# Multicomponent relativistic dissipative fluid dynamics from the Boltzmann
equation

Jan A. Fotakis<EMAIL_ADDRESS>Institut für Theoretische Physik,
Johann Wolfgang Goethe–Universität, Max-von-Laue-Str. 1, D–60438 Frankfurt am
Main, Germany  Etele Molnár  Institut für Theoretische Physik, Johann Wolfgang
Goethe–Universität, Max-von-Laue-Str. 1, D–60438 Frankfurt am Main, Germany
Incubator of Scientific Excellence–Centre for Simulations of Superdense
Fluids, University of Wrocław, pl. M. Borna 9, PL-50204 Wrocław, Poland  Harri
Niemi  Institut für Theoretische Physik, Johann Wolfgang Goethe–Universität,
Max-von-Laue-Str. 1, D–60438 Frankfurt am Main, Germany  Department of
Physics, University of Jyväskylä, P.O. Box 35, FI-40014 University of
Jyväskylä, Finland  Helsinki Institute of Physics, P.O. Box 64, FI-00014
University of Helsinki, Finland  Carsten Greiner  Institut für Theoretische
Physik, Johann Wolfgang Goethe–Universität, Max-von-Laue-Str. 1, D–60438
Frankfurt am Main, Germany  Dirk H. Rischke  Institut für Theoretische Physik,
Johann Wolfgang Goethe–Universität, Max-von-Laue-Str. 1, D–60438 Frankfurt am
Main, Germany  Helmholtz Research Academy Hesse for FAIR, Campus Riedberg,
Max-von-Laue-Str. 12, D-60438 Frankfurt am Main, Germany

###### Abstract

We derive multicomponent relativistic second-order dissipative fluid dynamics
from the Boltzmann equations for a reactive mixture of $N_{\text{spec}}$
particle species with $N_{q}$ intrinsic quantum numbers (e.g. electric charge,
baryon number, and strangeness) using the method of moments. We obtain the
continuity equations for multiple conserved charges as well as the
conservation equations for the total energy and momentum in the single-fluid
approximation. These $4+N_{q}$ conservation laws are closed by deriving the
second-order equations of motion for the dissipative quantities in the
$(10+4N_{q})$-moment approximation. The resulting fluid-dynamical equations
are formally similar to those of a single-component system, but feature
different thermodynamic relations and transport coefficients. We derive
general relations for all transport coefficients and compute them explicitly
in the ultrarelativistic limit.

##  I Introduction

Determining properties of strong-interaction matter from experimental data
measured in high-energy heavy-ion experiments at BNL-RHIC and CERN-LHC is
largely based on relativistic fluid-dynamical modeling, see, e.g. Refs. [1, 2,
3, 4, 5]. Consequently, relativistic fluid dynamics has become an
indispensable tool in the description of the dynamical evolution of
relativistic nuclear collisions.

The state of the art of fluid-dynamical modeling of relativistic nuclear
matter is based on the relativistic second-order dissipative fluid-dynamical
theory of Israel and Stewart [6]. This theory and method is based on the
pioneering works of Grad [7] and Müller [8] and was originally formulated for
a simple fluid, i.e. a fluid with a single conserved charge. On the other
hand, one of the basic features of the fluid created in high-energy nuclear
collisions is its multicomponent nature. For example, hadronic matter produced
in nuclear collisions consists of a multitude of different types of hadrons,
where each hadron species carries multiple intrinsic quantum numbers like
baryon number $B$, electric charge $Q$, and strangeness $S$. Therefore, a
multicomponent extension of relativistic fluid-dynamical theories that
explicitly accounts for multiple conserved charges is required for a proper
description of heavy-ion collisions.

Previous attempts to derive second-order fluid-dynamical equations of motion
for relativistic multicomponent mixtures include the pioneering work by
Prakash et al. [9], which is an extension of Israel-Stewart theory to
multicomponent mixtures, and the works by Monnai and Hirano [10, 11], which
generalize the equations of Ref. [9] by including additional second-order
terms as well as providing Onsager’s reciprocity relations [12, 13] for the
transport coefficients. More recent developments by Kikuchi et al. [14] apply
the renormalization-group method to rederive the second-order equations of
motion including additional second-order terms in the dissipative quantities
resulting from the non-linear part of the collision integral.

Furthermore it has been shown in Refs. [15, 16, 17, 18] that many features of
multicomponent systems depend on the detailed coupling between the diffusion
currents associated with different conserved charges. In addition, the mapping
between the state of the fluid and the corresponding momentum distribution of
particles plays an important role [19, 20].

In this work, we present a derivation of multicomponent relativistic second-
order dissipative fluid dynamics for a reactive mixture of $N_{\text{spec}}$
species with $N_{q}$ conserved quantum numbers by generalizing the method of
moments established for single-component systems by Denicol et al. [21]. By
summing the dynamical equations of motion describing the individual particle
species we obtain a reduced set of equations corresponding to the so-called
“single-fluid” description of a multicomponent fluid. We derive the continuity
equations for each conserved quantum charge as well as the conservation laws
for total energy and momentum in this single-fluid approximation. These
$4+N_{q}$ equations of motion are closed by providing second-order equations
of motion in the $(10+4N_{q})$-moment approximation for the dissipative
quantities. The latter equations are formally similar to the relaxation
equations of a single-component system but feature different transport
coefficients, which contain the microscopic interactions of all components.
Our approach reproduces the results of Ref. [21] for a single-component fluid
in the $14$-moment approximation, i.e. for $N_{\text{spec}}=N_{q}=1$.

This paper is organized as follows. In Sec. II, we introduce the Boltzmann
equation, thermodynamic quantities in local equilibrium, as well as fluid-
dynamical quantities, both in and out of equilibrium. The definition of the
local rest frame, the matching conditions, and the conservation equations are
also given. In Sec. III we derive the equations of motion for the irreducible
moments from the Boltzmann equation, linearize the collision term, and discuss
the Navier-Stokes limit and the order-of-magnitude approximation. Finally, the
second-order dissipative fluid-dynamical equations of motion in the
$(10+4N_{q})$-moment approximation are derived and discussed. We conclude this
work with a summary in the final section. Details of the calculations are
delegated to several appendices. They also contain explicit expressions for
all second-order transport coefficients, as well as an explicit calculation of
the transport coefficients in the ultrarelativistic limit.

Throughout this paper we adopt natural units, $\hbar=c=k_{B}=1$, and work in
flat Minkowski space-time with metric tensor
$g_{\mu\nu}=\text{diag}(1,-1,-1,-1)$. The time-like fluid four-velocity is
denoted by $u^{\mu}=\gamma(1,\mathbf{v})^{T}$, with normalization
$u^{\mu}u_{\mu}\equiv 1$, where $\mathbf{v}$ is the three-velocity and
$\gamma=(1-\mathbf{v}^{2})^{-1/2}$. In the local rest (LR) frame of the fluid,
$u_{\text{LR}}^{\mu}=(1,\mathbf{0})^{T}$. The rank-two projection operator
onto the three-space orthogonal to $u^{\mu}$ is defined as
$\Delta^{\mu\nu}\equiv g^{\mu\nu}-u^{\mu}u^{\nu}$. We define the projection of
any four-vector $A^{\mu}$ onto the three-dimensional subspace orthogonal to
$u^{\mu}$ as
$A^{\left\langle\mu\right\rangle}\equiv\Delta_{\nu}^{\mu}A^{\nu}$. The
generalization to projection tensors of rank $2\ell$, denoted by
$\Delta_{\nu_{1}\cdots\nu_{\ell}}^{\mu_{1}\cdots\mu_{\ell}}$, is constructed
using the elementary projection operator $\Delta^{\mu}_{\nu}$. The irreducible
symmetric, traceless, and orthogonal projection of a rank-$\ell$ tensor
$A^{\mu_{1}\cdots\mu_{\ell}}$ is denoted as
$A^{\left\langle\mu_{1}\cdots\mu_{\ell}\right\rangle}\equiv\Delta_{\nu_{1}\cdots\nu_{\ell}}^{\mu_{1}\cdots\mu_{\ell}}A^{\nu_{1}\cdots\nu_{\ell}}$.
For example, the rank-four symmetric, traceless, and orthogonal projection
operator is defined as
$\Delta_{\alpha\beta}^{\mu\nu}\equiv\frac{1}{2}(\Delta_{\alpha}^{\mu}\Delta_{\beta}^{\nu}+\Delta_{\beta}^{\mu}\Delta_{\alpha}^{\nu})-\frac{1}{3}\Delta^{\mu\nu}\Delta_{\alpha\beta}$,
hence
$A^{\left\langle\mu\nu\right\rangle}\equiv\Delta_{\alpha\beta}^{\mu\nu}A^{\alpha\beta}$.

The four-momentum of a particle of species $i$ is denoted by
$k_{i}^{\mu}=(k_{i}^{0},\mathbf{k}_{i})^{T}$, which is normalized to the
corresponding species rest mass squared, $k_{i}^{\mu}k_{i,\mu}=m_{i}^{2}$. The
energy of a particle of species $i$ is defined as
$E_{i,\mathbf{k}}=k_{i}^{\mu}u_{\mu}$, and coincides with the on-shell energy
$k_{i}^{0}=\sqrt{\mathbf{k}_{i}^{2}+m_{i}^{2}}$ in the LR frame of the fluid.
The orthogonal projection of the four-momentum is
$k_{i}^{\left\langle\mu\right\rangle}\equiv\Delta_{\nu}^{\mu}k_{i}^{\nu}$, and
in the LR frame it reduces to the three-momentum $\mathbf{k}_{i}$.

The comoving derivative $D\equiv u^{\mu}\partial_{\mu}$ of any four-vector
$A^{\mu}=(A^{0},\mathbf{A})^{T}$ is denoted by $\dot{A^{\mu}}\equiv
u^{\nu}\partial_{\nu}{A^{\mu}}=DA^{\mu}$, while the space-time four-gradient
is $\nabla_{\nu}A^{\mu}\equiv\Delta_{\nu}^{\alpha}\partial_{\alpha}A^{\mu}$.
Note that in the LR frame these relativistic space-time derivatives reduce to
the usual time and three-space derivatives, $\partial_{t}A^{\mu}$ and
$\mathbf{\nabla}\otimes\mathbf{A}$. Thus, the four-derivative is decomposed as
$\partial_{\mu}\equiv u_{\mu}D+\nabla_{\mu}$, hence the relativistic Cauchy-
Stokes decomposition reads, $\partial_{\mu}u_{\nu}\equiv
u_{\mu}\dot{u}_{\nu}+\nabla_{\mu}u_{\nu}=u_{\mu}\dot{u}_{\nu}+\frac{1}{3}\theta\Delta_{\mu\nu}+\sigma_{\mu\nu}+\omega_{\mu\nu}$.
Here we have defined the expansion scalar, $\theta\equiv\nabla_{\mu}u^{\mu}$,
the shear tensor
$\sigma^{\mu\nu}\equiv\nabla^{\left\langle\mu\right.}u^{\left.\nu\right\rangle}=\frac{1}{2}(\nabla^{\mu}u^{\nu}+\nabla^{\nu}u^{\mu})-\frac{1}{3}\theta\Delta^{\mu\nu}$,
and the vorticity
$\omega^{\mu\nu}\equiv\frac{1}{2}(\nabla^{\mu}u^{\nu}-\nabla^{\nu}u^{\mu})$,
such that $\sigma^{\mu\nu}u_{\nu}=\omega^{\mu\nu}u_{\nu}=0$.

Moreover, we label the conserved charge types in the system with the letter
$q$, which will be treated as an index running over $B$ (baryon number), $Q$
(electric charge), and $S$ (strangeness) for the case a strong-interaction
system. For notational convenience, we employ the following notation for the
sums over charge types,

$\displaystyle\sum_{q}^{\\{B,Q,S\\}}\equiv\sum_{q\,=\,B,Q,S}.$

##  II Reactive mixtures in kinetic theory and fluid dynamics

In this section we first introduce the Boltzmann equation for a reactive
mixture with special emphasis on the collision term. Before we discuss fluid-
dynamical quantities in local equilibrium and out of equilibrium, we study
thermodynamic quantities in local equilibrium. This is followed by a
discussion of the matching conditions and the choices for the local rest
frame. Finally, we list the conservation equations of second-order dissipative
fluid dynamics for a multicomponent fluid.

###  II.1 The Boltzmann equation for a reactive mixture

A mixture of $N_{\text{spec}}$ different (elementary) particle species (i.e.
different chemical components) is characterized by the single-particle
distribution functions for each particle species $i$,
$f\left(x,k_{i}\right)\equiv f_{i,\mathbf{k}}$, where we label the particle
species by a lower index $i$. The space-time evolution of the distribution
function of species $i$ is determined by the relativistic Boltzmann equation
[22, 23],

$k_{i}^{\mu}\partial_{\mu}f_{i,\mathbf{k}}\equiv
C_{i}\left(x,k_{i}\right)=\sum_{j\,=\,1}^{N_{\text{spec}}}C_{ij}[f]\;,$ (1)

where we neglect any external forces and assume binary collisions only for the
sake of simplicity. For binary inelastic, i.e. reactive collisions, the
initial and final particles species may be different, $i+j\rightarrow a+b$,
such that the collision term reads

$C_{ij}[f]=\frac{1}{2}\sum_{a,b\,=\,1}^{N_{\text{spec}}}\int\mathrm{d}K_{j}^{\prime}\mathrm{d}P_{a}\mathrm{d}P_{b}^{\prime}\left(W_{ab\rightarrow
ij}^{pp^{\prime}\rightarrow
kk^{\prime}}f_{a,\mathbf{p}}f_{b,\mathbf{p}^{\prime}}\tilde{f}_{i,\mathbf{k}}\tilde{f}_{j,\mathbf{k}^{\prime}}-W_{ij\rightarrow
ab}^{kk^{\prime}\rightarrow
pp^{\prime}}f_{i,\mathbf{k}}f_{j,\mathbf{k}^{\prime}}\tilde{f}_{a,\mathbf{p}}\tilde{f}_{b,\mathbf{p}^{\prime}}\right)\;,$
(2)

where $\tilde{f}_{i,\mathbf{k}}=1-a_{i}f_{i,\mathbf{k}}/g_{i}$, with
$a_{i}=\pm 1$ for fermions/bosons, and $a_{i}\rightarrow 0$ for classical
particles, respectively. Here, $g_{i}$ is the spin degeneracy of particle
species $i$, while the Lorentz-invariant integration measure is
$\mathrm{d}K_{i}=\mathrm{d}^{3}\mathbf{k}_{i}/[(2\pi)^{3}k_{i}^{0}]$. Here,
the factor $1/2$ corrects for the double counting when integrating over the
momenta of particles in the initial and final state [22].

The transition probabilities respect certain symmetry properties under
exchange of particles in the initial and final state, $W_{ij\rightarrow
ab}^{kk^{\prime}\rightarrow pp^{\prime}}=W_{ji\rightarrow
ba}^{k^{\prime}k\rightarrow p^{\prime}p}$, as well as the bilateral
normalization property of the microscopic processes [22],

$\displaystyle\sum_{a,b\,=\,1}^{N_{\text{spec}}}\int\mathrm{d}P_{a}\mathrm{d}P_{b}^{\prime}\,W_{ab\rightarrow
ij}^{pp^{\prime}\rightarrow
kk^{\prime}}=\sum_{a,b\,=\,1}^{N_{\text{spec}}}\int\mathrm{d}P_{a}\mathrm{d}P_{b}^{\prime}\,W_{ij\rightarrow
ab}^{kk^{\prime}\rightarrow pp^{\prime}}\;.$ (3)

In the absence of a reaction threshold, this relation is invariant under time
reversal, $W_{ij\rightarrow ab}^{kk^{\prime}\rightarrow
pp^{\prime}}=W_{ab\rightarrow ij}^{pp^{\prime}\rightarrow kk^{\prime}}$, i.e.
microscopic reversibility or detailed balance. In the case of a binary process
the transition probability is [24, 25]

$\displaystyle W_{ij\rightarrow ab}^{kk^{\prime}\rightarrow
pp^{\prime}}=\frac{(2\pi)^{4}}{16}|\mathcal{M}_{ij\rightarrow
ab}(\sqrt{s},\Omega)|^{2}\,\delta^{(4)}\left(k^{\mu}_{i}+k^{\prime\mu}_{j}-p^{\mu}_{a}-p^{\prime\mu}_{b}\right),$
(4)

where $|\mathcal{M}_{ij\rightarrow ab}(\sqrt{s},\Omega)|^{2}$ is the Lorentz-
invariant transition probability averaged over incoming and summed over
outgoing spin states. It only depends on the total center-of-momentum (CM)
energy squared,
$s\equiv(k^{\mu}_{i}+k^{\prime\mu}_{j})^{2}=(p^{\mu}_{a}+p^{\prime\mu}_{b})^{2}$
and the solid angle $\Omega$ under which outgoing particles are scattered with
respect to the direction of the incoming particles, while the
$\delta^{(4)}$-function ensures the conservation of energy and momentum in
each binary collision. The differential cross section in the CM frame, where
$\mathbf{k}_{i}+\mathbf{k}^{\prime}_{j}=\mathbf{p}_{a}+\mathbf{p}^{\prime}_{b}\equiv
0$ and $k^{0}_{i}+k^{\prime 0}_{j}=p^{0}_{a}+p^{\prime 0}_{b}\equiv\sqrt{s}$,
is defined via the invariant transition probability as

$\displaystyle\mathrm{d}\sigma_{ij\rightarrow
ab}(\sqrt{s},\Omega)=\frac{1}{64\pi^{2}s}\frac{p_{ab}}{p_{ij}}|\mathcal{M}_{ij\rightarrow
ab}(\sqrt{s},\Omega)|^{2}\,\mathrm{d}\Omega\;,$ (5)

where the incoming $p_{ij}$ and outgoing $p_{ab}$ momenta in the CM frame are

$\displaystyle
p_{ij}\equiv\frac{1}{2\sqrt{s}}\sqrt{\Big{(}s-(m_{i}+m_{j})^{2}\Big{)}\Big{(}s-(m_{i}-m_{j})^{2}\Big{)}}\;.$
(6)

We note that in the elastic limit $p_{ij}=p_{ab}$. We define the total
(integrated over angles) cross section as

$\displaystyle\sigma_{\text{tot},\,ij\rightarrow ab}\equiv
2\pi\gamma_{ab}\int\limits_{-1}^{1}\mathrm{d}\cos\vartheta\,\frac{\mathrm{d}\sigma_{ij\rightarrow
ab}(\sqrt{s},\Omega)}{\mathrm{d}\Omega}\;.$ (7)

Here, the symmetry factor $\gamma_{ab}\equiv 1-\delta_{ab}/2$ accounts for the
double counting when integrating over the momenta of indistinguishable
particles in the final state. In the case of isotropic scattering the
differential cross section is

$\displaystyle\frac{\mathrm{d}\sigma_{ij\rightarrow
ab}(\sqrt{s})}{\mathrm{d}\Omega}\overset{\text{isotropic}}{=}\frac{\sigma_{\text{tot},ij\rightarrow
ab}(\sqrt{s})}{4\pi\gamma_{ab}}\;.$ (8)

In the elastic limit the transition rates in Eq. (2) are defined as [22]

$\displaystyle W_{ij\rightarrow ab}^{kk^{\prime}\rightarrow
pp^{\prime}}\equiv\gamma_{ij}\left(\delta_{ia}\delta_{jb}+\delta_{ib}\delta_{ja}\right)W_{ij}^{kk^{\prime}\rightarrow
pp^{\prime}}\;,$ (9)

where

$\displaystyle W_{ij}^{kk^{\prime}\rightarrow
pp^{\prime}}\equiv\frac{(2\pi)^{4}}{16}|\mathcal{M}_{ij\rightarrow
ij}(\sqrt{s},\Omega)|^{2}\,\delta^{(4)}\left(k^{\mu}_{i}+k^{\prime\mu}_{j}-p^{\mu}_{i}-p^{\prime\mu}_{j}\right)\;.$
(10)

Note that in the above expression we have already introduced the symmetry
factor, even though we have not yet integrated over the momentum. For later
convenience, we can use Eqs. (5) and (8) to rewrite Eq. (9) in the case of
isotropic elastic scattering in terms of the total cross section defined in
Eq. (7),

$\displaystyle W_{ij\rightarrow ab}^{kk^{\prime}\rightarrow
pp^{\prime}}\equiv\left(\delta_{ia}\delta_{jb}+\delta_{ib}\delta_{ja}\right)(2\pi)^{6}s\frac{\sigma_{\text{tot},ij}}{4\pi}\,\delta^{(4)}\left(k^{\mu}_{i}+k^{\prime\mu}_{j}-p^{\mu}_{i}-p^{\prime\mu}_{j}\right)\;.$
(11)

###  II.2 Local equilibrium and associated thermodynamic quantities

In general the single-particle distribution function $f_{i,\mathbf{k}}$ for
every species $i$ can be decomposed into an equilibrium part,
$f_{i,\mathbf{k}}^{(0)}$, and an out-of-equilibrium part, $\delta
f_{i,\mathbf{k}}$, as

$f_{i,\mathbf{k}}=f_{i,\mathbf{k}}^{(0)}+\delta f_{i,\mathbf{k}}\;,$ (12)

where the local-equilibrium distribution function of species $i$ is given by
the Jüttner distribution function [26],

$f_{i,\mathbf{k}}^{\left(0\right)}=g_{i}\left[\exp\left(\frac{E_{i,\mathbf{k}}-\mu_{i}}{T}\right)+a_{i}\right]^{-1}\;.$
(13)

Due to detailed balance, the collision integral vanishes identically for the
local-equilibrium distribution function [22]. Here, $T\equiv 1/\beta$ is the
temperature and $\mu_{i}$ is the chemical potential of species $i$ as defined
in the local rest frame. The exact form of the non-equilibrium part of the
distribution function $\delta f_{i,\mathbf{k}}$ will be clarified later.

In various cases of interest such as in high-energy particle physics or
relativistic heavy-ion collisions there are inelastic collisions where the
particle number corresponding to a given species is not conserved due to
particle creation and annihilation processes (i.e. various chemical
reactions). Such strong-interaction matter is therefore described by a few
conserved intrinsic quantum numbers, such as electric charge, baryon number,
and strangeness.

This means that in local equilibrium the chemical potential $\mu_{i}$ of a
given particle $i$ may be expressed in terms of $N_{q}$ chemical potentials of
conserved quantum ”charges”,

$\mu_{i}\left(\left\\{\mu_{q}\right\\}\right)\equiv\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}\mu_{q}=B_{i}\mu_{B}+Q_{i}\mu_{Q}+S_{i}\mu_{S}\;,$
(14)

where
$\left\\{\mu_{q}\right\\}\equiv\left\\{\mu_{B},\mu_{Q},\mu_{S}\right\\}$, with
$\mu_{B}$, $\mu_{Q}$, and $\mu_{S}$, being the baryon, electric and
strangeness chemical potentials, respectively, while $B_{i}$, $Q_{i}$, and
$S_{i}$ are the baryon number, electric charge, and strangeness of the
respective particle species $i$.

Now we introduce the ratio of chemical potential over temperature
corresponding to particle species $i$ as $\alpha_{i}\equiv\mu_{i}/T$, as well
as the ratio of chemical potential over temperature for the conserved quantum
charges, $\alpha_{q}\equiv\mu_{q}/T$. Applying the chain rule, we obtain from
Eq. (14)

$\mathrm{d}\alpha_{i}\left(\left\\{\alpha_{q}\right\\}\right)\equiv\sum_{q}^{\left\\{B,Q,S\right\\}}\frac{\partial\alpha_{i}}{\partial\alpha_{q}}\mathrm{d}\alpha_{q}=\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}\mathrm{d}\alpha_{q}\;,$
(15)

where the intrinsic quantum number of particle species $i$ can also be
obtained as
$q_{i}\equiv\partial\alpha_{i}(\\{\alpha_{q^{\prime}}\\})/\partial\alpha_{q}$.
Note that the infinitesimal change in any variable $A$, here denoted by $dA$,
can be interchangeably used for the comoving derivative, $DA$, as well as the
space-time four-gradient, $\nabla^{\mu}A$.

In local thermodynamic equilibrium, we define the following rank-$n$ tensor
moments of given power $r\geq 0$ in energy $E_{i,\mathbf{k}}^{r}$ for any
given particle species $i$ as

$\mathcal{I}_{i,r}^{\mu_{1}\cdots\mu_{n}}\equiv\int\mathrm{d}K_{i}E_{i,\mathbf{k}}^{r}k_{i}^{\mu_{1}}\cdots
k_{i}^{\mu_{n}}f_{i,\mathbf{k}}^{(0)}=\left\langle
E_{\mathbf{k}}^{r}k^{\mu_{1}}\cdots k^{\mu_{n}}\right\rangle_{i,0}\;,$ (16)

where the angular brackets are the abbreviation of the integrals,

$\left\langle\cdots\right\rangle_{i,0}\equiv\int\mathrm{d}K_{i}\left(\cdots\right)_{i}f_{i,\mathbf{k}}^{\left(0\right)}\;.$
(17)

Following Ref. [6] we expand the equilibrium moments (16) in terms of the flow
velocity and the associated orthogonal projection operator, which leads to the
following expression,

$\mathcal{I}_{i,r}^{\mu_{1}\cdots\mu_{n}}=\sum_{m\,=\,0}^{\left[n/2\right]}\left(-1\right)^{m}\frac{n!}{2^{m}m!\left(n-2m\right)!}\Delta^{\left(\mu_{1}\mu_{2}\right.}\cdots\Delta^{\mu_{2m-1}\mu_{2m}}u^{\mu_{2m+1}}\cdots
u^{\left.\mu_{n}\right)}I_{i,r+n,m}\;,$ (18)

where $n$ and $m$ are natural numbers, while $\left[n/2\right]\leq n/2$
denotes the largest integer divisible by two. The coefficient
$n!/[2^{m}m!(n-2m)!]$ counts the number of distinct terms in the symmetrized
tensor products
$\Delta^{\left(\mu_{1}\mu_{2}\right.}\cdots\Delta^{\mu_{2m-1}\mu_{2m}}$
$\times u^{\mu_{2m+1}}\cdots u^{\left.\mu_{n}\right)}$. The coefficients
$I_{i,r+n,m}$ are thermodynamic integrals which only depend on $\alpha_{i}$
and $\beta$,

$I_{i,nm}\left(\alpha_{i},\beta\right)=\frac{\left(-1\right)^{m}}{\left(2m+1\right)!!}\left\langle
E_{\mathbf{k}}^{n-2m}\left(\Delta_{\mu\nu}k^{\mu}k^{\nu}\right)^{m}\right\rangle_{i,0}\;,$
(19)

where $\left(2m+1\right)!!\equiv\left(2m+1\right)!/(2^{m}m!)$ is the double
factorial of an odd integer.

The total derivative of the thermodynamic integrals with respect to the
variables $\alpha_{i}$ and $\beta$ reads

$\displaystyle\mathrm{d}I_{i,nm}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial
I_{i,nm}}{\partial\alpha_{i}}\right)_{\beta}\mathrm{d}\alpha_{i}+\left(\frac{\partial
I_{i,nm}}{\partial\beta}\right)_{\alpha_{i}}\mathrm{d}\beta$ (20)
$\displaystyle\equiv$ $\displaystyle
J_{i,nm}\mathrm{d}\alpha_{i}-J_{i,n+1,m}\mathrm{d}\beta,$

where we have defined the auxiliary thermodynamic integrals

$J_{i,nm}\left(\alpha_{i},\beta\right)\equiv\left(\frac{\partial
I_{i,nm}}{\partial\alpha_{i}}\right)_{\beta}=\frac{\left(-1\right)^{m}}{\left(2m+1\right)!!}\int\mathrm{d}K_{i}E_{i,\mathbf{k}}^{n-2m}\left(\Delta_{\mu\nu}k_{i}^{\mu}k_{i}^{\nu}\right)^{m}f_{i,\mathbf{k}}^{\left(0\right)}\tilde{f}_{i,\mathbf{k}}^{\left(0\right)}\;.$
(21)

An integration by parts with
$df_{i,\mathbf{k}}^{\left(0\right)}/dE_{i,\mathbf{k}}=-\beta
f_{i,\mathbf{k}}^{\left(0\right)}\tilde{f}_{i,\mathbf{k}}^{\left(0\right)}$
leads to the following relation between the thermodynamic integrals,

$\beta J_{i,nm}=I_{i,n-1,m-1}+\left(n-2m\right)I_{i,n-1,m}\;.$ (22)

Furthermore, with Eq. (15) we obtain from Eq. (20) an expression for the total
derivative of the thermodynamic integrals of species $i$ in terms of the
conserved quantum charges,

$\displaystyle\mathrm{d}I_{i,nm}$ $\displaystyle=$
$\displaystyle\sum_{q}^{\left\\{B,Q,S\right\\}}\left(\frac{\partial
I_{i,nm}}{\partial\alpha_{q}}\right)_{\beta}\mathrm{d}\alpha_{q}+\left(\frac{\partial
I_{i,nm}}{\partial\beta}\right)_{\\{\alpha_{q}\\}}\mathrm{d}\beta$ (23)
$\displaystyle\equiv$
$\displaystyle\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}J_{i,nm}\mathrm{d}\alpha_{q}-J_{i,n+1,m}\mathrm{d}\beta\;,$

from which follows

$\left(\frac{\partial
I_{i,nm}}{\partial\alpha_{q}}\right)_{\beta}=q_{i}J_{i,nm}\;.$ (24)

For later use we define the thermodynamic integrals summed over all particle
species, which we denote by $I_{nm}$ and $J_{nm}$. Similarly, the
thermodynamic integrals of conserved quantum charges, $I_{nm}^{q}$,
$J_{nm}^{q}$, as well as the auxiliary thermodynamic quantities,
$I_{nm}^{qq^{\prime}}$, $J_{nm}^{qq^{\prime}}$, are defined as follows,

$\displaystyle I_{nm}$
$\displaystyle\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}I_{i,nm}\;,\quad\quad\;\;J_{nm}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}J_{i,nm}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left(\frac{\partial
I_{i,nm}}{\partial\alpha_{i}}\right)_{\beta}\;,$ (25) $\displaystyle
I_{nm}^{q}$
$\displaystyle\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}I_{i,nm}\;,\quad\;\;\;J_{nm}^{q}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}J_{i,nm}=\left(\frac{\partial
I_{nm}}{\partial\alpha_{q}}\right)_{\beta}\;,$ (26) $\displaystyle
I_{nm}^{qq^{\prime}}$
$\displaystyle\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}q_{i}^{\prime}I_{i,nm}\;,\quad
J_{nm}^{qq^{\prime}}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}q_{i}^{\prime}J_{i,nm}=\left(\frac{\partial
I_{nm}^{q}}{\partial\alpha_{q^{\prime}}}\right)_{\beta}\;.$ (27)

Now using Eq. (23) together with the above definitions, the total differential
of the thermodynamic integrals summed over all particle species as well as
that of the thermodynamic integral of a specific conserved charge read

$\displaystyle\mathrm{d}I_{nm}\left(\left\\{\alpha_{q}\right\\},\beta\right)$
$\displaystyle=$
$\displaystyle\sum_{q}^{\left\\{B,Q,S\right\\}}J_{nm}^{q}\mathrm{d}\alpha_{q}-J_{n+1,m}\mathrm{d}\beta\;,$
(28)
$\displaystyle\mathrm{d}I_{nm}^{q}\left(\left\\{\alpha_{q}\right\\},\beta\right)$
$\displaystyle=$
$\displaystyle\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}J_{nm}^{qq^{\prime}}\mathrm{d}\alpha_{q^{\prime}}-J_{n+1,m}^{q}\mathrm{d}\beta\;.$
(29)

In terms of physical quantities, we identify the thermodynamic integral of
particle species $i$ with indices $n=1$ and $m=0$ as the particle density,
$n_{i}=I_{i,10}$, while that with indices $n=2$ and $m=0$ is the energy
density, $e_{i}=I_{i,20}$. From Eqs. (20) and (23) we then obtain the standard
thermodynamic relations

$\displaystyle\mathrm{d}n_{i}$ $\displaystyle\equiv$
$\displaystyle\left(\frac{\partial
n_{i}}{\partial\alpha_{i}}\right)_{\beta}\mathrm{d}\alpha_{i}+\left(\frac{\partial
n_{i}}{\partial\beta}\right)_{\alpha_{i}}\mathrm{d}\beta=\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}J_{i,10}\mathrm{d}\alpha_{q}-J_{i,20}\mathrm{d}\beta\;,$
(30) $\displaystyle\mathrm{d}e_{i}$ $\displaystyle\equiv$
$\displaystyle\left(\frac{\partial
e_{i}}{\partial\alpha_{i}}\right)_{\beta}\mathrm{d}\alpha_{i}+\left(\frac{\partial
e_{i}}{\partial\beta}\right)_{\alpha_{i}}\mathrm{d}\beta=\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}J_{i,20}\mathrm{d}\alpha_{q}-J_{i,30}\mathrm{d}\beta\;.$
(31)

From these results and Eqs. (25) – (27), or directly from Eq. (29), we may
express the total differential of the density of conserved charge $q$ as

$\displaystyle\mathrm{d}n_{q}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\mathrm{d}n_{i}=\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\frac{\partial
n_{q}}{\partial\alpha_{q^{\prime}}}\mathrm{d}\alpha_{q^{\prime}}+\frac{\partial
n_{q}}{\partial\beta}\mathrm{d}\beta$ (32) $\displaystyle=$
$\displaystyle\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}J_{10}^{qq^{\prime}}\mathrm{d}\alpha_{q^{\prime}}-J_{20}^{q}\mathrm{d}\beta\equiv\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\left(\mathcal{T}^{-1}\right)_{qq^{\prime}}\mathrm{d}\alpha_{q^{\prime}}+\left(\mathcal{T}^{-1}\right)_{q0}\mathrm{d}\beta\;,$

and the total differential of the energy density as

$\displaystyle\mathrm{d}e$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}\mathrm{d}e_{i}=\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\frac{\partial
e}{\partial\alpha_{q^{\prime}}}\mathrm{d}\alpha_{q^{\prime}}+\frac{\partial
e}{\partial\beta}\mathrm{d}\beta$ (33) $\displaystyle=$
$\displaystyle\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}J_{20}^{q^{\prime}}\mathrm{d}\alpha_{q^{\prime}}-J_{30}\mathrm{d}\beta\equiv\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\left(\mathcal{T}^{-1}\right)_{0q^{\prime}}\mathrm{d}\alpha_{q^{\prime}}+\left(\mathcal{T}^{-1}\right)_{00}\mathrm{d}\beta\;,$

where we have defined the following inverse matrix,

$\left(\mathcal{T}^{-1}\right)_{qq^{\prime}}\equiv\left(\begin{array}[]{cc}\displaystyle\frac{\partial
e}{\partial\beta}&\displaystyle\frac{\partial
e}{\partial\alpha_{q^{\prime}}}\\\ \displaystyle\frac{\partial
n_{q}}{\partial\beta}&\displaystyle\frac{\partial
n_{q}}{\partial\alpha_{q^{\prime}}}\\\
\end{array}\right)=\left(\begin{array}[]{cccc}-J_{30}&J_{20}^{B}&J_{20}^{Q}&J_{20}^{S}\\\
-J_{20}^{B}&J_{10}^{BB}&J_{10}^{BQ}&J_{10}^{BS}\\\
-J_{20}^{Q}&J_{10}^{QB}&J_{10}^{QQ}&J_{10}^{QS}\\\
-J_{20}^{S}&J_{10}^{SB}&J_{10}^{SQ}&J_{10}^{SS}\end{array}\right)\;.$ (34)

Equations (32) and (33) can be solved for $d\beta$ and $d\alpha_{q}$,

$\displaystyle\mathrm{d}\beta$ $\displaystyle=$
$\displaystyle\mathcal{T}_{00}\mathrm{d}e+\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\mathcal{T}_{0q^{\prime}}\mathrm{d}n_{q^{\prime}}\;,$
(35) $\displaystyle\mathrm{d}\alpha_{q}$ $\displaystyle=$
$\displaystyle\mathcal{T}_{q0}\mathrm{d}e+\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\mathcal{T}_{qq^{\prime}}\mathrm{d}n_{q^{\prime}}\;.$
(36)

Note that the relations (32) and (33), or equivalently (35) and (36), encode
the thermodynamic response of the medium to perturbations and contain
information about the chemical composition and/or the equation of state. These
thermodynamic relations will be used later in the equations of motion.

###  II.3 Equilibrium fluid-dynamical quantities

The equilibrium moments (16), for $r=0$ and for the tensor ranks $\ell=1$ and
$\ell=2$, define the partial particle four-current and energy-momentum tensor,

$\displaystyle N_{i,0}^{\mu}$ $\displaystyle\equiv$
$\displaystyle\mathcal{I}_{i,0}^{\mu}=\int\mathrm{d}K_{i}k_{i}^{\mu}f_{i,\mathbf{k}}^{\left(0\right)}\equiv\left\langle
k^{\mu}\right\rangle_{i,0}\;,$ (37) $\displaystyle T_{i,0}^{\mu\nu}$
$\displaystyle\equiv$
$\displaystyle\mathcal{I}_{i,0}^{\mu\nu}=\int\mathrm{d}K_{i}k_{i}^{\mu}k_{i}^{\nu}f_{i,\mathbf{k}}^{\left(0\right)}\equiv\left\langle
k^{\mu}k^{\nu}\right\rangle_{i,0}\;.$ (38)

The tensor decomposition (18) of these quantities with respect to an arbitrary
time-like normalized flow velocity $u^{\mu}$ and the projection operator
$\Delta^{\mu\nu}$ reads

$\displaystyle N_{i,0}^{\mu}$ $\displaystyle\equiv$ $\displaystyle
I_{i,10}u^{\mu}=n_{i}u^{\mu}\;,$ (39) $\displaystyle T_{i,0}^{\mu\nu}$
$\displaystyle\equiv$ $\displaystyle
I_{i,20}u^{\mu}u^{\nu}-I_{i,21}\Delta^{\mu\nu}=e_{i}u^{\mu}u^{\nu}-P_{i}\Delta^{\mu\nu}\;.$
(40)

Tensor-projecting these quantities leads to the partial particle density,
energy density, and pressure of species $i$,

$\displaystyle n_{i}$ $\displaystyle\equiv$ $\displaystyle
N_{i,0}^{\mu}u_{\mu}=\left\langle
E_{\mathbf{k}}\right\rangle_{i,0}=I_{i,10}\equiv\mathcal{I}_{i,1}\;,$ (41)
$\displaystyle e_{i}$ $\displaystyle\equiv$ $\displaystyle
T_{i,0}^{\mu\nu}u_{\mu}u_{\nu}=\left\langle
E_{\mathbf{k}}^{2}\right\rangle_{i,0}=I_{i,20}\equiv\mathcal{I}_{i,2}\;,$ (42)
$\displaystyle P_{i}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{3}\,T_{i,0}^{\mu\nu}\Delta_{\mu\nu}=-\frac{1}{3}\left\langle\Delta_{\mu\nu}k^{\mu}k^{\nu}\right\rangle_{i,0}=I_{i,21}\equiv-\frac{1}{3}\left(m_{i}^{2}\,\mathcal{I}_{i,0}-\mathcal{I}_{i,2}\right)\;.$
(43)

The sum over all particle species $i$ of the partial equilibrium moments leads
to the total particle four-current, the conserved charge four-currents, and
the energy-momentum tensor of the mixture,

$\displaystyle N_{0}^{\mu}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}N_{i,0}^{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}n_{i}u^{\mu}\equiv
nu^{\mu}\;,$ (44) $\displaystyle N_{q,0}^{\mu}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}N_{i,0}^{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}n_{i}u^{\mu}\equiv
n_{q}u^{\mu}\;,$ (45) $\displaystyle T_{0}^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}T_{i,0}^{\mu\nu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left(e_{i}u^{\mu}u^{\nu}-P_{i}\Delta^{\mu\nu}\right)\equiv
eu^{\mu}u^{\nu}-P\Delta^{\mu\nu}\;.$ (46)

Note that in our case the conserved net-charge four-currents are simply the
electric, the baryon, and the strangeness four-current. The primary
thermodynamic quantities of the mixture, i.e. the total number density, net-
charge density, total energy density, and total pressure, are obtained by
summing over all constituents

$n=\sum_{i\,=\,1}^{N_{\text{spec}}}n_{i}\;,\quad
n_{q}=\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}n_{i}\;,\quad
e=\sum_{i\,=\,1}^{N_{\text{spec}}}e_{i}\;,\quad
P=\sum_{i\,=\,1}^{N_{\text{spec}}}P_{i}\;.$ (47)

The particle and net-charge number as well as the energy are extensive
thermodynamic quantities, while the total pressure of the mixture follows
Dalton’s law of partial pressures.111These relations hold for systems which
can be described by kinetic theory, however they are violated once the
Stoßzahlansatz [22] does not apply, i.e. when long-range interactions or
multi-particle correlations become non-negligible. An equation of state
determines these thermodynamic quantities as functions of temperature and
chemical potentials, i.e. $n_{q}=n_{q}\left(T,\mu_{B},\mu_{Q},\mu_{S}\right)$,
$e=e\left(T,\mu_{B},\mu_{Q},\mu_{S}\right)$, and
$P=P\left(T,\mu_{B},\mu_{Q},\mu_{S}\right)$.

Note that in local thermodynamic equilibrium the individual particle four-
currents, $N_{0,i}^{\mu}=n_{i}u^{\mu}$, as well as the energy current of
species $i$, $T_{0,i}^{\mu\nu}u_{\nu}=e_{i}u^{\mu}$, are parallel to each
other. Therefore, all of these currents lead to the same local rest frame of
the fluid. Out of equilibrium the fluid-dynamical four-velocity can no longer
be uniquely defined. Nonetheless, without any loss of generality, a common
flow velocity tied to a chosen local rest frame can still be defined. The
difference of fluid-dynamical quantities from their local-equilibrium form
will be discussed next.

###  II.4 Out-of-equilibrium fluid-dynamical quantities

Out of equilibrium, the distribution function differs from its local-
equilibrium form by $\delta
f_{i,\mathbf{k}}=f_{i,\mathbf{k}}-f_{i,\mathbf{k}}^{\left(0\right)}$.
Introducing a similar notation for the momentum integrals as in Eq. (17),

$\left\langle\cdots\right\rangle_{i,\delta}\equiv\int\mathrm{d}K_{i}\left(\cdots\right)_{i}\delta
f_{i,\mathbf{k}}\;,$ (48)

and

$\left\langle\cdots\right\rangle_{i}\equiv\int\mathrm{d}K_{i}\left(\cdots\right)_{i}f_{i,\mathbf{k}}=\left\langle\cdots\right\rangle_{i,0}+\left\langle\cdots\right\rangle_{i,\delta}\;,$
(49)

cf. Refs. [21, 27], we define the irreducible moments of tensor-rank $\ell$
and energy-rank $r$ of the deviation of the single-particle distribution
function from equilibrium for a given particle species $i$,

$\rho_{i,r}^{\mu_{1}\cdots\mu_{\ell}}\equiv\Delta_{\nu_{1}\cdots\nu_{\ell}}^{\mu_{1}\cdots\mu_{\ell}}\int\mathrm{d}K_{i}E_{i,\mathbf{k}}^{r}k_{i}^{\mu_{1}}\cdots
k_{i}^{\mu_{\ell}}\delta f_{i,\mathbf{k}}=\left\langle
E_{\mathbf{k}}^{r}k^{\left\langle\mu_{1}\right.}\cdots
k^{\left.\mu_{\ell}\right\rangle}\right\rangle_{i,\delta}\;.$ (50)

Furthermore, we expand the distribution function $f_{i,\mathbf{k}}$ around
$f^{(0)}_{i,\mathbf{k}}$ as in Ref. [21],

$\delta f_{i,\mathbf{k}}\equiv
f_{i,\mathbf{k}}^{(0)}\tilde{f}_{i,\mathbf{k}}^{(0)}\phi_{i,\mathbf{k}}=f_{i,\mathbf{k}}^{(0)}\tilde{f}_{i,\mathbf{k}}^{(0)}\sum_{\ell\,=\,0}^{\infty}\sum_{n\,=\,0}^{N_{\ell}}\rho_{i,n}^{\mu_{1}\cdots\mu_{\ell}}k_{i,\left\langle\mu_{1}\right.}\cdots
k_{\left.i,\mu_{\ell}\right\rangle}\mathcal{H}_{i,\mathbf{k}n}^{(\ell)}\;,$
(51)

where the irreducible tensors orthogonal to the four-flow are
$k_{i}^{\left\langle\mu_{1}\right.}\cdots
k_{i}^{\left.\mu_{\ell}\right\rangle}=\Delta^{\mu_{1}\cdots\mu_{\ell}}_{\nu_{1}\cdots\nu_{\ell}}k_{i}^{\nu_{1}}\cdots
k_{i}^{\nu_{\ell}}$. These tensors form a complete and orthogonal basis in
momentum space. The coefficient $\mathcal{H}_{i,\mathbf{k}n}^{(\ell)}$ is a
polynomial in energy of order $N_{\ell}$ defined as

$\mathcal{H}_{i,\mathbf{k}n}^{(\ell)}=\frac{\left(-1\right)^{\ell}}{\ell!\
J_{i,2\ell,\ell}}\sum_{m\,=\,n}^{N_{\ell}}a_{i,mn}^{(\ell)}\sum_{r\,=\,0}^{m}a_{i,mr}^{\left(\ell\right)}E_{i,\mathbf{k}}^{r}\;.$
(52)

In principle, the expansion in polynomials in energy is an infinite series,
i.e. $N_{\ell}\rightarrow\infty$. However, we have already introduced
parameters $N_{\ell}<\infty$ (for each $\ell\geq 0$) at this point since we
will truncate the series later on in order to derive a fluid-dynamical theory.
The coefficients $a_{i,mn}^{(\ell)}$ are calculated via the Gram-Schmidt
orthogonalization procedure and can be expressed in terms of thermodynamic
integrals, see Ref. [21] for more details.

Thus, similarly to the equilibrium moments we define the out-of-equilibrium
particle four-current and energy-momentum tensor for particle species $i$ as

$\displaystyle N_{i}^{\mu}$ $\displaystyle\equiv$
$\displaystyle\int\mathrm{d}K_{i}k_{i}^{\mu}\left(f_{i,\mathbf{k}}^{(0)}+\delta
f_{i,\mathbf{k}}\right)=N_{i,0}^{\mu}+\rho_{i,0}^{\mu}=\left\langle
k^{\mu}\right\rangle_{i,0}+\left\langle
k^{\mu}\right\rangle_{i,\delta}\equiv\left\langle k^{\mu}\right\rangle_{i}\;,$
(53) $\displaystyle T_{i}^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\int\mathrm{d}K_{i}k_{i}^{\mu}k_{i}^{\nu}\left(f_{i,\mathbf{k}}^{(0)}+\delta
f_{i,\mathbf{k}}\right)=T_{i,0}^{\mu\nu}+\rho_{i,0}^{\mu\nu}=\left\langle
k^{\mu}k^{\mu}\right\rangle_{i,0}+\left\langle
k^{\mu}k^{\nu}\right\rangle_{i,\delta}\equiv\left\langle
k^{\mu}k^{\nu}\right\rangle_{i}\;,$ (54)

where $N_{i,0}^{\mu}$ and $T_{i,0}^{\mu\nu}$ were defined in Eqs. (39) and
(40), respectively.

The tensor decompositions with respect to an arbitrary time-like normalized
flow velocity $u^{\mu}$, summed over all particle species, lead to the total
fluid-dynamical quantities of the mixture,

$\displaystyle N^{\mu}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}N_{i}^{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left[\left(n_{i}+\rho_{i,1}\right)u^{\mu}+V_{i}^{\mu}\right]$
(55) $\displaystyle\equiv$ $\displaystyle\left(n+\delta
n\right)u^{\mu}+V^{\mu}\;,$ $\displaystyle N_{q}^{\mu}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}N_{i}^{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left[q_{i}\left(n_{i}+\rho_{i,1}\right)u^{\mu}+q_{i}V_{i}^{\mu}\right]$
(56) $\displaystyle\equiv$ $\displaystyle\left(n_{q}+\delta
n_{q}\right)u^{\mu}+V_{q}^{\mu}\;,$ $\displaystyle T^{\mu\nu}$
$\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}T_{i}^{\mu\nu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left[\left(e_{i}+\rho_{i,2}\right)u^{\mu}u^{\nu}-\left(P_{i}+\Pi_{i}\right)\Delta^{\mu\nu}+2W_{i}^{\left(\mu\right.}u^{\left.\nu\right)}+\pi_{i}^{\mu\nu}\right]$
(57) $\displaystyle\equiv$ $\displaystyle\left(e+\delta
e\right)u^{\mu}u^{\nu}-\left(P+\Pi\right)\Delta^{\mu\nu}+2W^{\left(\mu\right.}u^{\left.\nu\right)}+\pi^{\mu\nu}\;.$

The net-particle density, the net-charge density, the energy density, and the
isotropic pressure of the out-of-equilibrium mixture are

$\displaystyle n+\delta n$ $\displaystyle\equiv$ $\displaystyle
N^{\mu}u_{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left\langle
E_{\mathbf{k}}\right\rangle_{i}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}\left(n_{i}+\rho_{i,1}\right)\;,$
(58) $\displaystyle n_{q}+\delta n_{q}$ $\displaystyle\equiv$ $\displaystyle
N_{q}^{\mu}u_{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\left\langle
E_{\mathbf{k}}\right\rangle_{i}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\left(n_{i}+\rho_{i,1}\right)\;,$
(59) $\displaystyle e+\delta e$ $\displaystyle\equiv$ $\displaystyle
T^{\mu\nu}u_{\mu}u_{\nu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left\langle
E_{\mathbf{k}}^{2}\right\rangle_{i}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}\left(e_{i}+\rho_{i,2}\right)\;,$
(60) $\displaystyle P+\Pi$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{3}T^{\mu\nu}\Delta_{\mu\nu}=-\sum_{i\,=\,1}^{N_{\text{spec}}}\frac{1}{3}\left\langle\Delta_{\mu\nu}k^{\mu}k^{\nu}\right\rangle_{i}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left[P_{i}-\frac{1}{3}\left(m_{i}^{2}\rho_{i,0}-\rho_{i,2}\right)\right]\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}\left(P_{i}+\Pi_{i}\right)\;,$
(61)

with an equation of state relating the equilibrium quantities. Note that the
latter were defined in Eq. (47), while the partial pressure appearing in the
last equation was defined in Eq. (43). Hence, it follows that the non-
equilibrium correction to the pressure, the so-called bulk viscous pressure of
particle species $i$, is

$\Pi_{i}\equiv-\frac{1}{3}\left\langle\Delta_{\alpha\beta}k^{\alpha}k^{\beta}\right\rangle_{i,\delta}=-\frac{1}{3}\left(m_{i}^{2}\rho_{i,0}-\rho_{i,2}\right)\;.$
(62)

The net-particle diffusion, the net-charge diffusion, and the energy-momentum
diffusion currents are

$\displaystyle V^{\mu}$ $\displaystyle\equiv$
$\displaystyle\Delta_{\nu}^{\mu}N^{\nu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left\langle
k^{\left\langle\mu\right\rangle}\right\rangle_{i}=\sum_{i\,=\,1}^{N_{\text{spec}}}\rho_{i,0}^{\mu}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}V_{i}^{\mu}\;,$
(63) $\displaystyle V_{q}^{\mu}$ $\displaystyle\equiv$
$\displaystyle\Delta_{\nu}^{\mu}N_{q}^{\nu}=\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\left\langle
k^{\left\langle\mu\right\rangle}\right\rangle_{i}=\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\rho_{i,0}^{\mu}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}V_{i}^{\mu}\;,$
(64) $\displaystyle W^{\mu}$ $\displaystyle\equiv$
$\displaystyle\Delta_{\alpha}^{\mu}T^{\alpha\beta}u_{\beta}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left\langle
E_{\mathbf{k}}k^{\left\langle\mu\right\rangle}\right\rangle_{i}=\sum_{i\,=\,1}^{N_{\text{spec}}}\rho_{i,1}^{\mu}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}W_{i}^{\mu}\;.$
(65)

Finally, the shear-stress tensor of the mixture is

$\displaystyle\pi^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\Delta_{\alpha\beta}^{\mu\nu}T^{\alpha\beta}=\sum_{i\,=\,1}^{N_{\text{spec}}}\left\langle
k^{\left\langle\mu\right.}k^{\left.\nu\right\rangle}\right\rangle_{i}=\sum_{i\,=\,1}^{N_{\text{spec}}}\rho_{i,0}^{\mu\nu}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}\pi_{i}^{\mu\nu}\;.$
(66)

Equations (58) – (66) represent the fluid-dynamical fields of the mixture,
which (similarly as in chemical solutions) is a combination of multiple
particle species and where the number of particles of an individual species
may or may not be conserved. Originally, these fields constitute
$14N_{\text{spec}}$ variables (10 for each energy-momentum tensor
$T_{i}^{\mu\nu}$ and 4 for each particle current $N_{i}^{\mu}$). We assume
that the mixture can be treated as a single fluid such that its space-time
evolution can be entirely determined in terms of the total energy-momentum
tensor $T^{\mu\nu}$ and the charge four-currents $N^{\mu}_{q}$. This approach
reduces the number of unknown fluid-dynamical fields to $10+4N_{q}$.222Note
that, naively counting the number of unknowns, there are actually $15+5N_{q}$
degrees of freedom: 5 degrees of freedom (d.o.f.s) for the $N_{q}$ charge
four-currents ($n_{q}$, $\delta n_{q}$, and 3 components of $V^{\mu}_{q}$),
and 4 scalar ($e$, $\delta e$, $P$, $\Pi$), 6 vector ($u^{\mu}$ and
$W^{\mu}$), and 5 tensor d.o.f.s ($\pi^{\mu\nu}$) for the energy-momentum
tensor. However, one d.o.f. is reduced by the equation of state,
$P=P(e,\\{n_{q}\\})$. Furthermore, $4+N_{q}$ additional d.o.f.s are reduced by
the matching conditions, see Sec. II.5.

Further, the out-of-equilibrium part of the distribution function $\delta
f_{i,\mathbf{k}}$ was expanded in terms of an infinite set of independent
irreducible moments $\rho_{i,n}^{\mu_{1}\cdots\mu_{\ell}}$, each of which
obeys an equation of motion derived from the relativistic Boltzmann equation
(1) (see Sec. III). The dissipative fluid-dynamical fields, Eqs. (62) – (66),
are defined in terms of (some of) these moments, and thus are solutions of
these equations of motion. The crucial step in the derivation of fluid
dynamics using the method of moments is to truncate the infinite set of
equations of motion for the irreducible moments, and thus also the series in
Eq. (51), in a well-defined manner. To this end, the sums over tensor rank
$\ell$ and over powers of energy $n$ in Eq. (51) are truncated. The latter is
already implied by the truncation parameter $N_{\ell}$, which depends on the
respective tensor rank $\ell$ of the moment. The lowest possible truncation in
$\ell$ and $n$ is to account for the lowest-order irreducible moments which
explicitly appear in the energy-momentum tensor (57) and the charge four-flow
(56), namely $\rho_{i,0}$, $\rho_{i,1}$, $\rho_{i,2}$, $\rho^{\mu}_{i,0}$,
$\rho^{\mu}_{i,1}$, and $\rho^{\mu\nu}_{i,0}$. This leads to the truncation
$\ell\leq 2$, and $N_{0}=2$, $N_{1}=1$, and $N_{2}=0$ in the series in Eq.
(51). This is the so-called $(10+4N_{q})$-moment approximation. As mentioned
above, these moments, however, further depend on other moments, which may also
be of higher tensor rank (e.g. $\ell>2$). In Sec. III.3 we discuss how to
further truncate the set of equations of motion.

###  II.5 Matching conditions and local rest frames

In local equilibrium the thermodynamic state of matter is completely
determined by a few scalar fields, namely a common temperature $T$ and the
chemical potentials of the constituent particle species $\alpha_{i}$, which
are in turn given by the chemical potentials of the conserved charges
$\alpha_{q}$ via Eq. (14). A common way to determine these thermodynamic
variables in an arbitrary state (which is not too far from local equilibrium)
is to demand that the net-charge densities and the total energy density are
the same as in some fictitious local-equilibrium reference state. These are
the so-called Landau matching conditions [28],

$\displaystyle\left(N_{q}^{\mu}-N_{q,0}^{\mu}\right)u_{\mu}$
$\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\left(N_{i}^{\mu}-N_{i,0}^{\mu}\right)u_{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\rho_{i,1}\equiv\delta
n_{q}=0\;,$ (67)
$\displaystyle\left(T^{\mu\nu}-T_{0}^{\mu\nu}\right)u_{\mu}u_{\nu}$
$\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}\left(T_{i}^{\mu\nu}-T_{i,0}^{\mu\nu}\right)u_{\mu}u_{\nu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\rho_{i,2}\equiv\delta
e=0\;,$ (68)

where
$\left(N^{\mu}-N_{0}^{\mu}\right)u_{\mu}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}\left(N_{i}^{\mu}-N_{i,0}^{\mu}\right)u_{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\rho_{i,1}\equiv\delta
n\neq 0$, since the total number of particles is not necessarily conserved.
Furthermore, Landau’s matching condition for the energy, Eq. (68), also leads
to a simplification of the bulk viscous pressure in Eqs. (61) and (62),

$\Pi\equiv-\frac{1}{3}\sum_{i\,=\,1}^{N_{\text{spec}}}m_{i}^{2}\rho_{i,0}=\sum_{i\,=\,1}^{N_{\text{spec}}}\Pi_{i}\;.$
(69)

We note that using the matching conditions (67) and (68), we can express some
scalar moments by the others, and thus reduce the number of scalar moments of
the multicomponent mixture by $N_{q}+1$.

The number of independent unknowns is further reduced once we choose a local
rest frame, or equivalently a definition for the fluid-dynamical flow
velocity. The definition of Landau and Lifshitz [28] leads to the so-called
Landau frame, or energy frame, and requires that the total energy-momentum
diffusion current of the mixture is zero,

$W^{\mu}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}W_{i}^{\mu}=0\;.$ (70)

This directly implies that $T^{\mu\nu}u_{\nu}=eu^{\mu}$, meaning that the flow
velocity $u^{\mu}$ is the time-like eigenvector of the energy-momentum tensor
with eigenvalue $e$. This choice reduces the total number of unknowns by three
and leads to additional constraints between the remaining species-specific
vector fields $W_{i}^{\mu}$, i.e. there are only $N_{\text{spec}}-1$
independent energy-momentum diffusion fluxes in the mixture of
$N_{\text{spec}}$ different species. Note that Eq. (70) also implies that if
the fluid consists of a single component, i.e. $N_{\text{spec}}=1$, there is
no energy-momentum diffusion present in this frame. Unless stated otherwise,
the Landau frame is our choice for the local rest frame.

More traditionally, one may use Eckart’s definition [29] to fix the local rest
frame by demanding that the overall diffusion of one of the conserved net
charges, say that of charge $q^{\star}$, in the mixture is zero,

$\tilde{V}_{q^{\star}}^{\mu}\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}\tilde{V}_{q^{\star},i}^{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}q^{\star}_{i}\tilde{V}_{i}^{\mu}=0\;,$
(71)

where quantities in this particular $q^{\star}$-charge frame are denoted by a
tilde. However, in high-energy heavy-ion collisions, where there are multiple
conserved charges, which are not necessarily non-vanishing in all regions of
space-time, the definition of the rest frame according to Eckart is less
suitable.

Nevertheless, in case of a single non-vanishing conserved charge $q$, the
Landau and Eckart reference frames are essentially equivalent, hence choosing
one over the other is a matter of taste. Namely, the energy-momentum diffusion
current in the Eckart frame can be related to the charge diffusion current in
the Landau frame via

$\tilde{W}^{\mu}\equiv-h_{q}V_{q}^{\mu}\;,$ (72)

where we introduced the enthalpy per charge, $h_{q}=(e+P)/n_{q}$, see Appendix
B for more details.

###  II.6 Conservation equations

Due to the fact that in binary collisions the net charges as well as the
energy and momentum of particles are conserved, the equations of fluid
dynamics of a mixture are derived from the Boltzmann equation (1) as

$\displaystyle\partial_{\mu}N_{q}^{\mu}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\partial_{\mu}N_{i}^{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\int\mathrm{d}K_{i}C_{i}=0\;,$
(73) $\displaystyle\partial_{\mu}T^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}\partial_{\mu}T_{i}^{\mu\nu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\int\mathrm{d}K_{i}k^{\nu}_{i}C_{i}=0\;,$
(74)

where there are $N_{q}$ independent charge-conservation laws. Note that due to
inelastic collisions the number of particles of species $i$ is no longer
conserved and the individual particle species satisfy rate equations,
$\partial_{\mu}N_{i}^{\mu}\neq 0$. On the other hand, for purely elastic
collisions the numbers of particles are conserved, and the momentum integral
over each partial collision term vanishes separately,
$\partial_{\mu}N_{i}^{\mu}=0$.

With Eq. (56) the $N_{q}$ charge-conservation equations (73) assume the form

$\displaystyle\partial_{\mu}N_{q}^{\mu}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}D\left(n_{i}+\rho_{i,1}\right)+\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\left(n_{i}+\rho_{i,1}\right)\theta+\sum_{i\,=\,1}^{N_{\text{spec}}}q_{i}\partial_{\mu}V_{i}^{\mu}$
(75) $\displaystyle=$ $\displaystyle
Dn_{q}+n_{q}\theta+\partial_{\mu}V_{q}^{\mu}=0\;,$

where in the last step we used the first Landau matching condition (67).

The conservation of energy of the mixture is obtained by projecting Eq. (74)
onto $u_{\nu}$ and inserting Eq. (57),

$\displaystyle u_{\nu}\partial_{\mu}T^{\mu\nu}$
$\displaystyle\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}D\left(e_{i}+\rho_{i,2}\right)+\sum_{i\,=\,1}^{N_{\text{spec}}}\left(e_{i}+\rho_{i,2}+P_{i}+\Pi_{i}\right)\theta$
$\displaystyle+\sum_{i\,=\,1}^{N_{\text{spec}}}\partial_{\mu}W_{i}^{\mu}-\sum_{i\,=\,1}^{N_{\text{spec}}}W_{i}^{\mu}Du_{\mu}-\sum_{i\,=\,1}^{N_{\text{spec}}}\pi_{i}^{\mu\nu}\sigma_{\mu\nu}$
$\displaystyle=De+\left(e+P+\Pi\right)\theta-\pi^{\mu\nu}\sigma_{\mu\nu}=0\;,$
(76)

where we have imposed the second Landau matching condition (68) and also fixed
the local rest frame according to Landau’s convention, Eq. (70).

Using these conservation equations to replace $dn_{q}$ and $de$ in Eqs. (35)
and (36) leads to the comoving derivatives of the inverse temperature and the
charge chemical potentials multiplied by the inverse temperature,

$\displaystyle D\beta$ $\displaystyle=$
$\displaystyle-\mathcal{T}_{00}\left[\left(e+P+\Pi\right)\theta-\pi^{\mu\nu}\sigma_{\mu\nu}\right]-\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\mathcal{T}_{0q^{\prime}}\left[n_{q^{\prime}}\theta+\partial_{\mu}V_{q^{\prime}}^{\mu}\right]\;,$
(77) $\displaystyle D\alpha_{q}$ $\displaystyle=$
$\displaystyle-\mathcal{T}_{q0}\left[\left(e+P+\Pi\right)\theta-\pi^{\mu\nu}\sigma_{\mu\nu}\right]-\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\mathcal{T}_{qq^{\prime}}\left[n_{q^{\prime}}\theta+\partial_{\mu}V_{q^{\prime}}^{\mu}\right]\;.$
(78)

Finally, projecting Eq. (74) onto $\Delta_{\beta}^{\mu}$ (which gives the
momentum conservation of the mixture), and using Eqs. (68) and (70) leads to

$\displaystyle\Delta_{\beta}^{\mu}\partial_{\alpha}T^{\alpha\beta}$
$\displaystyle\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}\left(e_{i}+P_{i}+\Pi_{i}\right)Du^{\mu}-\nabla^{\mu}\sum_{i\,=\,1}^{N_{\text{spec}}}\left(P_{i}+\Pi_{i}\right)+\Delta_{\beta}^{\mu}\partial_{\alpha}\sum_{i\,=\,1}^{N_{\text{spec}}}\pi_{i}^{\alpha\beta}$
$\displaystyle=\left(e+P+\Pi\right)Du^{\mu}-\nabla^{\mu}\left(P+\Pi\right)+\Delta_{\beta}^{\mu}\partial_{\alpha}\pi^{\alpha\beta}=0\;.$
(79)

This leads to an expression for the acceleration $Du^{\mu}$ of the fluid.
Noting that in local equilibrium the Gibbs-Duhem relation holds in the form

$\beta\mathrm{d}P=\sum_{q}^{\left\\{B,Q,S\right\\}}n_{q}\mathrm{d}\alpha_{q}-(e+P)\mathrm{d}\beta\;,$
(80)

we obtain

$Du^{\mu}=-\frac{1}{\beta}\nabla^{\mu}\beta+\frac{1}{\beta(e+P)}\sum_{q}^{\left\\{B,Q,S\right\\}}n_{q}\nabla^{\mu}\alpha_{q}-\frac{1}{e+P}\left(\Pi
Du^{\mu}-\nabla^{\mu}\Pi+\Delta_{\beta}^{\mu}\partial_{\alpha}\pi^{\alpha\beta}\right)\;.$
(81)

Let us recount the unknown variables and equations of the mixture. There are
$N_{q}$ conservation equations (75) for $n_{q}$ and $V_{q}^{\mu}$,
representing $4N_{q}$ variables. The conservation of energy and momentum of
the mixture provides the four equations (76) and (79), entailing $e$, $\Pi$,
$u^{\mu}$, and $\pi^{\mu\nu}$, which represent 10 variables in total, since
the equation of state already defines the pressure in terms of $e$ and the
$n_{q}$’s. Thus, in a dissipative mixture of $N_{q}$ conserved charges we have
only $4+N_{q}$ conservation equations for a total of $10+4N_{q}$ unknown
fields. The additional equations for the dissipative fields $\Pi$,
$V_{q}^{\mu}$, and $\pi^{\mu\nu}$ will be derived from the Boltzmann equation
in the next section.

##  III Second-order dissipative fluid-dynamical equations of motion

In this section, we first derive the equations of motion for the irreducible
tensor moments for particle species $i$. For a single-component fluid, these
equations were first given in Ref. [21]. Here, we extend them towards
multicomponent fluids. Then we linearize the collision term and discuss the
Navier-Stokes limit as well as the order-of-magnitude approximation, which
provides a simple, and yet effective way to close the set of equations of
motion. Finally, we derive and discuss the second-order dissipative fluid-
dynamical equations of motion.

###  III.1 Equations of motion for the irreducible moments

The equations of motion for the irreducible moments
$\rho_{i,r}^{\mu_{1}\cdots\mu_{\ell}}$ are obtained directly from the
Boltzmann equation (1) by multiplying it with
$E_{i,\mathbf{k}}^{r}k_{i}^{\langle\mu_{1}}\cdots k_{i}^{\mu_{\ell}\rangle}$,
integrating over momentum space, and taking the comoving derivative. Then,
projecting onto $\Delta^{\mu_{1}\cdots\mu_{\ell}}_{\nu_{1}\cdots\nu_{\ell}}$,
we obtain the equations of motion for the irreducible moments, i.e. an
equation for the comoving derivative
$\dot{\rho}_{i,r}^{\left\langle\mu_{1}\cdots\mu_{\ell}\right\rangle}\equiv\Delta^{\mu_{1}\cdots\mu_{\ell}}_{\nu_{1}\cdots\nu_{\ell}}D\rho_{i,r}^{\nu_{1}\cdots\nu_{\ell}}$.
The irreducible moments of the collision term (2) are defined as

$C_{i,r-1}^{\left\langle\mu_{1}\cdots\mu_{\ell}\right\rangle}\equiv\Delta_{\nu_{1}\cdots\nu_{\ell}}^{\mu_{1}\cdots\mu_{\ell}}\sum_{j\,=\,1}^{N_{\text{spec}}}\int\mathrm{d}K_{i}E_{i,\mathbf{k}}^{r-1}\,k_{i}^{\nu_{1}}\cdots
k_{i}^{\nu_{\ell}}C_{ij}\left[f\right]\;.$ (82)

After some calculation the equation of motion for the irreducible moments of
tensor-rank zero reads

$\displaystyle\dot{\rho}_{i,r}-C_{i,r-1}=$
$\displaystyle\;\alpha_{i,r}^{\left(0\right)}\theta-\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\left(J_{i,r+1,0}\mathcal{T}_{0q^{\prime}}-\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}J_{i,r0}\mathcal{T}_{qq^{\prime}}\right)\left(\nabla_{\mu}V_{q^{\prime}}^{\mu}-V_{q^{\prime}}^{\mu}\dot{u}_{\mu}\right)$
$\displaystyle+\frac{\theta}{3}\left[m_{i}^{2}\left(r-1\right)\rho_{i,r-2}-\left(r+2\right)\rho_{i,r}-3\left(J_{i,r+1,0}\mathcal{T}_{00}-\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}J_{i,r0}\mathcal{T}_{q0}\right)\Pi\right]$
$\displaystyle+r\rho_{i,r-1}^{\mu}\dot{u}_{\mu}-\nabla_{\mu}\rho_{i,r-1}^{\mu}+\left[\left(r-1\right)\rho_{i,r-2}^{\mu\nu}+\left(J_{i,r+1,0}\mathcal{T}_{00}-\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}J_{i,r0}\mathcal{T}_{q0}\right)\pi^{\mu\nu}\right]\sigma_{\mu\nu}\;.$
(83)

This equation is different from Eq. (35) of Ref. [21], because thermodynamic
relations are modified in mixtures with multiple conserved charges as compared
to a single-component fluid. Similarly, the transport coefficient
$\alpha_{i,r}^{\left(0\right)}$ has additional contributions given by the sums
$\sum_{q}^{\left\\{B,Q,S\right\\}}$ when compared to Eq. (42) of Ref. [21],

$\displaystyle\alpha_{i,r}^{\left(0\right)}=$
$\displaystyle-\left[I_{i,r0}+\left(r-1\right)I_{i,r1}+\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\left(J_{i,r+1,0}\mathcal{T}_{0q^{\prime}}-\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}J_{i,r0}\mathcal{T}_{qq^{\prime}}\right)n_{q^{\prime}}\right]$
$\displaystyle-\left(J_{i,r+1,0}\mathcal{T}_{00}-\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}J_{i,r0}\mathcal{T}_{q0}\right)\left(e+P\right)\;.$
(84)

The equation of motion for the irreducible moments of tensor-rank one is very
similar to Eq. (36) of Ref. [21], except for the first-order gradient term
$\sum_{q}^{\left\\{B,Q,S\right\\}}\alpha_{i,r,q}^{\left(1\right)}\nabla^{\mu}\alpha_{q}$,

$\displaystyle\dot{\rho}_{i,r}^{\left\langle\mu\right\rangle}-C_{i,r-1}^{\left\langle\mu\right\rangle}=$
$\displaystyle\;\sum_{q}^{\left\\{B,Q,S\right\\}}\alpha_{i,r,q}^{\left(1\right)}\nabla^{\mu}\alpha_{q}+r\rho_{i,r-1}^{\mu\nu}\dot{u}_{\nu}-\frac{1}{3}\nabla^{\mu}\left(m_{i}^{2}\rho_{i,r-1}-\rho_{i,r+1}\right)+\alpha_{i,r}^{h}\nabla^{\mu}\Pi$
$\displaystyle-\Delta_{\lambda}^{\mu}\left(\nabla_{\nu}\rho_{i,r-1}^{\lambda\nu}+\alpha_{i,r}^{h}\partial_{\nu}\pi^{\nu\lambda}\right)+\frac{1}{3}\left[m_{i}^{2}\left(r-1\right)\rho_{i,r-2}^{\mu}-\left(r+3\right)\rho_{i,r}^{\mu}\right]\theta$
$\displaystyle+\frac{1}{3}\left[m_{i}^{2}r\rho_{i,r-1}-\left(r+3\right)\rho_{i,r+1}-3\alpha_{i,r}^{h}\Pi\right]\dot{u}^{\mu}+\rho_{i,r,\nu}\omega^{\mu\nu}$
$\displaystyle+\frac{1}{5}\left[m_{i}^{2}\left(2r-2\right)\rho_{i,r-2,\nu}-\left(2r+3\right)\rho_{i,r,\nu}\right]\sigma^{\mu\nu}+\left(r-1\right)\rho_{i,r-2}^{\mu\nu\lambda}\sigma_{\nu\lambda}\;,$
(85)

where the transport coefficients are

$\displaystyle\alpha_{i,r,q}^{\left(1\right)}=q_{i}J_{i,r+1,1}+\alpha_{i,r}^{h}\frac{n_{q}}{\beta}\;,\qquad\alpha_{i,r}^{h}=-\beta\frac{J_{i,r+2,1}}{e+P}\;.$
(86)

Finally, the equation of motion for the irreducible moments of tensor-rank two
is formally unchanged from Eq. (37) of Ref. [21], only the species-specific
index $i$ is introduced here,

$\displaystyle\dot{\rho}_{i,r}^{\left\langle\mu\nu\right\rangle}-C_{i,r-1}^{\langle\mu\nu\rangle}=$
$\displaystyle\;2\alpha_{i,r}^{\left(2\right)}\sigma^{\mu\nu}+\frac{2}{15}\left[m_{i}^{4}\left(r-1\right)\rho_{i,r-2}-\left(2r+3\right)m_{i}^{2}\rho_{i,r}+\left(r+4\right)\rho_{i,r+2}\right]\sigma^{\mu\nu}$
$\displaystyle+\frac{2}{5}\dot{u}^{\left\langle\mu\right.}\left[m_{i}^{2}r\rho_{i,r-1}^{\left.\nu\right\rangle}-\left(r+5\right)\rho_{i,r+1}^{\left.\nu\right\rangle}\right]-\frac{2}{5}\nabla^{\left\langle\mu\right.}\left(m_{i}^{2}\rho_{i,r-1}^{\left.\nu\right\rangle}-\rho_{i,r+1}^{\left.\nu\right\rangle}\right)$
$\displaystyle+r\rho_{i,r-1}^{\mu\nu\lambda}\dot{u}_{\lambda}-\Delta_{\alpha\beta}^{\mu\nu}\nabla_{\lambda}\rho_{i,r-1}^{\alpha\beta\lambda}+\left(r-1\right)\rho_{i,r-2}^{\mu\nu\lambda\kappa}\sigma_{\lambda\kappa}+2\rho_{i,r}^{\lambda\left\langle\mu\right.}\omega_{\left.{}\right.\lambda}^{\left.\nu\right\rangle}$
$\displaystyle+\frac{1}{3}\left[m_{i}^{2}\left(r-1\right)\rho_{i,r-2}^{\mu\nu}-\left(r+4\right)\rho_{i,r}^{\mu\nu}\right]\theta+\frac{2}{7}\left[m_{i}^{2}\left(2r-2\right)\rho_{i,r-2}^{\lambda\left\langle\mu\right.}-\left(2r+5\right)\rho_{i,r}^{\lambda\left\langle\mu\right.}\right]\sigma_{\lambda}^{\left.\nu\right\rangle}\;,$
(87)

where the coefficient $\alpha_{i,r}^{\left(2\right)}$ is also formally the
same as in Eq. (44) of Ref. [21],

$\alpha_{i,r}^{\left(2\right)}=\left(r-1\right)I_{i,r+2,2}+I_{i,r+2,1}\;.$
(88)

These are the equations of motion for the irreducible moments up to tensor-
rank two for any particle species $i$. One can show that in the case of a
single-component fluid they reduce to the equations found in Ref. [21].

Furthermore, note that since the conserved quantities in fluid dynamics
contain no tensors of rank higher than two, the higher-rank tensors,
$\rho_{i,r}^{\mu_{1}\cdots\mu_{\ell}}=0$ for $\ell\geq 3$, in the equations of
motion (85) and (87) will be neglected in the following (see Sec. III.3).

###  III.2 The linearized collision integral

Further progress requires the linearization of the collision integral (2) in
the quantities $\phi_{i,\mathbf{k}}=\delta
f_{i,\mathbf{k}}/[f_{i,\mathbf{k}}^{(0)}\tilde{f}_{i,\mathbf{k}}^{(0)}]$, such
that it simplifies to

$C_{i}\left(x,k\right)\simeq\frac{1}{2}\sum_{j,a,b\,=\,1}^{N_{\text{spec}}}\int\mathrm{d}K_{j}^{\prime}\mathrm{d}P_{a}\mathrm{d}P_{b}^{\prime}\,W_{ij\rightarrow
ab}^{kk^{\prime}\rightarrow
pp^{\prime}}f_{i,\mathbf{k}}^{\left(0\right)}f_{j,\mathbf{k}^{\prime}}^{\left(0\right)}\tilde{f}^{\left(0\right)}_{a,\mathbf{p}}\tilde{f}^{\left(0\right)}_{b,\mathbf{p}^{\prime}}\left(\phi_{a,\mathbf{p}}+\phi_{b,\mathbf{p}^{\prime}}-\phi_{i,\mathbf{k}}-\phi_{j,\mathbf{k}^{\prime}}\right)\;,$
(89)

where the bilateral normalization condition holds and the equality
$f^{(0)}_{i,\bf{k}}f^{(0)}_{j,\bf{k}^{\prime}}\tilde{f}^{\left(0\right)}_{a,\mathbf{p}}\tilde{f}^{\left(0\right)}_{b,\mathbf{p}^{\prime}}=f^{(0)}_{a,\bf{p}}f^{(0)}_{b,\bf{p}^{\prime}}\tilde{f}^{\left(0\right)}_{i,\mathbf{k}}\tilde{f}^{\left(0\right)}_{j,\mathbf{k}^{\prime}}$
was used [22, 23].

Using the linearized collision integral (89) one can show that the
corresponding irreducible moments (82) of the collision integral can be
expressed in terms of a linear combination of irreducible moments,
$\rho_{i,r}^{\nu_{1}\cdots\nu_{\ell}}$, in a similar way as in Eq. (50) of
Ref. [21],

$\displaystyle C_{i,r-1}^{\langle\mu_{1}\cdots\mu_{\ell}\rangle}$
$\displaystyle\equiv$
$\displaystyle-\sum_{s\,=\,1}^{N_{\text{spec}}}\sum_{n\,=\,0}^{N_{\ell}}\sum_{m\,=\,0}^{\infty}\left(\mathcal{A}_{is,rn}\right)_{\nu_{1}\cdots\nu_{m}}^{\mu_{1}\cdots\mu_{\ell}}\rho_{s,n}^{\nu_{1}\cdots\nu_{m}},$
(90) $\displaystyle=$
$\displaystyle-\sum_{s\,=\,1}^{N_{\text{spec}}}\sum_{n\,=\,0}^{N_{\ell}}\left(\mathcal{A}_{is,rn}^{\left(\ell\right)}\right)\rho_{s,n}^{\mu_{1}\cdots\mu_{\ell}}\;.$

Here, we have defined the following tensors

$\displaystyle\left(\mathcal{A}_{is,rn}\right)^{\mu_{1}\cdots\mu_{\ell}\nu_{1}\cdots\nu_{m}}=$
$\displaystyle\;\frac{1}{2}\sum_{j,a,b\,=\,1}^{N_{\text{spec}}}\int\mathrm{d}K_{i}\mathrm{d}K_{j}^{\prime}\mathrm{d}P_{a}\mathrm{d}P_{b}^{\prime}W_{ij\rightarrow
ab}^{kk^{\prime}\rightarrow
pp^{\prime}}f_{i,\mathbf{k}}^{\left(0\right)}f_{j,\mathbf{k}^{\prime}}^{\left(0\right)}\tilde{f}^{\left(0\right)}_{a,\mathbf{p}}\tilde{f}^{\left(0\right)}_{b,\mathbf{p}^{\prime}}E_{i,\mathbf{k}}^{r-1}k_{i}^{\left\langle\mu_{1}\right.}\cdots
k_{i}^{\left.\mu_{\ell}\right\rangle}$
$\displaystyle\times\left[\delta_{si}\mathcal{H}_{s,\mathbf{k}n}^{(m)}k_{s}^{\left\langle\nu_{1}\right.}\cdots
k_{s}^{\left.\nu_{m}\right\rangle}+\delta_{sj}\mathcal{H}_{s,\mathbf{k}^{\prime}n}^{(m)}k_{s}^{\prime\left\langle\nu_{1}\right.}\cdots
k_{s}^{\prime\left.\nu_{m}\right\rangle}\right.$
$\displaystyle\quad\left.-\delta_{sa}\mathcal{H}_{s,\mathbf{p}n}^{(m)}p_{s}^{\left\langle\nu_{1}\right.}\cdots
p_{s}^{\left.\nu_{m}\right\rangle}-\delta_{sb}\mathcal{H}_{s,\mathbf{p}^{\prime}n}^{(m)}p_{s}^{\prime\left\langle\nu_{1}\right.}\cdots
p_{s}^{\prime\left.\nu_{m}\right\rangle}\right]\;,$ (91)

where the polynomials $\mathcal{H}_{s,\mathbf{k}n}^{(m)}$ were defined in Eq.
(52). These tensors can be decomposed and projected, hence one finally obtains
the collision matrix

$\mathcal{A}_{is,rn}^{\left(\ell\right)}=\frac{1}{2\ell+1}\Delta_{\mu_{1}\cdots\mu_{\ell}}^{\nu_{1}\cdots\nu_{\ell}}\left(\mathcal{A}_{is,rn}\right)_{\nu_{1}\cdots\nu_{\ell}}^{\mu_{1}\cdots\mu_{\ell}}\;,$
(92)

cf. Eq. (A18) of Ref. [21]. Note that in the case of a single-component
system, i.e. $N_{\text{spec}}=1$, the above equation reduces to the diagonal
components of $\mathcal{A}_{is,rn}^{\left(\ell\right)}$. Hence, the particle
species labels $i$ and $s$ can be dropped and we simply recover the result of
Ref. [21],
$C_{i,r-1}^{\langle\mu_{1}\cdots\mu_{\ell}\rangle}\overset{\text{single}}{\rightarrow}C_{r-1}^{\langle\mu_{1}\cdots\mu_{\ell}\rangle}=-\sum_{n\,=\,0}^{N_{\ell}}\mathcal{A}_{rn}^{\left(\ell\right)}\rho_{n}^{\mu_{1}\cdots\mu_{\ell}}$
.

The inverse of the collision matrix defines the so-called relaxation-time
matrix,

$\tau_{si,nr}^{\left(\ell\right)}\equiv\left(\mathcal{A}^{-1}\right)^{\left(\ell\right)}_{si,nr}\;,$
(93)

where the matrix elements are proportional to the inverse of the mean free
path between collisions $\lambda^{-1}_{\textrm{mfp}}$. Therefore, multiplying
both sides of Eq. (90) by the relaxation-time matrix we obtain an important
relation between the moments of the linearized collision integral and the
irreducible moments,

$\sum_{i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{\ell}}\tau_{si,nr}^{\left(\ell\right)}C_{i,r-1}^{\left\langle\mu_{1}\cdots\mu_{\ell}\right\rangle}\equiv-\sum_{i,s^{\prime}\,=\,1}^{N_{\text{spec}}}\sum_{r,n^{\prime}\,=\,0}^{N_{\ell}}\tau_{si,nr}^{\left(\ell\right)}\mathcal{A}_{is^{\prime},rn^{\prime}}^{\left(\ell\right)}\rho_{s^{\prime},n^{\prime}}^{\mu_{1}\cdots\mu_{\ell}}=-\rho_{s,n}^{\mu_{1}\cdots\mu_{\ell}}.$
(94)

The infinite set of equations of motions for the irreducible moments contains
infinitely many degrees of freedom. In order to close the equations of motion
for the mixture treated as a single fluid, the number of degrees of freedom
must be reduced and infinite sums must be truncated at some finite number.

One of the key features of transient fluid dynamics is that the corresponding
equations of motion have a single time scale that controls the transient
behavior, e.g. relaxation towards equilibrium. On the other hand, the
Boltzmann equation has infinitely many of such time scales. Even in a single-
component gas the modes corresponding to the same tensor rank $\ell$ are
coupled to each other and their dynamics depends on these infinitely many
scales. In the case of a mixture the situation is even more complicated as the
modes corresponding to different particle species are also coupled. The reason
for this is that due to interactions between particles of different species
the corresponding moments are correlated, see Refs. [30, 31].

One possible way to reduce the number of degrees of freedom would be to
generalize the approach of Ref. [21] developed for a single-component system,
and diagonalize the collision matrix to determine the slowest microscopic time
scales, i.e. the relaxation times that are relevant in our approximation, and
the corresponding modes that dominate the long-time dynamics of the fluid in
the transient regime. This has the advantage that the relaxation times that
appear in the equations of motion could be explicitly identified as real
microscopic time scales. However, the downside of this method is the
appearance of terms that are of second or higher order in gradients, denoted
as $\mathcal{O}({\rm Kn}^{2})$ in Ref. [21]. These terms can violate stability
and causality of the theory. In principle, this can be cured by introducing
further independent dynamical variables, cf. for instance Ref. [32], but this
is beyond the scope of the current work. Instead, as discussed below, we will
employ a slightly simpler approach, where the problematic $\mathcal{O}({\rm
Kn}^{2})$ terms do not appear.

###  III.3 The Navier-Stokes limit and the order-of-magnitude approximation

The explicit relation between the irreducible tensor of a given rank and the
corresponding fluid-dynamical gradients can be derived by multiplying Eqs.
(83), (85), and (87) by the corresponding relaxation-time matrices,
$\tau_{si,nr}^{\left(\ell\right)}$, and using Eq. (94). In this way, the
following equations of motion for the irreducible moments of tensor rank
$\ell=0,\,1$, and 2 are obtained,

$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{0}}\tau_{si,nr}^{\left(0\right)}\dot{\rho}_{i,r}+\rho_{s,n}$
$\displaystyle=$ $\displaystyle-\zeta_{s,n}\theta+\mathcal{O}(2)\;,$ (95)
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{1}}\tau_{si,nr}^{\left(1\right)}\dot{\rho}_{i,r}^{\left\langle\mu\right\rangle}+\rho_{s,n}^{\mu}$
$\displaystyle=$
$\displaystyle\sum_{q}^{\left\\{B,Q,S\right\\}}\kappa_{s,n,q}\nabla^{\mu}\alpha_{q}+\mathcal{O}(2)\;,$
(96)
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,nr}^{\left(2\right)}\dot{\rho}_{i,r}^{\left\langle\mu\nu\right\rangle}+\rho_{s,n}^{\mu\nu}$
$\displaystyle=$ $\displaystyle 2\eta_{s,n}\sigma^{\mu\nu}+\mathcal{O}(2)\;,$
(97)

where $\mathcal{O}(2)$ denote all remaining second- and higher-order terms
from the corresponding equations of motion for the irreducible moments. These
are terms which are at least of quadratic order in the Knudsen number,
$\mathcal{O}(\mathrm{Kn}^{2})$, or in the inverse Reynolds number,
$\mathcal{O}(\mathrm{Rn^{-2}})$, or of linear order in their product,
$\mathcal{O}(\mathrm{Kn}\,\mathrm{Rn^{-1}})$. Here, we have defined the
species-specific bulk-viscosity, diffusion, and shear-viscosity coefficients
as

$\displaystyle\zeta_{s,n}$ $\displaystyle\equiv$
$\displaystyle-\sum_{i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{0}}\tau_{si,nr}^{\left(0\right)}\alpha_{i,r}^{\left(0\right)}\;,$
(98) $\displaystyle\kappa_{s,n,q}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{1}}\tau_{si,nr}^{\left(1\right)}\alpha_{i,r,q}^{\left(1\right)}\;,$
(99) $\displaystyle\eta_{s,n}$ $\displaystyle\equiv$
$\displaystyle\sum_{i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,nr}^{\left(2\right)}\alpha_{i,r}^{\left(2\right)}\;.$
(100)

Note that the definition of the bulk-viscosity coefficient differs by a factor
of $-m^{2}_{s}/3$ compared to Eq. (63) of Ref. [21].

From here on, we will make the assumption that the irreducible moments
$\rho_{i,r}$, $\rho_{i,r}^{\left\langle\mu\right\rangle}$, and
$\rho_{i,r}^{\left\langle\mu\nu\right\rangle}$ are of the same order,
irrespective of the particle species. This implies that the sum over all
species of these irreducible moments, i.e. $\rho_{r}$,
$\rho_{r}^{\left\langle\mu\right\rangle}$, and
$\rho_{r}^{\left\langle\mu\nu\right\rangle}$ are also of the same order as the
species-specific irreducible moments.

The so-called “order-of-magnitude approximation” is based on the first-order
solution of the moment equations (95) – (97), which is equivalent to the
Navier-Stokes limit. In this limit, the irreducible moments are algebraically
related to terms of first order in Knudsen number, also called thermodynamic
forces, multiplied by the corresponding transport coefficients,

$\displaystyle\rho_{s,n}$ $\displaystyle=$
$\displaystyle-\zeta_{s,n}\theta+\mathcal{O}(2)\;,$ (101)
$\displaystyle\rho_{s,n}^{\mu}$ $\displaystyle=$
$\displaystyle\sum_{q}^{\left\\{B,Q,S\right\\}}\kappa_{s,n,q}\nabla^{\mu}\alpha_{q}+\mathcal{O}(2)\;,$
(102) $\displaystyle\rho_{s,n}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\eta_{s,n}\sigma^{\mu\nu}+\mathcal{O}(2)\;,$ (103)

while all tensor moments of rank higher than two are at least of second order,
$\rho_{s,n}^{\mu_{1}\cdots\mu_{\ell}}\sim\mathcal{O}(2)$ for any $\ell>2$, see
Ref. [21] for details.

In principle, all irreducible moments are of first order in inverse Reynolds
number, $\mathcal{O}(\mathrm{Rn^{-1}})$, and thus formally independent of the
power counting in Knudsen number. The order-of-magnitude approximation, which
is based on the Navier-Stokes limit (101) – (103), assumes that the
irreducible moments are of first order in Knudsen number,
$\mathcal{O}(\mathrm{Kn})$, i.e. the regime where the Knudsen and the inverse
Reynolds numbers are of the same magnitude. This defines a power-counting
scheme, similar to the one described in Ref. [21], which helps to assign a
certain order to the various terms in the equations of motion. Then, all
$\mathcal{O}(2)$ terms on the right-hand sides of Eqs. (95) – (97), as well as
the comoving derivatives on the left-hand sides are of second order in Knudsen
number. The order-of-magnitude approximation is very similar to the “order-of-
magnitude method” in non-relativistic fluid dynamics [33].

Using the approximation (101) – (103) while summing over all particle species,
and for the moment omitting $\mathcal{O}(2)$ terms, we obtain the Navier-
Stokes relations for the mixture. From Eq. (69) together with Eq. (101) the
total bulk viscous pressure of the mixture reads

$\Pi\equiv-\sum_{s\,=\,1}^{N_{\text{spec}}}\frac{m_{s}^{2}}{3}\rho_{s,0}=\sum_{s\,=\,1}^{N_{\text{spec}}}\frac{m_{s}^{2}}{3}\zeta_{s,0}\theta\equiv-\zeta\theta\;.$
(104)

Similarly, we obtain from Eq. (64) together with Eq. (102) for the conserved
charge currents

$V_{q}^{\mu}\equiv\sum_{s\,=\,1}^{N_{\text{spec}}}q_{s}\rho_{s,0}^{\mu}=\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\sum_{s\,=\,1}^{N_{\text{spec}}}q_{s}\kappa_{s,0,q^{\prime}}\nabla^{\mu}\alpha_{q^{\prime}}\equiv\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\kappa_{qq^{\prime}}\nabla^{\mu}\alpha_{q^{\prime}}\;.$
(105)

Finally, the shear-stress tensor of the mixture follows from Eq. (66) together
with Eq. (103),

$\pi^{\mu\nu}\equiv\sum_{s\,=\,1}^{N_{\text{spec}}}\rho_{s,0}^{\mu\nu}=\sum_{s\,=\,1}^{N_{\text{spec}}}2\eta_{s,0}\sigma^{\mu\nu}\equiv
2\eta\sigma^{\mu\nu}\;.$ (106)

Note that the first-order thermodynamic forces are the same for all particle
species and for the mixture. Due to this fact, we obtain the first-order
transport coefficients of the mixture: the bulk viscosity $\zeta$, the
diffusion coefficients $\kappa_{qq^{\prime}}$, and the shear viscosity $\eta$,

$\zeta\equiv-\sum_{s\,=\,1}^{N_{\text{spec}}}\frac{m_{s}^{2}}{3}\zeta_{s,0}\;,\qquad\kappa_{qq^{\prime}}\equiv\sum_{s\,=\,1}^{N_{\text{spec}}}q_{s}\kappa_{s,0,q^{\prime}}\;,\qquad\eta\equiv\sum_{s\,=\,1}^{N_{\text{spec}}}\eta_{s,0}\;.$
(107)

Therefore, using Eqs. (101) – (103) together with the relativistic Navier-
Stokes relations (104) – (106) we readily obtain the following algebraic
relations between the species-specific irreducible moments and the primary
dissipative quantities of the mixture,

$\displaystyle\rho_{s,n}$ $\displaystyle=$
$\displaystyle\bar{\zeta}_{s,n}\Pi+\mathcal{O}(2)\;,$ (108)
$\displaystyle\rho_{s,n}^{\mu}$ $\displaystyle=$
$\displaystyle\sum_{q}^{\left\\{B,Q,S\right\\}}\bar{\kappa}_{s,n}^{\left(q\right)}V_{q}^{\mu}+\mathcal{O}(2)\;,$
(109) $\displaystyle\rho_{s,n}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\bar{\eta}_{s,n}\pi^{\mu\nu}+\mathcal{O}(2)\;,$ (110)

where we introduced the normalized transport coefficients for each species,

$\bar{\zeta}_{s,n}=\frac{\zeta_{s,n}}{\zeta}\;,\qquad\bar{\kappa}_{s,n}^{\left(q\right)}=\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\kappa_{s,n,q^{\prime}}\left(\kappa^{-1}\right)_{q^{\prime}q}\;,\qquad\bar{\eta}_{s,n}=\frac{\eta_{s,n}}{\eta}\;.$
(111)

Here, $\left(\kappa^{-1}\right)_{q^{\prime}q}$ is the inverse of the
diffusion-coefficient matrix defined in Eq. (107). We will use Eqs. (108) –
(110) to close the equations of motion (83) – (87) in the next section.

###  III.4 The equations of motion in $\left(10+4N_{q}\right)$-moment
approximation

In the $\left(10+4N_{q}\right)$-moment approximation, we truncate the infinite
set of moment equations (83), (85), and (87) in the following way. We first
multiply these equations with the corresponding relaxation-time matrices
$\tau_{si,nr}^{\left(\ell\right)}$. Equation (85) is also multiplied by
$q_{s}$, and all equations are summed over species. In what follows, we only
consider the set of equations for $n=0$. In all terms we then substitute
$\rho_{i,r}$, $\rho_{i,r}^{\mu}$, and $\rho_{i,r}^{\mu\nu}$ by the dissipative
quantities $\Pi,\,V_{q}^{\mu}$, and $\pi^{\mu\nu}$ using Eqs. (108) – (110).
Note that in this substitution the $\mathcal{O}(2)$ terms in Eqs. (108) –
(110) become $\mathcal{O}(3)$ terms and can be neglected together with other
higher-order terms. In this way, we finally obtain a closed set of equations
of motion for the dissipative quantities $\Pi,\,V_{q}^{\mu}$, and
$\pi^{\mu\nu}$. Further, as discussed in Sec. II.4, the
$\left(10+4N_{q}\right)$-moment approximation is the lowest-order truncation
of the series in Eq. (51), where $\ell\leq 2$, and $N_{0}=2$, $N_{1}=1$, and
$N_{2}=0$. It should be noted that, for $\ell\leq 2$, such a truncation in
powers of energy (i.e. $N_{\ell}<\infty$) neglects infinitely many
contributions of order $\mathcal{O}(\text{Kn})$ in Eq. (51). As shown in Ref.
[21], however, systematically increasing the parameters $N_{0}$, $N_{1}$, and
$N_{2}$ the values of the corresponding transport coefficients exhibit rapid
convergence.

The resulting equation of motion for the bulk viscous pressure is

$\tau_{\Pi}\dot{\Pi}+\Pi=-\zeta\theta-\delta_{\Pi\Pi}\,\Pi\theta+\lambda_{\Pi\pi}\,\pi^{\mu\nu}\sigma_{\mu\nu}-\sum_{q}^{\left\\{B,Q,S\right\\}}\ell_{\Pi
V}^{(q)}\,\nabla_{\mu}V_{q}^{\mu}-\sum_{q}^{\left\\{B,Q,S\right\\}}\tau_{\Pi
V}^{(q)}\,V_{q}^{\mu}\dot{u}_{\mu}-\sum_{q,q^{\prime}}^{\left\\{B,Q,S\right\\}}\lambda_{\Pi
V}^{(q,q^{\prime})}\,V_{q}^{\mu}\nabla_{\mu}\alpha_{q^{\prime}}\;,$ (112)

where we have defined the relaxation time and the bulk-viscosity coefficient
as

$\displaystyle\tau_{\Pi}$ $\displaystyle=$
$\displaystyle-\sum_{s,i\,=\,1}^{N_{\text{spec}}}\frac{m_{s}^{2}}{3}\sum_{r\,=\,0}^{N_{0}}\tau_{si,0r}^{\left(0\right)}\bar{\zeta}_{i,r}\;,$
(113) $\displaystyle\zeta$ $\displaystyle=$
$\displaystyle-\sum_{s\,=\,1}^{N_{\text{spec}}}\frac{m_{s}^{2}}{3}\zeta_{s,0}\equiv\sum_{s,i\,=\,1}^{N_{\text{spec}}}\frac{m_{s}^{2}}{3}\sum_{r\,=\,0}^{N_{0}}\tau_{si,0r}^{\left(0\right)}\alpha_{i,r}^{\left(0\right)}\;.$
(114)

All second-order transport coefficients appearing in Eq. (112) are listed in
Appendix C.1.

Similarly, the equations of motion for the charge diffusion currents read

$\displaystyle\sum_{q}^{\left\\{B,Q,S\right\\}}\tau_{q^{\prime}q}\,\dot{V}_{q}^{\left\langle\mu\right\rangle}+V_{q^{\prime}}^{\mu}$
$\displaystyle=\sum_{q}^{\left\\{B,Q,S\right\\}}\kappa_{q^{\prime}q}\,\nabla^{\mu}\alpha_{q}-\sum_{q}^{\left\\{B,Q,S\right\\}}\tau_{q^{\prime}q}\,V_{q,\nu}\omega^{\nu\mu}-\sum_{q}^{\left\\{B,Q,S\right\\}}\delta_{VV}^{(q^{\prime},q)}\,V_{q}^{\mu}\theta-\sum_{q}^{\left\\{B,Q,S\right\\}}\lambda_{VV}^{(q^{\prime},q)}\,V_{q,\nu}\sigma^{\mu\nu}$
$\displaystyle-\ell_{V\Pi}^{(q^{\prime})}\,\nabla^{\mu}\Pi+\ell_{V\pi}^{(q^{\prime})}\,\Delta^{\mu\nu}\nabla_{\lambda}\pi_{\nu}^{\lambda}+\tau_{V\Pi}^{(q^{\prime})}\,\Pi\dot{u}^{\mu}-\tau_{V\pi}^{(q^{\prime})}\,\pi^{\mu\nu}\dot{u}_{\nu}$
$\displaystyle+\sum_{q}^{\left\\{B,Q,S\right\\}}\lambda_{V\Pi}^{(q^{\prime},q)}\,\Pi\nabla^{\mu}\alpha_{q}-\sum_{q}^{\left\\{B,Q,S\right\\}}\lambda_{V\pi}^{(q^{\prime},q)}\,\pi^{\mu\nu}\nabla_{\nu}\alpha_{q}\;,$
(115)

where the relaxation-time matrix and the diffusion-coefficient matrix are

$\displaystyle\tau_{q^{\prime}q}$ $\displaystyle=$
$\displaystyle\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{1}}q^{\prime}_{s}\tau_{si,0r}^{\left(1\right)}\bar{\kappa}^{(q)}_{i,r}\;,$
(116) $\displaystyle\kappa_{q^{\prime}q}$ $\displaystyle=$
$\displaystyle\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{1}}q^{\prime}_{s}\tau_{si,0r}^{\left(1\right)}\alpha_{i,r,q}^{\left(1\right)}\;,$
(117)

and the second-order transport coefficients are listed in Appendix C.2.

The diffusion-coefficient matrix $\kappa_{qq^{\prime}}$ has been evaluated for
several hadronic and partonic systems in Refs. [16, 17, 18]. In general, this
matrix couples the diffusion current of a specific charge to _all_ gradients
of the charge chemical potentials via
$V_{q^{\prime}}^{\mu}\sim\sum_{q}^{\left\\{B,Q,S\right\\}}\kappa_{q^{\prime}q}\,\nabla^{\mu}\alpha_{q}+\mathcal{O}(2)$.
Due to this coupled diffusion, the density gradients in one charge could lead
to the local separation in another charge, as demonstrated in Ref. [16]. From
the above equations of motion one can see that such a coupling is also present
in various second-order terms.

The equation of motion for the shear-stress tensor follows in a similar
manner,

$\displaystyle\tau_{\pi}\dot{\pi}^{\left\langle\mu\nu\right\rangle}+\pi^{\mu\nu}$
$\displaystyle=2\eta\sigma^{\mu\nu}+2\tau_{\pi}\,\pi_{\lambda}^{\left\langle\mu\right.}\omega^{\left.\nu\right\rangle\lambda}-\delta_{\pi\pi}\,\pi^{\mu\nu}\theta-\tau_{\pi\pi}\,\pi^{\lambda\left\langle\mu\right.}\sigma_{\lambda}^{\left.\nu\right\rangle}+\lambda_{\pi\Pi}\,\Pi\sigma^{\mu\nu}$
$\displaystyle-\sum_{q}^{\left\\{B,Q,S\right\\}}\tau_{\pi
V}^{(q)}\,V_{q}^{\left\langle\mu\right.}\dot{u}^{\left.\nu\right\rangle}+\sum_{q}^{\left\\{B,Q,S\right\\}}\ell_{\pi
V}^{(q)}\,\nabla^{\left\langle\mu\right.}V_{q}^{\left.\nu\right\rangle}+\sum_{q,q^{\prime}}^{\left\\{B,Q,S\right\\}}\lambda_{\pi
V}^{(q,q^{\prime})}\,V_{q}^{\left\langle\mu\right.}\nabla^{\left.\nu\right\rangle}\alpha_{q^{\prime}}\;,$
(118)

where the relaxation time and the coefficient of the shear viscosity are given
by

$\displaystyle\tau_{\pi}$ $\displaystyle=$
$\displaystyle\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,0r}^{\left(2\right)}\bar{\eta}_{i,r}\;,$
(119) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,0r}^{\left(2\right)}\alpha_{i,r}^{\left(2\right)}\;,$
(120)

while the remaining second-order transport coefficients are given in Appendix
C.3. The equations of motion (112), (115), and (118) are of relaxation type
and are identical to those found in Refs. [10, 11, 14]. For more details we
refer to the discussion in Appendix A.

As a simple example of a relativistic multicomponent system, we discuss an
ultrarelativistic, ideal gas with elastic, isotropic hard-sphere interactions
and multiple conserved charges in Appendix F. While the transport coefficients
cannot be further reduced to simple and convenient forms, one may easily prove
that one obtains well-known results in the single-component limit [21].

##  IV Conclusions and Outlook

In this paper, we have presented the derivation of relativistic second-order
dissipative fluid dynamics for multicomponent systems in the
$(10+4N_{q})$-moment approximation from the relativistic Boltzmann equation
using the method of moments. Starting from the relativistic Boltzmann equation
for a multicomponent system we have obtained the equations of motion for the
irreducible moments for particle species $i$. In the single-fluid
approximation for the mixture, the sum of the dynamical equations of motion
reduces to $4+N_{q}$ conservation equations that are closed by providing
$6+3N_{q}$ relaxation-type equations of motion for the dissipative quantities.
In such a mixture, where the constituents in general carry multiple quantum
charges (e.g. a proton carrying electric charge as well as baryon number), the
equation of state depends on multiple chemical potentials and temperature.
With the help of a new approximation scheme, the so-called order-of-magnitude
approximation, we have derived a second-order dissipative theory that does not
contain terms of second order in Knudsen number, which are known to render the
equations of motion parabolic and thus acausal [21]. Furthermore, the
irreducible moments of the deviation of the single-particle distribution of
each particle species from equilibrium are directly proportional to the total
bulk viscous pressure $\Pi$, the conserved charge-diffusion currents
$V^{\mu}_{q}$, and the total shear-stress tensor $\pi^{\mu\nu}$ via Eqs. (108)
– (110).

Similar to other works which treat multicomponent systems [10, 11, 14], in
this theory the existence of multiple conserved charges is manifest in the
equations of motion Eqs. (112), (115), and (118). As expected we obtained
exactly the same equations of motion as found in earlier works [10, 11, 14].

Further, the coupled charge transport becomes explicit in the appearance of
mixing terms in the equations of motions, e.g. a dissipative current ($\Pi$,
$V^{\mu}_{q}$, $\pi^{\mu\nu}$) is coupled to _any other_ gradient in chemical
potential or diffusion current. As a prominent example, instead of a diffusive
Navier-Stokes term with only one diffusion coefficient $\kappa$ as in a
single-component system, we obtain a Navier-Stokes term entailing a matrix of
diffusion coefficients $\kappa_{qq^{\prime}}$, which explicitly couples every
diffusion current to all gradients in chemical potential. The appearance of a
charge-coupled Navier-Stokes term and potential implications for the transport
of charge was discussed in Ref. [16] in the case of relativistic nuclear
matter.

The advantage of our derivation compared to other theories is that it yields
explicit expressions for the transport coefficients in terms of the linearized
collision term. Since the mutual interactions of all particle species is
contained in the collision term, the multicomponent nature of the mixture is
naturally encoded in the transport coefficients.

In the future, this theory will be used to revisit the transport of coupled
charge in heavy-ion collisions initiated in Ref. [16] in a more realistic
manner. Especially, we expect that it will be relevant for the discussion of
physics of compressed baryonic matter at the future FAIR and NICA facilities
or for the interpretation of recent results of the isobar run at RHIC. We
expect that the coupling of diffusion currents or the charge gradients to the
bulk viscous pressure and the shear-stress tensor may be important in future
studies. Now that the explicit expressions of the transport coefficients have
been derived, they can be evaluated for nuclear systems. To this end,
equations of state from lattice QCD for non-vanishing chemical potentials may
be used [34, 35]. At the same time, Eq. (51) provides an expression for the
so-called $\delta f$-correction needed for the freeze-out of the system at the
end of the fluid-dynamical phase during the simulation of a heavy-ion
collision.

## Acknowledgments

The authors thank G.S. Denicol and P. Huovinen for fruitful discussions. They
acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through the CRC-TR 211 “Strong-interaction matter under
extreme conditions” – project number 315477589 – TRR 211. J.A.F. acknowledges
support from the Helmholtz Graduate School for Heavy-Ion Research. E. M. was
also supported by the program Excellence Initiative–Research University of the
University of Wrocław of the Ministry of Education and Science. D.H.R. is
supported by the State of Hesse within the Research Cluster ELEMENTS (Project
ID 500/10.006). This project is supported by the European Research Council
under project ERC-2018-ADG-835105 YoctoLHC. This publication is part of a
project that has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement STRONG – 2020 - No
824093. This research was funded as a part of the CoE in Quark Matter of the
Academy of Finland.

##  Appendix A Comparison to other works

In this appendix we perform a comparison of the second-order relaxation
equations found in this paper, Eqs. (112), (115), and (118), to earlier
derivations of Monnai and Hirano [10] and Kikuchi, Tsumura, and Kunihiro [14].

Our second-order relaxation equation (112) for the bulk viscosity contains 8
terms in total, while Eq. (69) of Ref. [10] contains 13 terms. Here, we recall
this equation noting that $u^{\mu}_{E}=u^{\mu}$, while $J=q$ and
$K=q^{\prime}$,

$\displaystyle\Pi$ $\displaystyle=$
$\displaystyle-\zeta\nabla_{\mu}u^{\mu}_{E}-\tau_{\Pi}D\Pi+\chi_{\Pi\Pi}^{c}\Pi\nabla_{\mu}u^{\mu}_{E}+\chi_{\Pi\pi}\pi_{\mu\nu}\nabla^{\langle\mu}u^{\nu\rangle}_{E}$
(121) $\displaystyle+$ $\displaystyle\sum_{J,K}\chi_{\Pi
V_{J}}^{aK}V^{J}_{\mu}\nabla^{\mu}\frac{\mu_{K}}{T}+\sum_{J}\chi_{\Pi
V_{J}}^{c}V^{J}_{\mu}Du^{\mu}_{E}+\sum_{J}\chi_{\Pi
V_{J}}^{d}\nabla^{\mu}V^{J}_{\mu}$ $\displaystyle+$
$\displaystyle\sum_{J}\chi_{\Pi\Pi}^{aJ}\Pi
D\frac{\mu_{J}}{T}+\chi_{\Pi\Pi}^{b}\Pi D\frac{1}{T}-\zeta_{\Pi\delta
e}D\frac{1}{T}+\sum_{J}\zeta_{\Pi\delta
n_{J}}D\frac{\mu_{J}}{T}+\sum_{J}\chi_{\Pi
V_{J}}^{b}V^{J}_{\mu}\nabla^{\mu}\frac{1}{T}\;.$

The difference between this equation and ours is due to the difference in the
thermodynamic forces and the way the comoving derivatives and space-time four-
gradients are employed. In our derivation the comoving derivatives,
$D\beta=D\frac{1}{T}$ and $D\alpha_{q}=D\frac{\mu_{J}}{T}$, do not appear
explicitly since they were replaced using Eqs. (77) and (78), while the space-
time four-gradient of the inverse temperature, $\nabla^{\mu}\beta$, is given
by Eq. (81). The terms that are expressed differently are in the third-line of
the above equation (121). Now, collecting these various terms one can show
that Eq. (121) reduces to Eq. (112).

Similarly, we recall Eq. (77) of Ref. [14], which contains 11 terms,

$\displaystyle\Pi$
$\displaystyle=-\zeta\theta-\tau_{\Pi}\frac{\partial}{\partial\tau}\Pi-\sum_{A\,=\,1}^{M}\ell^{A}_{\Pi
J}\nabla\cdot
J_{A}+\kappa_{\Pi\Pi}\Pi\theta+\kappa_{\Pi\pi}\pi_{\rho\sigma}\sigma^{\rho\sigma}$
$\displaystyle+b_{\Pi\Pi\Pi}\Pi^{2}+\sum_{A,B\,=\,1}^{M}b_{\Pi
JJ}^{AB}J_{A}^{\rho}J_{B,\rho}+b_{\Pi\pi\pi}\pi^{\rho\sigma}\pi_{\rho\sigma}$
$\displaystyle+\sum_{A\,=\,1}^{M}\kappa^{(1)A}_{\Pi
J}J_{A,\rho}\nabla^{\rho}T+\sum_{A,B\,=\,1}^{M}\kappa^{(2)BA}_{\Pi
J}J_{A,\rho}\nabla^{\rho}\frac{\mu_{B}}{T}\;,$ (122)

where we note that $V_{A}=V_{q}$ while $A=q$ and $B=q^{\prime}$. From these
the terms in the second line are of second order in dissipative quantities,
i.e. of second order in inverse Reynolds number, originating from the non-
linear part of the collision integral. Note that such second-order terms were
also obtained in Refs. [21, 36], but are neglected in our study. The remaining
2 terms are formally the same, which can be seen using Eq. (81).

The relaxation equation (115) for the conserved charge current contains 12
terms, while Eq. (70) of Ref. [10] listed below contains 19 terms,

$\displaystyle V_{J}^{\mu}$ $\displaystyle=$
$\displaystyle\kappa_{V_{J}}\nabla^{\mu}\frac{\mu_{J}}{T}+\sum_{K\neq
J}\kappa_{V_{J}V_{K}}\nabla^{\mu}\frac{\mu_{K}}{T}-\tau_{V_{J}}\Delta^{\mu\nu}DV^{J}_{\nu}-\sum_{K\neq
J}\tau_{V_{J}V_{K}}\Delta^{\mu\nu}DV^{K}_{\nu}$ (123) $\displaystyle+$
$\displaystyle\sum_{K}\chi_{V_{J}V_{K}}^{c}V_{K}^{\mu}\nabla_{\nu}u^{\nu}_{E}+\sum_{K}\chi_{V_{J}V_{K}}^{d}V_{K}^{\nu}\nabla_{\nu}u^{\mu}_{E}+\sum_{K}\chi_{V_{J}V_{K}}^{e}V_{K}^{\nu}\nabla^{\mu}u_{\nu}^{E}$
$\displaystyle+$
$\displaystyle\sum_{K}\chi_{V_{J}\pi}^{aK}\pi^{\mu\nu}\nabla_{\nu}\frac{\mu_{K}}{T}+\chi_{V_{J}\pi}^{c}\pi^{\mu\nu}Du_{\nu}^{E}+\chi_{V_{J}\pi}^{d}\Delta^{\mu\nu}\nabla^{\rho}\pi_{\nu\rho}+\sum_{K}\chi_{V_{J}\Pi}^{aK}\Pi\nabla^{\mu}\frac{\mu_{K}}{T}+\chi_{V_{J}\Pi}^{c}\Pi
Du^{\mu}_{E}+\chi_{V_{J}\Pi}^{d}\nabla^{\mu}\Pi$ $\displaystyle+$
$\displaystyle\chi_{V_{J}\pi}^{b}\pi^{\mu\nu}\nabla_{\nu}\frac{1}{T}+\chi_{V_{J}\Pi}^{b}\Pi\nabla^{\mu}\frac{1}{T}+\sum_{K,L}\chi_{V_{J}V_{K}}^{aL}V_{K}^{\mu}D\frac{\mu_{L}}{T}+\sum_{K}\chi_{V_{J}V_{K}}^{b}V_{K}^{\mu}D\frac{1}{T}+\kappa_{V_{J}W}\bigg{(}\frac{1}{T}Du_{E}^{\mu}+\nabla^{\mu}\frac{1}{T}\bigg{)}\;.$

After closer inspection, we observe that the first line contains two sums that
are equivalent to our sums over charges, while the additional 4 terms in the
last line can be incorporated into already existing terms. Furthermore, the
last remaining term, $\kappa_{V_{q}W}(\beta Du^{\mu}+\nabla^{\mu}\beta)$, may
be expressed using Eq. (81), and hence is fully accounted for in our approach.

On the other hand, Eq. (78) of Ref. [14] contains 14 terms, of which the last
2 are of second order in inverse Reynolds number, while the other terms are
formally the same,

$\displaystyle J_{A}^{\mu}$
$\displaystyle=\sum_{B\,=\,1}^{M}\lambda_{AB}\frac{T^{2}}{h^{2}}\nabla^{\mu}\frac{\mu_{B}}{T}-\sum_{B\,=\,1}^{M}\tau_{J}^{AB}\Delta^{\mu\rho}\frac{\partial}{\partial\tau}J_{B,\rho}-\ell^{A}_{J\Pi}\nabla^{\mu}\Pi-\ell^{A}_{J\pi}\Delta^{\mu\rho}\nabla_{\nu}{\pi^{\nu}}_{\rho}$
$\displaystyle+\kappa^{(1)A}_{J\Pi}\Pi\nabla^{\mu}T+\sum_{B\,=\,1}^{M}\kappa^{(2)AB}_{J\Pi}\Pi\nabla^{\mu}\frac{\mu_{B}}{T}+\sum_{B\,=\,1}^{M}\kappa^{(1)AB}_{JJ}J^{\mu}_{B}\theta+\sum_{B\,=\,1}^{M}\kappa^{(2)AB}_{JJ}J_{B,\rho}\sigma^{\mu\rho}+\kappa^{(3)AB}_{JJ}J_{B,\rho}\omega^{\mu\rho}$
$\displaystyle+\kappa^{(1)A}_{J\pi}\pi^{\mu\rho}\nabla_{\rho}T+\sum_{B\,=\,1}^{M}\kappa^{(2)AB}_{J\pi}\pi^{\mu\rho}\nabla_{\rho}\frac{\mu_{B}}{T}+\sum_{B\,=\,1}^{M}b^{AB}_{J\Pi
J}\Pi J_{B}^{\mu}+\sum_{B\,=\,1}^{M}b^{AB}_{JJ\pi}J_{B,\rho}\pi^{\rho\mu}\;.$
(124)

The relaxation equation (118) for the shear-stress tensor contains 10 terms,
while Eq. (71) of Ref. [10] contains 12 terms

$\displaystyle\pi^{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\eta\nabla^{\langle\mu}u^{\nu\rangle}_{E}-\tau_{\pi}D\pi^{\langle\mu\nu\rangle}+\chi_{\pi\pi}^{c}\pi^{\mu\nu}\nabla_{\rho}u^{\rho}_{E}+\chi_{\pi\pi}^{d}\pi^{\rho\langle\mu}\nabla_{\rho}u^{\nu\rangle}_{E}+\chi_{\pi\Pi}\Pi\nabla^{\langle\mu}u^{\nu\rangle}_{E}$
(125) $\displaystyle+$ $\displaystyle\sum_{J}\chi_{\pi
V_{J}}^{c}V_{J}^{\langle\mu}Du^{\nu\rangle}_{E}+\sum_{J}\chi_{\pi
V_{J}}^{d}\nabla^{\langle\mu}V_{J}^{\nu\rangle}+\sum_{J,K}\chi_{\pi
V_{J}}^{aJ}V_{J}^{\langle\mu}\nabla^{\nu\rangle}\frac{\mu_{K}}{T}$
$\displaystyle+$ $\displaystyle\sum_{J}\chi_{\pi
V_{J}}^{b}V_{J}^{\langle\mu}\nabla^{\nu\rangle}\frac{1}{T}+\sum_{J}\chi_{\pi\pi}^{aJ}\pi^{\mu\nu}D\frac{\mu_{J}}{T}+\chi_{\pi\pi}^{b}\pi^{\mu\nu}D\frac{1}{T}\;.$

Here the last 3 terms that are expressed using $D\beta$, $D\alpha_{q}$, and
$\nabla^{\mu}\beta$, may once again be incorporated into other terms.

Finally, Eq. (79) of Ref. [14] contains 13 terms. The last 3 are of second
order in inverse Reynolds number, while the remaining 10 terms are formally
similar to ours,

$\displaystyle\pi^{\mu\nu}$
$\displaystyle=2\eta\sigma^{\mu\nu}-\tau_{\pi}\Delta^{\mu\nu\rho\sigma}\frac{\partial}{\partial\tau}\pi_{\rho\sigma}-\sum_{A\,=\,1}^{M}\ell^{A}_{\pi
J}\nabla^{\langle\mu}J^{\nu\rangle}_{A}+\kappa_{\pi\Pi}\Pi\sigma^{\mu\nu}+\sum_{A\,=\,1}^{M}\kappa^{(1)A}_{\pi
J}J^{\langle\mu}_{A}\nabla^{\nu\rangle}T+\sum_{A,B\,=\,1}^{M}\kappa^{(2)BA}_{\pi
J}J^{\langle\mu}_{A}\nabla^{\nu\rangle}\frac{\mu_{B}}{T}$
$\displaystyle+\kappa^{(1)}_{\pi\pi}\pi^{\mu\nu}\theta+\kappa^{(2)}_{\pi\pi}\pi^{\lambda\langle\mu}{\sigma^{\nu\rangle}}_{\lambda}+\kappa^{(3)}_{\pi\pi}\pi^{\lambda\langle\mu}{\omega^{\nu\rangle}}_{\lambda}+b_{\pi\Pi\pi}\Pi\pi^{\mu\nu}+\sum_{A,B\,=\,1}^{M}b^{AB}_{\pi
JJ}J^{\langle\mu}_{A}J^{\nu\rangle}_{B}+b_{\pi\pi\pi}\pi^{\lambda\langle\mu}{\pi^{\nu\rangle}}_{\lambda}\;.$
(126)

##  Appendix B Eckart frame

In most textbooks and relevant publications the local rest frame and the fluid
four-velocity are chosen according to Eckart [29], since this choice
intuitively follows the non-relativistic interpretation of physical
quantities. On the other hand, all our results are given relative to the local
rest frame of Landau. In this appendix, we will elaborate on the differences.

We may choose to define a different time-like normalized flow vector,
$\tilde{u}^{\mu}$, and hence a local frame of reference different from the
previously chosen local rest frame (the Landau frame) given by $u^{\mu}$. The
$\tilde{u}$-frame is related to the $u$-frame by a Lorentz transformation. If
we assume that the difference between the frame vectors is small,
$\tilde{u}^{\mu}-u^{\mu}\sim\mathcal{O}(1)$, we may write

$\tilde{u}^{\mu}=u^{\mu}+w^{\mu}+\mathcal{O}(2)\;.$ (127)

Computing the normalization of $\tilde{u}^{\mu}$ up to order $\mathcal{O}(1)$,

$\tilde{u}^{\mu}\tilde{u}_{\mu}=u^{\mu}u_{\mu}+2u^{\mu}w_{\mu}+w^{\mu}w_{\mu}=1+2u^{\mu}w_{\mu}+\mathcal{O}(2)\;,$
(128)

and demanding that $\tilde{u}^{\mu}$ is also normalized, we conclude that
$w^{\mu}$ must be orthogonal to $u^{\mu}$, $u^{\mu}w_{\mu}=0$. The projection
operator onto the three-space orthogonal to $\tilde{u}^{\mu}$ is

$\tilde{\Delta}^{\mu\nu}\equiv
g^{\mu\nu}-\tilde{u}^{\mu}\tilde{u}^{\nu}=\Delta^{\mu\nu}-2u^{\left(\mu\right.}w^{\left.\nu\right)}-w^{\mu}w^{\nu}=\Delta^{\mu\nu}-2u^{\left(\mu\right.}w^{\left.\nu\right)}+\mathcal{O}(2)\;.$
(129)

The tensor decomposition of the primary fluid-dynamical quantities with
respect to $\tilde{u}^{\mu}$ leads to results similar to the tensor
decompositions listed in Eqs. (56) – (57),

$\displaystyle N^{\mu}_{q}$ $\displaystyle\equiv$
$\displaystyle\tilde{n}_{q}\tilde{u}^{\mu}+\tilde{V}_{q}^{\mu}=\tilde{n}_{q}\left(u^{\mu}+w^{\mu}\right)+\tilde{V}_{q}^{\mu}\;,$
(130) $\displaystyle T^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\tilde{e}\,\tilde{u}^{\mu}\tilde{u}^{\nu}-(\tilde{P}+\tilde{\Pi})\tilde{\Delta}^{\mu\nu}+\tilde{\pi}^{\mu\nu}+2\tilde{W}^{\left(\mu\right.}\tilde{u}^{\left.\nu\right)}$
(131) $\displaystyle=$
$\displaystyle\tilde{e}\,u^{\mu}u^{\nu}-(\tilde{P}+\tilde{\Pi})\Delta^{\mu\nu}+\tilde{\pi}^{\mu\nu}+2(\tilde{e}+\tilde{P})w^{\left(\mu\right.}u^{\left.\nu\right)}+2\tilde{W}^{\left(\mu\right.}u^{\left.\nu\right)}+\mathcal{O}(2)\;,$

where we have explicitly applied Landau’s matching conditions from Eqs. (67),
(68). We have also made use of Eq. (127) and the fact that all dissipative
quantities are $\sim\mathcal{O}(1)$. The physical quantities follow from
similar projection operations as in Eqs. (58) – (66).

Furthermore, one can show that neglecting corrections of order
$\mathcal{O}(2)$, the net-particle density, conserved net-charge density,
energy density, equilibrium and bulk pressures, as well the shear-stress
components are equal in both local rest frames,

$\tilde{n}_{q}=n_{q}\;,\quad\tilde{e}=e\;,\quad\tilde{P}=P\;,\quad\tilde{\Pi}=\Pi\;,\quad\tilde{\pi}^{\mu\nu}=\pi^{\mu\nu}\;.$
(132)

Note that the quantities without tilde are taken in the Landau frame and are
not identical with the corresponding quantities when the frame is not yet
specified.

The choice of the local rest frame changes the diffusion currents orthogonal
to the flow velocity, i.e. the so-called peculiar velocities. In Eckart’s
definition of the local rest frame, for _one_ conserved charge $q$ it is
required that

$\displaystyle N_{q}^{\mu}\equiv
n_{q}\tilde{u}^{\mu}\quad\textrm{and}\quad\tilde{V}^{\mu}_{q}\equiv 0\;,$
(133)

where $n_{q}=N_{q}^{\mu}\tilde{u}_{\mu}$ and
$\tilde{V}_{q}^{\mu}=N_{q}^{\nu}\tilde{\Delta}^{\mu}_{\nu}$. This means that
according to the definition of Eckart, there is no net-charge diffusion
current of charge type $q$ in its own local rest frame. Now, comparing this
with the form of $N^{\mu}_{q}$ in the Landau frame leads to

$N_{q}^{\mu}\equiv
n_{q}\left(u^{\mu}+w^{\mu}\right)\overset{!}{=}n_{q}u^{\mu}+V_{q}^{\mu}\;.$
(134)

Thus, for an observer in Landau’s local rest frame the particles are diffusing
with peculiar four-velocities proportional to the net-charge diffusion
current,

$w^{\mu}=\frac{V_{q}^{\mu}}{n_{q}}\;.$ (135)

Since the dissipative quantity $V_{q}^{\mu}\sim\mathcal{O}(1)$, this is
consistent with our assumption that $w^{\mu}\sim\mathcal{O}(1)$. Up to terms
of order $\mathcal{O}(1)$, the energy-momentum tensor (131) reads

$\displaystyle T^{\mu\nu}$ $\displaystyle\equiv$ $\displaystyle
e\,\tilde{u}^{\mu}\tilde{u}^{\nu}-(P+\Pi)\tilde{\Delta}^{\mu\nu}+\pi^{\mu\nu}+2\tilde{W}^{\left(\mu\right.}\tilde{u}^{\left.\nu\right)}$
(136) $\displaystyle=$ $\displaystyle
e\,u^{\mu}u^{\nu}-(P+\Pi)\Delta^{\mu\nu}+\pi^{\mu\nu}+2(e+P)w^{\left(\mu\right.}u^{\left.\nu\right)}+2\tilde{W}^{\left(\mu\right.}u^{\left.\nu\right)}\;.$

Using this result together with Eq. (135) and comparing it to the energy-
momentum tensor in the Landau frame,

$\displaystyle
T^{\mu\nu}=e\,u^{\mu}u^{\nu}-(P+\Pi)\Delta^{\mu\nu}+\pi^{\mu\nu}\;,$ (137)

leads to the total energy-momentum diffusion current
$\tilde{W}^{\mu}=\Delta_{\alpha}^{\mu}T^{\alpha\beta}u_{\beta}$ in the Eckart
frame,

$\tilde{W}^{\mu}\equiv-\frac{e+P}{n_{q}}V_{q}^{\mu}\equiv-h_{q}V_{q}^{\mu}\;.$
(138)

This relates the total energy-momentum flux seen in the Eckart frame to the
diffusion flux of charge $q$ observed in the Landau frame.

Finally, we give the four-flow of the charges $q^{\prime}$, of which the
diffusion currents were _not_ chosen to vanish by Eckart’s choice (133) (i.e.
$q^{\prime}\neq q$). Using the fact that the charge densities $n_{q^{\prime}}$
are equal in both frames, we obtain the condition

$\displaystyle
N^{\mu}_{q^{\prime}}=n_{q^{\prime}}\left(u^{\mu}+w^{\mu}\right)+\tilde{V}^{\mu}_{q^{\prime}}\overset{!}{=}n_{q^{\prime}}u^{\mu}+V^{\mu}_{q^{\prime}}\;.$
(139)

Employing Eq. (135), we arrive at an expression for the diffusion currents as
observed in the Eckart frame:

$\displaystyle\tilde{V}^{\mu}_{q^{\prime}}=V^{\mu}_{q^{\prime}}-\frac{n_{q^{\prime}}}{n_{q}}V^{\mu}_{q}\;.$
(140)

Note that for the case $q=q^{\prime}$, we again recover the requirement by the
Eckart frame definition for the charge $q$, Eq. (133).

##  Appendix C Transport Coefficients

In this appendix, we list all second-order transport coefficients in the
equations of motion (112), (115), and (118).

###  C.1 The coefficients in the bulk viscosity equation

The second-order transport coefficients in the equation of motion (112) for
the bulk viscous pressure are

$\displaystyle\delta_{\Pi\Pi}$
$\displaystyle=\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{0}}\frac{m_{s}^{2}}{9}\tau_{si,0r}^{\left(0\right)}\left[m_{i}^{2}\left(r-1\right)\bar{\zeta}_{i,r-2}-\left(r+2\right)\bar{\zeta}_{i,r}-3\left(J_{i,r+1,0}\mathcal{T}_{00}-\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}J_{i,r,0}\mathcal{T}_{q0}\right)\right]$
$\displaystyle+\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{0}}\frac{m_{s}^{2}}{3}\tau_{si,nr}^{\left(0\right)}\sum_{q}^{\left\\{B,Q,S\right\\}}\frac{\partial\bar{\zeta}_{i,r}}{\partial\alpha_{q}}\left[\mathcal{T}_{q0}\left(e+P\right)+\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\mathcal{T}_{qq^{\prime}}n_{q^{\prime}}\right]$
$\displaystyle+\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{0}}\frac{m_{s}^{2}}{3}\tau_{si,nr}^{\left(0\right)}\frac{\partial\bar{\zeta}_{i,r}}{\partial\beta}\left[\mathcal{T}_{00}\left(e+P\right)+\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\mathcal{T}_{0q^{\prime}}n_{q^{\prime}}\right]\;,$
(141) $\displaystyle\lambda_{\Pi\pi}$
$\displaystyle=-\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{0}}\frac{m_{s}^{2}}{3}\tau_{si,0r}^{\left(0\right)}\left[\left(r-1\right)\bar{\eta}_{i,r-2}+J_{i,r+1,0}\mathcal{T}_{00}-\sum_{q}^{\left\\{B,Q,S\right\\}}q_{i}J_{i,r,0}\mathcal{T}_{q0}\right]\;,$
(142) $\displaystyle\ell_{\Pi V}^{(q)}$
$\displaystyle=-\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{0}}\frac{m_{s}^{2}}{3}\tau_{si,0r}^{\left(0\right)}\left(\bar{\kappa}_{i,r-1}^{\left(q\right)}+J_{i,r+1,0}\mathcal{T}_{0q}-\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}q_{i}^{\prime}J_{i,r,0}\mathcal{T}_{q^{\prime}q}\right)\;,$
(143) $\displaystyle\tau_{\Pi V}^{(q)}$
$\displaystyle=\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{0}}\frac{m_{s}^{2}}{3}\tau_{si,0r}^{\left(0\right)}\left(r\bar{\kappa}_{i,r-1}^{\left(q\right)}+\beta\frac{\partial\bar{\kappa}_{i,r-1}^{\left(q\right)}}{\partial\beta}+J_{i,r+1,0}\mathcal{T}_{0q}-\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}q_{i}^{\prime}J_{i,r,0}\mathcal{T}_{q^{\prime}q}\right)\;,$
(144) $\displaystyle\lambda_{\Pi V}^{(q,q^{\prime})}$
$\displaystyle=-\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{0}}\frac{m_{s}^{2}}{3}\tau_{si,0r}^{\left(0\right)}\left(\frac{\partial\bar{\kappa}_{i,r-1}^{\left(q\right)}}{\partial\alpha_{q^{\prime}}}+\frac{n_{q^{\prime}}}{e+P}\frac{\partial\bar{\kappa}_{i,r-1}^{\left(q\right)}}{\partial\beta}\right)\;.$
(145)

###  C.2 The coefficients in the charge diffusion equations

The coefficients in the equations of motion (115) for the charge diffusion
currents are:

$\displaystyle\delta_{VV}^{(q^{\prime},q)}$
$\displaystyle=-\sum_{s,i\,=\,1}^{N_{\text{spec}}}q_{s}^{\prime}\sum_{r\,=\,0}^{N_{1}}\tau_{si,0r}^{\left(1\right)}\frac{1}{3}\left[m_{i}^{2}\left(r-1\right)\bar{\kappa}_{i,r-2}^{\left(q\right)}-\left(r+3\right)\bar{\kappa}_{i,r}^{\left(q\right)}\right]$
$\displaystyle-\sum_{s,i\,=\,1}^{N_{\text{spec}}}q_{s}^{\prime}\sum_{r\,=\,0}^{N_{1}}\tau_{si,0r}^{\left(1\right)}\sum_{q^{\prime\prime}}^{\left\\{B,Q,S\right\\}}\frac{\partial\bar{\kappa}_{i,r}^{\left(q\right)}}{\partial\alpha_{q^{\prime\prime}}}\left[\mathcal{T}_{q^{\prime\prime}0}\left(e+P\right)+\sum_{q^{\prime\prime\prime}}^{\left\\{B,Q,S\right\\}}\mathcal{T}_{q^{\prime\prime}q^{\prime\prime\prime}}n_{q^{\prime\prime\prime}}\right]$
$\displaystyle-\sum_{s,i\,=\,1}^{N_{\text{spec}}}q_{s}^{\prime}\sum_{r\,=\,0}^{N_{1}}\tau_{si,0r}^{\left(1\right)}\frac{\partial\bar{\kappa}_{i,r}^{\left(q\right)}}{\partial\beta}\left[\mathcal{T}_{00}\left(e+P\right)+\sum_{q^{\prime\prime}}^{\left\\{B,Q,S\right\\}}\mathcal{T}_{0q^{\prime\prime}}n_{q^{\prime\prime}}\right]\;,$
(146) $\displaystyle\lambda_{VV}^{(q^{\prime},q)}$
$\displaystyle=-\sum_{s,i\,=\,1}^{N_{\text{spec}}}q_{s}^{\prime}\sum_{r\,=\,0}^{N_{1}}\tau_{si,0r}^{\left(1\right)}\frac{1}{5}\left[m_{i}^{2}\left(2r-2\right)\bar{\kappa}_{i,r-2}^{(q)}-\left(2r+3\right)\bar{\kappa}_{i,r}^{(q)}\right]\;,$
(147) $\displaystyle\ell_{V\Pi}^{(q^{\prime})}$
$\displaystyle=\sum_{s,i\,=\,1}^{N_{\text{spec}}}q_{s}^{\prime}\sum_{r\,=\,0}^{N_{1}}\tau_{si,0r}^{\left(1\right)}\left[\frac{1}{3}\left(m_{i}^{2}\bar{\zeta}_{i,r-1}-\bar{\zeta}_{i,r+1}\right)-\alpha_{i,r}^{h}\right]\;,$
(148) $\displaystyle\ell_{V\pi}^{(q^{\prime})}$
$\displaystyle=-\sum_{s,i\,=\,1}^{N_{\text{spec}}}q_{s}^{\prime}\sum_{r\,=\,0}^{N_{1}}\tau_{si,0r}^{\left(1\right)}\left(\bar{\eta}_{i,r-1}+\alpha_{i,r}^{h}\right)\;,$
(149) $\displaystyle\tau_{V\Pi}^{(q^{\prime})}$
$\displaystyle=\sum_{s,i\,=\,1}^{N_{\text{spec}}}q_{s}^{\prime}\sum_{r\,=\,0}^{N_{1}}\tau_{si,0r}^{\left(1\right)}\frac{1}{3}\left[m_{i}^{2}r\bar{\zeta}_{i,r-1}-\left(r+3\right)\bar{\zeta}_{i,r+1}-3\alpha_{i,r}^{h}+m_{i}^{2}\beta\frac{\partial\bar{\zeta}_{i,r-1}}{\partial\beta}-\beta\frac{\partial\bar{\zeta}_{i,r+1}}{\partial\beta}\right]\;,$
(150) $\displaystyle\tau_{V\pi}^{(q^{\prime})}$
$\displaystyle=-\sum_{s,i\,=\,1}^{N_{\text{spec}}}q_{s}^{\prime}\sum_{r\,=\,0}^{N_{1}}\tau_{si,0r}^{\left(1\right)}\left(r\bar{\eta}_{i,r-1}+\beta\frac{\partial\bar{\eta}_{i,r-1}}{\partial\beta}+\alpha_{i,r}^{h}\right)\;,$
(151) $\displaystyle\lambda_{V\Pi}^{(q^{\prime},q)}$
$\displaystyle=-\sum_{s,i\,=\,1}^{N_{\text{spec}}}q_{s}^{\prime}\sum_{r\,=\,0}^{N_{1}}\tau_{si,0r}^{\left(1\right)}\frac{1}{3}\left[m_{i}^{2}\left(\frac{\partial\bar{\zeta}_{i,r-1}}{\partial\alpha_{q}}+\frac{n_{q}}{e+P}\frac{\partial\bar{\zeta}_{i,r-1}}{\partial\beta}\right)-\frac{\partial\bar{\zeta}_{i,r+1}}{\partial\alpha_{q}}-\frac{n_{q}}{e+P}\frac{\partial\bar{\zeta}_{i,r+1}}{\partial\beta}\right]\;,$
(152) $\displaystyle\lambda_{V\pi}^{(q^{\prime},q)}$
$\displaystyle=\sum_{s,i\,=\,1}^{N_{\text{spec}}}q_{s}^{\prime}\sum_{r\,=\,0}^{N_{1}}\tau_{si,0r}^{\left(1\right)}\left(\frac{\partial\bar{\eta}_{i,r-1}}{\partial\alpha_{q}}+\frac{n_{q}}{e+P}\frac{\partial\bar{\eta}_{i,r-1}}{\partial\beta}\right)\;.$
(153)

###  C.3 The coefficients in the shear-stress equation

The coefficients in the equation of motion (118) for the shear-stress tensor
are:

$\displaystyle\delta_{\pi\pi}$
$\displaystyle=-\frac{1}{3}\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,0r}^{\left(2\right)}\left[m_{i}^{2}\left(r-1\right)\bar{\eta}_{i,r-2}-\left(r+4\right)\bar{\eta}_{i,r}\right]$
$\displaystyle-\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,0r}^{\left(2\right)}\left\\{\sum_{q}^{\left\\{B,Q,S\right\\}}\frac{\partial\bar{\eta}_{i,r}}{\partial\alpha_{q}}\left[\mathcal{T}_{q0}\left(e+P\right)+\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\mathcal{T}_{qq^{\prime}}n_{q^{\prime}}\right]+\frac{\partial\bar{\eta}_{i,r}}{\partial\beta}\left[\mathcal{T}_{00}\left(e+P\right)+\sum_{q^{\prime}}^{\left\\{B,Q,S\right\\}}\mathcal{T}_{0q^{\prime}}n_{q^{\prime}}\right]\right\\}\;,$
(154) $\displaystyle\tau_{\pi\pi}$
$\displaystyle=-\frac{2}{7}\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,0r}^{\left(2\right)}\left[m_{i}^{2}\left(2r-2\right)\bar{\eta}_{i,r-2}-\left(2r+5\right)\bar{\eta}_{i,r}\right]\;,$
(155) $\displaystyle\lambda_{\pi\Pi}$
$\displaystyle=\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,0r}^{\left(2\right)}\frac{2}{15}\left[m_{i}^{4}\left(r-1\right)\bar{\zeta}_{i,r-2}-\left(2r+3\right)m_{i}^{2}\bar{\zeta}_{i,r}+\left(r+4\right)\bar{\zeta}_{i,r+2}\right]\;,$
(156) $\displaystyle\tau_{\pi V}^{(q)}$
$\displaystyle=-\frac{2}{5}\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,0r}^{\left(2\right)}\left[m_{i}^{2}r\
\bar{\kappa}_{i,r-1}^{\left(q\right)}-\left(r+5\right)\bar{\kappa}_{i,r+1}^{\left(q\right)}+m_{i}^{2}\beta\frac{\partial\bar{\kappa}_{i,r-1}^{\left(q\right)}}{\partial\beta}-\beta\frac{\partial\bar{\kappa}_{i,r+1}^{\left(q\right)}}{\partial\beta}\right]\;,$
(157) $\displaystyle\ell_{\pi V}^{(q)}$
$\displaystyle=-\frac{2}{5}\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,0r}^{\left(2\right)}\left(m_{i}^{2}\bar{\kappa}_{i,r-1}^{\left(q\right)}-\bar{\kappa}_{i,r+1}^{\left(q\right)}\right)\;,$
(158) $\displaystyle\lambda_{\pi V}^{(q,q^{\prime})}$
$\displaystyle=-\frac{2}{5}\sum_{s,i\,=\,1}^{N_{\text{spec}}}\sum_{r\,=\,0}^{N_{2}}\tau_{si,0r}^{\left(2\right)}\left[m_{i}^{2}\left(\frac{\partial\bar{\kappa}_{i,r-1}^{\left(q\right)}}{\partial\alpha_{q^{\prime}}}+\frac{n_{q^{\prime}}}{e+P}\frac{\partial\bar{\kappa}_{i,r-1}^{\left(q\right)}}{\partial\beta}\right)-\frac{\partial\bar{\kappa}_{i,r+1}^{\left(q\right)}}{\partial\alpha_{q^{\prime}}}-\frac{n_{q^{\prime}}}{e+P}\frac{\partial\bar{\kappa}_{i,r+1}^{\left(q\right)}}{\partial\beta}\right]\;.$
(159)

##  Appendix D Comparison to a single-component fluid

In previous works the transport coefficients in the equations of motion were
calculated in a similar manner as explained in this paper. In order to
facilitate a comparison and for the sake of completeness we recall the
notational convention and provide some basic relations.

For an arbitrary function of energy, $F(E_{i,\mathbf{k}})$, the irreducible
tensors satisfy the following orthogonality condition [22],

$\displaystyle\int\mathrm{d}K_{i}F(E_{i,\mathbf{k}})k^{\langle\mu_{1}}_{i}\cdots
k^{\mu_{\ell}\rangle}_{i}k^{\langle\nu_{1}}_{i}\cdots
k^{\nu_{n}\rangle}_{i}=\frac{\ell!\,\delta_{\ell
n}}{(2\ell+1)!!}\Delta^{\mu_{1}\cdots\mu_{\ell}\nu_{1}\cdots\nu_{n}}\int\mathrm{d}K_{i}F(E_{i,\mathbf{k}})\left(\Delta_{\alpha\beta}k^{\alpha}_{i}k^{\beta}_{i}\right)^{\ell}\;.$
(160)

Therefore, for a given species $i$, any irreducible moment of tensor rank
$\ell$ of arbitrary order $r\leq 0$ may be expressed as a linear combination
of irreducible moments of the same tensor rank $\ell$, but with different
power of energy $n$ as

$\rho_{i,r}^{\mu_{1}\cdots\mu_{\ell}}=\sum_{n\,=\,0}^{N_{\ell}}\rho_{i,n}^{\mu_{1}\cdots\mu_{\ell}}\mathcal{F}_{i,-r,n}^{\left(\ell\right)}\;.$
(161)

where for $r,n\geq 0$, $\mathcal{F}_{-r,n}^{\left(\ell\right)}=\delta_{rn}$.
Therefore, for $r\rightarrow-r$, we obtain

$\rho_{i,-r}^{\mu_{1}\cdots\mu_{\ell}}=\sum_{n\,=\,0}^{N_{\ell}}\rho_{i,n}^{\mu_{1}\cdots\mu_{\ell}}\mathcal{F}_{i,r,n}^{\left(\ell\right)}\;,$
(162)

where using Eqs. (21) and (52) we defined the following coefficient similar to
Eq. (66) of Ref. [21],

$\displaystyle\mathcal{F}_{i,\pm r,n}^{\left(\ell\right)}$
$\displaystyle\equiv$
$\displaystyle\frac{\ell!}{\left(2\ell+1\right)!!}\int\mathrm{d}K_{i}E_{i,\mathbf{k}}^{\mp
r}\mathcal{H}_{i,\mathbf{k}n}^{\left(\ell\right)}\left(\Delta_{\alpha\beta}k_{i}^{\alpha}k_{i}^{\beta}\right)^{\ell}f^{(0)}_{i,\mathbf{k}}\tilde{f}^{(0)}_{i,\mathbf{k}}$
(163) $\displaystyle=$
$\displaystyle\sum_{n^{\prime}\,=\,n}^{N_{\ell}}\sum_{m\,=\,0}^{n^{\prime}}\frac{J_{i,\mp
r+m+2\ell,\ell}}{J_{i,2\ell,\ell}}a_{i,n^{\prime}n}^{\left(\ell\right)}a^{(\ell)}_{i,n^{\prime}m}\;.$

Therefore, using these results one can also show that the expansion
coefficients are related as

$\displaystyle\sum_{m\,=\,0}^{n^{\prime}}\frac{J_{i,r+m+2\ell,\ell}}{J_{i,2\ell,\ell}}a_{i,n^{\prime}n}^{\left(\ell\right)}a^{(\ell)}_{i,n^{\prime}m}=\delta_{rn^{\prime}}\delta_{rn}\;,$
(164)

which in the case of a single-component system is equivalent to the matrix
equation provided in Appendix E of Ref. [21].

Truncating these expressions in the $(10+4N_{q})$-moment approximation, hence
using Eq. (162) with the summation limits $N_{0}=2,\,N_{1}=1,\,N_{2}=0$ for
the various tensor ranks, we obtain the following relations,

$\displaystyle\rho_{i,-r}$
$\displaystyle\equiv\sum_{n\,=\,0}^{N_{0}}\rho_{i,n}\mathcal{F}_{i,r,n}^{\left(0\right)}\approx-\frac{3}{m_{i}^{2}}\Pi_{i}\mathcal{F}_{i,r,0}^{\left(0\right)}+\rho_{i,1}\mathcal{F}_{i,r,1}^{\left(0\right)}+\rho_{i,2}\left(\frac{1}{m_{i}^{2}}\mathcal{F}_{i,r,0}^{\left(0\right)}+\mathcal{F}_{i,r,2}^{\left(0\right)}\right)\;,$
(165) $\displaystyle\rho_{i,-r}^{\mu}$
$\displaystyle\equiv\sum_{n\,=\,0}^{N_{1}}\rho_{i,n}^{\mu}\mathcal{F}_{i,r,n}^{\left(1\right)}\approx
V_{i}^{\mu}\mathcal{F}_{i,r,0}^{\left(1\right)}+W_{i}^{\mu}\mathcal{F}_{i,r,1}^{\left(1\right)}\;,$
(166) $\displaystyle\rho_{i,-r}^{\mu\nu}$
$\displaystyle\equiv\sum_{n\,=\,0}^{N_{2}}\rho_{i,n}^{\mu\nu}\mathcal{F}_{i,r,n}^{\left(2\right)}\approx\pi_{i}^{\mu\nu}\mathcal{F}_{i,r,0}^{\left(2\right)}\;,$
(167)

Furthermore, using Eqs. (165) – (167) for irreducible moments with positive
$r$ we obtain similar relations by replacing $-r\rightarrow r$ .

On the other hand, summing Eqs. (165) – (167) over species, the irreducible
moments of the mixture also lead to the expressions for a single-component
fluid in the Landau frame,

$\displaystyle\rho_{-r}$
$\displaystyle\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}\rho_{i,-r}=\sum_{i\,=\,1}^{N_{\text{spec}}}\sum_{n\,=\,0}^{N_{0}}\rho_{i,n}\mathcal{F}_{i,r,n}^{\left(0\right)}\overset{\text{single}}{\rightarrow}-\frac{3}{m^{2}}\Pi\gamma_{r}^{(0)}+\mathcal{O}(1)\;,$
(168) $\displaystyle\rho_{-r}^{\mu}$
$\displaystyle\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}\rho_{i,-r}^{\mu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\sum_{n\,=\,0}^{N_{1}}\rho_{i,n}^{\mu}\mathcal{F}_{i,r,n}^{\left(1\right)}\overset{\text{single}}{\rightarrow}V^{\mu}\gamma_{r}^{(1)}+\mathcal{O}(1)\;,$
(169) $\displaystyle\rho_{-r}^{\mu\nu}$
$\displaystyle\equiv\sum_{i\,=\,1}^{N_{\text{spec}}}\rho_{i,-r}^{\mu\nu}=\sum_{i\,=\,1}^{N_{\text{spec}}}\sum_{n\,=\,0}^{N_{2}}\rho_{i,n}^{\mu\nu}\mathcal{F}_{i,r,n}^{\left(2\right)}\overset{\text{single}}{\rightarrow}\pi^{\mu\nu}\gamma_{r}^{(2)}+\mathcal{O}(\mathrm{1})\;,$
(170)

where the coefficients $\gamma^{(\ell)}_{r}$ are listed in Ref. [21] for the
case of a single-component fluid, $i=N_{\text{spec}}=1$.

Notice that moments with negative power of energy are expressed as a linear
combination of moments with positive $r$ which represent the coupling between
moments even for simple fluids with a single conserved charge. On the other
hand in mixtures the summations over all particle species lead to further
couplings, which renders the above expressions rather difficult to use. To
circumvent this we have introduced the order-of-magnitude approximation in
Sec. III.3 to express the negative moments and their sums.

Furthermore, in order to compare the irreducible moments to Eqs. (108) –
(110), we have to demand

$\Pi\bar{\zeta}_{1,-r}=\Pi\gamma_{r}^{(0)}\;,\quad\pi^{\mu\nu}\bar{\eta}_{1,-r}=\pi^{\mu\nu}\gamma_{r}^{(2)}\;,$
(171)

and since we are dealing with a single charge, say $q$, then
$V_{q}^{\mu}=qV^{\mu}$ and hence

$V_{q}^{\mu}\bar{\kappa}_{1,-r}^{\left(q\right)}=qV^{\mu}\gamma_{r}^{(1)}\;.$
(172)

##  Appendix E Performing the collision integrals

In order to evaluate the transport coefficients first we need to calculate the
irreducible moments (90) of the collision term (2). These moments are related
to the entries of the collision matrix defined in Eqs. (91) – (92). For the
sake of convenience we define the following tensor, similarly as in Ref. [21],

$\displaystyle\mathcal{X}_{sijab,r}^{\mu_{1}\cdots\mu_{\ell}\nu_{1}\cdots\nu_{\ell+m}}\equiv\mathcal{L}_{sijab,r}^{\mu_{1}\cdots\mu_{\ell}\nu_{1}\cdots\nu_{\ell+m}}-\mathcal{G}_{sijab,r}^{\mu_{1}\cdots\mu_{\ell}\nu_{1}\cdots\nu_{\ell+m}}\;,$
(173)

where the loss term is

$\displaystyle\mathcal{L}_{sijab,r}^{\mu_{1}\cdots\mu_{\ell}\nu_{1}\cdots\nu_{\ell+m}}$
$\displaystyle\equiv\int\mathrm{d}K_{i}\mathrm{d}K^{\prime}_{j}f^{(0)}_{i,\mathbf{k}}f^{(0)}_{j,\mathbf{k^{\prime}}}E^{r-1}_{i,\mathbf{k}}k^{\mu_{1}}_{i}\cdots
k^{\mu_{\ell}}_{i}\Big{(}k_{i}^{\nu_{1}}\cdots
k_{i}^{\nu_{\ell+m}}\delta_{is}+{k^{\prime}}_{j}^{\nu_{1}}\cdots{k^{\prime}}_{j}^{\nu_{\ell+m}}\delta_{js}\Big{)}$
$\displaystyle\times\int\mathrm{d}P_{a}\mathrm{d}P^{\prime}_{b}\,W_{ij\rightarrow
ab}^{kk^{\prime}\rightarrow
pp^{\prime}}\,\tilde{f}^{(0)}_{a,\mathbf{p}}\tilde{f}^{(0)}_{b,\mathbf{p^{\prime}}}\;,$
(174)

while the gain term is

$\displaystyle\mathcal{G}_{sijab,r}^{\mu_{1}\cdots\mu_{\ell}\nu_{1}\cdots\nu_{\ell+m}}$
$\displaystyle\equiv\int\mathrm{d}K_{i}\mathrm{d}K^{\prime}_{j}f^{(0)}_{i,\mathbf{k}}f^{(0)}_{j,\mathbf{k^{\prime}}}E^{r-1}_{i,\mathbf{k}}k^{\mu_{1}}_{i}\cdots
k^{\mu_{\ell}}_{i}$
$\displaystyle\times\int\mathrm{d}P_{a}\mathrm{d}P^{\prime}_{b}\,W_{ij\rightarrow
ab}^{kk^{\prime}\rightarrow
pp^{\prime}}\tilde{f}^{(0)}_{a,\mathbf{p}}\tilde{f}^{(0)}_{b,\mathbf{p^{\prime}}}\Big{(}p_{a}^{\nu_{1}}\cdots
p_{a}^{\nu_{\ell+m}}\delta_{as}+{p^{\prime}}_{b}^{\nu_{1}}\cdots{p^{\prime}}_{b}^{\nu_{\ell+m}}\delta_{bs}\Big{)}\;.$
(175)

Therefore, once we have evaluated the corresponding $\mathcal{X}$-tensors, the
elements of the collision matrix $\mathcal{A}^{(\ell)}_{is,rn}$ in Eq. (92)
can be obtained by calculating the following projections:

$\displaystyle\mathcal{A}^{(\ell)}_{is,rn}$
$\displaystyle=\frac{1}{2}\sum_{n^{\prime}\,=\,n}^{N_{\ell}}\sum_{m\,=\,0}^{n^{\prime}}\frac{(-1)^{\ell}\,a^{(\ell)}_{s,n^{\prime}n}a^{(\ell)}_{s,n^{\prime}m}}{\ell!\,(2\ell+1)\,J_{s,2\ell,\ell}}\sum_{j,a,b\,=\,1}^{N_{\text{spec}}}u_{\nu_{\ell+1}}\cdots
u_{\nu_{\ell+m}}\Delta^{\mu_{1}\cdots\mu_{\ell}}_{\nu_{1}\cdots\nu_{\ell}}\left(\mathcal{X}_{sijab,r}\right)_{\mu_{1}\cdots\mu_{\ell}}^{\nu_{1}\cdots\nu_{\ell+m}}\;.$
(176)

In order to evaluate the $\mathrm{d}P_{a}\mathrm{d}P^{\prime}_{b}$ integrals
of the loss term it is useful to choose the center-of-momentum (CM) frame to
perform the integration over the transition rate
$W_{ij\,\rightarrow\,ab}^{kk^{\prime}\rightarrow pp^{\prime}}$. The total
momentum involved in binary collisions, $P^{\mu}_{T}\equiv
k^{\mu}_{i}+k^{\prime\mu}_{j}$, defines the Mandelstam variable

$\displaystyle s$
$\displaystyle\equiv\left(k^{\mu}_{i}+k^{\prime\mu}_{j}\right)^{2}=\left(p^{\mu}_{a}+p^{\prime\mu}_{b}\right)^{2}\equiv
P^{\mu}_{T}P_{T,\mu}\;.$ (177)

The CM frame is defined such that

$\displaystyle\sqrt{s}$ $\displaystyle\equiv k^{0}_{i}+k^{\prime
0}_{j}=p^{0}_{a}+p^{\prime 0}_{b}\equiv P^{0}_{T},$ (178) $\displaystyle 0$
$\displaystyle\equiv\mathbf{k}_{i}+\mathbf{k}^{\prime}_{j}=\mathbf{p}_{a}+\mathbf{p}^{\prime}_{b}\equiv\mathbf{P}_{T}\;.$
(179)

In the following, we use the following substitutions,

$\displaystyle x\equiv p^{0}_{a}+p^{\prime
0}_{b},\qquad\frac{\mathrm{d}x}{x}=\frac{|\mathbf{p}|\,\mathrm{d}|\mathbf{p}|}{p^{0}_{a}p^{\prime
0}_{b}}\;,$ (180)

and

$|\mathbf{p}|=\frac{1}{2x}\sqrt{\big{(}x^{2}-m_{+}\big{)}\big{(}x^{2}-m_{-}\big{)}}\;,$
(181)

where $|\mathbf{p}|=|\mathbf{p}_{a}|=|\mathbf{p}^{\prime}_{b}|$, and
$m_{\pm}\equiv\big{(}m_{a}\pm m_{b}\big{)}^{2}$. Therefore, the second
integral in Eq. (174) leads to

$\displaystyle\mathcal{P}_{ab}\equiv\int\mathrm{d}P_{a}\mathrm{d}P^{\prime}_{b}\,W_{ij\rightarrow
ab}^{kk^{\prime}\rightarrow
pp^{\prime}}\tilde{f}^{(0)}_{a,\mathbf{p}}\tilde{f}^{(0)}_{b,\mathbf{p^{\prime}}}=\frac{1}{2}\frac{1}{16(2\pi)^{2}}|\bar{\mathcal{M}}_{ij\,\rightarrow\,ab}|^{2}\tilde{f}^{(0)}_{a,\sqrt{s}}\tilde{f}^{(0)}_{b,\sqrt{s}}\sqrt{\left(1-\frac{m_{+}}{s}\right)\left(1-\frac{m_{-}}{s}\right)}\;,$
(182)

where we introduced the notation

$\displaystyle\tilde{f}^{(0)}_{i,\sqrt{s}}\equiv
1-\frac{a_{i}}{\exp\left(\beta\sqrt{m_{i}^{2}+\frac{s}{4}\left(1-\frac{m_{+}}{s}\right)\left(1-\frac{m_{-}}{s}\right)}-\alpha_{i}\right)+a_{i}}\;.$
(183)

Now using this result the remaining integral in Eq. (174) can be calculated.
For later use we define the angle-integrated transition probability:

$\displaystyle|\bar{\mathcal{M}}_{ij\,\rightarrow\,ab}(\sqrt{s})|^{2}\equiv\int_{S^{2}}\mathrm{d}\Omega\,|\mathcal{M}_{ij\,\rightarrow\,ab}(\sqrt{s},\Omega)|^{2}=2\pi\int_{0}^{\pi}\mathrm{d}\vartheta\,\sin\vartheta|\mathcal{M}_{ij\,\rightarrow\,ab}(\sqrt{s},\vartheta)|^{2}\;,$
(184)

where $\vartheta$ is the scattering angle in the reaction plane defined as

$\cos\vartheta=\frac{\left(k^{\mu}-k^{\prime\mu}\right)\left(p_{\mu}-p^{\prime}_{\mu}\right)}{\left(k^{\mu}-k^{\prime\mu}\right)^{2}}\;.$
(185)

The integral in Eq. (175) is more tedious. Here, we restrict ourselves to
isotropic scattering processes, hence all integrals only depend on the
normalized total momentum
$\tilde{P}^{\mu}_{T}\equiv\left(k^{\mu}_{i}+{k^{\prime}}^{\mu}_{j}\right)/\sqrt{s}$.
We also introduce the corresponding projection operator
$\Delta^{\mu\nu}_{P}=g^{\mu\nu}-\tilde{P}^{\mu}_{T}\tilde{P}^{\nu}_{T}$
orthogonal to the total momentum, $\Delta^{\mu\nu}_{P}\tilde{P}_{T,\nu}=0$.
Therefore, similarly to the thermodynamic integrals in Eq. (18), we decompose
the integrals in terms of the normalized total momentum and the associated
orthogonal projection operator,

$\displaystyle\Theta^{\nu_{1}\cdots\nu_{n}}_{a}$
$\displaystyle\equiv\int\mathrm{d}P_{a}\mathrm{d}P^{\prime}_{b}\,W_{ij\rightarrow
ab}^{kk^{\prime}\rightarrow
pp^{\prime}}\tilde{f}^{(0)}_{a,\mathbf{p}}\tilde{f}^{(0)}_{b,\mathbf{p^{\prime}}}\,p^{\nu_{1}}_{a}\cdots
p^{\nu_{n}}_{a}$
$\displaystyle=\sum_{m\,=\,0}^{[n/2]}(-1)^{m}\frac{2^{-m}n!}{m!(n-2m)!}\Delta^{(\nu_{1}\nu_{2}}_{P}\cdots\Delta^{\nu_{2m-1}\nu_{2m}}_{P}\tilde{P}^{\nu_{2m+1}}_{T}\cdots\tilde{P}^{\nu_{n})}_{T}C_{a,nm}\;,$
(186)

where the coefficients are

$\displaystyle C_{a,nm}$
$\displaystyle\equiv\frac{(-1)^{m}}{(2m+1)!!}\int\mathrm{d}P_{a}\mathrm{d}P^{\prime}_{b}\,W_{ij\rightarrow
ab}^{kk^{\prime}\rightarrow
pp^{\prime}}\tilde{f}^{(0)}_{a,\mathbf{p}}\tilde{f}^{(0)}_{b,\mathbf{p^{\prime}}}\,\left(\Delta_{P,\,\alpha\beta}\,p^{\alpha}_{a}p^{\beta}_{a}\right)^{m}\left(\tilde{P}_{T,\,\alpha}\,p^{\alpha}_{a}\right)^{n-2m}$
$\displaystyle=\frac{(-1)^{m}}{(2m+1)!!}\,\frac{1}{16(2\pi)^{2}}|\bar{\mathcal{M}}_{ij\,\rightarrow\,ab}|^{2}\tilde{f}^{(0)}_{a,\sqrt{s}}\tilde{f}^{(0)}_{b,\sqrt{s}}\,\frac{s^{n/2}}{2^{n+1}}\left[\left(1-\frac{m_{+}}{s}\right)\left(1-\frac{m_{-}}{s}\right)\right]^{\frac{2m+1}{2}}$
$\displaystyle\qquad\times\left[\frac{4m_{a}^{2}}{s}+\left(1-\frac{m_{+}}{s}\right)\left(1-\frac{m_{-}}{s}\right)\right]^{\frac{n-2m}{2}}\;.$
(187)

##  Appendix F Classical, ultrarelativistic system with hard-sphere
interactions in $(10+4N_{q})$-moment approximation

In this appendix we evaluate the transport coefficients of the theory in the
$(10+4N_{q})$-moment approximation ($N_{0}=2$, $N_{1}=1$, and $N_{2}=0$) for a
classical ($a_{i}\rightarrow 0$), ultrarelativistic ($m_{i}/T\rightarrow 0$)
multicomponent gas with elastic hard-sphere interactions, for which the
transition rate is given by Eq. (11). Equivalently, in Eqs. (182) and (E) we
can just replace

$\displaystyle\frac{1}{16(2\pi)^{2}}|\bar{\mathcal{M}}_{ij\,\rightarrow\,ab}|^{2}=\left(\delta_{ia}\delta_{jb}+\delta_{ib}\delta_{ja}\right)s\sigma_{\text{tot}}\;,$
(188)

with $\sigma_{\text{tot},ij}\equiv\sigma_{\text{tot}}=\text{const}$. With this
replacement, we obtain for Eqs. (182) and (E)

$\displaystyle\mathcal{P}_{ab}$
$\displaystyle=\frac{1}{2}\Big{(}\delta_{ia}\delta_{jb}+\delta_{ib}\delta_{ja}\Big{)}s\,\sigma_{\text{tot}}\;,$
(189) $\displaystyle C_{a,nm}$
$\displaystyle=\frac{1}{2}\Big{(}\delta_{ia}\delta_{jb}+\delta_{ib}\delta_{ja}\Big{)}\frac{(-1)^{m}}{(2m+1)!!}\frac{s^{(n+2)/2}}{2^{n}}\sigma_{\text{tot}}\;.$
(190)

Furthermore, according to Eqs. (47),
$P=\sum\limits_{i\,=\,1}^{N_{\text{spec}}}P_{i}$ is the total pressure of the
system, and it fulfills the ideal gas laws $e=3P$ and $n_{i}=\beta P_{i}$.

###  F.1 Collision matrix and its inverse

In the following we evaluate the collision matrix (176) for the vector and
tensor moments. The scalar moments play no role since their transport
coefficients are proportional to mass, and therefore vanish in the
ultrarelativistic case. In these calculations we make use of the
ultrarelativistic limit of Eq. (177), leading to
$s=2k_{i,\mu}{k^{\prime}}^{\mu}_{j}$.

Furthermore, in the classical limit, we express the thermodynamic integrals in
terms of the partial pressure of the respective species,

$\displaystyle
J_{i,nm}=I_{i,nm}=\frac{1}{2}\frac{(n+1)!}{(2m+1)!!}\frac{1}{\beta^{n-2}}P_{i}\;,$
(191)

where the thermodynamic integrals are related by

$I_{i,n+2,m}=m_{i}^{2}I_{i,nm}+\left(2m+3\right)I_{i,n+2,m+1}\;.$ (192)

Following Eq. (164) the relevant expansion coefficients in the
$(10+4N_{q})$-moment approximation are:

$\displaystyle a^{(\ell)}_{i,00}$ $\displaystyle=1,\quad\forall\,\ell\geq
0\;,$ (193) $\displaystyle a^{(1)}_{i,11}$
$\displaystyle=\frac{\beta}{2}\;,\quad
a^{(1)}_{i,10}=-\frac{4}{\beta}a^{(1)}_{i,11}=-2\;.$ (194)

#### Tensor moments ($\ell=2$):

The only relevant contraction of the $\mathcal{X}$-tensor reads

$\displaystyle\sum_{j,a,b\,=\,1}^{N_{\text{spec}}}\Delta_{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}\mathcal{X}^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}_{sijab,r}=\frac{4}{9}\sigma_{\text{tot}}\sum_{j\,=\,1}^{N_{\text{spec}}}\left[\big{(}2\delta_{is}-\delta_{js}\big{)}I_{i,r+4,0}I_{j,10}+\frac{2}{3}\big{(}\delta_{is}+\delta_{js}\big{)}I_{i,r+3,0}I_{j,20}\right]\;,$
(195)

and therefore the elements of the collision matrix (176) for the tensor
moments read in $(10+4N_{q})$-moment approximation (note that $N_{2}=0$)

$\displaystyle\mathcal{A}^{(2)}_{is,00}$
$\displaystyle=\frac{1}{2}\frac{a^{(2)}_{s,00}a^{(2)}_{s,00}}{10\,I_{s,42}}\sum_{j,a,b\,=\,1}^{N_{\text{spec}}}\Delta_{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}\mathcal{X}^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}_{sijab,0}$
$\displaystyle=\frac{\sigma_{\text{tot}}}{45I_{s,42}}\sum_{j\,=\,1}^{N_{\text{spec}}}\left[\big{(}2\delta_{is}-\delta_{js}\big{)}I_{i,40}I_{j,10}+\frac{2}{3}\big{(}\delta_{is}+\delta_{js}\big{)}I_{i,30}I_{j,20}\right]\;.$
(196)

Expressing this in terms of the pressure, we obtain

$\displaystyle\mathcal{A}^{(2)}_{is,00}=\frac{\sigma_{\text{tot}}\beta}{5}\sum_{j\,=\,1}^{N_{\text{spec}}}\frac{P_{i}P_{j}}{P_{s}}\big{(}4\delta_{is}-\delta_{js}\big{)}=\frac{\sigma_{\text{tot}}\beta}{5}\left(4\delta_{is}P-P_{i}\right)\;,$
(197)