# Diffusion matrix associated with the diffusion processes of multiple
conserved charges in a hot and dense hadronic matter

Arpan Das<EMAIL_ADDRESS>Institute of Nuclear Physics Polish Academy of
Sciences, PL-31-342 Kraków, Poland  Hiranmaya Mishra<EMAIL_ADDRESS>Theory
Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India
Ranjita K. Mohapatra<EMAIL_ADDRESS>Department of Physics, Banki
Autonomous College, Cuttack 754008, India

( )

###### Abstract

Bulk matter produced in heavy ion collisions has multiple conserved quantum
numbers like baryon number, strangeness and electric charge. The diffusion
process of these charges can be described by a diffusion matrix describing the
interdependence of diffusion of different charges. The diffusion coefficient
matrix is estimated here from the Boltzmann kinetic theory for the hadronic
phase within relaxation time approximation. In the derivation for the same, we
impose the Landau-Lifshitz conditions of fit. This leads to e.g. the diagonal
diffusion coefficients to be manifestly positive definite. The explicit
calculations are performed within the ambit of hadron resonance gas model with
and without excluded volume corrections. It is seen that the off-diagonal
components can be significant to affect the charge diffusion in a fluid with
multiple conserved charges. The excluded volume correction effects is seen to
be not significant in the estimation of the elements of the diffusion matrix.

##  I Introduction

Relativistic heavy-ion collision experiments, e.g. the Relativistic Heavy Ion
Collider (RHIC) and the Large Hadron Collider (LHC) give us an unique
opportunity to understand the properties of strongly interacting matter
governed by the laws of Quantum Chromodynamics (QCD). Experimental data at
RHIC and LHC indicates a transient phase of deconfined quarks and gluons also
known as the quark-gluon plasma (QGP). Quark-gluon plasma produced in the
initial stage of high multiplicity heavy-ion collision undergoes subsequent
space-time evolution and eventually results in a chemically and thermally
equilibrated hadronic medium. The strongly interacting medium so produced
shows collective motion and relativistic hydrodynamics along with a modeling
of the early stage and freeze-out of hadrons have become the important tool
for modeling relativistic heavy-ion collisions Romatschke:2017ejr ;
Florkowski:2010zz ; Gale:2013da ; Jeon:2015dfa ; Jaiswal:2016hex . Various
dissipative effects and the related transport coefficients, e.g., shear
($\eta$) and bulk ($\zeta$) viscosity, etc., can play a significant role in
the hydrodynamical evolution of the strongly interacting medium. Indeed, it
has been argued that a small value of the kinematic viscosity i.e. small value
of shear viscosity to entropy density ratio $(\eta/s)$, which is in accordance
with the Kovtun-Son-Starinet (KSS) bound of $(\eta/s)=1/(4\pi)$ obtained using
gauge gravity duality (AdS/CFT correspondence), can explain the flow data
Heinz:2013th ; Romatschke:2017ejr ; Kovtun:2004de . It may be emphasized that
QCD is not conformal in nature and the deviation of the conformality is
encoded in the bulk viscosity $\zeta$ of the strongly interacting medium
Gavin:1985ph ; Hosoya:1983xm ; Dobado:2012zf ; Sasaki:2008fg ; Sasaki:2008um ;
Karsch:2007jc ; Finazzo:2014cna ; Wiranata:2009cz ; Jeon:1995zm . Similar to
the shear viscosity, bulk viscosity also plays a very important role in the
viscous hydrodynamic description of the QCD medium Noronha-Hostler:2013gga ;
Ryu:2015vwa ; Ryu:2017qzn ; Vujanovic:2019yih . Other than the shear and bulk
viscosities, electrical conductivity ($\sigma_{el}$) also becomes important in
the magneto-hydrodynamic description of the QCD matter. This is due to the
fact that in the noncentral heavy-ion collision, a large magnetic field of the
order of $m_{\pi}^{2}$ is also expected to be produced. For phenomenological
manifestation of such magnetic fields on the strongly interacting medium
requires that a strong magnetic field survives for at least a few Fermi proper
time in the plasma. The crucial parameter that determines the time evolution
of magnetic field in the medium is the electrical conductivity Tuchin:2010gx ;
Tuchin:2010vs ; Inghirami:2016iru ; Das:2017qfi ; Greif:2016skc ;
Greif:2014oia ; Puglisi:2014pda ; Puglisi:2014sha ; Cassing:2013iz ;
Steinert:2013fza ; Aarts:2014nba ; Aarts:2007wj ; Amato:2013naa ; Gupta:2003zh
; Burnier:2012ts ; Ding:2010ga ; Kaczmarek:2013dya ; Marty:2013ita . Therefore
these transport coefficients play a very important role for a comprehensive
understanding of the hot and dense QCD medium produced in heavy-ion
collisions.

The study of the QCD phase structure both at finite temperature and chemical
potential is the goal of ultra relativistic heavy-ion collisions. While
lattice QCD (LQCD) studies indicate the phase transition from hadronic matter
to quark gluon matter at high temperature and at vanishing or small values of
baryon chemical potential is a crossover Aoki:2006we , it is expected that at
large baryon densities and low temperatures such a transition is most likely a
first order transition Asakawa:1989bq ; Ejiri:2008xt . Therefore, in the phase
diagram of strong interaction in the plane of baryon chemical potential and
temperature, one expects the first order line ending in a critical end point
(CEP). Fluctuation of the conserved charges plays an important role to find
the critical point Stephanov:1999zu ; Hatta:2003wn ; Asakawa:2000wh ;
Jeon:1999gr ; Ejiri:2005wq ; Kitazawa:2013bta ; Skokov:2012ds ; Pal:2020ucy .
It has been argued that event-by-event fluctuations of conserved quantities
such as net baryon number, net electric charge, and net strangeness are a
possible signal of the QGP formation and quark-hadron phase transition
Asakawa:2000wh ; Jeon:1999gr . Moreover, due to the rapid expansion of the
fireball fluctuations originated in the QGP phase may survive until the
freeze-out and can be used as a signal of the QGP formation in the early
stages of the relativistic heavy ion collisions Asakawa:2000wh ; Jeon:1999gr .
In the context of conserved charge fluctuations diffusion plays an important
role as the time evolution of conserved charges is caused by diffusion process
Asakawa:2015ybt ; Asakawa:2000wh ; Jeon:2000wg ; Shuryak:2000pd ;
Pratt:2019pnd .

In nonrelativistic systems the diffusion process is described by Fick’s law
which relates the diffusion current $(\Delta\vec{J}_{q})$ corresponding to
charge $q$, originated from the spatial inhomogeneity of the related charge
density $n_{q}(t,\vec{x})$. Explicitly,
$\Delta\vec{J}_{q}=-\kappa_{q}\vec{\nabla}n_{q}(t,\vec{x})$. The diffusion
coefficient ($\kappa_{q}$) is a dissipative transport coefficient that
characterizes the reaction strength of this thermal force Fotakis:2019nbq . If
we consider baryon number fluctuation associated with the baryon number
conservation then the effect of diffusion is not expected to be significant
due to almost vanishing net baryon density at mid rapidity region in ultra-
relativistic heavy-ion collisions at RHIC and LHC Monnai:2012jc . However,
with the Beam Energy Scan (BES) at RHIC, systematic low energy nuclear
collisions have been going on to investigate the phase diagram of nuclear
matter at finite net baryon density STAR:2010vob ; Mohanty:2011nm ;
Mitchell:2012mx . Furthermore, the heavy-ion collisions at Facility for
Antiproton and Ion Research (FAIR) at Darmstadt and in Nuclotron-based Ion
Collider fAcility (NICA) at Dubna one expects a baryon-rich medium
Odyniec:2013kna ; STAR:2017sal ; Friman:2011zz . With the decreasing collision
energy, the net baryon density increases and the diffusion processes are
expected to play an increasingly important role in the dissipative dynamics of
the evolution of hot and dense matter.

For a relativistic system the Fick’s law can be expressed as, $\Delta
J^{i}_{q}=\kappa_{qq}D^{i}(\mu_{q}/T)$. Due to the presence of multiple
conserved charges in QCD, i.e. baryon number ($B$), strangeness ($S$),
electric charge ($Q$), etc., the simple Fick’s law as above, now gets
modified. Since the strongly interacting particles, e.g. hadrons and quarks
can carry more than one of these conserved charges, the diffusion current of
each charge will no longer solely depends on the gradient of that specific
charge. Since the gradients of every single charge density can generate a
diffusion current of any other charge the diffusion currents of the conserved
charges must get coupled to each other. Therefore, in the presence of multiple
conserved charges, one has a generalized Fick’s law,

$\displaystyle\begin{pmatrix}\Delta J^{i}_{B}\\\ \Delta J^{i}_{Q}\\\ \Delta
J^{i}_{S}\end{pmatrix}=\begin{pmatrix}\kappa_{BB}&\kappa_{BQ}&\kappa_{BS}\\\
\kappa_{QB}&\kappa_{QQ}&\kappa_{QS}\\\
\kappa_{SB}&\kappa_{SQ}&\kappa_{SS}\end{pmatrix}\begin{pmatrix}D^{i}\alpha_{B}\\\
D^{i}\alpha_{Q}\\\ D^{i}\alpha_{S}\end{pmatrix}.$

In the above, $\alpha_{q}=\mu_{q}/T$ with $q=B,S,Q$ and
$D=u^{\mu}\partial_{\mu}$; $D^{\mu}=\partial^{\mu}-u^{\mu}D$ and
$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$ is the projector orthogonal to
fluid four velocity $u^{\mu}$. $\kappa_{qq^{\prime}}$ denotes the
multicomponent diffusion matrix. The dynamics of the thermal potentials, i.e.
$\alpha_{B}$, $\alpha_{Q}$, $\alpha_{S}$ and the corresponding currents in
heavy-ion collisions are not often rigorously explored. It has been argued in
Refs. Monnai:2012jc ; Greif:2017byw  that for low collision energies the
influence of diffusion currents on the hydrodynamical evolution of the net-
charge currents can be significant. Therefore not only the estimation of the
diagonal components of the diffusion matrix elements, i.e. $\kappa_{BB}$,
$\kappa_{QQ}$, $\kappa_{SS}$, but also the estimation of the off-diagonal
terms, i.e. $\kappa_{BQ}$, $\kappa_{QB}$, $\kappa_{BS}$, $\kappa_{SB}$,
$\kappa_{QS}$, $\kappa_{SQ}$ are also very important. In the present
investigation we estimate the diffusion matrix element for the hadronic medium
modeled by the hadron resonance gas model.

At chemical freeze-out the hadron resonance gas (HRG) model describes quite
well the hadronic phase of the strongly interacting medium created in heavy
ion collisions Braun-Munzinger:2003pwq ; Andronic:2005yp . In its simplest
form assuming the freezeout of strange and nonstrange particles on same
footing, HRG model has only two parameters $T$ and $\mu_{B}$. In general the
thermodynamics of interacting hadrons can be nontrivial, however, in the
presence of narrow resonances it has been argued that the thermodynamics of
interacting gas of hadrons can be approximated by the non-interacting gas of
hadrons and resonances PhysRev.187.345 ; PhysRevD.10.694 . Due to the small
number of parameters and simple structure HRG model and its various extensions
have been well explored regarding the experimental result of the thermal
abundance of different particle ratios in the heavy ion collisions Braun-
Munzinger:2001hwo ; Cleymans:1999st ; Becattini:2000jw ; Cleymans:2004pp ;
Andronic:2008gu , thermodynamics Karsch:2003zq ; Braun-Munzinger:2015hba ,
conserved charge fluctuations Nahrgang:2014fza ; Bhattacharyya:2013oya ;
Garg:2013ata ; PhysRevD.86.034509 ; Begun:2006jf , as well as transport
coefficients for hadronic matter Greif:2016skc ; Puglisi:2014pda ;
Puglisi:2014sha ; Prakash:1993bt ; Wiranata:2012br ; PhysRevC.83.014906 ;
Khvorostukhin:2010aj ; PhysRevC.86.054902 ; PhysRevC.77.024911 ;
PhysRevC.86.024913 ; PhysRevC.85.014908 ; PhysRevC.88.068201 ;
PhysRevC.89.045201 ; Wiranata:2014kva ; Wiranata:2012vv ; Tawfik:2010mb ;
PhysRevLett.103.172302 ; Kadam:2014cua ; Kadam:2014xka ; Ghosh:2014yea ;
Demir:2014kda ; PhysRevC.90.025202 ; PhysRevC.97.055204 ; PhysRevC.84.054911 ;
PhysRevD.90.094014 ; Bass:1998ca ; PhysRevC.92.035203 ; PhysRevD.99.014015 ;
Das:2021qii ; Mohapatra:2019mcl .

Initial investigations on the diffusion matrix of the strongly interacting
matter have been discussed in Refs. Greif:2017byw ; Fotakis:2021diq . Explicit
expressions of the diffusion matrix have been derived within the classical
kinetic theory approach, both in the first order Chapman-Enskog expansion as
well as in the relaxation time approximation. In the present investigation, we
also follow the classical kinetic theory approach to obtain the expression of
the diffusion matrix ($\kappa_{qq^{\prime}}$), keeping in mind the Landau-
Lifshitz matching condition in the local rest frame. The expression of the
diffusion matrix so obtained in the present work is significantly different
from the expression of $\kappa_{qq^{\prime}}$ as given in Refs. Greif:2017byw
; Fotakis:2021diq , e.g. the diagonal components of the diffusion matrix are
not manifestly positive definite in Refs. Greif:2017byw ; Fotakis:2021diq ;
Fotakis:2019nbq . In the present work, we explicitly show that the diagonal
component of the diffusion matrix ($\kappa_{qq}$) is positive definite. It
should be noted that the positivity of various components is not so obvious.
Signs of various transport coefficients in the context of dissipative
hydrodynamics can be obtained by demanding entropy production.

The paper is organized in the following manner, in Sec. (II) we discuss the
formalism of diffusion matrix element within the framework of the Boltzmann
kinetic equation and attempt to solve for the distribution functions within
relaxation time approximation. In Sec. (III) we briefly discuss the hadron
resonance gas model including the ideal HRG model as well as its extension to
the excluded volume HRG model. In Sec. (IV) we present the estimation of
diffusion coefficient for the hadron resonance gas model. Finally, we
summarize and draw the conclusion of our investigations in Sec. (V).

##  II Formalism

Let us consider the covariant Boltzmann equation in the absence of external
force,

$\displaystyle p_{a}\cdot\partial f_{a}(x,p)=\mathcal{C}_{a}=(u\cdot
p_{a})\sum_{b,c,d}\int\frac{d^{3}p_{b}}{(2\pi)^{3}}\frac{d^{3}p_{c}}{(2\pi)^{3}}\frac{d^{3}p_{c}}{(2\pi)^{3}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\frac{1}{1+\delta_{cd}}\left(f_{c}f_{d}-f_{a}f_{b}\right)W(a,b|c,d),$
(1)

where,

$\displaystyle
W(a,b|c,d)=\frac{(2\pi)^{4}\delta^{4}(p_{a}+p_{b}-p_{c}-p_{d})}{2E_{a}2E_{b}2E_{c}2E_{d}}|\mathcal{M}(a,b|c,d)|^{2}.$
(2)

Here $f_{a}$ is the out of equilibrium distribution function of species
$`a^{\prime}$, $\mathcal{C}$ is the collision term and $\mathcal{M}$ is the
transition matrix element for the process $a+b\rightarrow c+d$. The factor
$1/(1+\delta_{cd})$ takes into account the possibility of identical particles.
The scalar product is the usual dot product $a\cdot
b=a^{\mu}b_{\mu}=g_{\mu\nu}a^{\mu}b^{\nu}$; $g_{\mu\nu}$ is the flat space-
time metric with signature $(+,-,-,-)$. Note that without introducing too many
notations for momentum vector, throughout the manuscript we use the same
notation for momentum four vector and the magnitude of the momentum three
vector, i.e. $p^{\mu}\equiv(E_{p},\vec{p})$ is the momentum four vector with
$E_{p}=\sqrt{p^{2}+m^{2}}$ where $p\equiv|\vec{p}|$ is the magnitude of the
momentum three vector.

We consider the system under consideration to be initially in global
equilibrium. Small perturbation of the single-particle distribution function
from equilibrium is generated due to small gradients in the thermal
potentials. This perturbation away from the equilibrium generates a diffusion
current in the corresponding charges. To obtain a solution of the Boltzmann
equation, we write the distribution function in powers of the Knudsen number
and truncate such an expansion in the lowest order as Fotakis:2019nbq ;
Fotakis:2021diq ; Chakraborty:2010fr ,

$\displaystyle f_{a}(x,p)=f_{a}^{(0)}(p)\left(1+\phi^{a}(x,p)\right)$ (3)

where we have pushed the space-time dependence to the fluctuating part through
$\phi^{a}$, and the equilibrium distribution function is given by,

$\displaystyle f_{a}^{(0)}$ $\displaystyle=g_{a}\exp(-\beta u\cdot
p_{a}+\beta\sum_{q}q_{a}\mu_{q})$ $\displaystyle=g_{a}\exp(-\beta u\cdot
p_{a}+\alpha_{a}),$ (4)

where $u^{\mu}$ is the fluid four velocity with respect to which the invariant
$p\cdot u$ is measured, $\beta=1/T$ is the inverse of temperature of the
medium in the rest frame and $g_{a}$ is the degeneracy factor. Here,
$\alpha^{a}=\sum_{q}q_{a}\alpha_{q}\equiv\beta\sum_{q}q_{a}\mu_{q}$,
$\alpha_{q}=\beta\mu_{q}$ and $\mu_{q}$ corresponding to different chemical
potentials (e.g. $q=B,S,Q$).

Next, to identify the transport coefficients, we write down the structure of
the energy-momentum tensor $T^{\mu\nu}$ and the conserved current
$J_{q}^{\mu}$ corresponding to a conserved charge $q$. In terms of temperature
$T$, chemical potential $\mu_{q}$ and flow velocity $u^{\mu}$, these are given
as

$\displaystyle T^{\mu\nu}=-Pg^{\mu\nu}+\omega u^{\mu}u^{\nu}+\Delta
T^{\mu\nu},$ (5)

and

$\displaystyle J_{q}^{\mu}=n_{q}u^{\mu}+\Delta J_{q}^{\mu}.$ (6)

We shall be taking $u^{\mu}$ as the velocity of energy flow normalized as
$u_{\mu}u^{\mu}=1$. Further, in the above $P$, $\varepsilon$ is the pressure
and energy density respectively and $\omega=\varepsilon+P$ is the enthalpy.
$\Delta T^{\mu\nu}$ is the dissipative correction to the energy-momentum
tensor due to viscosity,

$\displaystyle\Delta
T^{\mu\nu}=\eta\left(D^{\mu}u^{\nu}+D^{\nu}u^{\mu}+\frac{2}{3}\Delta^{\mu\nu}\theta\right)-\zeta\Delta^{\mu\nu}\theta$
(7)

and, the dissipative contribution to the conserved current is given as,

$\displaystyle\Delta
J_{q}^{\mu}=\kappa_{qq^{\prime}}D^{\mu}\alpha_{q^{\prime}},$ (8)

which is a relativistic form of Fick’s law generalised to different conserved
charges $q$ with $\kappa_{qq^{\prime}}$ being the diffusion matrix
coefficient. The diffusion current is generated by the gradient in the thermal
potential $\alpha_{q}$. In the above, $\theta=\partial\cdot u$ is the
expansion scalar; $D=u^{\mu}\partial_{\mu}$; $D^{\mu}=\partial^{\mu}-u^{\mu}D$
and $\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$ is the projector orthogonal to
$u^{\mu}$.

Having defined the transport coefficients from the structure of the energy-
momentum tensor and the currents associated with the conserved charges, we
next express these quantities in terms of the microscopic distribution
functions obtained by solving the Boltzmann equation. The idea is to then
identify the dissipation coefficients in terms of the distribution functions
by comparing the corresponding tensor structures associated with the
$T^{\mu\nu}$ and $J^{\mu}_{q}$. The energy-momentum tensor is written as,

$\displaystyle
T^{\mu\nu}=\sum_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p^{\mu}_{a}p^{\nu}_{a}}{E_{a}}f_{a},$
(9)

and the current corresponding to a conserved charge $q$ given as

$\displaystyle
J_{q}^{\mu}=\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p^{\mu}_{a}}{E_{a}}f_{a}.$
(10)

Before proceeding further, we make a comment here, that in the present case,
we are not considering here, the possibility of including mean-field effects
or the medium dependent masses which can be generalized easily using the
methods used in e.g. Refs. Albright:2015fpa ; Albright:2015edp ; Deb:2016myz ;
Chakraborty:2010fr .

Now, let us note that the non-equilibrium part $\phi_{a}$ of the distribution
function $f_{a}$ leads to the non-equilibrium contributions $\Delta
T^{\mu\nu}$ and $\Delta J^{\mu}_{q}$. This means $\phi_{a}$ should have the
same tensor structure as $\Delta T^{\mu\nu}$ and $\Delta J^{\mu}_{q}$. Thus we
take the form of $\phi_{a}$ as Albright:2015fpa ,

$\displaystyle\phi_{a}=-A_{a}\theta-\sum_{q}B_{a}^{q}p_{a}^{\mu}D_{\mu}\alpha_{q}+C_{a}p_{a}^{\mu}p_{a}^{\nu}\Sigma_{\mu\nu}$
(11)

where, the functions $A_{a}$, $B_{a}^{q}$ and $C_{a}$ are functions of
magnitude of momentum and
$\Sigma_{\mu\nu}=D_{\mu}u_{\nu}+D_{\nu}u_{\mu}+\frac{2}{3}\Delta_{\mu\nu}\theta$.
This leads to,

$\displaystyle\Delta J^{\mu}_{q}=$
$\displaystyle\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{\mu}}{E_{a}}f_{a}^{(0)}\phi_{a},$
$\displaystyle=$
$\displaystyle\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{\mu}}{E_{a}}f_{a}^{(0)}$
$\displaystyle\times\bigg{(}-A_{a}\theta-\sum_{q^{\prime}}B_{a}^{q^{\prime}}p_{a}^{\sigma}D_{\sigma}\alpha_{q^{\prime}}+C_{a}p_{a}^{\sigma}p_{a}^{\rho}\Sigma_{\sigma\rho}\bigg{)}.$
(12)

In $\Delta J_{q}^{i}$, terms with $A_{a}$ and $C_{a}$ would give rise to odd
momentum integration. Therefore,

$\displaystyle\Delta J_{q}^{i}=$
$\displaystyle\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{i}}{E_{a}}f_{a}^{(0)}(-)B^{q^{\prime}}_{a}p_{a}^{\rho}D_{\rho}\alpha_{q^{\prime}}$
$\displaystyle=$
$\displaystyle\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{2}}{3E_{a}}f_{a}^{(0)}B^{q^{\prime}}_{a}D^{i}\alpha_{q^{\prime}}$
$\displaystyle=$ $\displaystyle\kappa_{qq^{\prime}}D^{i}\alpha_{q^{\prime}},$
(13)

here the diffusion matrix $\kappa_{qq^{\prime}}$ can be identified as,

$\displaystyle\kappa_{qq^{\prime}}=\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{2}}{3E_{a}}f_{a}^{(0)}B^{q^{\prime}}_{a}.$
(14)

To solve for the functions $A_{a}$, $B_{a}^{q}$, and $C_{a}$, we use the
Boltzmann equation within the Chapman-Enskog approximation. This corresponds
to expanding both the sides of the Boltzmann equation to first order in
$\phi_{a}$. Let us start with the L.H.S of the Boltzmann Eq. (1). Using
$\partial_{\mu}=u_{\mu}D+D_{\mu}$, we have

$\displaystyle p_{a}^{\mu}\partial_{\mu}f_{a}^{(0)}=$ $\displaystyle
p_{a}^{\mu}(u_{\mu}D+D_{\mu})f_{a}^{(0)}$ $\displaystyle=$
$\displaystyle(E_{a}D+p^{\mu}_{a}D_{\mu})f_{a}^{(0)}.$ (15)

Using the expression of the equilibrium distribution function as given in Eq.
(4) we obtain,

$\displaystyle Df_{a}^{(0)}=f_{a}^{(0)}\bigg{(}-E_{a}D\beta-\beta
p_{a}^{\mu}Du_{\mu}+D\alpha^{a}\bigg{)}.$ (16)

Furthermore, $Du_{\mu}$ can be estimated by considering the conservation of
energy-momentum tensor in ideal hydrodynamics, i.e.
$\partial_{\mu}T_{(0)}^{\mu\nu}=0$ where,

$\displaystyle T^{\mu\nu}_{(0)}=(\varepsilon+P)u^{\mu}u^{\nu}-Pg^{\mu\nu}.$
(17)

Taking the projection of $\partial_{\mu}T^{\mu\nu}_{(0)}=0$ in the direction
orthogonal to $u^{\mu}$ one gets,

$\displaystyle Du^{\alpha}=\frac{1}{\omega}D^{\alpha}P.$ (18)

Further, using the thermodynamics we obtain Ollitrault:2007du ,

$\displaystyle
D_{\alpha}\varepsilon=TD_{\alpha}s+\sum_{q}\mu_{q}D_{\alpha}n_{q},$ (19)

and,

$\displaystyle P=-\varepsilon+Ts+\sum_{q}\mu_{q}n_{q},$
$\displaystyle\implies$ $\displaystyle
D_{\alpha}P=-D_{\alpha}\varepsilon+TD_{\alpha}s+(D_{\alpha}T)s$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\sum_{q}\mu_{q}D_{\alpha}n_{q}+\sum_{q}(D_{\alpha}\mu_{q})n_{q}.$
(20)

Using Eqs. (19) and (20) we get,

$\displaystyle D_{\alpha}P$
$\displaystyle=\frac{\omega}{T}D_{\alpha}T-\sum_{q}n_{q}\mu_{q}\frac{D_{\alpha}T}{T}+\sum_{q}n_{q}D_{\alpha}\mu_{q}.$
(21)

Note that,

$\displaystyle
T\sum_{q}n_{q}D_{\alpha}\alpha_{q}=T\sum_{q}n_{q}D_{\alpha}(\mu_{q}/T)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\sum_{q}\left(n_{q}D_{\alpha}\mu_{q}-n_{q}\frac{\mu_{q}}{T}D_{\alpha}T\right).$
(22)

Therefore, Eqs. (21) and (22) gives us,

$\displaystyle
D_{\alpha}P=\frac{\omega}{T}D_{\alpha}T+\sum_{q}Tn_{q}D_{\alpha}\alpha_{q},$
(23)

leading to,

$\displaystyle\beta
Du^{\alpha}=-D^{\alpha}\beta+\sum_{q}\frac{n_{q}}{\omega}D^{\alpha}\alpha_{q}.$
(24)

Using the expression of $Du^{\alpha}$ as given in Eq. (24) in Eq. (16) we get,

$\displaystyle Df_{a}^{(0)}$
$\displaystyle=-f_{a}^{(0)}\bigg{(}E_{a}D\beta+p_{a}^{\mu}\bigg{(}-D_{\mu}\beta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{q}\frac{n_{q}}{\omega}D_{\mu}\alpha_{q}\bigg{)}-D\alpha^{a}\bigg{)}.$
(25)

Again,

$\displaystyle D_{\mu}f_{a}^{(0)}=f_{a}^{(0)}\bigg{(}-D_{\mu}\beta(u\cdot
p_{a})-\beta p^{\alpha}_{a}D_{\mu}u_{\alpha}+D_{\mu}\alpha^{a}\bigg{)}.$ (26)

Therefore,

$\displaystyle p_{a}^{\mu}D_{\mu}f_{a}^{(0)}$
$\displaystyle=-f_{a}^{(0)}\bigg{(}E_{a}p_{a}^{\mu}D_{\mu}\beta+\beta
p^{\mu}_{a}p^{\alpha}_{a}D_{\mu}u_{\alpha}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-p^{\mu}_{a}D_{\mu}\alpha^{a}\bigg{)}.$
(27)

The second term within the parenthesis on the right hand side can be expressed
as,

$\displaystyle p_{a}^{\mu}p_{a}^{\alpha}D_{\mu}u_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2}p_{a}^{\mu}p_{a}^{\alpha}\bigg{(}D_{\mu}u_{\alpha}+D_{\alpha}u_{\mu}-\frac{2}{3}\Delta_{\alpha\mu}\theta\bigg{)}+\frac{1}{3}p_{a}^{\mu}p_{a}^{\alpha}\Delta_{\alpha\mu}\theta$
$\displaystyle=$
$\displaystyle\frac{1}{2}p_{a}^{\mu}p_{a}^{\alpha}\Sigma_{\mu\alpha}+\frac{1}{3}p_{a}^{\mu}p_{a}^{\alpha}\Delta_{\alpha\mu}\theta,$
(28)

so that,

$\displaystyle p_{a}^{\mu}D_{\mu}f_{a}^{(0)}$
$\displaystyle=-f_{a}^{(0)}\bigg{(}E_{a}p_{a}^{\mu}D_{\mu}\beta+\frac{\beta}{2}p_{a}^{\mu}p_{a}^{\alpha}\Sigma_{\mu\alpha}$
$\displaystyle+\frac{\beta}{3}p_{a}^{\mu}p_{a}^{\alpha}\Delta_{\alpha\mu}\theta-p^{\mu}_{a}D_{\mu}\alpha^{a}\bigg{)},$
(29)

using Eqs. (25) and (29), Eq. (15) can be simplified to,

$\displaystyle p_{a}^{\mu}\partial_{\mu}f_{a}^{(0)}$
$\displaystyle=E_{a}Df_{a}^{(0)}+p_{a}^{\mu}D_{\mu}f_{a}^{(0)}$
$\displaystyle=-f_{a}^{(0)}\bigg{[}E_{a}^{2}D\beta-E_{a}D\alpha^{a}$
$\displaystyle~{}~{}+\beta
p_{a}^{\mu}p_{a}^{\alpha}\bigg{(}\frac{1}{2}\Sigma_{\mu\alpha}+\frac{1}{3}\Delta_{\mu\alpha}\theta\bigg{)}$
$\displaystyle~{}~{}+p_{a}^{\mu}\sum_{q}\bigg{(}\frac{E_{a}n_{q}}{\omega}-q_{a}\bigg{)}D_{\mu}\alpha_{q}\bigg{]}.$
(30)

In the following, we shall confine our attention to diffusion only. In such a
case, only the third term in the square bracket of Eq. (30) will be relevant
and we write,

$\displaystyle\frac{p_{a}^{\mu}\partial_{\mu}f_{a}^{(0)}}{E_{a}}\simeq-
f_{a}^{(0)}\frac{p_{a}^{\mu}}{E_{a}}\sum_{q}\bigg{(}\frac{E_{a}n_{q}}{\omega}-q_{a}\bigg{)}D_{\mu}\alpha_{q}.$
(31)

Next let us consider the R.H.S of the Boltzmann equation (Eq. (1)). In the
Chapman-Enskog approximation, the collision term can be written as,

$\displaystyle\mathcal{C}_{a}=$ $\displaystyle
E_{a}f_{a}^{(0)}\sum_{b,c,d}\frac{1}{1+\delta_{cd}}\int\frac{d^{3}p_{b}}{(2\pi)^{3}}\frac{d^{3}p_{c}}{(2\pi)^{3}}\frac{d^{3}p_{d}}{(2\pi)^{3}}f_{b}^{(0)}$
$\displaystyle W(a,b|c,d)\left(\phi_{c}+\phi_{d}-\phi_{a}-\phi_{b}\right).$
(32)

Let us note that the collision term vanishes in equilibrium due to the
detailed balance condition which follows from the energy momentum conserving
delta function. Therefore in equilibrium
$f_{a}^{(0)}f_{b}^{(0)}=f_{c}^{(0)}f_{d}^{(0)}$. As for the L.H.S, we shall
confine our attention to diffusive process only so that
$\phi_{a}\simeq-\sum_{q}B_{a}^{q}p_{a}^{\mu}D_{\mu}\alpha_{q}$. Equating the
coefficients of $D_{\mu}\alpha_{q}$ from Eqs. (31) and (32) leads to,

$\displaystyle\frac{p_{a}^{\mu}}{E_{a}}\left(\frac{E_{q}n_{q}}{\omega}-q_{a}\right)=\sum_{b,c,d}\int\frac{d^{3}p_{b}}{(2\pi)^{3}}\frac{d^{3}p_{c}}{(2\pi)^{3}}\frac{d^{3}p_{d}}{(2\pi)^{3}}f_{b}^{(0)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\frac{1}{1+\delta_{cd}}W(a,b|c,d)(B_{c}^{q}p^{\mu}_{c}+B_{d}^{q}p^{\mu}_{d}-B_{a}^{q}p^{\mu}_{a}-B_{b}^{q}p^{\mu}_{b}).$
(33)

The above is an integral equation for the functions $B_{a}^{q}$’s which depend
upon the magnitude of the momentum $p_{a}$.

Let us note that if we have a particular solution, denoted as
$B_{a-part}^{q}$, we can generate another solution $B_{a-part}^{q}-b^{q}$,
where $b_{q}$ is a constant independent of the species $a$. The reason is that
the scattering term conserves the charge number. This can be easily seen by
making a substitution $B_{a}^{q}=B_{a-part}^{q}-b^{q}$ in Eq.(33). This calls
for additional constraints to select an unique solution. This arbitrariness is
associated with the freedom to define the local rest frame or, equivalently,
the flow velocity $u^{\mu}$. In the Landau Lifshitz definition, $u^{\mu}$ is
the velocity of the flow of energy. On the other hand, the Eckart definition
of flow velocity corresponds to flow of conserved charge. Choosing a specific
frame is called the condition of fit. To choose the Landau Lifshitz frame, we
impose the condition in the local rest frame as Albright:2015edp ;
Albright:2015fpa

$\displaystyle\Delta T^{0i}=0,\quad\quad\Delta J^{0}_{q}=0.$ (34)

The second condition is trivially satisfied in the local rest frame, which can
be understood from Eq. (12). In the local rest frame $u^{\mu}=(1,0,0,0)$, so
that $\theta$ and $\Sigma^{\mu\nu}$ vanish in the local rest frame.
Furthermore, in the local rest frame $D^{\mu}$ operator has only spatial
components. This makes the integrand of $\Delta J^{0}_{q}$ an odd function of
momentum. Therefore $\Delta J^{0}_{q}=0$ in the local rest frame.

The first condition, i.e. $\Delta T^{0i}=0$ means

$\displaystyle\sum_{q}\sum_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}p_{a}^{i}(-)B_{a}^{q}p_{a}^{j}D_{j}\alpha_{q}f_{a}^{(0)}=0,$
(35)

With $B_{a}^{q}=B_{a-part}^{q}-b^{q}$, the above condition reduces to,

$\displaystyle\sum_{q}\sum_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}p_{a}^{i}(B_{a-part}^{q}-b^{q})p_{a}^{j}D_{j}\alpha_{q}f_{a}^{(0)}=0,$
$\displaystyle\implies$
$\displaystyle\sum_{q}b_{q}\sum_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}p_{a}^{i}p_{a}^{j}D_{j}\alpha_{q}f_{a}^{(0)}$
$\displaystyle=\sum_{q}\sum_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}p_{a}^{i}p_{a}^{j}B_{a-part}^{q}D_{j}\alpha_{q}f_{a}^{(0)}$
(36)

Comparing the coefficient of $D_{j}\alpha_{q}$ from both sides of the above
equation we obtain,

$\displaystyle
b_{q}\sum_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}p_{a}^{2}f_{a}^{(0)}=\sum_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}p_{a}^{2}B_{a-part}^{q}f_{a}^{(0)}.$
(37)

The quantity multiplying $b_{q}$ in the L.H.S of the above equation is just
$3T\omega$ (for a detailed derivation see Appendix (A)). This leads to,

$\displaystyle
b^{q}=\frac{1}{3T\omega}\sum_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}p_{a}^{2}B_{a-part}^{q}f_{a}^{(0)}.$
(38)

Now in Eq.(14), we substitute $B_{a}^{q}=B_{a-part}^{q}-b^{q}$, so that the
diffusion coefficient is given by,

$\displaystyle\kappa_{qq^{\prime}}=$
$\displaystyle\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{2}}{3E_{a}}f_{a}^{(0)}\left(B_{a-part}^{q^{\prime}}-b^{q^{\prime}}\right)$
$\displaystyle=$
$\displaystyle\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{2}}{3E_{a}}f_{a}^{(0)}B_{a-part}^{q^{\prime}}$
$\displaystyle-b^{q^{\prime}}\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{2}}{3E_{a}}f_{a}^{(0)}.$
(39)

Further, it can be shown that the factor multiplying $b^{q^{\prime}}$ in Eq.
(39) is $n_{q}T$ (for a detailed derivation see Appendix (B)), i.e.

$\displaystyle\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{2}}{3E_{a}}f_{a}^{(0)}=n_{q}T.$
(40)

Using Eqs. (38) and (40) in Eq. (39) we obtain,

$\displaystyle\kappa_{qq^{\prime}}=\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{2}}{3E_{a}}f_{a}^{(0)}B_{a-part}^{q^{\prime}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\frac{n_{q}}{3\omega}\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}p_{a}^{2}f_{a}^{(0)}B_{a-part}^{q^{\prime}}$
$\displaystyle=\sum_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{2}}{3E_{a}}\left(q_{a}-\frac{n_{q}E_{a}}{\omega}\right)f_{a}^{(0)}B_{a-part}^{q^{\prime}}.$
(41)

Thus once we know $B_{a-part}^{q^{\prime}}$, we have the expression for the
diffusivity as above. In this study, we use the relaxation time approximation,
where it is assumed that all particles are in equilibrium except for the
species $`a^{\prime}$ in the Boltzmann equation for $f_{a}$, i.e
$B_{b-part}=0=B_{c-part}=B_{d-part}$ in Eq. (33) then we have,

$\displaystyle\bigg{(}q_{a}-\frac{E_{a}n_{q}}{\omega}\bigg{)}\frac{p_{a}^{\mu}}{E_{a}}$
$\displaystyle=\frac{1}{1+\delta_{cd}}\int\frac{d^{3}p_{b}}{(2\pi)^{3}}\frac{d^{3}p_{c}}{(2\pi)^{3}}\frac{d^{3}p_{d}}{(2\pi)^{3}}$
$\displaystyle~{}~{}~{}\times f_{b}^{(0)}W(a,b|c,d)B_{a-part}^{q}p_{a}^{\mu}.$
(42)

In the above equation, we can identify the energy-dependent relaxation time of
particle $`a^{\prime}$ as,

$\displaystyle\tau_{a}^{-1}(E_{a})=\frac{1}{1+\delta_{cd}}\int\frac{d^{3}p_{b}}{(2\pi)^{3}}\frac{d^{3}p_{c}}{(2\pi)^{3}}\frac{d^{3}p_{d}}{(2\pi)^{3}}f_{b}^{(0)}W(a,b|c,d).$
(43)

Therefore from Eq. (42) we get,

$\displaystyle
B_{a-part}^{q}=\frac{\tau_{a}}{E_{a}}\bigg{(}q_{a}-\frac{E_{a}n_{q}}{\omega}\bigg{)}.$
(44)

Substituting the above in the expression for $\kappa_{qq^{\prime}}$ as in
Eq.(41) we have,

$\displaystyle\kappa_{qq^{\prime}}=\sum_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{2}}{3E_{a}^{2}}\left(q_{a}-\frac{n_{q}E_{a}}{\omega}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\tau_{a}\left(q^{\prime}_{a}-\frac{n_{q}^{\prime}E_{a}}{\omega}\right)f_{a}^{(0)}.$
(45)

This makes the expression of $\kappa_{qq}$ positive definite. Also note that
$\kappa_{qq^{\prime}}$ is symmetric with respect to the change
$q\leftrightarrow q^{\prime}$ PhysRev.37.405 ; PhysRev.38.2265 ;
Fotakis:2019nbq . A few comments about the various approximation methods in
the calculation of various transport coefficients are in order here. Several
methods, e.g. the Relaxation Time Approximation (RTA), the Chapman-Enskog (CE)
method, etc. have been discussed in the literature for the calculation of the
transport coefficients. The Chapman-Enskog approach is a variational approach
and in this method depending on the order of approximation used one can obtain
solutions with an arbitrary accuracy Wiranata:2012br . The relaxation time
approach on the other hand is based on an ansatz for the collision kernel in
the Boltzmann equation. Such an ansatz for the collision kernel does not allow
to have control of the precision of the approximation. The relaxation time
approximation is incompatible with microscopic and macroscopic conservation
laws. Novel approaches have been proposed to overcome such problems, e.g.
PhysRev.94.511 ; Rocha:2021zcw . Nonetheless, the qualitative nature of
various transport coefficients as obtain in the relaxation time approximation
is similar to the Chapman-Enskog approach and the RTA approach has been even
more often used due to its simplicity. Various quantities, e.g. enthalpy,
number density, relaxation time that enters in the Eq. (45) can be obtained
for the hadronic matter modeled by the hadron resonance gas model.

##  III Hadron resonance gas model

The ideal hadron resonance gas model (IHRG) model which is based on the
Dashen, Ma, and Bernstein theorem PhysRev.187.345 ; PhysRevD.10.694 ,
indicates that a dilute system of strongly interacting matter can be described
by a gas of free hadrons and resonances. It should be emphasized that the
strong nuclear force has attractive as well as repulsive parts. Both the long-
range attraction and the short-range repulsion are important for a consistent
description of the strongly interacting matter PhysRevC.88.024902 . The
attractive part of the interaction is taken care of by the resonances. This
free gas of hadrons and resonances constitutes the ideal hadron resonance gas
model (IHRG) model. However, the repulsive nature of the nuclear force is not
manifested by the presence of resonances Andronic:2012ut . The repulsive part
of the strong nuclear force is incorporated through the excluded-volume
effects Rischke:1991ke ; Andronic:2012ut . The excluded-volume HRG (EVHRG)
model has been used in the hydrodynamical models of nucleus-nucleus collision
Hama:2004rr ; PhysRevC.82.044904 , to study the correlation and fluctuation of
conserved charges Bhattacharyya:2013oya , viscous coefficients of hadronic
matter Kadam:2015xsa , etc.

The grand canonical partition function of an ideal hadron resonance gas (IHRG)
model can be written as Braun-Munzinger:1994ewq ; Braun-Munzinger:2003pwq

$\displaystyle\ln Z^{id}=\sum_{a}\ln Z_{a}^{id},$ (46)

where the sum $`a^{\prime}$ is over all the hadrons and resonances,
$`id^{\prime}$ indicates the non interacting hadron resonance gas. The
partition function of the $`a^{\prime}$th species,

$\displaystyle\ln
Z_{a}^{id}=\pm\frac{Vg_{a}}{2\pi^{2}}\int_{0}^{\infty}dp~{}p^{2}\ln[1\pm\exp(-\beta(E_{a}-\mu_{a}))].$
(47)

Here $V$ is the volume of the system, $g_{a}$ is the degeneracy factor,
$E_{a}$ is the single-particle energy, $\mu_{a}=\sum_{q}q_{a}\mu_{q}\equiv
B_{a}\mu_{B}+S_{a}\mu_{S}+Q_{a}\mu_{Q}$ is the chemical potential,
$B_{a},S_{a},Q_{a}$ are respectively the baryon number, strangeness number and
electric charge. The $(+)$ and $(-)$ sign corresponds to fermions and bosons
respectively. Note that we present our results only for the Boltzmann limit.
Due to the conservation of different quantum numbers like baryon number,
charge, and strangeness various chemical potentials, e.g. $\mu_{B}$,
$\mu_{S}$, $\mu_{Q}$ are not independent. However, for simplicity, we assume
$\mu_{S}=0=\mu_{Q}$. Once the partition function is known all the
thermodynamic quantities of the system, e.g. pressure ($P^{id}$), net number
density associated with conserved charges ($n_{q}^{id}$), energy density
($\varepsilon^{id}$) can be obtained using various thermodynamic relations
Braun-Munzinger:1994ewq ; Braun-Munzinger:2003pwq ; Andronic:2012ut .

Let us now discuss the excluded volume HRG (EVHRG) model. To consider the
short-range repulsive hadron-hadron interaction in the EVHRG model the
geometrical size of the hadrons are explicitly incorporated as the excluded
volume correction Hagedorn:1980kb ; Rischke:1991ke ; Cleymans:1992jz ;
PhysRevC.56.2210 . A thermodynamically consistent excluded volume HRG model
the pressure can be written as,

$\displaystyle
P^{ex}(T,\mu_{1},\mu_{2},...)=\sum_{a}P_{a}^{id}(T,\tilde{\mu}_{1},\tilde{\mu}_{2},..),$
(48)

where the chemical potential of the $`a^{\prime}$th particle is,

$\displaystyle\tilde{\mu}_{a}=\mu_{a}-V^{ex}_{a}P^{ex}(T,\mu_{1},\mu_{2},..)$
(49)

where $V^{ex}_{a}=(16/3)\pi R_{a}^{3}$ is the excluded volume for the
$`a^{\prime}$ th hadron. In an iterative procedure one can get the total
pressure $P^{ex}(T,\mu_{1},\mu_{2},...)$. Other thermodynamic quantities which
can be obtained from the pressure using appropriate thermodynamic relations
can be given as,

$\displaystyle
n^{ex}=\sum_{a}\frac{n^{id}_{a}(T,\tilde{\mu}_{a})}{1+\sum_{b}V^{ex}_{b}n_{b}^{id}(T,\tilde{\mu}_{b})},$
(50) $\displaystyle
s^{ex}=\sum_{a}\frac{s^{id}_{a}(T,\tilde{\mu}_{a})}{1+\sum_{b}V^{ex}_{b}n_{b}^{id}(T,\tilde{\mu}_{b})},$
(51)
$\displaystyle\varepsilon^{ex}=\sum_{a}\frac{\varepsilon^{id}_{a}(T,\tilde{\mu}_{a})}{1+\sum_{b}V^{ex}_{b}n_{b}^{id}(T,\tilde{\mu}_{b})}.$
(52)

In general, the hardcore radius of mesons and the baryons can be considered to
be different. However, the thermodynamics of the EVHRG model is not strongly
dependent on different values of the hardcore radius of the mesons and baryons
Bhattacharyya:2013oya . Therefore we consider here same hardcore radius for
the mesons and baryons. The remaining unknown quantity in the expression of
$\kappa_{qq^{\prime}}$ is the relaxation time. In general relaxation time
depends on the energy and the momentum of the particles involved in the
scattering process. However for simplicity, one can integrate energy-dependent
relaxation time over equilibrium distribution functions to get the thermal
averaged relaxation time PhysRevC.92.035203 ; Das:2021qii . Without going into
the details of calculating the energy averaged relaxation time we give here
the important equations only. The thermal averaged relaxation time $\tau_{a}$
of the hadron species $``a"$ in terms of the scattering cross-section can be
expressed as,

$\displaystyle\tau_{a}^{-1}=\sum_{b}n_{b}\langle\sigma_{ab}v_{ab}\rangle,$
(53)

here $n_{b}$ denotes the number density of particle $``b"$ and
$\langle\sigma_{ab}v_{ab}\rangle$ represents thermal averaged cross section.
In Eq. (53) the sum is over all the hadrons and its resonances. For hard
sphere scattering in the Boltzmann approximation the thermal averaged cross
section can be expressed as PhysRevC.92.035203 ,

$\displaystyle\langle\sigma_{ab}v_{ab}\rangle$
$\displaystyle=\frac{\sigma}{8Tm_{a}^{2}m_{b}^{2}K_{2}(m_{a}/T)K_{2}(m_{b}/T)}$
$\displaystyle\times\int_{(m_{a}+m_{b})^{2}}^{\infty}ds\times\frac{[s-(m_{a}-m_{b})^{2}]}{\sqrt{s}}$
$\displaystyle\times[s-(m_{a}+m_{b})^{2}]K_{1}(\sqrt{s}/T),$ (54)

here the hard sphere scattering cross section can be expressed as $\sigma=4\pi
R^{2}$. $R$ is the radius of hadrons. Once the thermal averaged relaxation
time is known for each hadron species, different components of the diffusion
matrix can be obtained using Eq. (45).

Figure 1: Variation of dimensionless ratio $\kappa_{BB}/T^{2}$ with
temperature ($T$) and the baryon chemical potential ($\mu_{B}$). Here we have
assumed $\mu_{Q}=0=\mu_{S}$. Physically $\kappa_{BB}$ is the measure of the
baryon number current generation due to gradient in $\beta\mu_{B}$. Among all
the hadrons baryonic contribution is dominant over mesonic contribution in
$\kappa_{BB}/T^{2}$. Estimated values of $\kappa_{BB}/T^{2}$ are almost
similar in IHRG and EVHRG models. Different curves are overlapping. Figure 2:
Variation of dimensionless ratio $-\kappa_{BS}/T^{2}$ with temperature ($T$)
and the baryon chemical potential ($\mu_{B}$) for $\mu_{Q}=0=\mu_{S}$. For the
range of thermodynamic parameters considered here $\kappa_{BS}$ is always
negative. Physically $\kappa_{BS}$ is the measure of the diffusive coupling
between the baryon number and the strangeness number. Among all the hadrons
baryons contribute dominantly in $\kappa_{BS}/T^{2}$. Values of
$\kappa_{BS}/T^{2}$ as obtained in IHRG and EVHRG models are very similar with
overlapping curves. Figure 3: Variation of dimensionless ratio
$\kappa_{BQ}/T^{2}$ with temperature ($T$) and the baryon chemical potential
($\mu_{B}$) for $\mu_{Q}=0=\mu_{S}$. $\kappa_{BQ}$ is the measure of the
diffusive coupling between the baryon and electric charge. Among all the
hadrons leading contribution in $\kappa_{BQ}/T^{2}$ comes from the baryons.
Estimated values of $\kappa_{BQ}/T^{2}$ in IHRG and EVHRG models are very
similar with overlapping curves. Figure 4: Variation of diagonal component
$\kappa_{SS}/T^{2}$ with temperature ($T$) and the baryon chemical potential
($\mu_{B}$). Here also we consider the scenario with $\mu_{Q}=0=\mu_{S}$.
$\kappa_{SS}$ is the measure of the strangeness number current due to the
gradient in $\beta\mu_{S}$. Since the lightest strange hadrons are mesons,
therefore mesonic contribution is dominant in $\kappa_{SS}/T^{2}$ with respect
to the baryonic contribution. Estimated values of $\kappa_{SS}/T^{2}$ in IHRG
and EVHRG models are very similar with overlapping curves. Figure 5: Variation
of dimensionless ratio $\kappa_{SQ}/T^{2}$ with temperature ($T$) and the
baryon chemical potential ($\mu_{B}$) for $\mu_{Q}=0=\mu_{S}$. $\kappa_{SQ}$
is the measure of the diffusive coupling between the strangeness charge and
electric charge. Mesons contribute predominantly in $\kappa_{SQ}/T^{2}$.
Different curves for IHRG and EVHRG are overlapping. Figure 6: Variation of
diagonal component $\kappa_{QQ}/T^{2}$ with temperature ($T$) and the baryon
chemical potential ($\mu_{B}$). $\kappa_{QQ}$ is the measure of the electrical
current due to the gradient in $\beta\mu_{Q}$. For the hadron resonance gas
leading contribution to $\kappa_{QQ}/T^{2}$ comes from the mesons with respect
to the hadrons. Similar to other elements of the diffusion matrix,
$\kappa_{QQ}/T^{2}$ also has similar values in the IHRG and EVHRG model.

##  IV results and discussions

We present the results for the diffusion matrix elements
$\kappa_{qq^{\prime}}$ with temperature and baryon chemical potential. For
simplicity, we assume that electric chemical potential and the strangeness
chemical potential is zero, i.e. $\mu_{Q}=0=\mu_{S}$. For HRG, we consider all
the hadrons and their resonances up to a mass cutoff $\Lambda=2.6$ GeV, as is
listed in Ref. ParticleDataGroup:2008zun . Also for a detailed list of hadrons
and their resonances, we refer to Appendix A of Ref. PhysRevC.90.024915 .
These apart radii of the hard spheres also enter in the calculation of
relaxation time. We consider a uniform radius $R=0.5$ fm for all the hadrons
PhysRevC.92.035203 ; Braun-Munzinger:1999hun .

In Fig. (1) we show the variation of the diagonal component of the diffusion
matrix associated with the baryon number current, i.e. $\kappa_{BB}$ with
temperature and baryon chemical potential. For $\mu_{B}=0$ and 300 MeV the
dimensionless quantity $\kappa_{BB}/T^{2}$ increases with temperature. However
for $\mu_{B}=600$ MeV, $\kappa_{BB}/T^{2}$ decreases with temperature. Since
here we consider $\mu_{Q}=0=\mu_{S}$, for vanishing value of baryon chemical
potential the variation of $\kappa_{BB}/T^{2}$ can be understood in the
following manner. For $\mu_{B}=\mu_{Q}=\mu_{S}=0$, the net baryon number
density vanishes, i.e. $n_{B}=0$. Note that with temperature the relaxation
time of hadrons decreases. On the other hand with temperature the contribution
coming from the distribution function increase due to the increase in the
baryon numbers. This increase in the distribution function wins over the
decrease in the relaxation time, giving rise to an increasing behavior of
$\kappa_{BB}/T^{2}$ with temperature. However for $\mu_{B}\neq 0$, in the
expression of $\kappa_{BB}$, $n_{B}/\omega$ increases with temperature
Das:2019pqd  along with the decreases in the relaxation time with temperature.
For a sufficiently large value of the baryon chemical potential, this decrease
in the relaxation time is predominant giving rise to a decreasing behavior of
$\kappa_{BB}/T^{2}$ with temperature for $\mu_{B}=600$ MeV.

Further in the relatively low-temperature range $\kappa_{BB}/T^{2}$ increases
with $\mu_{B}$. Note that for the hadron resonance gas with $\mu_{B}$,
$n_{B}/\omega$ increases Das:2019pqd  along with the increase in the
distribution function. Again the relaxation time of various hadron species
decreases with the baryon chemical potential. Such an increase in
$n_{B}/\omega$ and distribution function dominates over the decreases in the
relaxation time giving rise to an overall increasing trend. However, for a
sufficiently high-temperature range, the variation of $\kappa_{BB}/T^{2}$ with
the baryon chemical potential is nonmonotonic. This nonmonotonic variation is
predominantly due to the decrease in the relaxation time with temperature and
baryon chemical potential. Note that results for $\kappa_{BB}/T^{2}$ as
obtained in the IHRG, as well as EVHRG, are almost similar. This is a generic
feature for other components of the diffusion matrix. Various thermodynamic
quantities, e.g. pressure, energy density, and number density are different in
EVHRG as compared to the IHRG model. But the thermodynamic quantity that
enters into the expression of $\kappa_{qq}$ is $n_{q}/\omega$. $n_{q}/\omega$
does not change significantly in the EVHRG as compared to the IHRG, giving
rise to an almost similar variation of the diffusion matrix elements with
temperature and baryon chemical potential.

In Fig. (2) we show the variation of $-\kappa_{BS}/T^{2}$ with temperature
($T$) and the baryon chemical potential ($\mu_{B}$). Physically a nonvanishing
value of $\kappa_{BS}$ indicates the generation of baryon current due to the
gradient in the number density of hadrons containing strangeness quantum
number. Further note that $\kappa_{BS}$ is negative for the range of
temperature and baryon chemical potential considered here. For $\mu_{B}=0$
this can be understood in the following way. For $\mu_{B}=\mu_{S}=\mu_{Q}=0$,
$n_{B}=n_{S}=0$ and $\kappa_{BS}$ is proportional to the (baryon number
$\times$ strangeness number). Now in the baryon sector, baryon number and
strangeness number of various baryons are opposite in sign giving rise to a
negative value of $\kappa_{BS}$. Also for finite $\mu_{B}$ baryon contributes
dominantly in $\kappa_{BS}$ due to an increase in the number density of
baryons. The opposite sign of the baryon number and the strangeness number for
baryons give rise to an overall negative sign to $\kappa_{BS}$. Similar to
$\kappa_{BB}/T^{2}$, the off-diagonal component $-\kappa_{BS}/T^{2}$ also
shows strong temperature dependence for $\mu_{B}=0,300$ MeV. Such a behaviour
is again predominantly due to increase in the equilibrium distribution
function. The nonmonotonic variation of $-\kappa_{BS}/T^{2}$ with $\mu_{B}$ is
again due to various $\mu_{B}$ dependent factors, e.g. relaxation time,
$n_{B}/\omega$, $n_{S}/\omega$ and the equilibrium distribution function.

In Fig. (3) we show the variation $\kappa_{BQ}/T^{2}$ with temperature ($T$)
and the baryon chemical potential ($\mu_{B}$). Similar to $\kappa_{BS}$ a
nonvanishing value of $\kappa_{BQ}$ indicates gradient in the baryon number
density can give rise to electric current. Further note that unlike
$\kappa_{BS}$, $\kappa_{BQ}$ is positive for the range of temperature and
baryon chemical potential considered here. Such behavior is easy to understand
for $\mu_{B}=0$ case. As discussed earlier for $\mu_{B}=\mu_{S}=\mu_{Q}=0$,
$n_{B}=n_{S}=0$ and $\kappa_{BQ}$ is proportional to the (baryon number
$\times$ electric charge). Now in the baryon sector for the lightest baryon
and antibaryons, baryon number and electric charge has the same sign giving
rise to a positive value of $\kappa_{BQ}$. Similarly one can also explain the
overall positive sign for $\kappa_{BQ}/T^{2}$at finite $\mu_{B}$. Variation of
$\kappa_{BQ}/T^{2}$ with temperature and baryon chemical potential is very
similar to $\kappa_{BB}/T^{2}$. Such variation of $\kappa_{BQ}/T^{2}$ with
temperature and baryon chemical can be qualitatively understood by looking
into the behavior of relaxation time, $n_{B}/\omega$, $n_{Q}/\omega$ and the
equilibrium distribution function with $T$ and $\mu_{B}$.

In Figs. (4) and (5) we present the results for the diffusion coefficients
associated with the strangeness current originated due to gradient in
$\beta\mu_{S}$ and $\beta\mu_{Q}$ respectively. According to Eq. (45) the
diagonal component $\kappa_{SS}/T^{2}$ is always positive. The off-diagonal
component $\kappa_{SQ}/T^{2}$ is also positive for the range of temperature
and baryon chemical potential considered here. Both $\kappa_{SS}/T^{2}$ and
$\kappa_{SQ}/T^{2}$ shows nonmonotonic variation with temperature and baryon
chemical potential. Such nonmontonic variation of $\kappa_{SS}/T^{2}$ and
$\kappa_{SQ}/T^{2}$ is rather convoluted as various factors, e.g. the
relaxation time, distribution function, $n_{S}/\omega$ and $n_{Q}/\omega$
depends upon temperature and baryon chemical potential. Although the behaviour
of $\kappa_{BB}/T^{2}$, $\kappa_{BS}/T^{2}$ and $\kappa_{BQ}/T^{2}$ as
presented in Figs. (1), (2) and (3) respectively, are similar to the results
presented in the Refs. Fotakis:2019nbq ; Greif:2017byw , the nonmonotonic
variation of $\kappa_{SS}/T^{2}$ and $\kappa_{SQ}/T^{2}$ is different from the
results presented in the Refs. Fotakis:2019nbq ; Greif:2017byw .

Finally, in Fig. (6) we show the variation of $\kappa_{QQ}/T^{2}$ with $T$ and
$\mu_{B}$. As argued in Refs. Fotakis:2019nbq ; Greif:2017byw , here also for
$\mu_{B}=0$, $\kappa_{QQ}/T^{2}\equiv\sigma_{el}/T$. $\sigma_{el}$ is the
electrical conductivity of the medium in the kinetic theory approach
Das:2019wjg . Although for non vanishing values of baryon chemical potential
$\kappa_{QQ}/T^{2}\neq\sigma_{el}/T$, but the qualitative behaviour of
$\kappa_{QQ}/T^{2}$ and $\sigma_{el}/T$ are similar even for $\mu_{B}\neq 0$
Das:2019wjg . Among all the other hadrons pions and protons contribute
dominantly in $\sigma_{el}/T$ as well as in $\kappa_{QQ}/T^{2}$. As argued in
Ref. Das:2019wjg  among various temperature and baryon chemical potential
dependent quantities, due to the decrease of the relaxation time of hadrons
with $T$ and $\mu_{B}$, $\sigma_{el}/T$ or $\kappa_{QQ}/T^{2}$ also decreases.

##  V conclusion

In the present investigation, we discuss the diffusion matrix associated with
the various conserved quantities. Using the classical kinetic theory within
the relaxation time approximation we obtained an analytical expression of the
diffusion matrix ($\kappa_{qq^{\prime}}$). The diagonal components of the
diffusion matrix are always positive, on the other hand, the off-diagonal
components can be negative as well as positive. Using the hadron resonance gas
model within the hard-sphere scattering approximation we estimated various
elements of $\kappa_{qq^{\prime}}$ for the hadronic medium produced in heavy-
ion collisions. Knowledge of the diffusion processes and the diffusion
currents is very important in the context of the bulk evolution of the
strongly interacting plasma, e.g. the cross-coupling between the diffusion
currents can dynamically generate non-zero net strangeness, even if it is
initially zero Fotakis:2019nbq . Further, the order of magnitude values of the
off-diagonal components of the diffusion matrix is not at all negligible as
compared to the diagonal elements. Therefore, it is of paramount importance to
computing the full table of diffusion coefficients for consistent fluid
dynamical simulations where the diffusion is also taken under consideration.
In this paper, we have ignored the effect of the mean-field or medium
modification on the constituent of the plasma. Such mean-field effects should
be included in the kinetic theory description as it might be quite important
across the QCD transition scale.

###### Acknowledgements.

The work of AD is supported by the Polish National Science Center Grant No.
2018/30/E/ST2/00432. AD would like to thank Guru Prakash Kadam for important
discussions on the excluded volume hadron resonance gas model.

## Appendix A

Let us start with the following integration for a single species,

$\displaystyle\mathcal{I}_{1}$
$\displaystyle=\int\frac{d^{3}p}{(2\pi)^{3}}p^{2}\exp(-\beta\sqrt{p^{2}+m^{2}})$
$\displaystyle=\frac{m^{5}}{2\pi^{2}}\int_{0}^{\infty}dy~{}\cosh{y}~{}(\sinh{y})^{4}\exp(-\beta
m\cosh{y})$
$\displaystyle=\frac{m^{5}}{2\pi^{2}}\int_{0}^{\infty}dy~{}((\cosh{y})^{5}-2(\cosh{y})^{3}+\cosh{y})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\exp(-\beta
m\cosh{y})$
$\displaystyle=\frac{m^{5}}{2\pi^{2}}\bigg{(}\mathcal{I}_{1a}-2\mathcal{I}_{1b}+\mathcal{I}_{1c}\bigg{)},$
(55)

here,

$\displaystyle\mathcal{I}_{1a}=\int_{0}^{\infty}dy(\cosh{y})^{5}\exp(-\beta
m\cosh{y}),$ (56)
$\displaystyle\mathcal{I}_{1b}=\int_{0}^{\infty}dy(\cosh{y})^{3}\exp(-\beta
m\cosh{y}),$ (57)
$\displaystyle\mathcal{I}_{1c}=\int_{0}^{\infty}dy\cosh{y}\exp(-\beta
m\cosh{y}).$ (58)

Using the integral representation of the Modified Bessel Function of the
second kind ($K_{n}(x)$) the integral it is easy to show that,

$\displaystyle\mathcal{I}_{1c}=K_{1}(\beta m),$ (59)
$\displaystyle\mathcal{I}_{1b}=\frac{\partial^{2}K_{1}(\beta
m)}{\partial(\beta m)^{2}}=\frac{1}{\beta m}K_{0}(\beta m)+\frac{2}{(\beta
m)^{2}}K_{1}(\beta m)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+K_{1}(\beta
m)$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}=\frac{1}{\beta m}K_{2}(\beta
m)+K_{1}(\beta m),$ (60)
$\displaystyle\mathcal{I}_{1a}=\frac{\partial^{4}K_{1}(\beta
m)}{\partial(\beta m)^{4}}$ $\displaystyle=\frac{12}{(\beta m)^{3}}K_{0}(\beta
m)+\frac{2}{\beta m}K_{0}(\beta m)+\frac{24}{(\beta m)^{4}}K_{1}(\beta m)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\frac{7}{(\beta m)^{2}}K_{1}(\beta
m)+K_{1}(\beta m)$ $\displaystyle=\frac{3}{(\beta m)^{2}}K_{3}(\beta
m)+\frac{2}{\beta m}K_{2}(\beta m)+K_{1}(\beta m).$ (61)

Eqs. (59)-(61) allows us to write the integral $\mathcal{I}_{1}$ as,

$\displaystyle\mathcal{I}_{1}=\frac{3m^{3}T^{2}}{2\pi^{2}}K_{3}(\beta m).$
(62)

Now the energy density and pressure of a single particle species of mass $m$
at finite temperature $T$ can be expressed as Florkowski:2014sfa ,

$\displaystyle\varepsilon=\frac{m^{2}T}{2\pi^{2}}\bigg{(}3TK_{2}(\beta
m)+mK_{1}(\beta m)\bigg{)},$ (63)

and,

$\displaystyle P=\frac{m^{2}T^{2}}{2\pi^{2}}K_{2}(\beta m).$ (64)

Therefore, the enthalpy ($\omega$) can be expressed as,

$\displaystyle\omega$ $\displaystyle=\varepsilon+P$
$\displaystyle=\frac{4m^{2}T^{2}}{2\pi^{2}}K_{2}(\beta
m)+\frac{m^{3}T}{2\pi^{2}}K_{1}(\beta m)$
$\displaystyle=\frac{m^{3}T}{2\pi^{2}}K_{3}(\beta
m)=\frac{1}{3T}\mathcal{I}_{1}.$ (65)

The above expression can be generalized to multiple particle species.

## Appendix B

We give here a short derivation of Eq.(40). Without loss of generality,let us
consider only the baryon number conservation and a single baryon and it’s
antibaryon species. In this case the integral,

$\displaystyle\mathcal{I}_{2}$
$\displaystyle=\sum_{a}q_{a}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{p_{a}^{2}}{3E_{a}}\exp(-\beta\sqrt{p_{a}^{2}+m_{a}^{2}})\exp(\beta
q_{a}\mu_{q})$
$\displaystyle=\int\frac{d^{3}p}{(2\pi^{3})}\frac{p^{2}}{3E}\exp(-\beta\sqrt{p^{2}+m^{2}})2\sinh(\beta\mu_{B})$
$\displaystyle=2\sinh(\beta\mu_{B})\frac{m^{4}}{6\pi^{2}}\int_{0}^{\infty}dy(\sinh{y})^{4}\exp(-\beta
m\cosh{y})$
$\displaystyle=2\sinh(\beta\mu_{B})\frac{m^{4}}{6\pi^{2}}(\mathcal{I}_{2a}-2\mathcal{I}_{2b}+\mathcal{I}_{2c}),$
(66)

here,

$\displaystyle\mathcal{I}_{2a}=\int_{0}^{\infty}dy(\cosh{y})^{4}\exp(-\beta
m\cosh{y}),$ (67)
$\displaystyle\mathcal{I}_{2b}=\int_{0}^{\infty}dy(\cosh{y})^{2}\exp(-\beta
m\cosh{y}),$ (68)
$\displaystyle\mathcal{I}_{1c}=\int_{0}^{\infty}dy\exp(-\beta m\cosh{y}).$
(69)

Using the integral representation of the Modified Bessel Function of the
second kind ($K_{n}(x)$) it can be shown that,

$\displaystyle\mathcal{I}_{2c}=K_{0}(\beta m),$ (70)
$\displaystyle\mathcal{I}_{2b}=K_{0}(\beta m)+\frac{1}{\beta m}K_{1}(\beta
m),$ (71) $\displaystyle\mathcal{I}_{2a}=\frac{3}{(\beta m)^{2}}K_{2}(\beta
m)+K_{2}(\beta m).$ (72)

Therefore,

$\displaystyle\mathcal{I}_{2}=2\sinh(\beta\mu_{B})\frac{m^{2}T^{2}}{2\pi^{2}}K_{2}(\beta
m).$ (73)

Net baryon number density can be expressed as,

$\displaystyle n_{B}$
$\displaystyle=2\sinh{\beta\mu_{B}}\int\frac{d^{3}p}{(2\pi)^{3}}\exp(-\beta
m\sqrt{p^{2}+m^{2}})$
$\displaystyle=2\sinh{\beta\mu_{B}}\frac{m^{2}T}{2\pi^{2}}K_{2}(\beta m).$
(74)

Therefore,

$\displaystyle\mathcal{I}_{2}=n_{B}T.$ (75)

The above expression can be easily generalized to include other conserved
charges.

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