# A consistent kinetic Fokker-Planck model for gas mixtures

Marlies Pirner

Abstract: Rarefied gas dynamics is usually described by the Boltzmann
equation. Unfortunately, the expense of evaluating this operator can be very
prohibitive. This made it worthwhile to look for approximations that convey
essentially an equivalent amount of physical information. One widely known
approximative collision operator is the Bathnagar-Gross-Krook (BGK) operator.
However, recently, the Foker-Planck approximation has become increasingly
popular. Nevertheless, the modeling of gas mixtures in the context of the
kinetic Fokker-Planck equation has so far only been addressed in a very few
papers. In this paper, we propose a general multi-species Fokker-Planck model.
We prove consistency of our model: conservation properties, positivity of all
temperatures, H-Theorem and the shape of equilibrium as Maxwell distributions
with the same mean velocity and temperature. Moreover, we derive the usual
macroscopic equations.

##  1 Introduction

Rarefied gas dynamics for gas mixtures consisting of $N$ species is usually
described by the Boltzmann equation with binary interactions as in [12],
chapter 6.2

$\displaystyle\partial_{t}f_{i}+v\cdot\nabla_{x}f_{i}=Q_{ii}(f_{i},f_{i})+\sum_{j=1,j\neq
i}^{N}Q_{ij}(f_{i},f_{j})$

Here, $f_{i}(x,v,t)>0,~{}i=1,...N$ is the distribution function of species $i$
where $x\in\mathbb{R}^{d}$ and $v\in\mathbb{R}^{d}$ are the phase space
variables in dimension $d\geq 1$ and $t\geq 0$ the time. The collision
operator on the right-hand side consists of a term $Q_{ii}$ describing the
interactions of particles of the species with itself and a sum of collision
operators $Q_{ij},i\neq j$ describing the interactions of particles of the
species $i$ with particles of species $j$. Unfortunately, the expense of
evaluating the Boltzmann operator can be prohibitive. Therefore one looks for
approximations that convey essentially an equivalent amount of physical
information. The most widely known of such an approximation is the Bhatnagar-
Gross-Krook (BGK) model [3]. The purpose of the BGK relaxation operator is to
provide an approximation of the Boltzmann collision operator that is more
computationally tractable, but still maintains important structural
properties. In particular, it has the same collision invariants as the
Boltzmann operator (which lead to conservation of the number of particles,
momentum, and energy) and it satisfies an H-Theorem. In the multi-species
case, these requirements are not as straight-forward to satisfy, but it can be
done. There are many BGK models for gas mixtures proposed in the literature
[23, 30, 18, 16, 34, 25, 20, 11, 1], many of which satisfy these basic
requirements and, in addition, are able to match some prescribed relaxation
rates and/or transport coefficients that come from more complicated physics
models or from experiment. Many of these approaches have been extended to
accommodate ellipsoid statistical (ES-BGK) models, polyatomic molecules,
chemical reactions, velocity dependent collision frequencies or quantum gases;
see for example [9, 10, 26, 36, 19, 27, 28, 4, 5, 29, 24]. Many of them have
the following structural similarity. Just like the Boltzmann equation for gas
mixtures contains a sum of collision terms on the right-hand side, they also
have a sum of BGK-type interaction terms in the relaxation operator.

$\displaystyle\begin{split}Q^{BGK}_{i}(f_{1},...f_{N})=\nu_{11}n_{1}(M_{1}-f_{1})+\sum_{i=1,i\neq
j}^{N}\nu_{ij}n_{i}(M_{ij}-f_{i})\end{split}$ (1)

Here, $M_{i}$ and $M_{ij}$ denote locally Maxwell distribution functions of
the form

$\displaystyle
M_{i}(x,v,t)=\frac{n_{i}}{\sqrt{2\pi\frac{T_{i}}{m_{i}}}^{3}}\exp({-\frac{|v-u_{i}|^{2}}{2\frac{T_{i}}{m_{i}}}}),~{}i=1,..,N$
(2) $\displaystyle
M_{ij}(x,v,t)=\frac{n_{ij}}{\sqrt{2\pi\frac{T_{ij}}{m_{j}}}^{d}}\exp({-\frac{|v-u_{ij}|^{2}}{2\frac{T_{ij}}{m_{i}}}}),~{}i\neq
j,~{}i,j=1,...N$ (3)

where $\nu_{ij}n_{i}$ denote the collision frequencies of species $i$ with
species $j$. The parameters $\nu_{ij}$ are assumed to be positive and only
depend on $x$ and $t$. The parameters $n_{i},n_{ij},u_{i},u_{ij},T_{i},T_{ij}$
are determined such that we have the following conservation properties:
conservation of mass, momentum and energy of the individual species in
interaction with the species itself:

  1. 1.

$\int Q_{ii}^{BGK}(f_{i},f_{i})dv=0\quad\text{for}\quad i=1,...,N,$

  2. 2.

$\int m_{i}vQ^{BGK}_{ii}(f_{i},f_{i})dv=0\quad\text{ for}\quad i=1,...,N$

  3. 3.

$\int m_{i}|v|^{2}Q^{BGK}_{ii}(f_{i},f_{i})dv=0\quad\text{ for}\quad
i=1,...,N.$

Conservation of total mass, momentum and energy

  1. 1.

$\int Q^{BGK}_{ij}(f_{i},f_{j})dv=0$ for $i,j=1,2,$

  2. 2.

$\int(m_{1}vQ^{BGK}_{12}(f_{1},f_{2})+m_{2}vQ^{BGK}_{21}(f_{2},f_{1}))dv=0,$

  3. 3.

$\int(m_{1}|v|^{2}Q^{BGK}_{12}(f_{1},f_{2})+m_{2}|v|^{2}Q^{BGK}_{21}(f_{2},f_{1}))dv=0.$

For this, we relate the distribution functions to macroscopic quantities by
mean-values of $f_{i}$

$\displaystyle\begin{split}\int f_{i}(v)\begin{pmatrix}1\\\ v\\\
m_{i}|v-u_{i}|^{2}\\\ \end{pmatrix}dv=:\begin{pmatrix}n_{i}\\\ n_{i}u_{i}\\\
3n_{i}T_{i}\end{pmatrix},\end{split}$ (4)

where $n_{i}$ is the number density, $u_{i}$ the mean velocity and $T_{i}$ the
temperature which is related to the pressure $p_{i}$ by $p_{i}=n_{i}T_{i}$.
Note that in this paper we shall write $T_{i}$ instead of $k_{B}T_{i}$, where
$k_{B}$ is Boltzmann’s constant. A general BGK model for gas mixtures which
contains most of the BGK models for gas mixtures in the literature is provided
in [25]. We will introduce this model briefly in the following. If we assume

$\displaystyle n_{12}=n_{1}\quad\text{and}\quad n_{21}=n_{2}$ (5)

in (3), we have conservation of the number of particles, see Theorem 2.1 in
[25]. If we further assume that $u_{12}$ is a linear combination of $u_{1}$
and $u_{2}$

$\displaystyle u_{12}=\delta u_{1}+(1-\delta)u_{2},\quad\delta\in\mathbb{R},$
(6)

then we have conservation of total momentum provided that

$\displaystyle
u_{21}=u_{2}-\frac{m_{1}}{m_{2}}\varepsilon(1-\delta)(u_{2}-u_{1}),$ (7)

see Theorem 2.2 in [25]. If we further assume that $T_{12}$ is of the
following form

$\displaystyle\begin{split}T_{12}&=\alpha
T_{1}+(1-\alpha)T_{2}+\gamma|u_{1}-u_{2}|^{2},\quad 0\leq\alpha\leq
1,\gamma\geq 0,\end{split}$ (8)

then we have conservation of total energy provided that

$\displaystyle\begin{split}T_{21}=\left[\frac{1}{d}\varepsilon
m_{1}(1-\delta)\left(\frac{m_{1}}{m_{2}}\varepsilon(\delta-1)+\delta+1\right)-\varepsilon\gamma\right]|u_{1}-u_{2}|^{2}\\\
+\varepsilon(1-\alpha)T_{1}+(1-\varepsilon(1-\alpha))T_{2},\end{split}$ (9)

see Theorem 2.3 in [25]. In order to ensure the positivity of all
temperatures, the parameters $\delta$ and $\gamma$ are restricted to

$\displaystyle
0\leq\gamma\leq\frac{m_{1}}{d}(1-\delta)\left[(1+\frac{m_{1}}{m_{2}}\varepsilon)\delta+1-\frac{m_{1}}{m_{2}}\varepsilon\right],$
(10)

and

$\displaystyle\frac{\frac{m_{1}}{m_{2}}\varepsilon-1}{1+\frac{m_{1}}{m_{2}}\varepsilon}\leq\delta\leq
1,$ (11)

see Theorem 2.5 in [25].

Moreover, it can be shown that this model satisfies an H-Theorem, see Theorem
2.4 in [25], meaning that we have the following inequality

$\sum_{i=1,..N}\int Q^{BGK}_{ij}(f_{i},f_{j})\log f_{i}dv\leq 0$

with equality if and only if $f_{i},f_{j}$ are Maxwell distributions with the
same mean velocity and temperature.

In the following, we will briefly motivate the meaning and possible choices of
the free parameters $\alpha,\delta,\gamma$, for more details see [32]. One
possibility is that one can choose the parameters such that one can generate
special cases in the literature [23, 30, 2, 16, 34, 13, 18, 11, 20]. For
instance if one chooses $\varepsilon=1$, $\delta=\frac{m_{1}}{m_{1}+m_{2}}$,
$\alpha=\frac{m_{1}^{2}+m_{2}^{2}}{(m_{1}+m_{2})^{2}}$ and
$\gamma=\frac{m_{1}m_{2}}{(m_{1}+m_{2})^{2}}\frac{m_{2}}{d}$, one obtains the
model by Hamel in [30]. In [19] also such relaxation parameters are used to
fix in the continuum limit Fick‘s law for diffusion velocities and Newton’s
law for viscous stress in the relevant set of Navier-Stokes equations.

Another possibility is to choose the parameters in a way such that the
macroscopic exchange terms of momentum and energy can be matched in a certain
way for example that they coincide with the ones for the Boltzmann equation.
For this, we first present the macroscopic equations with exchange terms of
the BGK model (1). If we multiply the BGK model for gas mixtures by
$1,m_{j}v,m_{j}\frac{|v|^{2}}{2}$ and integrate with respect to $v$, we obtain
the following macroscopic conservation laws

$\\\ \partial_{t}n_{1}+\nabla_{x}\cdot(n_{1}u_{1})=0,\\\
\partial_{t}n_{2}+\nabla_{x}\cdot(n_{2}u_{2})=0,\\\
\partial_{t}(m_{1}n_{1}u_{1})+\nabla_{x}\cdot\int m_{1}v\otimes
vf_{1}(v)dv+\nabla_{x}\cdot(m_{1}n_{1}u_{1}\otimes u_{1})=f_{m_{1,2}},\\\
\partial_{t}(m_{2}n_{2}u_{2})+\nabla_{x}\cdot\mathbb{P}_{2}+\nabla_{x}\cdot(m_{2}n_{2}u_{2}\otimes
u_{2})=f_{m_{2,1}},\\\
\partial_{t}\left(\frac{m_{1}}{2}n_{1}|u_{1}|^{2}+\frac{3}{2}n_{1}T_{1}\right)+\nabla_{x}\cdot\int
m_{1}|v|^{2}vf_{1}(v)dv=F_{E_{1,2}},\\\
\partial_{t}\left(\frac{m_{2}}{2}n_{2}|u_{2}|^{2}+\frac{3}{2}n_{2}T_{2}\right)+\nabla_{x}\cdot\int
m_{2}|v|^{2}vf_{2}(v)dv=F_{E_{2,1}},\\\ $

with exchange terms $f_{m_{i,j}}$ and $F_{E_{i,j}}$ given by

$\displaystyle f_{m_{1,2}}$
$\displaystyle=-f_{m_{2,1}}=m_{1}\nu_{12}n_{1}n_{2}(1-\delta)(u_{2}-u_{1}),$
$\displaystyle F_{m_{1,2}}$ $\displaystyle=-F_{m_{2,1}}$
$\displaystyle=\left[\nu_{12}\frac{1}{2}n_{1}n_{2}m_{1}(\delta-1)(u_{1}+u_{2}+\delta(u_{1}-u_{2}))+\frac{1}{2}\nu_{12}n_{1}n_{2}\gamma(u_{1}-u_{2})\right]\cdot(u_{1}-u_{2})$
$\displaystyle+\frac{3}{2}\varepsilon\nu_{21}n_{1}n_{2}(1-\alpha)(T_{2}-T_{1}).$

Here, one can observe a physical meaning of $\alpha$ and $\delta$. We see that
$\alpha$ and $\delta$ show up in the exchange terms of momentum and energy as
parameters in front of the relaxation of $u_{1}$ towards $u_{2}$ and $T_{1}$
towards $T_{2}$. So they determine, together with the collision frequencies,
the speed of relaxation of the mean velocities and the temperatures to a
common value.

Here now, as it is done in section 4.1 in [20] or section 4 in [32] and
compare the relaxation rates in the space-homogeneous case to the relaxation
rates for the space-homogeneous Boltzmann equation. In [20], they find values
for $\nu_{kj}$ such that either the relaxation rate for the mean velocities or
the relaxation for the temperatures coincides with the corresponding rate of
the Boltzmann equation. But using the free parameters $\alpha$, $\delta$ and
$\gamma$ one is able to match both of the relaxation rates at the same time.

Another approximate model is offered by a Fokker-Planck collision term

$\displaystyle
Q_{i}^{FP}(f_{1},...f_{N})=c_{ii}\text{div}(\nabla_{v}(\frac{T_{i}}{m_{i}}f_{i})+(v-u_{i})f_{i})+\sum_{j=1}^{N}c_{ij}\text{
div}(\nabla_{v}(\frac{T_{ij}}{m_{i}}f_{i})+(v-u_{ij})f_{i})$ (12)

The quantity $c_{ij}$ is a friction constant, see [37] for a motivation of the
one species case. Concerning the literature on multi-species Fokker Planck
models, there are less results than in the BGK case, but the interest in
multi-species Fokker-Planck models has been increased more and more recently.
Models for gas mixtures can be found in [15, 7, 17, 21] for different choices
of $u_{ij},T_{ij}$. We will discuss this later in section 2.3. The diffusion
limit of a kinetic Fokker-Planck system for charged particles towards the
Nernst-Planck equations was proved in [38]. Furthermore, in [15, 22], the
limit of vanishing electron-ion mass ratios for non-homogeneous kinetic
Fokker-Planck systems was investigated. In [7], the authors provide the first
existence analysis of a multi-species Fokker-Planck system of the shape above.
The works [17, 21] provide an extended Fokker-Planck model for hard-spheres
gas mixtures with to be able to also capture correct diffusion coefficients,
mixture viscosity and heat conductivity coefficients in the hydrodynamic
regime of the Navier-Stokes equations.

The aim of this paper is to present a general multi-species Fokker-Planck
model with collision terms of the shape (12) with free parameters
$\alpha,\delta,\gamma$ similar as this is done in [25] for the multi-species
BGK model. More concrete, we want to characterize for which choice of the
parameters $u_{ij},T_{ij}$ the conservation properties are satisfied, all
temperatures are positive and we have an H-Theorem. The models [15, 7, 17, 21]
can be shown to be a special case of this model presented here. This provides
the possibility to create different exchange terms of momentum and energy in
the macroscopic equations.

The outline of this paper is as follows: in section 2 we present the model for
two species and prove conservation properties of this model in section 2.1,
positivity of all temperatures in 2.2 and an H-Theorem in section 2.4. In
section 2.3 we derive macroscopic equations and discuss several special cases
in the literature [15, 7, 17, 21].

##  2 General multi-species Fokker-Planck model

In this section, we present a general multi-species Fokker-Planck model and
consider for which choice of $u_{ij},T_{ij}$, we have conservation properties,
an entropy inequality, the expected shape in equilibrium (Maxwell distribution
with common mean velocity and temperature) and positivity of all temperatures.
For simplicity, we present this model for two-species, but everything can be
extended to a general number of $N$ species, since we made the assumption of
only considering binary interactions. So in the rest of the paper, we consider
the following system of Fokker-Planck equations

$\displaystyle\partial_{t}f_{1}+v\cdot\nabla_{x}f_{1}=c_{11}n_{1}\text{
div}(\nabla_{v}(\frac{T_{1}}{m_{1}}f_{1})+(v-u_{1})f_{1})+c_{12}n_{2}\text{div}(\nabla_{v}(\frac{T_{12}}{m_{1}}f_{1})+(v-u_{12})f_{1})$
$\displaystyle\partial_{t}f_{1}+v\cdot\nabla_{x}f_{1}=c_{22}n_{2}\text{
div}(\nabla_{v}(\frac{T_{2}}{m_{2}}f_{2})+(v-u_{2})f_{2})+c_{21}n_{1}\text{div}(\nabla_{v}(\frac{T_{21}}{m_{2}}f_{2})+(v-u_{21})f_{2})$

To be flexible in choosing the relation ship between the constants
$c_{12},c_{21}$, we now assume the relationship

$\displaystyle c_{12}=\varepsilon~{}c_{21},\varepsilon\leq 1\text{ and
}\varepsilon\frac{m_{1}}{m_{2}}\leq 1$ (13)

Note, that the assumption on $\varepsilon$ covers the two common cases in the
literature for $\varepsilon$ which are $\varepsilon=\frac{m_{2}}{m_{1}}$ and
$\varepsilon=1$ if the notation of $1$ and $2$ is chosen in a suitable way.

###  2.1 Conservation properties

This section shows how the macroscopic quantities $u_{ij},T_{ij}$ in the
interspecies Maxwell distributions have to be chosen in order to ensure the
macroscopic conservation properties. We note that the mass is automatically
conserved.

######  Theorem 1 (Conservation of total momentum).

Assume the condition (13) for the collision frequencies and that $u_{12}$ is a
linear combination of $u_{1}$ and $u_{2}$

$\displaystyle u_{12}=\delta u_{1}+(1-\delta)u_{2},\quad\delta\in\mathbb{R}.$
(14)

Then we have conservation of total momentum

$\int m_{1}v[Q^{FP}_{11}(f_{1},f_{1})+Q^{FP}_{12}(f_{1},f_{2})]dv+\int
m_{2}v[Q^{FP}_{22}(f_{2},f_{2})+Q^{FP}_{21}(f_{2},f_{1})]dv=0,$

provided that

$\displaystyle
u_{21}=u_{2}-(1-\delta)\varepsilon\frac{m_{1}}{m_{2}}(u_{2}-u_{1}).$ (15)

###### Proof.

The flux of momentum of species $1$ is given by

$\displaystyle\begin{split}f_{m_{1,2}}:&=m_{1}c_{11}n_{1}\int v\text{
div}(\nabla_{v}(\frac{T_{1}}{m_{1}}f_{1})+(v-u_{1})f_{1})dv\\\
&+c_{12}m_{1}n_{2}\int v\text{
div}(\nabla_{v}(\frac{T_{12}}{m_{1}}f_{1})+(v-u_{12})f_{1})dv\\\
&=-m_{1}c_{12}n_{2}\int(\nabla_{v}(\frac{T_{12}}{m_{1}}f_{1})+(v-u_{12})f_{1})dv\\\
&=m_{1}c_{12}n_{1}n_{2}(u_{12}-u_{1})=m_{1}c_{12}n_{1}n_{2}(1-\delta)(u_{2}-u_{1}).\end{split}$
(16)

The flux of momentum of species $2$ is given by

$\displaystyle f_{m_{2,1}}=m_{2}c_{21}n_{2}n_{1}(u_{21}-u_{2}).$ (17)

In order to get conservation of momentum we therefore need

$\displaystyle
m_{1}c_{12}n_{1}n_{2}(1-\delta)(u_{2}-u_{1})+m_{2}c_{21}n_{1}n_{2}(u_{21}-u_{2})=0,$

which holds provided $u_{21}$ satisfies (15) ∎

######  Remark 1.

If we write $\tilde{\delta}=1-\frac{m_{1}}{m_{2}}\varepsilon(1-\delta)$ we
obtain a similar structure for $u_{21}$ as for $u_{12}$

$u_{21}=\tilde{\delta}u_{2}+(1-\tilde{\delta})u_{1}.$

######  Theorem 2 (Conservation of total energy).

Assume conditions (14) and (15) and assume that $T_{12}$ is of the following
form

$\displaystyle\begin{split}T_{12}=\alpha
T_{1}+(1-\alpha)T_{2}+\gamma|u_{1}-u_{2}|^{2},\quad 0\leq\alpha\leq
1,\gamma\geq 0.\end{split}$ (18)

Then we have conservation of total energy

$\int\frac{m_{1}}{2}|v|^{2}(Q^{FP}_{11}(f_{1},f_{1})+Q^{FP}_{12}(f_{1},f_{2}))dv+\int\frac{m_{2}}{2}|v|^{2}(Q^{FP}_{22}(f_{2},f_{2})+Q^{FP}_{21}(f_{2},f_{1}))dv=0,$

provided that

$\displaystyle\begin{split}T_{21}=\left[\frac{1}{d}\varepsilon
m_{1}(1-\delta)-\varepsilon\gamma\right]|u_{1}-u_{2}|^{2}+\varepsilon(1-\alpha)T_{1}+(1-\varepsilon(1-\alpha))T_{2}.\end{split}$
(19)

###### Proof.

Using the energy flux of species $1$

$\displaystyle F_{E_{1,2}}$
$\displaystyle:=c_{11}n_{1}\int\frac{m_{1}}{2}|v|^{2}\text{
div}(\nabla_{v}(\frac{T_{1}}{m_{1}}f_{1})+(v-u_{1}))f_{1}dv$
$\displaystyle+c_{12}n_{2}\int\frac{m_{1}}{2}|v|^{2}\text{
div}(\nabla_{v}(\frac{T_{12}}{m_{1}}f_{1})+(v-u_{12}))f_{1}dv$
$\displaystyle=-c_{12}n_{2}m_{1}\int
v\cdot(\nabla_{v}(\frac{T_{21}}{m_{1}}f_{1})+(v-u_{12}))f_{1}dv$
$\displaystyle=c_{12}n_{2}m_{1}d\int\frac{T_{12}}{m_{1}}f_{1}dv-
c_{12}n_{2}m_{1}\int v\cdot(v-u_{12})f_{1}dv$
$\displaystyle=c_{12}n_{2}dn_{1}T_{12}-c_{12}n_{2}m_{1}(dn_{1}\frac{T_{1}}{m_{1}}+n_{1}|u_{1}|^{2}-n_{1}u_{1}\cdot
u_{12})$
$\displaystyle=c_{12}n_{2}dn_{1}T_{12}-c_{12}n_{2}n_{1}(dT_{1}+m_{1}u_{1}\cdot(u_{1}-u_{12}))$
$\displaystyle=c_{12}n_{1}n_{2}d(T_{12}-T_{1})-c_{12}m_{1}n_{1}n_{2}u_{1}\cdot(u_{1}-u_{12})$
$\displaystyle=c_{12}n_{1}n_{2}(1-\alpha)(T_{2}-T_{1})-c_{12}m_{1}n_{2}n_{1}((1-\delta)u_{1}\cdot(u_{1}-u_{2})+\gamma|u_{1}-u_{2}|^{2})$

where we used (14) and (18). Analogously the energy flux of species $2$
towards $1$ is

$\displaystyle
F_{E_{2,1}}=c_{21}m_{2}n_{1}n_{2}(u_{2}\cdot(u_{21}-u_{2}))+d~{}c_{21}n_{1}n_{2}(T_{21}-T_{2})$
$\displaystyle=c_{21}m_{2}n_{1}n_{2}(1-\delta)\frac{m_{1}}{m_{2}}\varepsilon(u_{2}\cdot(u_{1}-u_{2}))+d~{}c_{21}n_{1}n_{2}(T_{21}-T_{2})$

Here, we substituted $u_{21}$ with (15). Adding these two terms, we see that
the total energy is conserved provided that $T_{21}$ is given by (19). ∎

######  Remark 2.

We have $0\leq 1-\varepsilon(1-\alpha)\leq 1$ and
$0\leq\varepsilon(1-\alpha)\leq 1$, so that in (19) the two terms with the
temperatures are also a convex combination of $T_{1}$ and $T_{2}$.

###  2.2 Positivity of the temperatures

######  Theorem 3.

Assume that $f_{1}(x,v,t),f_{2}(x,v,t)>0$. Then all temperatures $T_{1}$,
$T_{2}$, $T_{12}$ given by (18) and $T_{21}$ given by (19) are positive
provided that

$\displaystyle 0\leq\gamma\leq\frac{m_{1}}{d}(1-\delta)$ (20)

###### Proof.

$T_{1}$ and $T_{2}$ are positive as integrals of positive functions. $T_{12}$
is positive because by construction it is a convex combination of $T_{1}$ and
$T_{2}$. For $T_{21}$ we consider the coefficients in front of
$|u_{1}-u_{2}|^{2}$, $T_{1}$ and $T_{2}$. The term in front of $T_{1}$ is
positive by definition. The positivity of the term in front of $T_{2}$ is
equivalent to the condition $\alpha\geq 1-\frac{1}{\varepsilon}$, which is
satisfied since $\varepsilon\leq 1$, the positivity of the term in front of
$|u_{1}-u_{2}|^{2}$ is equivalent to the condition (20). ∎

######  Remark 3.

According to the definition of $\gamma$, $\gamma$ is a non-negative number, so
the right-hand side of the inequality in (20) must be non-negative. This
condition is equivalent to

$\displaystyle\delta\leq 1.$ (21)

###  2.3 Macroscopic equations and exchange terms of momentum and energy

In this section, we deal with macroscopic equations, exchange terms of
momentum and energy, and special cases in the literature. With a specific
choice of the parameters we can generate special cases in the literature [15,
7, 17]. For instance, in [15] the mean mixture velocities and temperatures are
chosen to be

$\displaystyle u_{12}=u_{21}=\frac{u_{1}+u_{2}}{2};\quad
T_{12}=T_{21}=\frac{m_{2}T_{1}+m_{1}T_{2}}{m_{1}+m_{2}}+\frac{m_{1}m_{2}}{m_{1}+m_{2}}\frac{1}{2d}|u_{1}-u_{2}|^{2}$

so we can generate this model by choosing

$\displaystyle\alpha=\frac{m_{2}}{m_{1}+m_{2}},\quad\delta=\frac{1}{2},\quad\gamma=\frac{1}{2d}\frac{m_{1}m_{2}}{m_{1}+m_{2}}$

With the choice of

$\displaystyle\alpha=\frac{c_{12}n_{2}}{c_{12}n_{2}+c_{21}n_{1}},\delta=\frac{c_{12}m_{1}n_{1}}{c_{12}m_{1}n_{1}+c_{12}m_{2}n_{2}},\gamma=\frac{c_{12}m_{1}n_{1}c_{21}m_{2}n_{2}}{d(c_{12}n_{1}+c_{21}n_{2})(c_{21}n_{1}n_{1}+c_{12}m_{2}n_{2})}$

we can generate $u_{12},u_{21},T_{12},T_{21}$ as in [7] given by

$\displaystyle u_{12}$
$\displaystyle=u_{21}=\frac{c_{21}m_{1}n_{1}u_{1}+c_{12}m_{2}n_{2}u_{2}}{c_{12}m_{2}n_{2}+c_{21}m_{1}n_{1}},$
$\displaystyle T_{12}$
$\displaystyle=T_{21}=\frac{c_{21}n_{1}T_{1}+c_{12}n_{2}T_{2}}{c_{12}n_{2}+c_{21}n_{1}}+\frac{c_{12}m_{1}n_{1}c_{21}m_{2}n_{2}}{d(c_{12}n_{1}+c_{21}n_{2})(c_{21}n_{1}n_{1}+c_{12}m_{2}n_{2})}|u_{1}-u_{2}|^{2}$

Another possibility for example is to choose as mixture velocity the velocity
of the other species as it is done for example in [17] with $\delta=0$ to have

$\displaystyle u_{12}=u_{2},u_{21}=u_{1}.$

Another way to see the influence of the parameters is in the macroscopic
exchange terms of momentum and energy. For this, we first present the
macroscopic equations with exchange terms of the Fokker-Planck model (12). If
we multiply the Fokker-Planck model for gas mixtures by
$1,m_{j}v,m_{j}\frac{|v|^{2}}{2}$ and integrate with respect to $v$, we obtain
the following macroscopic conservation laws

$\\\ \partial_{t}n_{1}+\nabla_{x}\cdot(n_{1}u_{1})=0,\\\
\partial_{t}n_{2}+\nabla_{x}\cdot(n_{2}u_{2})=0,\\\
\partial_{t}(m_{1}n_{1}u_{1})+\nabla_{x}\cdot\int m_{1}v\otimes
vf_{1}(v)dv=f_{m_{1,2}},\\\ \partial_{t}(m_{2}n_{2}u_{2})+\nabla_{x}\cdot\int
m_{2}v\otimes vf_{2}(v)dv=f_{m_{2,1}},\\\
\partial_{t}\left(\frac{m_{1}}{2}n_{1}|u_{1}|^{2}+\frac{3}{2}n_{1}T_{1}\right)+\nabla_{x}\cdot\int
m_{1}|v|^{2}vf_{1}(v)dv=F_{E_{1,2}},\\\
\partial_{t}\left(\frac{m_{2}}{2}n_{2}|u_{2}|^{2}+\frac{3}{2}n_{2}T_{2}\right)+\nabla_{x}\cdot\int
m_{2}|v|^{2}vf_{2}(v)dv=F_{E_{2,1}},\\\ $

with exchange terms $f_{m_{i,j}}$ and $F_{E_{i,j}}$ given by

$\displaystyle\begin{split}f_{m_{1,2}}&=-f_{m_{2,1}}=m_{1}c_{12}n_{1}n_{2}(1-\delta)(u_{2}-u_{1}),\\\
F_{E_{1,2}}&=-F_{E_{2,1}}\\\
&=c_{12}n_{1}n_{2}m_{1}(1-\delta)u_{1}\cdot(u_{2}-u_{1})+\gamma\frac{d}{m_{1}}|u_{1}-u_{2}|^{2}\\\
&+dc_{12}n_{1}n_{2}(1-\alpha)(T_{2}-T_{1}).\end{split}$ (22)

###  2.4 H-theorem for mixtures

In this section we will prove an H-Theorem for the model (12). For this, we
make the following additional assumptions on the free parameters. We make the
stronger assumptions of (18), (20), (21).

$\displaystyle\frac{\varepsilon}{1+\varepsilon}\leq\alpha\leq
1,\quad\frac{\varepsilon}{1+\varepsilon}\leq\delta\leq
1,\quad(1-\delta)^{2}\frac{m_{1}}{d}\leq\gamma\leq(1-\delta)\frac{m_{1}}{d}\frac{\varepsilon}{1+\varepsilon}$
(23)

Moreover, in order to simplify the notation we define the following quantities

$\displaystyle\gamma_{1}:=(1-\delta)^{2}\frac{m_{1}}{d};\quad\gamma_{2}:=(1-\delta)^{2}\frac{m_{2}}{d}\varepsilon^{2}(\frac{m_{1}}{m_{2}})^{2},\quad\tilde{\gamma}:=\frac{m_{1}}{d}\varepsilon(1-\delta)-\varepsilon\gamma$
(24)

and the temperatures

$\displaystyle\bar{T}_{12}=\alpha
T_{1}+(1-\alpha)T_{2};\quad\bar{T}_{21}=\varepsilon(1-\alpha)T_{1}+(1-\varepsilon(1-\alpha))T_{2}$
(25)

We start with some lemmas which we will need later for the proof of the
H-Theorem.

######  Lemma 4.

Let $M_{12},M_{21}$ the two Maxwell distributions given by (3). Then we have

$\displaystyle\frac{T_{12}}{m_{1}}c_{12}n_{2}\int\frac{M_{12}^{2}}{f_{1}}\left(\frac{\nabla_{v}f_{1}M_{12}-\nabla_{v}M_{12}f_{1}}{M_{12}^{2}}\right)^{2}dv$
$\displaystyle+\frac{T_{21}}{m_{2}}c_{21}n_{1}\int\frac{M_{21}^{2}}{f_{2}}\left(\frac{\nabla_{v}f_{2}M_{21}-\nabla_{v}M_{21}f_{2}}{M_{21}^{2}}\right)^{2}dv$
$\displaystyle=\frac{T_{12}}{m_{1}}c_{12}n_{2}\int\frac{|\nabla_{v}f_{1}|^{2}}{f_{1}}dv+\frac{T_{21}}{m_{2}}c_{21}n_{1}\int\frac{|\nabla_{v}f_{2}|^{2}}{f_{2}}dv$
$\displaystyle+c_{12}n_{2}n_{1}d\frac{T_{1}+\frac{m_{1}}{d}(1-\delta)^{2}|u_{1}-u_{2}|^{2}}{T_{12}}+c_{21}n_{2}n_{1}d\frac{T_{2}+\frac{m_{2}}{d}\varepsilon^{2}(\frac{m_{1}}{m_{2}})^{2}(1-\delta)^{2}|u_{1}-u_{2}|^{2}}{T_{21}}$
$\displaystyle-2(1+\varepsilon)c_{21}n_{1}n_{2}d$

###### Proof.

We can compute

$\displaystyle\frac{T_{12}}{m_{1}}c_{12}n_{2}\int\frac{M_{12}^{2}}{f_{1}}\left(\frac{\nabla_{v}f_{1}M_{12}-\nabla_{v}M_{12}f_{1}}{M_{12}^{2}}\right)^{2}dv+\frac{T_{21}}{m_{2}}c_{21}n_{1}\int\frac{M_{21}^{2}}{f_{2}}\left(\frac{\nabla_{v}f_{2}M_{21}-\nabla_{v}M_{21}f_{2}}{M_{21}^{2}}\right)^{2}dv$
$\displaystyle=\frac{T_{12}}{m_{1}}c_{12}n_{2}\int\frac{|\nabla_{v}f_{1}|^{2}}{f_{1}}dv+\frac{T_{21}}{m_{2}}c_{21}n_{1}\int\frac{|\nabla_{v}f_{2}|^{2}}{f_{2}}dv$
$\displaystyle+c_{12}n_{2}\int\frac{|v-u_{12}|^{2}}{T_{12}/m_{1}}f_{1}dv+c_{21}n_{1}\int\frac{|v-u_{21}|^{2}}{T_{21}/m_{1}}f_{2}dv$
$\displaystyle+2c_{12}n_{2}\int\nabla_{v}f_{1}\cdot(v-u_{12})dv+2c_{21}n_{1}\int\nabla_{v}f_{2}\cdot(v-u_{21})dv$
$\displaystyle=\frac{T_{12}}{m_{1}}c_{12}n_{2}\int\frac{|\nabla_{v}f_{1}|^{2}}{f_{1}}dv+\frac{T_{21}}{m_{2}}c_{21}n_{1}\int\frac{|\nabla_{v}f_{2}|^{2}}{f_{2}}dv$
$\displaystyle+c_{12}n_{2}n_{1}d\frac{T_{1}+\frac{m_{1}}{d}|u_{1}-u_{12}|^{2}}{T_{12}}+c_{21}n_{2}n_{1}d\frac{T_{2}+\frac{m_{2}}{d}\varepsilon^{2}(\frac{m_{1}}{m_{2}})^{2}(1-\delta)^{2}|u_{2}-u_{21}|^{2}}{T_{21}}$
$\displaystyle-2(1+\varepsilon)c_{21}n_{1}n_{2}d$
$\displaystyle=\frac{T_{12}}{m_{1}}c_{12}n_{2}\int\frac{|\nabla_{v}f_{1}|^{2}}{f_{1}}dv+\frac{T_{21}}{m_{2}}c_{21}n_{1}\int\frac{|\nabla_{v}f_{2}|^{2}}{f_{2}}dv$
$\displaystyle+c_{12}n_{2}n_{1}d\frac{T_{1}+\frac{m_{1}}{d}(1-\delta)^{2}|u_{1}-u_{2}|^{2}}{T_{12}}++c_{21}n_{2}n_{1}d\frac{T_{2}+\frac{m_{2}}{d}\varepsilon^{2}(\frac{m_{1}}{m_{2}})^{2}(1-\delta)^{2}|u_{1}-u_{2}|^{2}}{T_{21}}$
$\displaystyle-2(1+\varepsilon)c_{21}n_{1}n_{2}d$

Here, we used $\nabla_{v}M_{12}=-\frac{v-u_{12}}{T_{12}/m_{1}}M_{12}$ and the
relationship (14) and (15) for $u_{12},u_{21}$. ∎

######  Lemma 5.

We assume the estimate for $\alpha$ in (23). Then we have

$\displaystyle\varepsilon
T_{1}\bar{T}_{21}+\bar{T}_{12}T_{2}\leq(1+\varepsilon)\bar{T}_{12}\bar{T}_{21}$
(26)

###### Proof.

If we insert the expressions for $\bar{T}_{12},\bar{T}_{21}$ given by (25) we
get that (26) is equivalent to

$\displaystyle(1-\alpha)\varepsilon(\alpha-(1-\alpha)\varepsilon)(T_{1}-T_{2})^{2}\geq
0$

This is true if $1\geq\alpha\geq\frac{\varepsilon}{1+\varepsilon}$ which we
assumed in (23). ∎

######  Lemma 6.

We assume (23). Then we have

$\displaystyle\varepsilon\gamma_{1}\tilde{\gamma}+\gamma\gamma_{2}\leq(1+\varepsilon)\gamma\tilde{\gamma}$

###### Proof.

If we insert the expressions for $\gamma_{1},\gamma_{2},\tilde{\gamma}$ given
by (24), we obtain

$\displaystyle-((1+\varepsilon)\gamma(-\varepsilon\gamma+(1-\delta)\varepsilon\frac{m_{1}}{d}))+(1-\delta)^{2}\varepsilon\frac{m_{1}}{d}(-\varepsilon\gamma+(1-\delta)\varepsilon\frac{m_{1}}{d})$
$\displaystyle+(1-\delta)^{2}\gamma\varepsilon^{2}\frac{m_{1}}{d}\frac{m_{1}}{m_{2}}\leq
0$

This inequality is true if we can prove separately

$\displaystyle\begin{split}\varepsilon\gamma(-\varepsilon\gamma+(1-\delta)\varepsilon\frac{m_{1}}{d})\geq(1-\delta)^{2}\varepsilon\frac{m_{1}}{d}(-\varepsilon\gamma+(1-\delta)\varepsilon\frac{m_{1}}{d})\\\
\varepsilon\gamma^{2}-(1-\delta)\varepsilon\frac{m_{1}}{d}\gamma+(1-\delta)^{2}\varepsilon^{2}\gamma\frac{m_{1}}{d}\frac{m_{1}}{m_{2}}\leq
0\end{split}$ (27)

We start with the first inequality. The factor
$-\varepsilon\gamma+(1-\delta)\varepsilon\frac{m_{1}}{d}$ is non-negative,
since this is the condition ensuring positivity of the temperatures (21).
Therefore, we get that $\gamma$ has to satisfy

$\displaystyle\gamma\geq(1-\delta)^{2}\frac{m_{1}}{d}$

as assumed in (23). This is possible and no restriction to the upper bound
ensuring the positivity (21), since we assumed $0\leq\delta\leq 1$.
Now, for the second inequality in (27), we divide by $\varepsilon\gamma$ to
get

$\displaystyle\gamma-(1-\delta)\frac{m_{1}}{d}+(1-\delta)^{2}\varepsilon\frac{m_{1}}{d}\frac{m_{1}}{m_{2}}\leq
0$

which is satisfied if

$\displaystyle\gamma\geq(1-\delta)\frac{m_{1}}{d}((1-\delta)\varepsilon\frac{m_{1}}{m_{2}}-1)$

This is satisfied due to the estimate on $\gamma$ in (23) and assumption (13).
∎

######  Lemma 7.

We assume (23). Then, we have

$\displaystyle\varepsilon
T_{1}\tilde{\gamma}|u_{1}-u_{2}|^{2}+\varepsilon\gamma_{1}|u_{1}-u_{2}|^{2}\bar{T}_{21}+\bar{T}_{12}\gamma_{2}|u_{1}-u_{2}|^{2}+\gamma|u_{1}-u_{2}|^{2}T_{2}$
$\displaystyle\leq(1+\varepsilon)\bar{T}_{12}\tilde{\gamma}|u_{1}-u_{2}|^{2}+(1+\varepsilon)\gamma|u_{1}-u_{2}|^{2}\bar{T}_{21}$

###### Proof.

We insert the expressions for $\bar{T}_{12},\bar{T}_{21}$ given by (25) and
get

$\displaystyle\varepsilon
T_{1}\tilde{\gamma}|u_{1}-u_{2}|^{2}+\varepsilon\gamma_{1}|u_{1}-u_{2}|^{2}(\varepsilon(1-\alpha)T_{1}+(1-\varepsilon(1-\alpha))T_{2})$
$\displaystyle+(\alpha
T_{1}+(1-\alpha)T_{2})\gamma_{2}|u_{1}-u_{2}|^{2}+\gamma|u_{1}-u_{2}|^{2}T_{2}$
$\displaystyle\leq(1+\varepsilon)(\alpha
T_{1}+(1-\alpha)T_{2})\tilde{\gamma}|u_{1}-u_{2}|^{2}$
$\displaystyle+(1+\varepsilon)\gamma|u_{1}-u_{2}|^{2}(\varepsilon(1-\alpha)T_{1}+(1-\varepsilon(1-\alpha))T_{2})$

We compare the coefficients in front of $T_{1}$ and $T_{2}$ and obtain the
inequalities

$\displaystyle\varepsilon\tilde{\gamma}+\varepsilon\gamma_{1}\varepsilon(1-\alpha)+\alpha\gamma_{2}$
$\displaystyle\leq(1+\varepsilon)\alpha\tilde{\gamma}+(1+\varepsilon)\gamma\varepsilon(1-\alpha)$
$\displaystyle\varepsilon\gamma_{1}(1-\varepsilon(1-\alpha))+(1-\alpha)\gamma_{2}+\gamma$
$\displaystyle\leq(1+\varepsilon)(1-\alpha)\tilde{\gamma}+(1+\varepsilon)(1-\varepsilon(1-\alpha))\gamma$

We start with the first inequality. According to the definition of
$\gamma_{1},\gamma_{2}$ given by (24) and the lower bound on $\gamma$ given by
(21) and assumption (13), we have

$\displaystyle\gamma_{1}\leq\gamma\text{ and }\gamma_{2}\leq\gamma$

Additionally, we observe that

$\displaystyle\tilde{\gamma}=\varepsilon(1-\delta)\frac{m_{1}}{d}-\varepsilon\gamma\geq\gamma$

since we assumed the stricter upper bound on $\gamma$ in (23). The stricter
upper bound on $\gamma$ is not a contradiction to the lower bound since we
assumed $\delta\geq\frac{\varepsilon}{1+\varepsilon}in\eqref{asspar}.$ All in
all, this leads to

$\displaystyle\varepsilon\gamma_{1}\varepsilon(1-\alpha)+\alpha\gamma_{2}\leq(\varepsilon^{2}(1-\alpha)+\alpha)\gamma=(\varepsilon^{2}(1-\alpha)+\varepsilon(1-\alpha))\gamma+(\alpha-\varepsilon(1-\alpha))\gamma$
$\displaystyle\leq(\varepsilon^{2}(1-\alpha)+\varepsilon(1-\alpha))\gamma+(\alpha-\varepsilon(1-\alpha))\tilde{\gamma}$

which corresponds to the first inequality. The last inequality is possible
since we assumed $\alpha\geq\frac{\varepsilon}{1+\varepsilon}$ in (23) and
therefore the coefficient $\alpha-\varepsilon(1-\alpha)$ is non-negative. In a
similar way, one can prove the second inequality. ∎

######  Theorem 8 (H-theorem for mixture).

Assume $f_{1},f_{2}>0$. Assume the relationship between the collision
frequencies (13) , the conditions for the interspecies Maxwellians (5) , (14),
(15), (18) and (19) with $\alpha,\delta\neq 1$, the positivity of the
temperatures (20) and the assumptions on the parameters (23), then

$\displaystyle\int(\ln f_{1})~{}Q^{FP}_{11}(f_{1},f_{1})$ $\displaystyle+(\ln
f_{1})~{}Q^{FP}_{12}(f_{1},f_{2})dv$ $\displaystyle+\int(\ln
f_{2})~{}Q^{FP}_{22}(f_{2},f_{2})+(\ln f_{2})~{}Q^{FP}_{21}(f_{2},f_{1})dv\leq
0,$

with equality if and only if $f_{1}$ and $f_{2}$ are Maxwell distributions
with equal velocity and temperature.

###### Proof.

The fact that $\int\ln f_{k}Q(f_{k},f_{k})dv\leq 0$, $k=1,2$ is shown in
proofs of the H-theorem of the single Fokker-Planck-model, for example in
[33]. In both cases we have equality if and only if $f_{1}=M_{1}$ and
$f_{2}=M_{2}$.

Let us define

$\displaystyle I:$ $\displaystyle=\int Q^{FP}_{12}(f_{1},f_{2})\ln
f_{1}dv+\int Q^{FP}_{21}(f_{2},f_{1})\ln f_{2}dv$ $\displaystyle=\int
c_{12}n_{1}\text{
div}(\nabla_{v}(\frac{T_{12}}{m_{1}}f_{1})+(v-u_{12})f_{1})\ln f_{1}dv$
$\displaystyle+\int
c_{21}n_{2}\text{div}(\nabla_{v}(\frac{T_{21}}{m_{2}}f_{2})+(v-u_{21})f_{2})\ln
f_{2}dv$

Integration by parts leads to

$\displaystyle I$ $\displaystyle=-\int
c_{12}n_{1}(\nabla_{v}(\frac{T_{12}}{m_{1}}f_{1})+(v-u_{12})f_{1})\frac{\nabla_{v}f_{1}}{f_{1}}dv$
$\displaystyle-\int
c_{21}n_{2}(\nabla_{v}(\frac{T_{21}}{m_{2}}f_{2})+(v-u_{21})f_{2})\frac{\nabla_{v}f_{2}}{f_{2}}dv$
$\displaystyle=\int
c_{12}n_{1}\frac{T_{12}}{m_{1}}f_{1}|\frac{\nabla_{v}f_{1}}{f_{1}}|^{2}-\int
c_{21}n_{2}\frac{T_{21}}{m_{2}}f_{2}|\frac{\nabla_{v}f_{2}}{f_{2}}|^{2}dv$
$\displaystyle-\int c_{12}n_{2}(v-u_{12})\cdot\nabla_{v}f_{1}dv-\int
c_{21}n_{1}(v-u_{21})\cdot\nabla_{v}f_{2}dv$ $\displaystyle=-\int
c_{12}n_{2}\frac{T_{12}}{m_{1}}f_{1}|\frac{\nabla_{v}f_{1}}{f_{1}}|^{2}dv-\int
c_{21}n_{2}\frac{T_{21}}{m_{2}}f_{2}|\frac{\nabla_{v}f_{2}}{f_{2}}|^{2}dv+c_{12}n_{2}n_{1}d+c_{21}n_{1}n_{2}d$

By using the relationship (13), we obtain

$\displaystyle I:=-\int
c_{12}n_{2}\frac{T_{12}}{m_{1}}f_{1}|\frac{\nabla_{v}f_{1}}{f_{1}}|^{2})dv-\int
c_{21}n_{2}\frac{T_{21}}{m_{2}}f_{2}|\frac{\nabla_{v}f_{2}}{f_{2}}|^{2})dv+c_{21}n_{2}n_{1}d(1+\varepsilon)$

By using lemma 4, we can write this as

$\displaystyle\begin{split}I&=-\frac{T_{12}}{m_{1}}c_{12}n_{2}\int\frac{M_{12}^{2}}{f_{1}}\left(\frac{\nabla_{v}f_{1}M_{12}-\nabla_{v}M_{12}f_{1}}{M_{12}^{2}}\right)^{2}dv\\\
&-\frac{T_{21}}{m_{2}}c_{21}n_{1}\int\frac{M_{21}^{2}}{f_{2}}\left(\frac{\nabla_{v}f_{2}M_{21}-\nabla_{v}M_{21}f_{2}}{M_{21}^{2}}\right)^{2}dv\\\
&+c_{12}n_{2}n_{1}d\frac{T_{1}+\frac{m_{1}}{d}(1-\delta)^{2}|u_{1}-u_{2}|^{2}}{T_{12}}+c_{21}n_{2}n_{1}d\frac{T_{2}+\frac{m_{2}}{d}\varepsilon^{2}(\frac{m_{1}}{m_{2}})^{2}(1-\delta)^{2}|u_{1}-u_{2}|^{2}}{T_{21}}\\\
&-(1+\varepsilon)c_{21}n_{1}n_{2}d\end{split}$ (28)

The first two terms are non-positive, so we get the claimed inequality if we
can prove

$\displaystyle
c_{12}n_{2}n_{1}d\left(T_{1}+\frac{m_{1}}{d}(1-\delta)^{2}|u_{1}-u_{2}|^{2}\right)T_{21}$
$\displaystyle+c_{21}n_{1}n_{2}dT_{12}\left(T_{2}+\frac{m_{2}}{d}(1-\delta)^{2}\varepsilon^{2}(\frac{m_{1}}{m_{2}})^{2}|u_{1}-u_{2}|^{2}\right)$
$\displaystyle\leq(1+\varepsilon)c_{21}n_{1}n_{2}dT_{12}T_{21}$

which is by using relationship (13) equivalent to

$\displaystyle\varepsilon\left(T_{1}+\frac{m_{1}}{d}(1-\delta)^{2}|u_{1}-u_{2}|^{2}\right)T_{21}+T_{12}\left(T_{2}+\frac{m_{2}}{d}(1-\delta)^{2}\varepsilon^{2}(\frac{m_{1}}{m_{2}})^{2}|u_{1}-u_{2}|^{2}\right)$
$\displaystyle\leq(1+\varepsilon)T_{12}T_{21}$

With the notation introduced in (24) and (26), we can write this as

$\displaystyle\gamma_{1}=\frac{m_{1}}{d}(1-\delta)^{2},\quad\gamma_{2}=\frac{m_{2}}{d}(1-\delta)^{2}\varepsilon^{2}(\frac{m_{1}}{m_{2}})^{2}$

and

$\displaystyle T_{12}=:\bar{T}_{12}+\gamma|u_{1}-u_{2}|^{2},\quad
T_{21}=:\bar{T}_{21}+\tilde{\gamma}|u_{1}-u_{2}|^{2}$

Then we get

$\displaystyle\varepsilon(T_{1}+\gamma_{1}|u_{1}-u_{2}|^{2})(\bar{T}_{21}+\tilde{\gamma}|u_{1}-u_{2}|^{2})+(\bar{T}_{12}+\gamma|u_{1}-u_{2}|^{2})(T_{2}+\gamma_{2}|u_{1}-u_{2}|^{2})$
$\displaystyle\leq(1+\varepsilon)(\bar{T}_{12}+\gamma|u_{1}-u_{2}|^{2})(\bar{T}_{21}+\tilde{\gamma}|u_{1}-u_{2}|^{2})$

This is equivalent to

$\displaystyle\varepsilon T_{1}\bar{T}_{21}+\varepsilon
T_{1}\tilde{\gamma}|u_{1}-u_{2}|^{2}+\varepsilon\gamma_{1}|u_{1}-u_{2}|^{2}\bar{T}_{21}+\varepsilon\gamma_{1}\tilde{\gamma}|u_{1}-u_{2}|^{4}+\bar{T}_{12}T_{2}$
$\displaystyle+\bar{T}_{12}\gamma_{2}|u_{1}-u_{2}|^{2}+\gamma|u_{1}-u_{2}|^{2}T_{2}+\gamma\gamma_{2}|u_{1}-u_{2}|^{4}$
$\displaystyle\leq(1+\varepsilon)(\bar{T}_{12}\bar{T}_{21}+\bar{T}_{12}\tilde{\gamma}|u_{1}-u_{2}|^{2}+\gamma|u_{1}-u_{2}|^{2}\bar{T}_{21}+\gamma\tilde{\gamma}|u_{1}-u_{2}|^{4}$

This is true if we have separately

$\displaystyle\varepsilon
T_{1}\bar{T}_{21}+\bar{T}_{12}T_{2}\leq(1+\varepsilon)\bar{T}_{12}\bar{T}_{21}$
(29)
$\displaystyle(\varepsilon\gamma_{1}\tilde{\gamma}+\gamma\gamma_{2})|u_{1}-u_{2}|^{4}\leq(1+\varepsilon)\gamma\tilde{\gamma}|u_{1}-u_{2}|^{4}$
(30) $\displaystyle\begin{split}\varepsilon
T_{1}\tilde{\gamma}|u_{1}-u_{2}|^{2}+\varepsilon\gamma_{1}|u_{1}-u_{2}|^{2}\bar{T}_{21}+\bar{T}_{12}\gamma_{2}|u_{1}-u_{2}|^{2}+\gamma|u_{1}-u_{2}|^{2}T_{2}\\\
\leq(1+\varepsilon)\bar{T}_{12}\tilde{\gamma}|u_{1}-u_{2}|^{2}+(1+\varepsilon)\gamma|u_{1}-u_{2}|^{2}\bar{T}_{21}\end{split}$
(31)

These three inequalities are satisfied according to lemmas 5, 6, 7. Then the
last four terms in (28) can be estimated by zero from above. So we obtain

$\displaystyle
I\leq-\frac{T_{12}}{m_{1}}c_{12}n_{2}\int\frac{M_{12}^{2}}{f_{1}}\left(\frac{\nabla_{v}f_{1}M_{12}-\nabla_{v}M_{12}f_{1}}{M_{12}^{2}}\right)^{2}dv$
$\displaystyle-\frac{T_{21}}{m_{2}}c_{21}n_{1}\int\frac{M_{21}^{2}}{f_{2}}\left(\frac{\nabla_{v}f_{2}M_{21}-\nabla_{v}M_{21}f_{2}}{M_{21}^{2}}\right)^{2}dv$
$\displaystyle=-\frac{T_{12}}{m_{1}}c_{12}n_{2}\int
f_{1}\left|\frac{M_{12}}{f_{1}}\nabla_{v}\left(\frac{f_{1}}{M_{12}}\right)\right|^{2}dv-\frac{T_{21}}{m_{2}}c_{21}n_{1}\int
f_{2}\left|\frac{M_{21}}{f_{2}}\nabla_{v}\left(\frac{f_{2}}{M_{21}}\right)\right|^{2}dv$
$\displaystyle=-\frac{T_{12}}{m_{1}}c_{12}n_{2}\int
f_{1}\left|\nabla_{v}\ln\frac{f_{1}}{M_{12}}\right|^{2}dv-\frac{T_{21}}{m_{2}}c_{21}n_{1}\int
f_{2}\left|\nabla_{v}\ln\frac{f_{2}}{M_{21}}\right|^{2}dv\leq 0$

with equality if and only if $f_{1}=M_{12}$ and $f_{2}=M_{21}$. This means the
equality is characterized by two Maxwell distributions. In addition, if we
compute the mean velocities of these expressions, we get in case of equality
$u_{1}=u_{12}=\delta u_{1}+(1-\delta)u_{2}$ which leads to $u_{1}=u_{2}$.
Similar, for the temperatures, we obtain $T_{1}=T_{2}$. ∎

Define the total entropy $H(f_{1},f_{2})=\int(f_{1}\ln f_{1}+f_{2}\ln
f_{2})dv$. We can compute

$\partial_{t}H(f_{1},f_{2})+\nabla_{x}\cdot\int(f_{1}\ln f_{1}+f_{2}\ln
f_{2})vdv=S(f_{1},f_{2}),$

by multiplying the Fokker-Planck equation for the species $1$ by $\ln f_{1}$,
the Fokker-Planck equation for the species $2$ by $\ln f_{2}$ and integrating
the sum with respect to $v$.

######  Corollary 8.1 (Entropy inequality for mixtures).

Assume $f_{1},f_{2}>0$. Assume a fast enough decay of $f_{1},f_{2}$ to zero
for $v\rightarrow\infty$. Assume relationship (13), the conditions (5) , (14),
(15), (18) and (19) with $\alpha,\delta\neq 1$, the positivity of the
temperatures (20) and the assumptions on the free parameters (23), then we
have the following entropy inequality

$\partial_{t}\left(\int f_{1}\ln f_{1}dv+\int f_{2}\ln
f_{2}dv\right)+\nabla_{x}\cdot\left(\int vf_{1}\ln f_{1}dv+\int vf_{2}\ln
f_{2}dv\right)\leq 0,$

with equality if and only if $f_{1}$ and $f_{2}$ are Maxwell distributions
with equal bulk velocity and temperature. Moreover at equilibrium the
interspecies Maxwellians $M_{12}$ and $M_{21}$ satisfy
$u_{12}=u_{2}=u_{1}=u_{21}$ and $T_{12}=T_{2}=T_{1}=T_{21}$.

We now explicitly specify the global equilibrium.

######  Theorem 9 (Equilibrium).

Assume $f_{1},f_{2}>0$. Assume relationship (13) , the conditions (5), (14),
(15), (18) and (19)and the positivity of the temperatures (20). Then
$Q^{FP}_{11}(f_{1},f_{1})+Q^{FP}_{12}(f_{1},f_{2})=0$ and
$Q^{FP}_{22}(f_{2},f_{2})+Q^{FP}_{21}(f_{2},f_{1})=0$, if and only if $f_{1}$
and $f_{2}$ are Maxwell distributions with equal mean velocity and
temperature.

###### Proof.

If $Q^{FP}_{11}(f_{1},f_{1})+Q^{FP}_{12}(f_{1},f_{2})=0$ and
$Q^{FP}_{22}(f_{2},f_{2})+Q^{FP}_{21}(f_{2},f_{1})=0$ , then $\ln
f_{1}~{}Q^{FP}_{11}(f_{1},f_{1})+\ln f_{1}~{}Q^{FP}_{12}(f_{1},f_{2})+\ln
f_{2}~{}Q^{FP}_{22}(f_{2},f_{2})+\ln f_{2}~{}Q^{FP}_{21}(f_{2},f_{1})=0$ and
so we have equality in the H-theorem ∎

##  3 Acknowledgements

Marlies Pirner was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587,
Mathematics Münster: Dynamics–Geometry–Structure, by the Alexander von
Humboldt foundation and the German Science Foundation DFG (grant no. PI
1501/2-1).

## Declarations

  * •

Competing Interests: Not applicable

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##  9 Appendix

Table 3: RQ1 results for Prior approaches. The best scores for each dataset
are shown in bold.

| Micro-avg Scores | Macro-avg scores | Training Time (s) | Prediction Time (s)
---|---|---|---|---
| P | R | F1 | P | R | F1
Dataset-1 |
AR-MINER | 0.36 | 0.36 | 0.36 | No micro or macro average | 261.71 | NA
SUR-MINER | 0.57 | 0.56 | 0.56 | 0.53 | 0.47 | 0.5 | 105.31 | 21.06
Ensemble | 0.71 | 0.7 | 0.7 | 0.7 | 0.6 | 0.65 | 115.84 | 23.17
DL+WE | 0.84 | 0.71 | 0.77 | 0.83 | 0.67 | 0.74 | 99.62 | 14.23
Dataset-2 |
AR-MINER | 0.49 | 0.32 | 0.39 | No micro or macro average | 81.17 | NA
SUR-MINER | 0.6 | 0.55 | 0.57 | 0.65 | 0.51 | 0.57 | 27.18 | 3.88
Ensemble | 0.8 | 0.69 | 0.74 | 0.7 | 0.66 | 0.68 | 29.9 | 4.98
DL+WE | 0.93 | 0.84 | 0.88 | 0.94 | 0.84 | 0.89 | 26.31 | 5.26
Dataset-3 |
AR-MINER | 0.44 | 0.36 | 0.4 | No micro or macro average | 74.62 | NA
SUR-MINER | 0.62 | 0.63 | 0.62 | 0.59 | 0.53 | 0.56 | 19.33 | 3.22
Ensemble | 0.77 | 0.75 | 0.76 | 0.69 | 0.71 | 0.7 | 21.46 | 4.29
DL+WE | 0.86 | 0.84 | 0.85 | 0.77 | 0.72 | 0.74 | 19.74 | 3.95
Dataset-4 |
AR-MINER | 0.45 | 0.33 | 0.38 | No micro or macro average | 71.96 | NA
SUR-MINER | 0.63 | 0.58 | 0.6 | 0.55 | 0.53 | 0.54 | 11.74 | 2.35
Ensemble | 0.84 | 0.74 | 0.79 | 0.82 | 0.7 | 0.76 | 13.5 | 2.7
DL+WE | 0.87 | 0.76 | 0.81 | 0.85 | 0.73 | 0.79 | 12.29 | 2.05
Dataset-5 |
AR-MINER | 0.53 | 0.35 | 0.42 | No micro or macro average | 57.02 | NA
SUR-MINER | 0.62 | 0.54 | 0.58 | 0.57 | 0.56 | 0.56 | 9.11 | 1.3
Ensemble | 0.86 | 0.76 | 0.81 | 0.84 | 0.81 | 0.82 | 10.93 | 2.19
DL+WE | 0.84 | 0.75 | 0.79 | 0.83 | 0.81 | 0.82 | 9.73 | 1.62
Dataset-6 |
AR-MINER | 0.47 | 0.41 | 0.44 | No micro or macro average | 71.91 | NA
SUR-MINER | 0.58 | 0.52 | 0.55 | 0.6 | 0.57 | 0.58 | 10.07 | 1.44
Ensemble | 0.75 | 0.73 | 0.74 | 0.85 | 0.81 | 0.83 | 11.98 | 1.71
DL+WE | 0.77 | 0.76 | 0.76 | 0.86 | 0.82 | 0.84 | 10.3 | 1.47
Dataset-7 | Merged Dataset
AR-MINER | 0.49 | 0.34 | 0.4 | No micro or macro average | 401.57 | NA
SUR-MINER | 0.66 | 0.55 | 0.6 | 0.64 | 0.6 | 0.62 | 156.54 | 31.31
Ensemble | 0.76 | 0.7 | 0.73 | 0.79 | 0.76 | 0.77 | 176.89 | 29.48
DL+WE | 0.84 | 0.78 | 0.81 | 0.82 | 0.74 | 0.78 | 155.66 | 25.94

Table 4: RQ1 results for PTMs. The best scores for each dataset are shown in
bold.

| Micro-avg Scores | Macro-avg scores | Training Time (s) | Prediction Time (s) | Time-diff (%)
---|---|---|---|---|---
| P | R | F1 | P | R | F1
Dataset-1 |
BERT | 0.79 | 0.74 | 0.764 | 0.76 | 0.69 | 0.723 | 345.8 | 19.21 | 34.34
ALBERT | 0.93 | 0.91 | 0.92 | 0.89 | 0.84 | 0.864 | 286.5 | 15.92 | 11.33
RoBERTa | 0.89 | 0.82 | 0.854 | 0.85 | 0.82 | 0.835 | 357.35 | 19.85 | 38.81
XLNET | 0.76 | 0.68 | 0.718 | 0.72 | 0.68 | 0.699 | 514.92 | 36.78 | 157.2
Dataset-2 |  |
BERT | 0.77 | 0.69 | 0.728 | 0.86 | 0.83 | 0.845 | 185.14 | 10.29 | 165.21
ALBERT | 0.96 | 0.88 | 0.918 | 0.94 | 0.89 | 0.914 | 119.93 | 6.66 | 71.65
RoBERTa | 0.9 | 0.85 | 0.874 | 0.79 | 0.74 | 0.764 | 202.44 | 11.25 | 189.95
XLNET | 0.85 | 0.77 | 0.808 | 0.87 | 0.78 | 0.823 | 345.57 | 24.68 | 536.08
Dataset-3 |  |
BERT | 0.93 | 0.91 | 0.92 | 0.9 | 0.82 | 0.858 | 156.9 | 6.82 | 111.8
ALBERT | 0.91 | 0.87 | 0.89 | 0.89 | 0.79 | 0.837 | 105.53 | 4.59 | 42.55
RoBERTa | 0.93 | 0.88 | 0.904 | 0.89 | 0.81 | 0.848 | 175.29 | 7.62 | 136.65
XLNET | 0.88 | 0.86 | 0.87 | 0.79 | 0.73 | 0.759 | 307.66 | 17.09 | 430.75
Dataset-4 |  |
BERT | 0.84 | 0.78 | 0.809 | 0.79 | 0.72 | 0.753 | 75.69 | 2.91 | 41.95
ALBERT | 0.85 | 0.81 | 0.83 | 0.72 | 0.65 | 0.683 | 53.66 | 2.06 | 0.49
RoBERTa | 0.93 | 0.9 | 0.915 | 0.87 | 0.81 | 0.839 | 87.61 | 3.37 | 64.39
XLNET | 0.78 | 0.75 | 0.765 | 0.8 | 0.76 | 0.779 | 149.36 | 7.11 | 246.83
Dataset-5 |  |
BERT | 0.88 | 0.81 | 0.844 | 0.8 | 0.77 | 0.785 | 74.52 | 2.57 | 97.69
ALBERT | 0.93 | 0.89 | 0.91 | 0.93 | 0.88 | 0.904 | 38.63 | 1.33 | 2.31
RoBERTa | 0.94 | 0.89 | 0.914 | 0.87 | 0.83 | 0.85 | 80.69 | 2.78 | 113.85
XLNET | 0.87 | 0.79 | 0.828 | 0.924 | 0.85 | 0.904 | 127.46 | 4.9 | 276.92
Dataset-6 |  |
BERT | 0.79 | 0.74 | 0.764 | 0.84 | 0.82 | 0.83 | 74.83 | 2.77 | 92.36
ALBERT | 0.8 | 0.77 | 0.785 | 0.93 | 0.86 | 0.894 | 48.74 | 1.81 | 25.69
RoBERTa | 0.82 | 0.77 | 0.794 | 0.93 | 0.91 | 0.92 | 81.2 | 3.01 | 109.03
XLNET | 0.73 | 0.66 | 0.693 | 0.93 | 0.85 | 0.888 | 143.15 | 6.22 | 331.94
Dataset-7 | Merged Dataset |
BERT | 0.91 | 0.84 | 0.874 | 0.88 | 0.8 | 0.844 | 574.32 | 24.97 | -3.74
ALBERT | 0.95 | 0.85 | 0.897 | 0.85 | 0.8 | 0.824 | 460.19 | 20.01 | -22.86
RoBERTa | 0.97 | 0.91 | 0.939 | 0.87 | 0.82 | 0.838 | 597.65 | 25.98 | 0.15
XLNET | 0.92 | 0.86 | 0.889 | 0.89 | 0.84 | 0.864 | 836.96 | 49.23 | 89.78

Table 5: RQ2 results for C-PTM-X1 (PTM-X1 = ALBERT). The best scores for each
dataset are bold.

| Micro-avg Scores | Macro-avg scores | Training Time (s) | Prediction Time (s) | Time-Diff (%)
---|---|---|---|---|---
| P | R | F1 | P | R | F1
Dataset-1 |
ALBERT (PTM-X1) | 0.93 | 0.91 | 0.92 | 0.89 | 0.84 | 0.864 | 286.5 | 15.92 | 11.33
C-ALBERT-BASE | 0.94 | 0.916 | 0.928 | 0.902 | 0.844 | 0.872 | 295.1 | 15.6 | 9.09
C-ALBERT-MEDIUM | 0.95 | 0.926 | 0.938 | 0.908 | 0.865 | 0.886 | 280.77 | 15.44 | 7.97
C-ALBERT-LARGE | 0.977 | 0.945 | 0.961 | 0.92 | 0.87 | 0.894 | 295.1 | 15.6 | 9.09
Dataset-2 |  |
ALBERT (PTM-X1) | 0.96 | 0.88 | 0.918 | 0.94 | 0.89 | 0.914 | 119.93 | 6.66 | 71.65
C-ALBERT-BASE | 0.972 | 0.888 | 0.928 | 0.947 | 0.894 | 0.92 | 117.53 | 6.79 | 75
C-ALBERT-MEDIUM | 0.987 | 0.906 | 0.945 | 0.958 | 0.903 | 0.93 | 123.53 | 6.86 | 76.8
C-ALBERT-LARGE | 0.991 | 0.91 | 0.955 | 0.978 | 0.933 | 0.955 | 121.13 | 6.86 | 76.8
Dataset-3 |  |
ALBERT (PTM-X1) | 0.91 | 0.87 | 0.886 | 0.8 | 0.75 | 0.774 | 105.53 | 4.59 | 42.55
C-ALBERT-BASE | 0.938 | 0.922 | 0.93 | 0.91 | 0.829 | 0.868 | 107.64 | 4.64 | 44.1
C-ALBERT-MEDIUM | 0.949 | 0.926 | 0.937 | 0.914 | 0.841 | 0.876 | 106.59 | 4.5 | 39.75
C-ALBERT-LARGE | 0.965 | 0.945 | 0.955 | 0.941 | 0.845 | 0.89 | 104.47 | 4.64 | 44.1
Dataset-4 |  |
ALBERT (PTM-X1) | 0.85 | 0.81 | 0.83 | 0.72 | 0.65 | 0.683 | 53.66 | 2.06 | 0.49
C-ALBERT-BASE | 0.94 | 0.907 | 0.923 | 0.875 | 0.821 | 0.847 | 54.2 | 2.02 | -1.46
C-ALBERT-MEDIUM | 0.945 | 0.92 | 0.932 | 0.886 | 0.832 | 0.858 | 53.12 | 2.04 | -0.49
C-ALBERT-LARGE | 0.976 | 0.945 | 0.96 | 0.9 | 0.841 | 0.87 | 52.59 | 2.08 | 1.46
Dataset-5 |  |
ALBERT (PTM-X1) | 0.93 | 0.89 | 0.91 | 0.93 | 0.88 | 0.904 | 38.63 | 1.33 | 2.31
C-ALBERT-BASE | 0.952 | 0.899 | 0.925 | 0.942 | 0.892 | 0.916 | 39.4 | 1.3 | 0
C-ALBERT-MEDIUM | 0.965 | 0.917 | 0.94 | 0.957 | 0.898 | 0.927 | 37.86 | 1.3 | 0
C-ALBERT-LARGE | 0.984 | 0.922 | 0.952 | 0.967 | 0.908 | 0.937 | 38.24 | 1.34 | 3.08
Dataset-6 |  |
ALBERT (PTM-X1) | 0.8 | 0.77 | 0.785 | 0.93 | 0.86 | 0.894 | 48.74 | 1.81 | 25.69
C-ALBERT-BASE | 0.831 | 0.775 | 0.802 | 0.938 | 0.921 | 0.929 | 49.71 | 1.83 | 27.08
C-ALBERT-MEDIUM | 0.842 | 0.791 | 0.816 | 0.95 | 0.924 | 0.937 | 49.71 | 1.77 | 22.92
C-ALBERT-LARGE | 0.86 | 0.799 | 0.828 | 0.963 | 0.943 | 0.953 | 49.71 | 1.83 | 27.08
Dataset-7 | Merged Dataset |
ALBERT (PTM-X1) | 0.95 | 0.85 | 0.897 | 0.85 | 0.8 | 0.824 | 460.19 | 20.01 | -22.86
C-ALBERT-BASE | 0.98 | 0.92 | 0.949 | 0.903 | 0.848 | 0.875 | 464.79 | 20.21 | -22.09
C-ALBERT-MEDIUM | 0.993 | 0.933 | 0.962 | 0.908 | 0.86 | 0.883 | 450.99 | 20.41 | -21.32
C-ALBERT-LARGE | 0.998 | 0.942 | 0.979 | 0.935 | 0.866 | 0.899 | 469.39 | 19.81 | -23.63

Table 6: RQ2 results for C-PTM-X2 (PTM-X2 = RoBERTa). The best scores are
bold.

| Micro-avg Scores | Macro-avg scores | Training Time (s) | Prediction Time (s) | Time Diff (%)
---|---|---|---|---|---
| P | R | F1 | P | R | F1
Dataset-1 |
RoBERTa (PTM-X2) | 0.89 | 0.82 | 0.854 | 0.85 | 0.82 | 0.835 | 357.35 | 19.85 | 38.81
C-RoBERTa-BASE | 0.949 | 0.925 | 0.937 | 0.903 | 0.85 | 0.876 | 360.92 | 19.65 | 37.41
C-RoBERTa-MEDIUM | 0.958 | 0.959 | 0.958 | 0.933 | 0.869 | 0.9 | 360.92 | 20.25 | 41.61
C-RoBERTa-LARGE | 0.991 | 0.982 | 0.996 | 0.97 | 0.911 | 0.94 | 364.5 | 19.45 | 36.01
Dataset-2 |  |
RoBERTa (PTM-X2) | 0.9 | 0.85 | 0.874 | 0.79 | 0.74 | 0.764 | 202.44 | 11.25 | 189.95
C-RoBERTa-BASE | 0.985 | 0.895 | 0.938 | 0.949 | 0.902 | 0.925 | 204.46 | 11.36 | 192.78
C-RoBERTa-MEDIUM | 0.962 | 0.914 | 0.937 | 0.989 | 0.938 | 0.963 | 200.42 | 11.03 | 184.28
C-RoBERTa-LARGE | 0.985 | 0.956 | 0.97 | 0.997 | 0.946 | 0.971 | 204.46 | 11.36 | 192.78
Dataset-3 |  |
RoBERTa (PTM-X2) | 0.93 | 0.88 | 0.904 | 0.89 | 0.81 | 0.848 | 175.29 | 7.62 | 136.65
C-RoBERTa-BASE | 0.953 | 0.931 | 0.942 | 0.925 | 0.845 | 0.883 | 173.54 | 7.7 | 139.13
C-RoBERTa-MEDIUM | 0.978 | 0.942 | 0.96 | 0.927 | 0.859 | 0.892 | 177.04 | 7.47 | 131.99
C-RoBERTa-LARGE | 0.995 | 0.986 | 0.99 | 0.965 | 0.884 | 0.923 | 177.04 | 7.47 | 131.99
Dataset-4 |  |
RoBERTa (PTM-X2) | 0.93 | 0.9 | 0.915 | 0.87 | 0.81 | 0.839 | 87.61 | 3.37 | 64.39
C-RoBERTa-BASE | 0.943 | 0.912 | 0.927 | 0.879 | 0.82 | 0.848 | 88.49 | 3.44 | 67.8
C-RoBERTa-MEDIUM | 0.966 | 0.94 | 0.953 | 0.897 | 0.841 | 0.868 | 89.36 | 3.4 | 65.85
C-RoBERTa-LARGE | 0.986 | 0.962 | 0.974 | 0.947 | 0.877 | 0.911 | 89.36 | 3.4 | 65.85
Dataset-5 |  |
RoBERTa (PTM-X2) | 0.94 | 0.89 | 0.914 | 0.87 | 0.83 | 0.85 | 80.69 | 2.78 | 113.85
C-RoBERTa-BASE | 0.95 | 0.917 | 0.933 | 0.948 | 0.901 | 0.924 | 79.08 | 2.84 | 118.46
C-RoBERTa-MEDIUM | 0.982 | 0.917 | 0.948 | 0.974 | 0.917 | 0.945 | 81.5 | 2.75 | 111.54
C-RoBERTa-LARGE | 0.997 | 0.949 | 0.972 | 0.974 | 0.947 | 0.96 | 79.08 | 2.81 | 116.15
Dataset-6 |  |
RoBERTa (PTM-X2) | 0.82 | 0.77 | 0.794 | 0.93 | 0.91 | 0.92 | 81.2 | 3.01 | 109.03
C-RoBERTa-BASE | 0.833 | 0.78 | 0.806 | 0.954 | 0.925 | 0.939 | 82.01 | 3.07 | 113.19
C-RoBERTa-MEDIUM | 0.869 | 0.802 | 0.834 | 0.961 | 0.945 | 0.953 | 79.58 | 2.95 | 104.86
C-RoBERTa-LARGE | 0.874 | 0.837 | 0.855 | 0.987 | 0.97 | 0.978 | 82.82 | 2.98 | 106.94
Dataset-7 | Merged Dataset |
RoBERTa (PTM-X2) | 0.97 | 0.91 | 0.939 | 0.87 | 0.82 | 0.838 | 597.65 | 25.98 | 0.15
C-RoBERTa-BASE | 0.98 | 0.921 | 0.95 | 0.903 | 0.856 | 0.879 | 585.7 | 25.46 | -1.85
C-RoBERTa-MEDIUM | 0.983 | 0.938 | 0.96 | 0.926 | 0.889 | 0.907 | 603.63 | 25.72 | -0.85
C-RoBERTa-LARGE | 0.995 | 0.975 | 0.984 | 0.949 | 0.895 | 0.921 | 591.67 | 26.5 | 2.16

Table 7: RQ3 results for Binary Classifications (Part i)

Datasets & Approaches | P | R | F1 | Training Time (s) | Prediction Time (s)
---|---|---|---|---|---
Dataset-1 | Randomly Selected Class: Bug Report
AR-MINER | 0.37 | 0.36 | 0.36 | 260.92 | NA
SUR-MINER | 0.58 | 0.56 | 0.57 | 105.63 | 21.08
Ensemble | 0.73 | 0.72 | 0.72 | 115.61 | 23.05
DL+WE | 0.87 | 0.73 | 0.79 | 99.72 | 14.3
BERT | 0.81 | 0.75 | 0.78 | 344.42 | 19.29
ALBERT | 0.95 | 0.92 | 0.93 | 287.93 | 15.95
RoBERTa | 0.91 | 0.83 | 0.87 | 358.78 | 19.87
XLNET | 0.78 | 0.69 | 0.73 | 514.41 | 36.74
C-ALBERT-BASE | 0.95 | 0.95 | 0.95 | 293.92 | 15.66
C-ALBERT-MEDIUM | 0.96 | 0.95 | 0.95 | 281.89 | 15.52
C-ALBERT-LARGE | 0.97 | 0.98 | 0.97 | 295.99 | 15.62
C-RoBERTa-BASE | 0.98 | 0.95 | 0.96 | 361.64 | 19.55
C-RoBERTa-MEDIUM | 0.98 | 0.97 | 0.97 | 359.84 | 20.17
C-RoBERTa-LARGE | 1 | 0.99 | 0.99 | 366.32 | 19.37
Dataset-2 | Randomly Selected Class: Inquiry
AR-MINER | 0.5 | 0.32 | 0.39 | 80.93 | NA
SUR-MINER | 0.6 | 0.56 | 0.58 | 27.04 | 3.88
Ensemble | 0.81 | 0.7 | 0.75 | 29.78 | 4.99
DL+WE | 0.94 | 0.86 | 0.9 | 26.34 | 5.29
BERT | 0.8 | 0.7 | 0.75 | 184.58 | 10.32
ALBERT | 0.98 | 0.91 | 0.94 | 120.29 | 6.69
RoBERTa | 0.93 | 0.86 | 0.89 | 203.45 | 11.21
XLNET | 0.87 | 0.78 | 0.82 | 345.92 | 24.66
C-ALBERT-BASE | 0.91 | 0.94 | 0.92 | 117.77 | 6.8
C-ALBERT-MEDIUM | 0.92 | 0.95 | 0.93 | 124.15 | 6.87
C-ALBERT-LARGE | 0.92 | 0.98 | 0.95 | 120.65 | 6.87
C-RoBERTa-BASE | 1 | 0.92 | 0.96 | 203.85 | 11.39
C-RoBERTa-MEDIUM | 0.98 | 0.93 | 0.95 | 200.02 | 11.07
C-RoBERTa-LARGE | 1 | 0.99 | 0.99 | 204.66 | 11.33
Dataset-3 | Randomly Selected Class: Reliability
AR-MINER | 0.44 | 0.36 | 0.4 | 74.77 | NA
SUR-MINER | 0.63 | 0.65 | 0.64 | 19.43 | 3.21
Ensemble | 0.79 | 0.77 | 0.78 | 21.42 | 4.29
DL+WE | 0.87 | 0.86 | 0.86 | 19.76 | 3.97
BERT | 0.93 | 0.93 | 0.93 | 157.53 | 6.83
ALBERT | 0.93 | 0.88 | 0.9 | 105.95 | 4.59
RoBERTa | 0.95 | 0.89 | 0.92 | 175.64 | 7.66
XLNET | 0.89 | 0.89 | 0.89 | 307.04 | 17.16
C-ALBERT-BASE | 0.94 | 0.95 | 0.92 | 107.1 | 4.65
C-ALBERT-MEDIUM | 0.93 | 0.94 | 0.95 | 107.02 | 4.49
C-ALBERT-LARGE | 0.96 | 0.96 | 0.96 | 104.68 | 4.63
C-RoBERTa-BASE | 0.98 | 0.97 | 0.96 | 173.37 | 7.72
C-RoBERTa-MEDIUM | 1 | 0.97 | 0.98 | 177.93 | 7.5
C-RoBERTa-LARGE | 1 | 1 | 1.0 | 177.22 | 7.46
Dataset-4 | Randomly Selected Class: Feature Request
AR-MINER | 0.46 | 0.34 | 0.39 | 71.67 | NA
SUR-MINER | 0.64 | 0.59 | 0.61 | 11.76 | 2.35
Ensemble | 0.85 | 0.75 | 0.8 | 13.54 | 2.7
DL+WE | 0.88 | 0.78 | 0.83 | 12.23 | 2.05
BERT | 0.86 | 0.8 | 0.83 | 75.84 | 2.9
ALBERT | 0.86 | 0.83 | 0.84 | 53.93 | 2.06
RoBERTa | 0.96 | 0.93 | 0.94 | 87.87 | 3.36
XLNET | 0.79 | 0.78 | 0.78 | 148.91 | 7.14
C-ALBERT-BASE | 0.94 | 0.94 | 0.9 | 54.15 | 2.02
C-ALBERT-MEDIUM | 0.94 | 0.94 | 0.91 | 53.33 | 2.03
C-ALBERT-LARGE | 0.96 | 0.97 | 0.91 | 52.38 | 2.07
C-RoBERTa-BASE | 0.97 | 0.95 | 0.94 | 88.31 | 3.42
C-RoBERTa-MEDIUM | 1 | 0.97 | 0.98 | 89.27 | 3.39
C-RoBERTa-LARGE | 1 | 0.99 | 0.99 | 89 | 3.42

Table 8: RQ3 results for Binary Classifications (Part ii)

Datasets & Approaches | P | R | F1 | Training Time (s) | Prediction Time (s)
---|---|---|---|---|---
Dataset-5 | Randomly Selected Class: App Problem
AR-MINER | 0.54 | 0.35 | 0.42 | 56.96 | NA
SUR-MINER | 0.63 | 0.56 | 0.59 | 9.08 | 1.3
Ensemble | 0.88 | 0.77 | 0.82 | 10.94 | 2.18
DL+WE | 0.85 | 0.76 | 0.8 | 9.72 | 1.62
BERT | 0.91 | 0.82 | 0.86 | 74.45 | 2.58
ALBERT | 0.95 | 0.91 | 0.93 | 38.82 | 1.32
RoBERTa | 0.97 | 0.92 | 0.94 | 80.53 | 2.78
XLNET | 0.9 | 0.81 | 0.85 | 127.08 | 4.92
C-ALBERT-BASE | 0.91 | 0.94 | 0.92 | 39.6 | 1.29
C-ALBERT-MEDIUM | 0.93 | 0.97 | 0.95 | 37.82 | 1.3
C-ALBERT-LARGE | 0.94 | 0.98 | 0.96 | 38.35 | 1.35
C-RoBERTa-BASE | 0.98 | 0.92 | 0.95 | 79.48 | 2.83
C-RoBERTa-MEDIUM | 1 | 0.93 | 0.96 | 81.34 | 2.74
C-RoBERTa-LARGE | 1 | 0.97 | 0.98 | 79.48 | 2.82
Dataset-6 | Randomly Selected Class: Bug Report
AR-MINER | 0.48 | 0.42 | 0.45 | 71.98 | NA
SUR-MINER | 0.59 | 0.53 | 0.56 | 10.02 | 1.44
Ensemble | 0.76 | 0.75 | 0.75 | 11.92 | 1.72
DL+WE | 0.78 | 0.77 | 0.77 | 10.32 | 1.47
BERT | 0.81 | 0.75 | 0.78 | 75.13 | 2.76
ALBERT | 0.82 | 0.79 | 0.8 | 48.79 | 1.82
RoBERTa | 0.84 | 0.78 | 0.81 | 81.52 | 3.01
XLNET | 0.75 | 0.67 | 0.71 | 142.72 | 6.24
C-ALBERT-BASE | 0.79 | 0.81 | 0.8 | 49.96 | 1.83
C-ALBERT-MEDIUM | 0.81 | 0.84 | 0.82 | 49.91 | 1.78
C-ALBERT-LARGE | 0.82 | 0.86 | 0.84 | 49.51 | 1.82
C-RoBERTa-BASE | 0.85 | 0.79 | 0.82 | 82.17 | 3.08
C-RoBERTa-MEDIUM | 0.88 | 0.81 | 0.84 | 79.66 | 2.94
C-RoBERTa-LARGE | 0.89 | 0.84 | 0.86 | 82.65 | 2.99
Dataset-7 | Merged Dataset; Randomly Selected Class: Aspect Evaluation
AR-MINER | 0.51 | 0.35 | 0.42 | 402.77 | NA
SUR-MINER | 0.68 | 0.56 | 0.61 | 157.01 | 31.25
Ensemble | 0.78 | 0.71 | 0.74 | 177.6 | 29.45
DL+WE | 0.86 | 0.79 | 0.82 | 156.13 | 26.04
BERT | 0.93 | 0.86 | 0.89 | 573.75 | 25.04
ALBERT | 0.97 | 0.87 | 0.92 | 462.49 | 20.05
RoBERTa | 1.0 | 0.93 | 0.96 | 600.64 | 25.88
XLNET | 0.94 | 0.88 | 0.91 | 834.45 | 48.98
C-ALBERT-BASE | 0.94 | 0.98 | 0.96 | 463.4 | 20.11
C-ALBERT-MEDIUM | 0.96 | 0.99 | 0.97 | 450.09 | 20.51
C-ALBERT-LARGE | 0.96 | 1.0 | 0.98 | 467.98 | 19.79
C-RoBERTa-BASE | 0.99 | 0.93 | 0.96 | 584.53 | 25.33
C-RoBERTa-MEDIUM | 0.99 | 0.95 | 0.97 | 605.44 | 25.8
C-RoBERTa-LARGE | 1.0 | 0.98 | 0.99 | 592.26 | 26.47

Table 9: RQ3 results for Zero-Shot Classification

Dataset # | P | R | F1 | Prediction Time (s)
---|---|---|---|---
Dataset-1 |
AR-MINER | 0.36 | 0.36 | 0.36 | N.A.
ALBERT-NLI | 0.47 | 0.42 | 0.44 | 15.35
RoBERTa-NLI | 0.5 | 0.46 | 0.48 | 20.35
C-ALBERT-LARGE-NLI | 0.5 | 0.47 | 0.48 | 16.7
C-RoBERTa-LARGE-NLI | 0.59 | 0.53 | 0.56 | 20.66
RoBERTa-LARGE-MNLI | 0.64 | 0.6 | 0.62 | 19.1
Dataset-2 |
AR-MINER | 0.49 | 0.32 | 0.39 | N.A.
ALBERT-NLI | 0.46 | 0.42 | 0.44 | 6.98
RoBERTa-NLI | 0.5 | 0.47 | 0.48 | 10.85
C-ALBERT-LARGE-NLI | 0.51 | 0.48 | 0.49 | 6.47
C-RoBERTa-LARGE-NLI | 0.56 | 0.5 | 0.53 | 11.64
RoBERTa-LARGE-MNLI | 0.64 | 0.58 | 0.61 | 10.8
Dataset-3 |
AR-MINER | 0.44 | 0.36 | 0.4 | N.A.
ALBERT-NLI | 0.43 | 0.4 | 0.41 | 4.37
RoBERTa-NLI | 0.51 | 0.46 | 0.48 | 7.28
C-ALBERT-LARGE-NLI | 0.55 | 0.51 | 0.53 | 4.72
C-RoBERTa-LARGE-NLI | 0.61 | 0.57 | 0.59 | 7.95
RoBERTa-LARGE-MNLI | 0.71 | 0.65 | 0.68 | 7.78
Dataset-4 |
AR-MINER | 0.45 | 0.33 | 0.38 | N.A.
ALBERT-NLI | 0.42 | 0.38 | 0.4 | 2.00
RoBERTa-NLI | 0.51 | 0.47 | 0.49 | 3.22
C-ALBERT-LARGE-NLI | 0.47 | 0.42 | 0.44 | 2.12
C-RoBERTa-LARGE-NLI | 0.65 | 0.59 | 0.62 | 3.51
RoBERTa-LARGE-MNLI | 0.69 | 0.64 | 0.66 | 3.21
Dataset-5 |
AR-MINER | 0.53 | 0.35 | 0.42 | N.A.
ALBERT-NLI | 0.39 | 0.35 | 0.37 | 1.37
RoBERTa-NLI | 0.53 | 0.49 | 0.51 | 2.91
C-ALBERT-LARGE-NLI | 0.47 | 0.44 | 0.45 | 1.37
C-RoBERTa-LARGE-NLI | 0.59 | 0.54 | 0.56 | 2.72
RoBERTa-LARGE-MNLI | 0.74 | 0.67 | 0.70 | 2.9
Dataset-6 |
AR-MINER | 0.47 | 0.41 | 0.44 | N.A.
ALBERT-NLI | 0.4 | 0.36 | 0.38 | 1.73
RoBERTa-NLI | 0.43 | 0.4 | 0.41 | 2.95
C-ALBERT-LARGE-NLI | 0.43 | 0.41 | 0.42 | 1.88
C-RoBERTa-LARGE-NLI | 0.52 | 0.48 | 0.5 | 3.13
RoBERTa-LARGE-MNLI | 0.64 | 0.58 | 0.61 | 2.87
Dataset-7 | Merged Dataset
AR-MINER | 0.49 | 0.34 | 0.4 | N.A.
ALBERT-NLI | 0.47 | 0.44 | 0.45 | 19.11
RoBERTa-NLI | 0.58 | 0.53 | 0.55 | 24.94
C-ALBERT-LARGE-NLI | 0.48 | 0.44 | 0.46 | 19.39
C-RoBERTa-LARGE-NLI | 0.65 | 0.61 | 0.63 | 24.84
RoBERTa-LARGE-MNLI | 0.74 | 0.68 | 0.71 | 27.28

Table 10: RQ3 results for Multi-Task (Category) Classification

| Micro-Scores | Macro-Scores | Training Time (s) | Prediction Time (s)
---|---|---|---|---
| P | R | F1 | P | R | F1
Category Labeled Data |
BERT | 0.813 | 0.759 | 0.791 | 0.752 | 0.714 | 0.732 | 287.16 | 9.988
ALBERT | 0.883 | 0.865 | 0.86 | 0.842 | 0.785 | 0.823 | 230.095 | 12.006
RoBERTa | 0.844 | 0.782 | 0.81 | 0.736 | 0.693 | 0.698 | 239.06 | 10.392
XLNET | 0.798 | 0.717 | 0.753 | 0.822 | 0.721 | 0.779 | 418.48 | 19.692
C-ALBERT-BASE | 0.888 | 0.85 | 0.872 | 0.853 | 0.785 | 0.807 | 139.437 | 12.126
C-ALBERT-MEDIUM | 0.897 | 0.864 | 0.897 | 0.865 | 0.809 | 0.837 | 180.396 | 6.123
C-ALBERT-LARGE | 0.933 | 0.89 | 0.917 | 0.859 | 0.804 | 0.831 | 281.634 | 5.943
C-RoBERTa-BASE | 0.945 | 0.837 | 0.878 | 0.887 | 0.858 | 0.86 | 292.85 | 10.184
C-RoBERTa-MEDIUM | 0.894 | 0.86 | 0.877 | 0.942 | 0.894 | 0.904 | 301.815 | 12.86
C-RoBERTa-LARGE | 0.932 | 0.90 | 0.901 | 0.931 | 0.879 | 0.927 | 295.835 | 10.6

Table 11: RQ3 results for Multi-Task (Sentiment) Classification

| Micro-Scores | Macro-Scores | Training Time (s) | Prediction Time (s)
---|---|---|---|---
| P | R | F1 | P | R | F1
Sentiment Labeled Data |
BERT | 0.852 | 0.769 | 0.804 | 0.759 | 0.748 | 0.739 | 689.184 | 32.461
ALBERT | 0.886 | 0.889 | 0.877 | 0.848 | 0.806 | 0.822 | 552.228 | 26.013
RoBERTa | 0.862 | 0.805 | 0.83 | 0.75 | 0.711 | 0.74 | 836.71 | 31.176
XLNET | 0.808 | 0.737 | 0.787 | 0.849 | 0.754 | 0.794 | 1088.048 | 63.999
C-ALBERT-BASE | 0.908 | 0.883 | 0.884 | 0.868 | 0.796 | 0.838 | 604.227 | 26.273
C-ALBERT-MEDIUM | 0.913 | 0.878 | 0.89 | 0.879 | 0.829 | 0.845 | 541.188 | 26.533
C-ALBERT-LARGE | 0.93 | 0.911 | 0.918 | 0.892 | 0.823 | 0.858 | 610.207 | 23.772
C-RoBERTa-BASE | 0.958 | 0.861 | 0.904 | 0.913 | 0.858 | 0.903 | 644.27 | 33.098
C-RoBERTa-MEDIUM | 0.934 | 0.864 | 0.913 | 0.955 | 0.905 | 0.929 | 784.719 | 28.292
C-RoBERTa-LARGE | 0.964 | 0.934 | 0.93 | 0.972 | 0.907 | 0.935 | 828.338 | 31.800

Table 12: RQ3 results for Multi-Resource Classification (Trained on App
Reviews and Tested on Twitter)

Datasets & Approaches | P | R | F1 | Prediction Time (s)
---|---|---|---|---
Dataset-2 |
AR-Miner | 0.42 | 0.28 | 0.34 | N.A.
SUR-Miner | 0.61 | 0.56 | 0.58 | 39.22
Ensemble | 0.78 | 0.72 | 0.75 | 38.79
DL+WE | 0.85 | 0.78 | 0.81 | 27.79
BERT | 0.57 | 0.50 | 0.53 | 17.49
ALBERT | 0.69 | 0.65 | 0.67 | 9.99
RoBERTa | 0.67 | 0.60 | 0.63 | 18.0
XLNET | 0.64 | 0.56 | 0.60 | 44.42
C-ALBERT-BASE | 0.75 | 0.67 | 0.71 | 11.54
C-ALBERT-MEDIUM | 0.79 | 0.75 | 0.77 | 10.29
C-ALBERT-LARGE | 0.84 | 0.76 | 0.80 | 12.35
C-RoBERTa-BASE | 0.80 | 0.74 | 0.77 | 20.45
C-RoBERTa-MEDIUM | 0.82 | 0.77 | 0.79 | 19.85
C-RoBERTa-LARGE | 0.87 | 0.86 | 0.86 | 20.45

Table 13: RQ3 results for Multi-Resource Classification (Trained on Twitter
and Tested on App Reviews)

Datasets & Approaches | P | R | F1 | Prediction Time (s)
---|---|---|---|---
Dataset-2 |
AR-Miner | 0.36 | 0.24 | 0.29 | N.A
SUR-Miner | 0.66 | 0.59 | 0.62 | 25.1
Ensemble | 0.83 | 0.76 | 0.79 | 23.66
DL+WE | 0.89 | 0.83 | 0.86 | 16.4
BERT | 0.62 | 0.55 | 0.58 | 9.62
ALBERT | 0.73 | 0.72 | 0.72 | 6.29
RoBERTa | 0.70 | 0.64 | 0.67 | 11.7
XLNET | 0.69 | 0.62 | 0.65 | 28.43
C-ALBERT-BASE | 0.79 | 0.70 | 0.74 | 6.81
C-ALBERT-MEDIUM | 0.85 | 0.83 | 0.84 | 5.87
C-ALBERT-LARGE | 0.93 | 0.82 | 0.87 | 6.79
C-RoBERTa-BASE | 0.82 | 0.75 | 0.78 | 13.09
C-RoBERTa-MEDIUM | 0.89 | 0.82 | 0.85 | 11.12
C-RoBERTa-LARGE | 0.92 | 0.92 | 0.92 | 11.45


$\textstyle{\mathcal{O}(\bar{x}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{1}}$$\textstyle{\mathcal{O}(2\bar{x}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{1}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{1}}$$\textstyle{\mathcal{O}((r_{1}-1)\bar{x}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{1}}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{1}}$$\scriptstyle{U_{2}}$$\scriptstyle{U_{n}}$$\textstyle{\mathcal{O}(\bar{x}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{2}}$$\textstyle{\mathcal{O}(2\bar{x}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{2}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{2}}$$\textstyle{\mathcal{O}((r_{2}-1)\bar{x}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{2}}$$\textstyle{\mathcal{O}(\bar{c})}$$\textstyle{\ldots}$$\textstyle{\ldots}$$\textstyle{\mathcal{O}(\bar{x}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{n}}$$\textstyle{\mathcal{O}(2\bar{x}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{n}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{n}}$$\textstyle{\mathcal{O}((r_{n}-1)\bar{x}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{n}}$

######  Example 9.1.

Fix some numbers $b_{i}\in\mathbb{Z}$, $0\leqslant b_{i}\leqslant r_{i}-1$.
Consider the subcollection

(9.3) $\\{\mathcal{O},\mathcal{O}(b\bar{x}_{i})\\}_{i=1\ldots n,b=1\ldots
b_{i}}.$

of (9.1). Let $\mathcal{T}_{l}$ be the subcategory in
$D^{b}(\operatorname{\mathrm{coh}}\mathbb{X})$ generated by (9.3), and
$\SS_{l}=\mathcal{T}\cap\operatorname{\mathrm{coh}}\mathbb{X}$. Endomorphism
algebra of (9.3) is the path algebra of the quiver

(9.4)
$\textstyle{\mathcal{O}(\bar{x}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(b_{1}\bar{x}_{1})}$$\textstyle{Q_{l}\colon}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots}$$\textstyle{\mathcal{O}(\bar{x}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(b_{n}\bar{x}_{n})}$

The vertex-like collection in $\SS_{l}$ generating $\mathcal{T}_{l}$ is the
left dual exceptional collection to (9.3) (see [3] for the definition and
details):

(9.5)
$\textstyle{S_{1,b_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1,b_{1}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots}$$\textstyle{\mathcal{O}}$$\textstyle{S_{n,b_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{n,b_{n}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{n,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where
$S_{i,j}:=\operatorname{\mathrm{coker}}(\mathcal{O}((j-1)\bar{x}_{i})\to\mathcal{O}(j\bar{x}_{i}))$
is a torsion sheaf supported at $x_{i}$. Note that $S_{ij}\cong S_{i,j+r_{i}}$
and $\tau S_{ij}\cong S_{i,j-1}$. Arrows in (9.5) denote
$\operatorname{\textup{Ext}}^{1}$ spaces, hence the
$\operatorname{\textup{Ext}}$-quiver of (9.5) is $Q_{l}$, and
$\SS_{l}\cong\mathrm{mod}_{0}{-}\Bbbk Q_{l}\cong\mathrm{mod{-}}\Bbbk Q_{l}$.
Note that $\SS_{l}$ is in general a mixed subcategory.

Now consider the remaining part of (9.2):

(9.6) $\\{\mathcal{O}(b\bar{x}_{i}),\mathcal{O}(\bar{c})\\}_{i=1\ldots
n,b=b_{i}+1\ldots r_{i}-1}.$

Let $\mathcal{T}_{r}$ be the subcategory in
$D^{b}(\operatorname{\mathrm{coh}}\mathbb{X})$ generated by (9.6), and
$\SS_{r}=\mathcal{T}_{r}\cap\operatorname{\mathrm{coh}}\mathbb{X}$.
Endomorphism algebra of (9.6) is the path algebra of the quiver

(9.7)
$\textstyle{\mathcal{O}((b_{1}+1)\bar{x}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}((r_{1}-1)\bar{x}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{r}\colon}$$\textstyle{\ldots}$$\textstyle{\mathcal{O}(\bar{c})}$$\textstyle{\mathcal{O}((b_{n}+1)\bar{x}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}((r_{n}-1)\bar{x}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The vertex-like collection in $\SS_{r}$ generating $\mathcal{T}_{r}$ is
obtained from the right dual exceptional collection to (9.6) by mutating
$\mathcal{O}(\bar{c})$ to the right end:

(9.8)
$\textstyle{S_{1,r_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1,r_{1}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1,b_{1}+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots}$$\textstyle{\mathcal{O}(\bar{c}-\sum(r_{i}-b_{i}-1)\bar{x}_{i})}$$\textstyle{S_{n,r_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{n,r_{n}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{n,b_{n}+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The $\operatorname{\textup{Ext}}$-quiver of (9.8) is
$Q_{r}^{\operatorname{\mathrm{op}}}$ (and not $Q_{r}$), hence
$\SS_{r}\cong\mathrm{mod}_{0}{-}\Bbbk
Q_{r}^{\operatorname{\mathrm{op}}}\cong\mathrm{mod{-}}\Bbbk
Q_{r}^{\operatorname{\mathrm{op}}}$. Subcategory $\SS_{r}$ is also mixed.

######  Example 9.2.

In Example 9.1 let $n=4$, $r_{1}=\ldots=r_{4}=5$, $b_{1}=\ldots=b_{4}=2$. Then
quivers $Q_{1}$ and $Q_{2}^{\operatorname{\mathrm{op}}}$ are affine Dynkin
quivers of type $\widetilde{D}_{4}$. Categories $\mathcal{T}_{l}$ and
$\mathcal{T}_{r}$ are equivalent to $D^{b}_{0}(\Bbbk\widetilde{D}_{4})\cong
D^{b}(\mathrm{mod{-}}\Bbbk\widetilde{D}_{4})\cong
D^{b}(\operatorname{\mathrm{coh}}\mathbb{X}_{2,2,2})$, where
$\mathbb{X}_{2,2,2}$ is a domestic weighted projective line of type $2,2,2$.
Hence both components of semi-orthogonal decomposition

$D^{b}(\operatorname{\mathrm{coh}}\mathbb{X})=\langle\mathcal{T}_{l},\mathcal{T}_{r}\rangle$

are equivalent to the derived category of a weighted projective line, and are
not curve-like.

######  Example 9.3.

Assume that $n\geqslant 2$. In collection (9.1), make mutation of
$\mathcal{O}(\bar{c})$ to the left end. One gets an exceptional collection

(9.9)
$\\{\mathcal{O}(-\bar{c}+\sum_{i=1}^{n}(r_{i}-1)\bar{x}_{i}),\mathcal{O},\mathcal{O}(b\bar{x}_{i})\\}_{i=1\ldots
n,b=1\ldots r_{i}-1}.$

Denote
$\mathcal{L}:=\mathcal{O}(-\bar{c}+\sum_{i=1}^{n}(r_{i}-1)\bar{x}_{i})$, then
$(\mathcal{L},\mathcal{O})$ is an exceptional pair. One has

$\operatorname{\textup{Hom}}(\mathcal{L},\mathcal{O})\cong\operatorname{\textup{Hom}}(\mathcal{O},\mathcal{O}((1-n)\bar{c}+\sum_{i}\bar{x}_{i}))=0$

and

$\operatorname{\textup{Ext}}^{1}(\mathcal{L},\mathcal{O})\cong\operatorname{\textup{Hom}}(\mathcal{O},\mathcal{L}(\bar{\omega}))^{*}\cong\operatorname{\textup{Hom}}(\mathcal{O},\mathcal{O}((n-3)\bar{c}+\sum_{i}(r_{i}-2)\bar{x}_{i}))^{*}\cong\Bbbk^{n-2}.$

Therefore, pair $(\mathcal{L},\mathcal{O})$ is vertex-like and its
$\operatorname{\textup{Ext}}$-quiver is the $m$-Kronecker quiver
$\mathrm{K}_{m}$, where $m=n-2$. Hence
$\mathcal{T}=\langle\mathcal{L},\mathcal{O}\rangle\subset
D^{b}(\operatorname{\mathrm{coh}}\mathbb{X})$ is equivalent to
$D^{b}_{0}(\Bbbk\mathrm{K}_{m})$, this subcategory is generated by an
exceptional collection and therefore admissible. Also $\mathcal{T}$ is clearly
torsion-free. If $n=4$ then $m=2$ and $\mathcal{T}$ is equivalent to
$D^{b}(\operatorname{\mathrm{coh}}\mathbb{P}^{1})$ but is not curve-like.

Suppose in addition that all $r_{i}=2$. Then ${}^{\perp}\mathcal{T}$ is
generated by completely orthogonal collection of line bundles
$\mathcal{O}(\bar{x}_{1}),\ldots,\mathcal{O}(\bar{x}_{n})$ and hence is
torsion-free. In this case one has a semi-orthogonal decomposition of
$D^{b}(\operatorname{\mathrm{coh}}\mathbb{X})$ in two torsion-free
subcategories.

######  Example 9.4.

Let $X$ be a connected smooth projective curve of genus $g\geqslant 1$. Let
$L$ be the family of all line bundles of degree $0$ on $X$. Clearly, $L$ is a
vertex-like family. By Riemann-Roch Theorem, for any
$\mathcal{L}_{1},\mathcal{L}_{2}\in L$

$\dim\operatorname{\textup{Ext}}^{1}(\mathcal{L}_{1},\mathcal{L}_{2})=\begin{cases}g-1&\mathcal{L}_{1}\not\cong\mathcal{L}_{2},\\\
g&\mathcal{L}_{1}\cong\mathcal{L}_{2}.\end{cases}$

By Corollary 4.14, one has equivalences

$\langle L\rangle\cong D^{b}_{0}(\Bbbk Q),\quad\langle
L\rangle_{\operatorname{\mathrm{Ab}}}\cong\mathrm{mod}_{0}{-}\Bbbk Q,$

where $Q$ is the following quiver: its set of vertices has the cardinality of
$L$ (and of $\Bbbk$), any vertex has $g$ loops and any two distinct vertices
are connected by $g-1$ arrows in both directions.

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