}$. $\square$

5 Bounds on the collision frequency

This section concerns the proof of Theorem 3. Note that throughout the proof, $C>0$ will denote a generic positive constant and $\varphi _{\alpha }\left( I\right) =I^{\delta ^{\left( \alpha \right) }/2-1}$ for $\alpha \in \mathcal { I}$.

Proof

Under assumption (16) there [6] exist positive numbers $\nu _{-}$ and ${\widetilde{\nu }}_{+}$, for which $0<\nu _{-}< {\widetilde{\nu }}_{+}$, such that for any $\alpha \in {\mathcal {I}}$, with $\nu _{\alpha }=\nu _{m\alpha }+\nu _{c\alpha }$,

$$\begin{aligned} \nu _{-}\left( 1+\left| \varvec{\xi }\right| +{\textbf{1}}_{\alpha \in {\mathcal {I}}_{poly}}\sqrt{I}\right) ^{1-\eta }\le \nu _{m\alpha }\le \nu _{\alpha }\le \widetilde{\nu }_{+}\left( 1+\left| \varvec{\xi } \right| +{\textbf{1}}_{\alpha \in {\mathcal {I}}_{poly}}\sqrt{I}\right) ^{1-\eta }+\nu _{c\alpha }\text {.} \end{aligned}$$

Now, remind expression (10) for $\nu _{c\alpha }$,

$$\begin{aligned} W_{\beta \gamma }^{\alpha }= & {} C_{\beta \gamma }^{\alpha }\frac{\varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) \varphi _{\gamma }\left( I^{\prime }\right) }{{\mathcal {E}} _{\beta \gamma }^{\delta ^{\left( \beta \right) }/2}\left( {\mathcal {E}} _{\beta \gamma }^{*}\right) ^{\delta ^{\left( \gamma \right) }/2}}\frac{1 }{\left| \varvec{\xi }_{*}-\varvec{\xi }^{\prime }\right| E_{\alpha }^{\eta /2}}\overline{\varvec{\delta }}{\textbf{1}} _{{\widehat{\Delta }}_{\beta \gamma }^{\alpha }{\mathcal {E}}>0}{\textbf{1}} _{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }} \\{} & {} \text { with }\overline{\varvec{\delta }}=\varvec{\delta }_{1}\left( \left| \varvec{\xi }_{*}-\varvec{\xi }^{\prime }\right| -\sqrt{\widehat{\Delta }_{\beta \gamma }^{\alpha }{\mathcal {E}}} \right) \varvec{\delta }_{3}\left( \varvec{\xi }-\frac{m_{\beta } \varvec{\xi }_{*}+m_{\gamma }\varvec{\xi }^{\prime }}{m_{\alpha } }\right) \text { and} \\{} & {} \text { }{\widehat{\Delta }}_{\beta \gamma }^{\alpha }{\mathcal {E}}=\frac{ 2m_{\alpha }}{m_{\beta }m_{\gamma }}\left( I-I_{*}{\textbf{1}}_{\beta \in {\mathcal {I}}_{poly}}-I^{\prime }{\textbf{1}}_{\gamma \in {\mathcal {I}} _{poly}}-\Delta _{\beta \gamma }^{\alpha }\varepsilon _{0}\right) \text {.} \end{aligned}$$

Noting that $\dfrac{\sqrt{{\widehat{\Delta }}_{\beta \gamma }^{\alpha } {\mathcal {E}}}}{E_{\alpha }^{\eta /2}}\le CI^{\left( 1-\eta \right) /2}\le C\left( 1+\left| \varvec{\xi }\right| +\sqrt{I}\right) ^{1-\eta }$, we obtain that

$$\begin{aligned}{} & {} \int _{{\mathcal {Z}}_{\beta }\times {\mathcal {Z}}_{\gamma }}\frac{ W_{\beta \gamma }^{\alpha }}{\varphi _{\alpha }\left( I\right) } \,d{\textbf{Z}}_{*}d{\textbf{Z}}^{\prime }=C_{\beta \gamma }^{\alpha }\int _{0}^{{\mathcal {E}}_{\beta \gamma }^{*}}\int _{0}^{ {\mathcal {E}}_{\beta \gamma }}\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}_{+}\times {\mathbb {S}}^{2}}\frac{\varphi _{\beta }\left( I_{*}\right) \varphi _{\gamma }\left( I^{\prime }\right) }{{\mathcal {E}}_{\beta \gamma }^{\delta ^{\left( \beta \right) }/2}\left( {\mathcal {E}}_{\beta \gamma }^{*}\right) ^{\delta ^{\left( \gamma \right) }/2}} \\{} & {} \qquad \times \frac{\left| \varvec{\xi }_{*}- \varvec{\xi }^{\prime }\right| }{E_{\alpha }^{\eta /2}}\overline{ \varvec{\delta }}{\textbf{1}}_{{\widehat{\Delta }}_{\beta \gamma }^{\alpha } {\mathcal {E}}>0}{\textbf{1}}_{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }}d\left| \varvec{\xi }_{*}-\varvec{\xi }^{\prime }\right| d\frac{m_{\beta }\varvec{\xi }_{*}+m_{\gamma } \varvec{\xi }^{\prime }}{m_{\alpha }}d\varvec{\sigma }dI_{*}dI^{\prime } \\{} & {} \quad =4\pi \int _{0}^{{\mathcal {E}}_{\beta \gamma }^{*}}\frac{\left( I^{\prime }\right) ^{\delta ^{\left( \gamma \right) }/2-1}}{\left( {\mathcal {E}}_{\beta \gamma }^{*}\right) ^{\delta ^{\left( \gamma \right) }/2}}\int _{0}^{ {\mathcal {E}}_{\beta \gamma }}C_{\beta \gamma }^{\alpha }\frac{I_{*}^{\delta ^{\left( \beta \right) }/2-1}}{{\mathcal {E}}_{\beta \gamma }^{\delta ^{\left( \beta \right) }/2}}\frac{\sqrt{{\widehat{\Delta }}_{\beta \gamma }^{\alpha }{\mathcal {E}}}}{E_{\alpha }^{\eta /2}}{\textbf{1}}_{{\widehat{\Delta }} _{\beta \gamma }^{\alpha }{\mathcal {E}}>0}{\textbf{1}}_{E_{\alpha }\ge \kappa _{\beta \gamma }^{\alpha }}dI_{*}dI^{\prime } \\{} & {} \quad \le C\left( 1+\left| \varvec{\xi }\right| +\sqrt{I}\right) ^{1-\eta }\text {.} \end{aligned}$$

Furthermore,

$$\begin{aligned} W_{\alpha \gamma }^{\beta }= & {} C_{\alpha \gamma }^{\beta }\frac{m_{\alpha }m_{\gamma }}{m_{\beta }}\frac{ \varphi _{\alpha }\left( I\right) \varphi _{\beta }\left( I_{*}\right) \varphi _{\gamma }\left( I^{\prime }\right) }{{\mathcal {E}}_{\alpha \gamma }^{\delta ^{\left( \alpha \right) }/2}\left( {\mathcal {E}}_{\alpha \gamma }^{*}\right) ^{\delta ^{\left( \gamma \right) }/2}E_{\beta }^{\eta /2}} \\{} & {} \times \varvec{\delta }_{1}\left( I_{*}-{\widetilde{E}}_{\alpha \gamma }^{\prime }\right) \varvec{\delta }_{3}\left( \varvec{\xi } _{*}-\frac{m_{\alpha }\varvec{\xi }+m_{\gamma }\varvec{\xi } ^{\prime }}{m_{\beta }}\right) {\textbf{1}}_{E_{\beta }\ge \kappa _{\alpha \gamma }^{\beta }}\text {.} \end{aligned}$$

Noting that

$$\begin{aligned}{} & {} \frac{\left( {\widetilde{E}}_{\alpha \gamma }^{\prime }\right) ^{\delta ^{\left( \beta \right) }/2-1}}{{\mathcal {E}}_{\alpha \gamma }^{\delta ^{\left( \alpha \right) }/2}\left( {\mathcal {E}}_{\alpha \gamma }^{*}\right) ^{\delta ^{\left( \gamma \right) }/2}\left( E_{\alpha \gamma }^{\prime }\right) ^{\eta /2}}\le C\left( {\widetilde{E}} _{\alpha \gamma }^{\prime }\right) ^{\left( 1-\eta \right) /2} \\{} & {} \quad \le C\left( 1+\left| \varvec{\xi }\right| +{\textbf{1}}_{\alpha \in {\mathcal {I}}_{poly}}\sqrt{I}\right) ^{1-\eta }\left( 1+{\textbf{1}}_{\gamma \in {\mathcal {I}}_{poly}}I^{\prime }\right) \left( 1+\left| \varvec{ \xi }^{\prime }\right| ^{2}\right) \text {,} \end{aligned}$$

we obtain that

$$\begin{aligned}{} & {} \int _{{\mathcal {Z}}_{\beta }\times {\mathcal {Z}}_{\gamma }}\frac{M_{\gamma }^{\prime }}{\varphi _{\alpha }\left( I\right) \varphi _{\gamma }\left( I^{\prime }\right) }W_{\alpha \gamma }^{\beta }\,d {\textbf{Z}}_{*}d{\textbf{Z}}^{\prime } \\{} & {} \quad =C_{\alpha \gamma }^{\beta }\frac{m_{\alpha }m_{\gamma }}{ m_{\beta }}\int \nolimits _{{\mathbb {R}}^{3}\times {\mathbb {R}}_{+}} \frac{M_{\gamma }^{\prime }\left( {\widetilde{E}}_{\alpha \gamma }^{\prime }\right) ^{\delta ^{\left( \beta \right) }/2-1}}{{\mathcal {E}}_{\alpha \gamma }^{\delta ^{\left( \alpha \right) }/2}\left( {\mathcal {E}}_{\alpha \gamma }^{*}\right) ^{\delta ^{\left( \gamma \right) }/2}\left( E_{\alpha \gamma }^{\prime }\right) ^{\eta /2}}{\textbf{1}}_{I^{\prime }\le {\mathcal {E}} _{\alpha \gamma }^{*}}\,d\varvec{\xi }^{\prime }dI^{\prime } \\{} & {} \quad \le C\left( 1+\left| \varvec{\xi }\right| +{\textbf{1}}_{\alpha \in {\mathcal {I}}_{poly}}\sqrt{I}\right) ^{1-\eta } \int _{{\mathbb {R}}^{3}}\left( 1+\left| \varvec{\xi } ^{\prime }\right| ^{2}\right) e^{-m_{\gamma }\left| \varvec{\xi } ^{\prime }\right| ^{2}/(2k_{B}T)}d\varvec{\xi }^{\prime } \\{} & {} \qquad \times \int _{0}^{\infty }\left( 1+I^{\prime }\right) \left( I^{\prime }\right) ^{\delta ^{\left( \gamma \right) }/2-1}e^{-I^{\prime }/(k_{B}T)}dI^{\prime }=C\left( 1+\left| \varvec{\xi } \right| +{\textbf{1}}_{\alpha \in {\mathcal {I}}_{poly}}\sqrt{I}\right) ^{1-\eta }\text {.} \end{aligned}$$

Hence, there is a positive constant $\nu _{+}>0$ satisfying (18) for all $\alpha \in {\mathcal {I}}$. $\square$

6 Conclusions

This work extends a model—based on the Boltzmann equation—for a multicomponent gas (consisting of polyatomic and possibly also monatomic components) where the particles interact through elastic binary collisions, to include also chemical reactions in the form of recombination/association and dissociation (and so implicitly also other chemical reactions) under a microreversibility assumption. Collision invariants and equilibrium distributions are derived. Studying deviations of an equilibrium distribution a linearized collision operator and several properties of it is derived. The linear operator is proven to be a self-adjoint nonnegative operator that can be decomposed into a positive multiplication operator (the collision frequency) minus a compact operator. For hard-sphere and hard potential with cut-off like models, the linear operator is proven to be Fredholm and its domain is also obtained. Those properties are fundamental during the process of obtaining hydrodynamic equations from the Boltzmann equation, but also for several other problems for the (also fully nonlinear) Boltzmann equation. This work is a base for further studies on the Boltzmann equation for chemically reacting gases.