# Dynamic density correlations in baryon rich fluid using Mori-Zwanzig
projection operator method

Guruprasad Kadam<EMAIL_ADDRESS>School of Physical Sciences,
National Institute of Science Education and Research Bhubaneswar, HBNI, Jatni
752050, Odisha, India

###### Abstract

In this work we calculate the dynamic density correlations using Mori-Zwanzig
projection operator method. With a judicious choice of slow variables we
derive the evolution equations for these slow variables starting from
generalised Langevin equation. We get the hydrodynamic form of density
correlations function which consist of two acoustic peaks: also called
Brillouin peaks, and one thermal peak: also called Rayleigh peak. We then
estimate the dynamic density correlations near the critical point using
critical exponents extracted from the statistical bootstrap model of hadronic
matter. We find that the bulk viscosity contributes to the sound attenuation
at leading order $\sim|t|^{-\frac{5}{4}}$ while the thermal conductivity
contribute at sub-leading order $\sim|t|^{-\frac{3}{4}}$. On the other hand,
only the thermal conductivity contributes at leading order
$\sim|t|^{\frac{1}{4}}$ to the thermal diffusivity. We discuss the
implications of these results in search for QCD critical point in the heavy-
ion collision experiments.

###### pacs:

12.38.Mh, 12.39.-x, 11.30.Rd, 11.30.Er

##  I Introduction

Understanding the quantum chromodynamics (QCD) phase diagram is one of the
major challenge in the high energy particle physics and astrophysics
todayFukushima and Hatsuda (2011); Hands (2001); Shuryak (1996). Main
impediments in this pursuit are practical and conceptual problems of lattice
quantum chromodynamics (LQCD) to understand the phenomenologically relevant
part of the phase diagram at finite baryon density. LQCD suffers from so
called sign problem at finite chemical potential, however for small chemical
potentials there are few reliable results availableAoki  _et al._ (2006,
2009); Borsányi  _et al._ (2010); Borsanyi  _et al._ (2020). So one has to
resort to various effective models of QCD at relatively high baryon density,
$viz.$, Nambu-Jona-Lasinio model (NJL)Klevansky (1992); Hatsuda and Kunihiro
(1994), Quark-Meson Coupling model(QMC)Schaefer  _et al._ (2007), Hadron
Resonance Gas model (HRG)Venugopalan and Prakash (1992), Relativistic Mean-
Field model(RMF)Kadam and Mishra (2019); Pal  _et al._ (2021a, b), Statistical
Bootstrap Model (SBM)Hagedorn (1965); Frautschi (1971); Satz (1979) etc. In
the QCD phase diagram a particularly interesting point is the conjectured
critical end point (CEP) which is an end point of first order phase transition
line. A lot of theoretical studies has been carried out to find the possible
signatures of CEP Bluhm  _et al._ (2020); Stephanov (2004); Dore  _et al._
(2020); Mroczek  _et al._ (2021); An  _et al._ (2022); Bzdak  _et al._ (2020);
Stephanov (2011); Grossi  _et al._ (2021); Florio  _et al._ (2021); Datta  _et
al._ (2013); Gavai (2011); Kapusta  _et al._ (2021); Kapusta and Torres-Rincon
(2012) and on the experimental side, the Beam Energy Scan (BES) program has
been devoted at the Super Proton Synchrotron (SPS) and at the Relativistic
Heavy Ion Collider (RHIC) to search for possible signatures of the CEP
Aggarwal  _et al._ (2010); Luo and Xu (2017).

Fluid described by the theory of hydrodynamics belongs to a class of systems
in which non-equilibrium dynamics can be described using only a small number
of so-called effective ”slow” variablesLandau and Lifshitz (1987); Romatschke
(2010); Jeon and Heinz (2015). For instance, one can describe the dynamics of
a fluid with only few number of equations of motion: conservation equations
for energy density, momentum density, fluid velocity and, possibly, charge
density if there are any conserved charges, despite the microscopic degrees of
freedom are very large. It turns out that this description can be accommodated
into a non-equilibrium transport equation which consist of reversible,
irreversible and noise parts. While the former two describes slow dynamics,
the noise term contain the information about leftover fast degrees of freedom.
Langevin equation is archetypal example of this formalismZwanzig (2001);
Balakrishnan (2008); Mazenko (2008).

A powerful way to study these aspects is Mori-Zwanzig projection operator
formalismMori (1965); Nordholm and Zwanzig (1975). In this formalism one can
derive the non-equilibrium transport equation by coarse-graining procedure.
One first choose a set of slow variables describing long-time dynamics of a
system under consideration. Operators corresponding to a set of observables
forms a Hilbert space with appropriately defined scalar product. One then
introduces the projection operator which project onto subspace formed by the
slow variables. While the Mori–Zwanzig formalism has been traditionally
applied to non-relativistic systemsGrabert (2006); te Vrugt and Wittkowski
(2020), it has recently attracted a lot of attention in high energy
physicsKoide (2005); Koide and Kodama (2008); Huang  _et al._ (2011); Minami
and Hidaka (2013) and cosmology researchVrugt  _et al._ (2021).

In present work we use the time-independent Mori-Zwanzig projection operator
method to study the dynamic density correlations in a baryon rich fluid. In
the theory of non-equilibrium phenomena correlation functions are very
important quantities as they are directly measurable. In the linear response
theory the response functions are linearly related to the external
perturbations and the time-correlation functions express the propagation of
equilibrium fluctuations in the system. Typically, for a fluid described by
hydrodynamics the density correlation function consist of three peaks: two of
them corresponds to acoustic waves and one corresponds to thermal
fluctuations. It has been noted in previous studyMinami and Kunihiro (2010)
that for a relativistic fluid the width of acoustic peaks, also known as
Brillouin peaks, picks up a purely relativistic correction, while the Rayleigh
peak due to thermal fluctuations remains unchanged. We shall discuss these
aspects in this work. We shall further discuss the impact of critical
singularities in the thermodynamic functions and transport coefficients on the
Brillouin and Rayleigh peaks.

We organize the paper as follows. In section II we recapitulate the Mori-
Zwanzig projection operator method. In section III we present the derivation
of density correlation function using Mori-Zwanzig projection operator method.
In section IV we discuss the density correlation near the QCD critical point
within ambit of SBM and discuss the implications of our results in the context
of search of CEP in heavy-ion collision experiments. Finally, in section V we
summarize and conclude.

##  II Mori-Zwanzig projection operator method

Mori-Zwanzig projection operator method is based on coarse graining a system
with large number of microscopic degrees of freedom governed by hamiltonian
dynamics into fast degrees of freedom and relatively few effective slow
degrees of freedom using time independent projection operator. This procedure
leads to so called generalised Langevin equation which describes the time
evolution of a slow operator. While the slow dynamics is captured by so called
memory function, the fast dynamics is contained in the noise term. On the
microscopic scale the time dependence of an operator $\mathcal{\hat{O}}$ is
governed by Heisenberg equation of motion:

$\frac{\partial\mathcal{\hat{O}}(t)}{\partial
t}=i[\hat{H},\mathcal{\hat{O}}]\equiv i\mathcal{\hat{L}}\mathcal{\hat{O}}(t)$
(1)

where $\mathcal{\hat{L}}\equiv[\hat{H},...]$ is the Liouville operator.
$\hat{H}$ is the hamiltonian operator. The formal solution of Eq.(1) is given
as,

$\mathcal{\hat{O}}(t)=e^{it\mathcal{\hat{L}}}\mathcal{\hat{O}}(0),$ (2)

where $\mathcal{\hat{O}}(0)$ corresponds to initial time operator. We define
inner product of two operators $\mathcal{\hat{A}}$ and $\mathcal{\hat{B}}$
asMori (1965),

$(\mathcal{\hat{A}},\mathcal{\hat{B}})=\frac{1}{\beta}\int_{0}^{\beta}d\tau<\mathcal{\hat{A}}(-i\tau)\mathcal{\hat{B}}^{\dagger}>_{0},$
(3)

where $\beta=T^{-1}$. The equilibrium average is defined as,

$<\mathcal{\hat{O}}>_{0}=\text{tr}(\hat{\rho}_{0}\mathcal{\hat{O}}).$ (4)

For a grand canonical ensemble the density matrix $\rho_{0}$ is given by,

$\hat{\rho}_{0}(\beta,\mu)\equiv\frac{e^{-\beta(\hat{H}-\mu\hat{N})}}{\text{tr}\>e^{-\beta(\hat{H}-\mu\hat{N})}}.$
(5)

If it is possible to separate the time scale into long-time and short-time
scales then there exists a set of slowly varying operators (corresponding to
slow variables) which describes the slow dynamics. Let
$\\{\hat{\mathcal{A}}_{n}\\}=\\{\mathcal{\hat{A}}_{1},\mathcal{\hat{A}}_{2},...,\mathcal{\hat{A}}_{n}\\}$
be set of such slowly varying operators which are not necessarily orthogonal.
It is possible to define orthogonal quantity $\mathcal{\hat{A}}^{n}$ to
$\mathcal{\hat{A}}_{n}$ using metricHidaka (2013),

$g_{nm}({\bf{x-y}})\equiv(\mathcal{\hat{A}}_{n}(0,{\bf{x}}),\mathcal{\hat{A}}_{m}(0,{\bf{y}})).$
(6)

The operators orthogonal to $\mathcal{\hat{A}}_{n}$ are defined as,

$\mathcal{\hat{A}}^{n}(t,{\bf{x}})=\int
d^{3}y\>g^{nm}({\bf{x-y}})\mathcal{\hat{A}}_{m},$ (7)

where $g^{nm}$ is the inverse of $g_{nm}$ which coincides with the second
derivative of the effective action $\beta\Gamma_{\text{eff}}$, with respect to
$\mathcal{\hat{A}}_{n}$:

$g^{nm}({\bf{x-y}})=\frac{\delta^{2}\beta\Gamma_{\text{eff}}(\mathcal{\hat{A}}_{n})}{\delta\mathcal{\hat{A}}_{m}({\bf{y}})\mathcal{\hat{A}}_{n}^{\dagger}({\bf{x}})}$
(8)

where the effective action is given by the Legendre transformation of the
generating functional $W(J^{n})$:

$\Gamma_{\text{eff}}(\mathcal{\hat{A}}_{n})=W(J^{n})-\int
d^{3}xJ^{m}({\bf{x}})\frac{\delta W(J^{n})}{\delta J^{m}({\bf{x}})}$ (9)

with $W(J^{n})=-\frac{1}{\beta}\text{ln}\mathcal{Z}$. The partition function
$\mathcal{Z}$ is given by,

$\mathcal{Z}(\beta;J^{n})=\text{tr}\bigg{[}e^{-\beta(\hat{H}-\mu\hat{N})}\text{exp}\bigg{(}\int
d^{3}x\mathcal{\hat{A}}_{n}(0,{\bf{x}})J^{n}(\bf{x})\bigg{)}\bigg{]}.$ (10)

We finally define the projection operator $\mathcal{\hat{P}}$ acting on any
arbitrary operator $\mathcal{\hat{O}}(t,{\bf{x}})$ as,

$\mathcal{\hat{P}}\mathcal{\hat{O}}(t,{\bf{x}})\equiv\int
d^{3}z\>\mathcal{\hat{A}}_{n}(0,{\bf{z}})(\mathcal{\hat{O}}(t,{\bf{x}}),\mathcal{\hat{A}}^{n}(0,{\bf{z}})).$
(11)

Operator $\mathcal{\hat{P}}$ project out the slowly varying part of
$\mathcal{\hat{O}}$. We further define orthogonal projector
$\mathcal{\hat{Q}}\equiv 1-\mathcal{\hat{P}}$.

Now consider an operator identity:

$\frac{\partial}{\partial
t}e^{it\mathcal{\hat{L}}}=e^{it\mathcal{\hat{L}}}\mathcal{\hat{P}}\>i\mathcal{\hat{L}}+\int_{0}^{t}ds\>e^{i(t-s)\mathcal{\hat{L}}}\mathcal{\hat{P}}\>i\mathcal{\hat{L}}\>e^{it\mathcal{\hat{Q}}\mathcal{\hat{L}}}\>\mathcal{\hat{Q}}\>i\mathcal{\hat{L}}+e^{it\mathcal{\hat{Q}}\mathcal{\hat{L}}}\>\mathcal{\hat{Q}}\>i\mathcal{\hat{L}}$
(12)

Multiplying both sides of (12) by $\mathcal{\hat{A}}_{n}(0)$ we get

$\displaystyle\frac{\partial}{\partial t}\mathcal{\hat{A}}_{n}(t,{\bf{x}})$
$\displaystyle=$ $\displaystyle\int
d^{3}y\>iK_{n}^{(s)m}({\bf{x-y}})\mathcal{\hat{A}}_{m}(t,{\bf{y}})$
$\displaystyle-$ $\displaystyle\int_{0}^{\infty}ds\int
d^{3}y\>K_{n}^{(d)m}(t-s,{\bf{x-y}})\mathcal{\hat{A}}_{m}(s,{\bf{y}})+\mathcal{\hat{N}}_{n}(t,{\bf{x}}),$

where,

$\displaystyle iK_{n}^{(s)m}({\bf{x-y}})$ $\displaystyle\equiv$
$\displaystyle(i\mathcal{\hat{L}}\mathcal{\hat{A}}_{n}(0,{\bf{x}}),\mathcal{\hat{A}}^{m}(0,{\bf{y}}))$
(14) $\displaystyle K_{n}^{(d)m}(t-s,{\bf{x-y}})$ $\displaystyle\equiv$
$\displaystyle-\theta(t-s)(i\mathcal{\hat{L}}\mathcal{\hat{N}}_{n}(t,{\bf{x}}),\mathcal{\hat{A}}^{m}(s,{\bf{y}}))$
(15) $\displaystyle\mathcal{\hat{N}}_{n}(t,{\bf{x}})$ $\displaystyle\equiv$
$\displaystyle
e^{it\mathcal{\hat{Q}}\mathcal{\hat{L}}}\>\mathcal{\hat{Q}}\>i\mathcal{\hat{L}}\mathcal{\hat{A}}_{n}(0,{\bf{x}}).$
(16)

Eq.(II) is called generalized Langevin equationMori (1965). $iK_{n}^{(s)m}$ is
called static memory function and it captures the time-reversible change.
While, $K_{n}^{(d)m}$ is called dynamic memory function and it captures the
time-irreversible change and hence the dissipation in the system. Note that
$K_{n}^{(d)m}$ depends on a past time value of $\mathcal{\hat{A}}_{n}(s)$ for
$s<t$. The last term in the generalized Langevin equation $\mathcal{\hat{N}}$
is called the noise. It is only the memory function terms that contribute to
the slow dynamics and hence we neglect the noise term hereafter.

Taking Fourier transform of (II) we get,

$\displaystyle\frac{\partial}{\partial t}\mathcal{\hat{A}}_{n}(t,{\bf{k}})$
$\displaystyle=$
$\displaystyle\>iK_{n}^{(s)m}({\bf{k}})\mathcal{\hat{A}}_{m}(t,{\bf{k}})$
$\displaystyle-$
$\displaystyle\int_{0}^{\infty}ds\>K_{n}^{(d)m}(t-s,{\bf{k}})\mathcal{\hat{A}}_{m}(s,{\bf{k}})+\mathcal{\hat{N}}_{n}(t,{\bf{k}}).$

Eq.(II) can we written in more compact form as,

$\frac{\partial}{\partial
t}\mathcal{\hat{A}}(t)=\>i{\bf{K^{(s)}}}\mathcal{\hat{A}}(t)-\int_{0}^{\infty}ds\>{\bf{K^{(d)}}}(t-s){\bf{\mathcal{\hat{A}}}}(s)+\mathcal{\hat{N}}(t)$
(18)

In the Eq.(18) $\mathcal{\hat{A}}$ is a column matrix of order $n$ for a set
of ’$n$’ slow variables. memory matrices $\bf{K^{(s)}}$ and $\bf{K^{(d)}}$ are
$n\times n$ matrices.

##  III Density correlations for the equilibrium fluctuations

We shall now calculate the density correlations for the fluctuations about the
thermal equilibrium. Near the thermal equilibrium any thermodynamic quantity
can be written as,

$A(t,{\bf{x}})=A_{0}+\delta A(t,{\bf{x}}),$ (19)

where $A_{0}$ is the equilibrium value and $\delta A(t,{\bf{x}}$ fluctuation
about equilibrium. For our purpose we choose fluctuations in baryon number
density $\delta\hat{n}_{b}$, energy density $\delta\hat{\epsilon}$ and the
fluid velocity $\delta\hat{{\bf{v}}}$ as slow variables.

The generalised Langevin equation for $\delta\hat{n}_{b}$ reads:

$\displaystyle\frac{\partial}{\partial t}\delta\hat{n}_{b}(t,{\bf{x}})$
$\displaystyle=$ $\displaystyle\int
d^{3}y\>iK_{{n}_{b}}^{(s)m}({\bf{x-y}})\mathcal{\hat{A}}_{m}(t,{\bf{x}})$
$\displaystyle-$ $\displaystyle\int_{0}^{\infty}ds\int
d^{3}y\>K_{n_{b}}^{(d)m}(t-s,{\bf{x-y}})\mathcal{\hat{A}}_{m}(s,{\bf{x}})$

Using memory matrices calculated in the Appendix V we get,

$\frac{\partial\delta\hat{n}_{b}}{\partial
t}+\hat{n}_{b,0}\nabla\cdot\delta{\bf{\hat{v}}}-\kappa\bigg{(}\frac{\hat{n}_{b,0}T_{0}}{h_{0}}\bigg{)}^{2}\nabla^{2}\delta(\hat{\beta}\hat{\mu}_{b})=0.$
(21)

A similar calculation for $\delta\hat{e}$ and $\delta{\bf{v}}$ respectively
gives,

$h_{0}\frac{\partial\delta{\bf{\hat{v}}}}{\partial
t}+\nabla\delta\hat{P}-\eta\nabla^{2}\delta\hat{\bf{v}}-(\zeta+\frac{1}{3}\eta)\nabla(\nabla\cdot\delta{\bf{\hat{v}}})=0.$
(22)

$\frac{\partial\delta{\hat{\epsilon}}}{\partial
t}+h_{0}\nabla\cdot\delta\hat{{\bf{v}}}=0$ (23)

Using following thermodynamic relations:

$\displaystyle\delta\epsilon$ $\displaystyle=$ $\displaystyle
T_{0}\delta(n_{b}s)+\mu_{b,0}\delta n_{b},$ (24) $\displaystyle\delta P$
$\displaystyle=$ $\displaystyle n_{b,0}s_{0}\delta T+n_{b,0}\delta\mu_{B},$
(25) $\displaystyle h_{0}$ $\displaystyle=$ $\displaystyle
T_{0}n_{b,0}s_{0}+n_{b,0}\mu_{b,0},$ (26)

and choosing $\delta\hat{n}_{b}$ and $\delta\hat{T}$ as independent variables
we can rewrite Eqs. (21)-(23) in the form:

$\displaystyle\bigg{[}\frac{\partial}{\partial
t}-\kappa\bigg{(}\frac{T_{0}C_{s}^{2}\tilde{C}_{n}}{h_{0}\tilde{C}_{P}}\bigg{)}\nabla^{2}\bigg{]}\delta\hat{n_{b}}+\kappa\bigg{[}1-\frac{\tilde{C}_{n}C_{s}^{2}\alpha_{P}T_{0}}{\tilde{C}_{P}}\bigg{]}\nabla^{2}\delta\hat{T}+n_{b,0}\nabla\cdot\delta\hat{\bf{v}}$
$\displaystyle=$ $\displaystyle 0,$ (27)
$\displaystyle\bigg{(}\frac{h_{0}\tilde{C_{n}}C_{s}^{2}}{n_{b,0}\tilde{C}_{P}}\bigg{)}\nabla\delta\hat{n}_{b}+\bigg{(}\frac{h_{0}\tilde{C_{n}}C_{s}^{2}\alpha_{P}}{\tilde{C}_{P}}\bigg{)}\nabla\delta\hat{T}+\bigg{(}h_{0}\frac{\partial}{\partial
t}-\eta\nabla^{2}\bigg{)}\delta\hat{{\bf{v}}}-\bigg{(}\zeta+\frac{1}{3}\eta\bigg{)}\nabla(\nabla\cdot\delta\hat{\bf{v}})$
$\displaystyle=$ $\displaystyle 0,$ (28)
$\displaystyle\bigg{[}\bigg{(}\frac{h_{0}C_{s}^{2}\alpha_{P}\tilde{C}_{n}}{n_{b,0}\tilde{C}_{P}}\bigg{)}\frac{\partial}{\partial
t}-\kappa\bigg{(}\frac{\tilde{C}_{n}C_{s}^{2}\tilde{C}_{n}}{n_{b,0}\tilde{C}_{P}}\bigg{)}\nabla^{2}\bigg{]}\delta\hat{n}_{b}-\bigg{[}\frac{n_{b,0}\tilde{C}_{n}}{T_{0}}\frac{\partial}{\partial
t}+\kappa\bigg{(}\frac{\alpha_{P}C_{s}^{2}\tilde{C}_{n}}{\tilde{C}_{P}}-\frac{1}{T_{0}}\bigg{)}\nabla^{2}\bigg{]}\delta\hat{T}$
$\displaystyle=$ $\displaystyle 0$ (29)

where we have used following relations:

$\displaystyle\delta P$ $\displaystyle=$ $\displaystyle\bigg{(}\frac{\partial
P}{\partial n_{b}}\bigg{)}_{T}\delta n_{b}+\bigg{(}\frac{\partial P}{\partial
T}\bigg{)}_{n_{b}}\delta T,$ (30) $\displaystyle\delta s$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{\partial s}{\partial n_{b}}\bigg{)}_{T}\delta
n_{b}+\bigg{(}\frac{\partial s}{\partial T}\bigg{)}_{n_{b}}\delta T,$

together with the thermodynamic identitiesMinami and Kunihiro (2010):

$\displaystyle\bigg{(}\frac{\partial P}{\partial n_{b}}\bigg{)}_{T}$
$\displaystyle=$
$\displaystyle\frac{h_{0}C_{s}^{2}\tilde{C}_{n}}{\tilde{C}_{P}n_{b,0}},$ (31)
$\displaystyle\bigg{(}\frac{\partial P}{\partial T}\bigg{)}_{n_{b}}$
$\displaystyle=$
$\displaystyle\frac{h_{0}\alpha_{P}C_{s}^{2}\tilde{C}_{n}}{\tilde{C}_{P}},$
(32) $\displaystyle\bigg{(}\frac{\partial s}{\partial n_{b}}\bigg{)}_{T}$
$\displaystyle=$
$\displaystyle-\frac{h_{0}\alpha_{P}C_{s}^{2}\tilde{C}_{n}}{n_{b,0}^{2}\tilde{C}_{P}},$
(33) $\displaystyle\bigg{(}\frac{\partial s}{\partial T}\bigg{)}_{n_{b}}$
$\displaystyle=$ $\displaystyle\frac{\tilde{C}_{n}}{T_{0}}.$ (34)

It is convenient to transform Eqs.(27)-(29) using Laplace-Fourier transform:

$\tilde{f}(z,{\bf{k}})=\int_{-\infty}^{+\infty}d{\bf{x}}\>e^{i{\bf{k\cdot
x}}}\int_{0}^{\infty}dt\>e^{-zt}\>f(t,{\bf{x}})$ (35)

Eqs.(27)-(29) can now be written in a matrix form:

$\displaystyle X\begin{bmatrix}\delta\hat{n}_{b}(z,{\bf{k}})\\\
\delta\hat{T}(z,{\bf{k}})\\\ \delta\hat{{\bf{v_{\parallel}}}}(z,{\bf{k}})\\\
\delta\hat{{\bf{v_{\perp}}}}(z,{\bf{k}})\\\
\end{bmatrix}=\begin{bmatrix}\delta\hat{n}_{b}(0,{\bf{k}})\\\
\frac{n_{b,0}\tilde{C}_{n}}{T_{0}h_{0}}\delta\hat{T}(0,{\bf{k}})-\frac{\alpha_{P}\tilde{C}_{n}C_{S}^{2}}{n_{b,0}\tilde{C}_{P}}\delta\hat{n}_{b}(0,{\bf{k}})\\\
\delta\hat{{\bf{v_{\parallel}}}}(0,{\bf{k}})\\\
\frac{h_{0}}{zh_{0}+k^{2}\eta}\delta\hat{{\bf{v_{\perp}}}}(0,{\bf{k}})\\\
\end{bmatrix}$ (36)

or

$X\hat{A}_{n}(z,{\bf{k}})=\hat{A}_{n}(t=0,{\bf{k}})$ (37)

where,

$\displaystyle
X=\begin{bmatrix}z+\kappa\frac{T_{0}C_{s}^{2}\tilde{C}_{n}}{h_{0}\tilde{C}_{P}}k^{2}&\bigg{[}-\kappa\frac{n_{b,0}}{h_{0}}(1-\frac{\alpha_{P}\tilde{C}_{n}C_{S}^{2}T_{0}}{\tilde{C}_{P}})k^{2}\bigg{]}&in_{b,0}k&0\\\
i\frac{\tilde{C_{n}}C_{s}^{2}}{n_{b,0}\tilde{C}_{P}}k&i\frac{\tilde{C_{n}}C_{s}^{2}\alpha_{P}}{\tilde{C}_{P}}k&z+\Omega_{l}k^{2}&0\\\
\bigg{[}-\bigg{(}\frac{h_{0}C_{s}^{2}\alpha_{P}\tilde{C}_{n}}{n_{b,0}\tilde{C}_{P}}\bigg{)}z-\kappa\bigg{(}\frac{\tilde{C}_{n}C_{s}^{2}\tilde{C}_{n}}{n_{b,0}\tilde{C}_{P}}\bigg{)}k^{2}\bigg{]}&\frac{n_{b,0}\tilde{C}_{n}}{h_{0}T_{0}}\bigg{[}z+\frac{h_{0}\tilde{C}_{P}}{n_{b,0}\tilde{C}_{n}}D_{T}\bigg{(}1-\frac{T_{0}\alpha_{P}C_{s}^{2}\tilde{C}_{n}}{\tilde{C}_{P}}\bigg{)}k^{2}\bigg{]}&0&0\\\
0&0&0&1\\\ \end{bmatrix}$ (38)

where we have decomposed the fluid velocity parallel
($\hat{\bf{v}}_{\parallel}$) and perpendicular ($\hat{\bf{v}}_{\perp}$) to the
wave vector ${\bf{k}}$. The quantities $D_{T}$ and $\Omega_{l}$ are given by,

$\displaystyle D_{T}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{h_{0}\tilde{C}_{P}},$ (39)
$\displaystyle\Omega_{l}$ $\displaystyle=$
$\displaystyle\frac{1}{h_{0}}\bigg{(}\zeta+\frac{4}{3}\eta\bigg{)}.$ (40)

$D_{T}$ and $\Omega_{l}$ are, respectively, called thermal diffusivity and
longitudinal viscosity. Note that the transverse velocity fluctuation is
decoupled from the density fluctuation. So we shall ignore this mode in
present analysis. We Replace 4-dimensional matrix equation (37) by
3-dimensional matrix equation as,

$\mathcal{X}\mathcal{\hat{A}}_{n}(z,{\bf{k}})=\mathcal{\hat{A}}_{n}(t=0,{\bf{k}})$
(41)

which can be obtained from (36) by deleting bottom row.

Dynamical density correlation function is defined as,

$\mathcal{C}_{n_{b}n_{b}}(\omega,{\bf{k}})\equiv<\delta\hat{n}_{b}(\omega,{\bf{k}})\delta\hat{n}_{b}(t=0,{\bf{k}})>,$
(42)

where $\delta\hat{n}_{b}(\omega,{\bf{k}})$ is the Fourier transform of
$\delta\hat{n}_{b}(t,{\bf{k}})$ whereas later can be obtained by taking
inverse Laplace’s transform of $\delta\hat{n}_{b}(z,{\bf{k}})$ as,

$\delta\hat{n}_{b}(t,{\bf{k}})=\frac{1}{2\pi
i}\int_{\delta-i\infty}^{\delta-i\infty}dz\>e^{zt}\delta\hat{n}_{b}(z,{\bf{k}}).$
(43)

We first calculate $\delta\hat{n}_{b}(z,{\bf{k}})$ by inverting matrix
equation (41) as,

$\mathcal{\hat{A}}_{n}(z,{\bf{k}})=\mathcal{X}^{-1}\mathcal{\hat{A}}_{n}(t=0,{\bf{k}}).$
(44)

The Fourier-Laplace space density function $\delta\hat{n}_{b}(z,{\bf{k}})$ is
the (1,1) component of $\hat{A}_{n}$.

$\delta\hat{n}_{b}(z,{\bf{k}})=(\mathcal{\hat{A}}_{n}(z,{\bf{k}}))_{11}=\sum_{j=1}^{4}\mathcal{X}^{-1}_{1j}\mathcal{\hat{A}}_{n}(t=0,{\bf{k}})_{j1}.$
(45)

The inverse of $X$ can be easily obtained one we know its determinant. At the
leading order the determinant of $X$ is,

$\text{det}X=\frac{n_{b,0}\tilde{C}_{n}}{h_{0}T_{0}}\bigg{\\{}z^{3}+z^{2}k^{2}\bigg{[}\bigg{(}\frac{h_{0}\tilde{C}_{P}}{n_{0}\tilde{C}_{n}}+\tilde{C}_{P}C_{s}^{2}T_{0}-2C_{s}^{2}\alpha_{P}T_{0}\bigg{)}D_{T}+\Omega_{l}\bigg{]}+zk^{2}C_{s}^{2}+\mathcal{O}(k^{4})\bigg{\\}}.$
(46)

The Fourier-Laplace coefficient of density correlation (45) at the order
$\mathcal{O}(k^{2})$ becomes,

$\frac{\delta\hat{n}_{b}(z,{\bf{k}})}{\delta\hat{n}_{b}(0,{\bf{k}})}\simeq\frac{(z+\Gamma_{S}k^{2}+iC_{s}k)(z+\Gamma_{S}k^{2}-iC_{s}k)+zk^{2}D_{T}\tilde{C}_{n}C_{s}^{2}-k^{2}C_{s}^{2}\frac{\tilde{C}_{n}}{\tilde{C}_{P}}}{(z+\Gamma_{S}k^{2}+iC_{s}k)(z+\Gamma_{S}k^{2}-iC_{s}k)(z+\frac{h_{0}}{n_{b,0}}D_{T}k^{2})},$
(47)

where,

$\displaystyle\Gamma_{T}$ $\displaystyle=$
$\displaystyle\frac{h_{0}}{n_{b,0}}D_{T},$ (48) $\displaystyle\Gamma_{S}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{\\{}\Omega_{l}+D_{T}\bigg{[}\bigg{(}\frac{\tilde{C}_{P}}{\tilde{C}_{n}}-1\bigg{)}+C_{s}^{2}T_{0}(\tilde{C}_{P}-2\alpha_{P})\bigg{]}\bigg{\\}}.$
(49)

Taking inverse Fourier-Laplace transform and then averaging over thermal
equilibrium we finally obtain the dynamical density correlation function as,

$\displaystyle\mathcal{C}_{n_{b}n_{b}}(\omega,{\bf{k}})$ $\displaystyle=$
$\displaystyle<(\delta\hat{n}_{b}(t=0,{\bf{k}}))^{2}>\bigg{[}\bigg{(}1-\frac{\tilde{C}_{n}}{\tilde{C}_{P}}\bigg{)}\frac{2\Gamma_{T}k^{2}}{\omega^{2}+\Gamma_{T}^{2}k^{4}}$
$\displaystyle+$
$\displaystyle\frac{\tilde{C}_{n}}{\tilde{C}_{P}}\bigg{(}\frac{\Gamma_{S}k^{2}}{(\omega-
C_{s}k)^{2}+\Gamma^{2}k^{4}}+\frac{\Gamma_{S}k^{2}}{(\omega+C_{s}k)^{2}+\Gamma^{2}_{S}k^{4}}\bigg{)}\bigg{]}$

One can derive similar form using linearised hydrodynamic equations in the
Landau frameMinami and Kunihiro (2010). The density correlation function has
three peaks as in the non-relativistic caseMazenko (2008). The peak around
$\omega=0$ is called Rayleigh peak which corresponds to thermally induced
density fluctuations. We note that the width of Rayleigh peak $\Gamma_{T}$
(Eq.(48)) is the same as that of non-relativistic case except for the pre-
factor $\frac{h_{0}}{n_{b,0}}$. The two peaks at $\omega=\pm C_{s}k$
corresponds to the acoustic waves and they are called Brillioun peaks. Note
that there are additional terms (see the discussion below) in the width of
Brillioun peaks $\Gamma_{S}$ (Eq.(49)) which are purely relativisticMinami and
Kunihiro (2010). These terms are proportional to the thermal diffusivity.

To bring out the genuine relativistic effects in the density correlations
consider a gas of massless particles. In the massless limit $\zeta=0$ . Thus
only shear viscosity and thermal conductivity terms contribute to the sound
attenuation as,

$\Gamma_{S}=\frac{1}{2}\bigg{[}\frac{h_{0}}{n_{0}}D_{T}\bigg{(}\frac{\tilde{C}_{P}}{\tilde{C}_{n}}-1\bigg{)}+\frac{4}{3}\frac{\eta}{h_{0}}+C_{s}^{2}T_{0}\bigg{(}\frac{\kappa}{h_{0}}-\frac{2\alpha_{P}h_{0}}{n_{0}}D_{T}\bigg{)}\bigg{]}=\Gamma_{S,R}+\delta\Gamma_{\text{S,R}}$
(51)

where,

$\displaystyle\Gamma_{S,R}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}\frac{h_{0}}{n_{0}}D_{T}\bigg{(}\frac{\tilde{C}_{P}}{\tilde{C}_{n}}-1\bigg{)}+\frac{4}{3}\frac{\eta}{h_{0}}\bigg{]},$
(52) $\displaystyle\delta\Gamma_{\text{S,R}}$ $\displaystyle=$ $\displaystyle
C_{s}^{2}T_{0}\bigg{(}\frac{\kappa}{h_{0}}-\frac{2\alpha_{P}h_{0}}{n_{0}}D_{T}\bigg{)}$
(53)

$\Gamma_{S,R}$ is the relativistic counterpart of sound attenuation.
$\delta\Gamma_{\text{S,R}}$ is a purely relativistic correction. In the
extreme relativistic limit, i.e when $T_{0}$ is very large or when the kinetic
energy dominates the rest-mass energy, we can approximate the gas by its ideal
limit. In this limit
$\delta\Gamma_{\text{S,R}}\simeq-\frac{\kappa}{24n_{b,0}}$. On the other hand,
$\delta\Gamma_{\text{S,R}}\rightarrow 0$ in the non-relativistic limit
($T_{0}\rightarrow 0$ or when the rest-mass energy dominates the kinetic
energy). Thus the relativistic correction to the sound attenuation is always
negative. This implies that the Brillouin peaks are enhanced with the width
reduced by an amount $\delta\Gamma_{\text{R}}$ as compared to its non-
relativistic counterpart. On the other hand the thermal diffusivity is
practically remain unaltered for the relativistic systems and hence the the
strength and the width of the Rayleigh peaks are not affected by the
relativistic effects.

##  IV Density correlations near the critical point

In the $T\rightarrow T_{c}$ limit, any relevant thermodynamic quantity can be
separated into a regular part and a singular part. The singular part show
power law behaviour. This behaviour is universal and characterized by the
critical exponents, $\hat{\alpha}$, $\hat{\beta}$, $\hat{\gamma}$ and
$\hat{\nu}$. They are defined through the following power laws Huang (1987)
(in the limit $t\rightarrow 0^{-}$):

$\displaystyle C_{V}$ $\displaystyle=$
$\displaystyle\mathcal{C_{-}}\>|t|^{-\hat{\alpha}},$ (54)
$\displaystyle\tilde{n}_{b}$ $\displaystyle=$
$\displaystyle\mathcal{N_{-}}\>|t|^{\hat{\beta}},$ (55) $\displaystyle k_{T}$
$\displaystyle=$ $\displaystyle\mathcal{K}_{-}\>|t|^{-\hat{\gamma}},$ (56)
$\displaystyle\xi$ $\displaystyle=$ $\displaystyle\Xi_{-}\>|t|^{-\hat{\nu}},$
(57)

where, $\tilde{n}_{b}=1-\frac{n_{b}}{n_{b,c}}$ is the order parameter with
$n_{b,c}$ being critical density. $C_{V}$, $k_{T}$ and $\xi$ respectively
denote the specific heat, the isothermal compressibility and the correlation
length. $\mathcal{C_{-}}$, $\mathcal{N_{-}}$, $\mathcal{K}_{-}$ and $\Xi_{-}$
are the corresponding amplitudes from the hadronic side ($T<T_{c}$).

The thermodynamic properties of statistical bootstrap model (SBM) can be
derived using following partition functionKadam  _et al._ (2021):

$\text{ln}\mathcal{Z}_{H}(T,V,z_{b})=AVz_{b}\left(\frac{T}{2\pi}\right)^{3/2}\int_{M}^{\infty}dm\>m^{a+3/2}e^{\big{(}\frac{1}{T_{c}}-\frac{1}{T}\big{)}m}.$
(58)

where, $z_{b}=e^{\mu_{b}/T}$ is the fugacity. It can be shown that for
$a<-\frac{7}{2}$ SBM shows critical behaviourSatz (1979). It turns out that
the energy density and entropy remains finite as $T\rightarrow T_{c}$ whereas
all the higher order derivatives show singular behaviour. In fact, it is
straightforward to extract the power law behaviour given by Eqns (54)-(56).

Approaching the critical point the functional dependence of transport
coefficients on various thermodynamic quantities can be obtained by
dimensional analysis. We assume following ansatz (in the natural units) for
the shear viscosity ($\eta$), bulk viscosity ($\zeta$)Antoniou  _et al._
(2017) and thermal conductivity ($\kappa$):

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{T}{\xi^{2}C_{s}}\>\mathcal{F}^{(\eta)}\bigg{(}\frac{\tilde{C}_{p}}{\tilde{C}_{n}}\bigg{)},$
(59) $\displaystyle\zeta$ $\displaystyle=$ $\displaystyle h\xi
C_{s}\>\mathcal{F}^{(\zeta)}\bigg{(}\frac{\tilde{C}_{p}}{\tilde{C}_{n}}\bigg{)},$
(60) $\displaystyle\kappa$ $\displaystyle=$ $\displaystyle\frac{h\xi
T^{2}C_{s}}{\tilde{n}_{B}}\>\mathcal{F}^{(\kappa)}\bigg{(}\frac{\tilde{C}_{p}}{\tilde{C}_{n}}\bigg{)}$
(61)

Substituting power law behaviour of the thermodynamic quantities given by
Eqns.(54)-(57) in Eq.(59) one can obtain the singular behaviour of the
transport coefficients. The leading term is

$\displaystyle\eta$ $\displaystyle\sim$
$\displaystyle|t|^{-\hat{\gamma}+2\hat{\nu}+\hat{\alpha}/2},$ (62)
$\displaystyle\zeta$ $\displaystyle\sim$
$\displaystyle|t|^{-\hat{\gamma}-\hat{\nu}+3\hat{\alpha}/2},$ (63)
$\displaystyle\kappa$ $\displaystyle\sim$
$\displaystyle|t|^{\hat{-\beta}-\hat{\gamma}-\hat{\nu}+3\hat{\alpha}/2}$ (64)

Note that not all the critical exponents are independent but are related via.
scaling laws:

$\displaystyle 2-\hat{\alpha}$ $\displaystyle=$ $\displaystyle\hat{\nu}d$ (65)
$\displaystyle\hat{\alpha}+2\hat{\beta}+\hat{\gamma}$ $\displaystyle=$
$\displaystyle 2$ (66)

where $d$ is the number of space dimensions. For the choice $a=-4$Kadam  _et
al._ (2021); Antoniou and Kapoyannis (2003) we get the critical exponents of
SBM as:

$\hat{\alpha}=\frac{1}{2},\hskip 5.69046pt\hat{\beta}=0,\hskip
5.69046pt\hat{\gamma}=\frac{3}{2},\hskip 5.69046pt\hat{\nu}=\frac{1}{2}$ (67)

Substituting these critical exponents in (62) we get

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\eta_{-}\>|t|^{-\frac{1}{4}},$ (68) $\displaystyle\zeta$
$\displaystyle=$ $\displaystyle\zeta_{-}\>|t|^{-\frac{5}{4}},$
$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\kappa_{-}\>|t|^{-\frac{5}{4}},$

where $\eta_{-},\zeta_{-},\kappa_{-}$ are constants. Thus the bulk viscosity
and thermal conductivity dominates the dissipation near the critical point.
The thermal diffusivity behaves near the critical point as

$D_{T}\sim\frac{\kappa_{-}}{\mathcal{K}_{-}T_{0}}|t|^{\hat{\beta}-\hat{\nu}+3\hat{\alpha}/2}$
(69)

Thus,

$D_{T}\sim|t|^{\frac{1}{4}}$ (70)

where we have substituted the critical exponents (67) of SBM mode. This shows
that the width of Rayleigh peak narrows as the QCD critical point is
approached from hadronic side. Similarly the the leading order behaviour of
the sound attenuation coefficient is:

$\Gamma\sim
K_{1}\kappa_{-}|t|^{-\hat{\beta}-\hat{\gamma}-\hat{\nu}+\frac{5}{2}\hat{\alpha}}+K_{2}\zeta_{-}|t|^{-\hat{\gamma}-\hat{\nu}+\frac{3}{2}\hat{\alpha}}+K_{3}\eta_{-}|t|^{-\hat{\gamma}+2\hat{\nu}+\frac{\hat{\alpha}}{2}}$
(71)

where $K_{1},K_{2},K_{3}$ contains quantities which are not singular.
Substituting the critical exponents of SBM we see that the leading order
behaviour of the singularity near the critical point is dominated by the bulk
viscosity while the thermal conductivity contributes at subleading order.

$\Gamma\sim\zeta_{-}|t|^{-\frac{5}{4}}+\kappa_{-}|t|^{-\frac{3}{4}}$ (72)

Thus the width of Brillouin peak must diverge near at the critical point.
However, the strengths of the Rayleigh and Brillouin peaks is governed by the
ratio of two specific heats $\gamma=\frac{\tilde{C}_{P}}{\tilde{C}_{n}}$ which
behave in the limit $T\rightarrow T_{c}$ as

$\gamma\sim|t|^{-\hat{\gamma}+\hat{\alpha}}$ (73)

In the limit $T\rightarrow T_{c}$, $\gamma\rightarrow\infty$ and the density
correlation behaves as

$\frac{S_{n_{b}n_{b}}(\bf{k},\omega)}{<(\delta
n({\bf{k}},t=0))^{2}>}=\frac{2\Gamma_{T}k^{2}}{\omega^{2}+\Gamma_{T}^{2}k^{4}}$
(74)

Thus the strength of the Brillouin peaks is attenuated and only the Rayleigh
peak governs the density correlations near the critical point.

Let us discuss the implications of these results in the search for CP in
heavy-ion collision experiments. Near the critical point the width and the
strength of these modes depends on the singularities in static thermodynamic
quantities and the transport coefficients. The critical exponents calculated
within ambit of SBM are given by Eq.(67), while the dynamical critical
exponents can be read out from (68). We note that both the bulk viscosity
($\zeta$) and thermal conductivity ($\kappa$) rise very rapidly near the
critical point. Such rapid rise in the bulk viscosity has been estimated in
previous studiesKarsch  _et al._ (2008); Antoniou  _et al._ (2017); Kadam  _et
al._ (2021). This rapid rise in both $\zeta$ and $\kappa$ render divergence in
the width of the Brillouin peaks. However, as we have seen above, the
strengths of the Rayleigh and Brillouin peaks is governed by the ratio of two
specific heats $\gamma=\frac{\tilde{C}_{P}}{\tilde{C}_{n}}$ which diverge as
$T\rightarrow T_{c}$. Thus, the acoustic mode loose its strength near the
critical point and the divergent behaviour of the bulk viscosity may not be
observed in the density correlations. Nonetheless, rapid rise of bulk
viscosity near CEP should manifest itself through the decrease in average
transverse momentum of produced final state particles. Further, it should be
also increase the total multiplicity due to large entropy associated with a
large bulk viscosity.

##  V Summary and conclusion

To summarise, we have calculated the density correlations in a baryon rich
relativistic fluid from Mori-Zwanzig projection operator formalism. The
hydrodynamic form of density correlation can be obtained, with a judicious
choice of slow variables, using this formalism. The spectral function of
density correlation is found to consist of usual three peaks: two acoustic
modes at $\omega=\pm C_{s}k$ and one thermal mode at $\omega=0$. The width of
the Rayleigh peak is found to be the same as that in the non-relativistic
case, whereas the width of Brillouin peak consist of extra relativistic
correction which is negative and hence reduces its width.

We further discussed the behaviour of Rayleigh and Brillouin peaks near the
critical point. We extracted the singular behaviour of thermodynamic
quantities using statistical bootstrap model. The singular behaviour of
transport coefficient, shear and bulk viscosity as well as thermal
conductivity can be extracted using the ansatz for this quantities given by
Eq.(59). We found that the bulk viscosity contributes to the sound attenuation
at leading order $\sim|t|^{-\frac{5}{4}}$ while the thermal conductivity
contribute at sub-leading order $\sim|t|^{-\frac{3}{4}}$. On the other hand,
only the thermal conductivity contributes at leading order
$\sim|t|^{\frac{1}{4}}$ to the thermal diffusivity. We noted that the acoustic
mode loose its strength near the critical point and the divergent behaviour of
the bulk viscosity may not be observed in the density correlations. However,
it can be inferred from the average transverse momenta and multiplicities of
produced particles in heavy ion collisions.

In conclusion, density correlations may not be suitable observable to search
for the QCD critical point. However, as noted in Ref.Minami and Kunihiro
(2010) suppression or disappearance of Mach cone, which is related to the
existence of sound mode in a fluid, may indicate that system have passed close
to the critical point.

##  Appendix A Calculation of the memory matrices

We shall briefly discuss the calculation of the memory matrices. The details
can be found in Refs.Mazenko (2008); Minami and Hidaka (2013).

###  A.1 Static memory matrix

The static memory matrix can be written as,

$\displaystyle
i{\bf{K}}^{(s)}\equiv\begin{bmatrix}K^{(s),n}_{n}&K^{(s),e}_{n}&K^{(s),{\bf{p}}}_{n}\\\
K^{(s),n}_{e}&K^{(s),e}_{e}&K^{(s),{\bf{p}}}_{e}\\\
K^{(s),n}_{\bf{p}}&K^{(s),e}_{\bf{p}}&K^{(s),{\bf{p}}}_{\bf{p}}\\\
\end{bmatrix}$ (75)

where ${\bf{p}}=h_{0}{\bf{v}}$. From Eq.(14) we can write,

$\displaystyle iK_{{n}_{b}}^{(s){n}_{b}}({\bf{k}})$ $\displaystyle=$
$\displaystyle(i\mathcal{L}\delta\hat{n}_{b}({\bf{k}}),\mathcal{\hat{A}}^{{n}_{b}}(0))$
$\displaystyle=$
$\displaystyle(i\mathcal{L}\delta\hat{n}_{b}({\bf{k}}),\mathcal{\hat{A}}_{l}(0))g^{l{n}_{b}}({\bf{k}})$
$\displaystyle=$
$\displaystyle(i\mathcal{L}\delta\hat{n}_{b}({\bf{k}}),\delta\hat{n}_{b}(0))g^{{n}_{b}{n}_{b}}({\bf{k}})+(i\mathcal{L}\delta\hat{n}_{b}({\bf{k}}),\delta\hat{e}(0))g^{en_{b}}({\bf{k}})$
$\displaystyle+$
$\displaystyle(i\mathcal{L}\delta\hat{n}_{b}({\bf{k}}),\delta\hat{{\bf{p}}}(0))g^{{\bf{p}}n_{b}}({\bf{k}})$
$\displaystyle=$ $\displaystyle-
ik^{l}\big{[}g_{{j}^{l}_{b}{n}_{b}}({\bf{k}})g^{{n}_{b}{n}_{b}}({\bf{k}})+g_{j^{l}_{b}e}({\bf{k}})g^{en_{b}}({\bf{k}})+g_{j^{l}_{b}p^{j}}({\bf{k}})g^{p^{j}n_{b}}({\bf{k}})\big{]}$

where $j_{b}$ is the baryon current. If we assume that the equilibrium density
matrix $\hat{\rho}_{0}$ is time-translational invariant then
$\delta\hat{n}_{b}$ and $\delta\hat{e}$ are even functions of time while
$\delta\hat{p}^{i}$ and $\delta\hat{j}_{b}^{i}$ are odd. Hence the time
reversal symmetry implies that $\delta\hat{n}_{b}$ and $\delta\hat{T}$ do not
mix with $\delta\hat{p}^{i}$ and $\delta\hat{j}_{b}^{i}$. Thus Eq.(A.1)
implies that $iK_{{n}_{b}}^{(s){n}_{b}}=0$. This further implies that
$iK_{{n}_{b}}^{(s)e}$ also vanishes and only
$iK_{{n}_{b}}^{(s){\bf{p}}}({\bf{k}})$ survives.

$\displaystyle iK_{{n}_{b}}^{(s)p^{i}}({\bf{k}})$ $\displaystyle=$
$\displaystyle(i\mathcal{L}\delta\hat{n}_{b}({\bf{k}}),\delta\hat{v}_{l}(0))g^{p^{l}p^{i}}({\bf{k}})$
$\displaystyle=$ $\displaystyle-
ik^{j}(\delta\hat{j}^{j}_{b}({\bf{k}}),\delta\hat{v}^{l}(0))g^{p^{l}p^{i}}({\bf{k}})$
$\displaystyle=$ $\displaystyle-
ik^{j}g_{j_{b}^{j}p^{l}}({\bf{k}})g^{p^{l}p^{i}}({\bf{k}})$

In the low energy limit ${\bf{k}}\rightarrow 0$ we can expand
$g_{mn}({\bf{k}})$ around ${\bf{k}}=0$ as a power series as,

$g_{mn}({\bf{k}})=g_{mn}({\bf{0}})+{\bf{k}}\cdot\nabla_{{\bf{k}}}g_{mn}({\bf{k}})+\mathcal{O}(k^{2})$
(78)

At linear order only the leading term $g_{mn}({\bf{0}})$ contributes in (78).
Further, $g_{mn}({\bf{0}})$ satisfy following relations:

$g_{p^{i}p^{j}}({\bf{0}})=\int
d^{3}x(\hat{T}^{0i}(0,{\bf{x}}),\hat{T}^{0j}(0,{\bf{0}}))=\delta^{ij}T_{0}h_{0}$
(79)

$g_{p^{i}j_{b}^{j}}({\bf{0}})=\int
d^{3}x(\hat{T}^{0i}(0,{\bf{x}}),\hat{j}_{b}^{j}(0,{\bf{0}}))=\delta^{ij}T_{0}n_{0}$
(80)

Using Eqs.(79) and (80), Eq(A.1) reduces to

$iK_{{n}_{b}}^{(s)p^{i}}({\bf{k}})\simeq-ik^{i}T_{0}n_{0}$ (81)

We can similarly calculate $iK_{e}^{(s)m}$ and $iK_{v^{i}}^{(s)m}$. The static
memory matrix is finally written as

$\displaystyle i{\bf{K}}^{(s)}\equiv\begin{bmatrix}0&0&-ik^{i}n_{0}T_{0}\\\
0&0&-ik^{i}\\\ -ik^{i}n_{0}T_{0}&-ik^{i}n_{0}T_{0}&0\\\ \end{bmatrix}$ (82)

###  A.2 Dynamic memory matrix

The static memory matrix can be written as,

$\displaystyle
i{\bf{K}}^{(d)}\equiv\begin{bmatrix}K^{(d),n}_{n}&K^{(d),e}_{n}&K^{(d),{\bf{p}}}_{n}\\\
K^{(d),n}_{e}&K^{(d),e}_{e}&K^{(d),{\bf{p}}}_{e}\\\
K^{(d),n}_{\bf{p}}&K^{(d),e}_{\bf{p}}&K^{(d),{\bf{p}}}_{\bf{p}}\\\
\end{bmatrix}$ (83)

From Eq. (15) we write

$K_{n_{b}}^{(d)m}(t-s,{\bf{k}})\equiv-\theta(t-s)\big{(}i\mathcal{\hat{L}}e^{it\mathcal{\hat{Q}}\mathcal{\hat{L}}}\>\mathcal{\hat{Q}}\>i\mathcal{\hat{L}}\delta\hat{n}_{b}(0,{\bf{k}}),\mathcal{\hat{A}}^{m}(s,{\bf{0}})\big{)}$
(84)

In this case, $i\hat{\mathcal{L}}\delta\hat{e}=-{\bf{k}}\cdot\delta{\bf{p}}$,
and since $\delta{\bf{p}}$ is a slow variable, its orthogonal projection
should vanish. Thus $\hat{\mathcal{Q}}\>i\hat{\mathcal{L}}\delta\hat{e}=0$ and
hence $K_{n_{b}}^{(d)e}$ vanishes. Further,
$K_{n_{b}}^{(d){\bf{p}}}\simeq\mathcal{O}(k^{3})$, which we neglect. Thus,
only only $K_{n_{b}}^{(d)n_{b}}$ survives. It can be easily calculated in the
Markov approximation. The final expression is written as:

$K_{n_{b}}^{(d)n_{b}}({\bf{k}})\simeq{\bf{k^{2}}}\bigg{(}\frac{n_{0}T_{0}}{h_{0}}\bigg{)}^{2}\kappa$
(85)

where we have define the thermal conductivity coefficient as,

$\kappa=\bigg{(}\frac{h_{0}}{n_{0}T_{0}}\bigg{)}^{2}\int_{0}^{\infty}dt\int
d^{3}z\bigg{(}\delta{\bf{j}_{q}}(t,{\bf{x}}),\delta{\bf{j}_{q}}(0,{\bf{0}})\bigg{)}$
(86)

with the heat current defined as
$\delta{\bf{j}_{q}}\equiv\delta{\bf{j}_{b}}-\frac{n_{0}}{h_{0}}\delta{\bf{p}}$.
One can similarly carry out the calculation of other elements in the
$i{\bf{K}}^{(d)}$. The final result is:

$\displaystyle
i{\bf{K}}^{(d)}\equiv\begin{bmatrix}{\bf{k^{2}}}\tilde{\kappa}&0&\mathcal{O}(k^{3})\\\
K^{(d),n}_{e}&K^{(d),e}_{e}&K^{(d),{\bf{p}}}_{e}\\\
\mathcal{O}(k^{3})&0&T_{0}\tilde{\nu}k^{i}k^{j}+T_{0}\eta{\bf{k}}^{2}\delta^{ij}\\\
\end{bmatrix}$ (87)

where $\tilde{\kappa}=\bigg{(}\frac{n_{0}T_{0}}{h_{0}}\bigg{)}^{2}\kappa$ and
$\tilde{\nu}=\bigg{(}\zeta+\frac{1}{3}\eta\bigg{)}$.

## Acknowledgments

Guruprasad Kadam is financially supported by the DST-INSPIRE faculty award
under Grant No. DST/INSPIRE/04/2017/002293. G.K. thanks Amaresh Jaiswal for
useful discussion.

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