$\displaystyle\omega(\epsilon+P)u_{1}^{i}-k^{i}P_{1}+\frac{1}{3}\zeta
k^{i}\Big{[}i(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}\cdot\mathchoice{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$\scriptstyle
u_{1}$}}{\mbox{\boldmath$\scriptscriptstyle
u_{1}$}})+\frac{1}{3}\zeta\beta_{0}\omega(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}\cdot\mathchoice{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$\scriptstyle
u_{1}$}}{\mbox{\boldmath$\scriptscriptstyle u_{1}$}})\Big{]}$
$\displaystyle+i\eta\Big{[}k^{2}u^{i}_{1}+\frac{1}{3}k^{i}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}\cdot\mathchoice{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$\scriptstyle
u_{1}$}}{\mbox{\boldmath$\scriptscriptstyle
u_{1}$}})\Big{]}-2\eta^{2}\beta_{2}\omega\Big{[}k^{2}u^{i}_{1}+\frac{1}{3}k^{i}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}\cdot\mathchoice{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$\scriptstyle
u_{1}$}}{\mbox{\boldmath$\scriptscriptstyle u_{1}$}})\Big{]}$
$\displaystyle+\frac{nT\kappa}{(\epsilon+P)}(\mu_{1}-\alpha
T_{1})\Big{[}\frac{\alpha_{0}\zeta k^{2}}{3}k^{i}+\frac{4}{3}\alpha_{1}\eta
k^{2}k^{i}\Big{]}\,.$

Using the linearization technique, the number conservation equation in the
$\omega-k$ space becomes,

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\omega
n_{1}-n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}\cdot\mathchoice{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$\scriptstyle
u_{1}$}}{\mbox{\boldmath$\scriptscriptstyle u_{1}$}})-i\frac{n^{2}\kappa
Tk^{2}}{(\epsilon+P)^{2}}\Big{[}(\mu_{1}-\alpha
T_{1})(1+i\omega\kappa\beta_{1})\Big{]}$ (5.10)
$\displaystyle+\frac{1}{3}\frac{n\kappa
Tk^{2}}{(\epsilon+P)}\Big{[}\zeta\alpha_{0}+4\eta\alpha_{1}\Big{]}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}\cdot\mathchoice{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$\scriptstyle
u_{1}$}}{\mbox{\boldmath$\scriptscriptstyle u_{1}$}})\,.$

We have considered terms up to the first order in perturbation as

$\displaystyle
n\,u^{\mu}\to~{}(n+n_{1})(u^{\mu}+u_{1}^{\mu})\approx~{}nu^{\mu}+n_{1}u^{\mu}+nu_{1}^{\mu}\,.$
(5.11)

In deriving the Eqs.(5.8), (5.9), (5.10), we have used

$\displaystyle\partial_{\mu}\alpha$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Big{[}\frac{\mu+\mu_{1}}{T+T_{1}}\Big{]}~{}\approx\partial_{\mu}\Big{[}(\frac{\mu+\mu_{1}}{T})(1-\frac{T_{1}}{T})\Big{]}$
(5.12) $\displaystyle=$
$\displaystyle\partial_{\mu}\Big{[}\frac{\mu+\mu_{1}}{T}-\frac{\mu}{T^{2}}T_{1}\Big{]}=~{}\frac{1}{T}\big{[}\partial_{\mu}\mu_{1}-\alpha\partial_{\mu}T_{1}\big{]}~{},$

and,

$\displaystyle\nabla_{\mu}\alpha=\frac{1}{T}\big{[}\nabla_{\mu}\mu_{1}-\alpha\nabla_{\mu}T_{1}\big{]}\,.$
(5.13)

In Eqs. (5.8), (5.9) and (5.10), we have considered terms up to first order in
perturbations and neglected the higher order terms. Also, we have not
perturbed the different transport coefficients, as they are not hydrodynamical
variables. In LRF, we take them as constant in space and time, hence their
comoving derivative are zero. As mentioned, we have considered, $\eta\neq 0$,
$\kappa\neq 0$ and $\zeta$=0. We decompose the fluid velocity into directions
perpendicular and parallel to the direction of wave vector, $k$ as:

$\mathchoice{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$\scriptstyle
u_{1}$}}{\mbox{\boldmath$\scriptscriptstyle
u_{1}$}}=\mathchoice{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$\scriptstyle
u_{1}$}}{\mbox{\boldmath$\scriptscriptstyle
u_{1}$}}_{\bot}+\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}\cdot\mathchoice{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$\scriptstyle
u_{1}$}}{\mbox{\boldmath$\scriptscriptstyle u_{1}$}})/k^{2}\,.$ (5.14)

The modes propagating along the direction of $k$ are called longitudinal and
those perpendicular to $k$ are called transverse modes.

The quantities, $\epsilon_{1}$, $P_{1}$ and $\mu_{1}$ defined above can be
expressed in terms of the derivatives of the thermodynamic quantities (which
are connected to response functions like specific heat, etc.) as follows:

$\displaystyle\epsilon_{1}=\Big{(}\frac{\partial\epsilon}{\partial
T}\Big{)}_{n}T_{1}+\Big{(}\frac{\partial\epsilon}{\partial
n}\Big{)}_{T}n_{1}\,,$ $\displaystyle P_{1}=\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n}T_{1}+\Big{(}\frac{\partial P}{\partial n}\Big{)}_{T}n_{1}\,,$
$\displaystyle\mu_{1}=\Big{[}-\Big{(}\frac{\partial n}{\partial
T}\Big{)}_{n}T_{1}+n_{1}\Big{]}\Big{(}\frac{\partial\mu}{\partial
n}\Big{)}_{T}\,.$ (5.15)

###  5.4 Dispersion Relations

The dispersion relation ($\omega$ (frequency) as a function of $k$ (wave
vector)) plays an important role in determining the fate of the perturbation
in a fluid. The $\omega$ can be a real as well as an imaginary quantity. The
real part of the frequency, $\omega_{{\mathcal{R}}e}$, dictates the
possibility of the propagation of the wave in the medium. Whereas, the
imaginary part of the frequency, $\omega_{Im}$, decides the decay of the waves
along its propagation in the medium.

To evaluate the dispersion relation, we would like to write down a equation
that dictates the $k$ dependence of $\omega$. For this purpose we use Eqs.
(5.8), (5.9) and (5.10), and collected the coefficients
$(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}\cdot\mathchoice{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$u_{1}$}}{\mbox{\boldmath$\scriptstyle
u_{1}$}}{\mbox{\boldmath$\scriptscriptstyle u_{1}$}})$, $T_{1}$ and $n_{1}$ to
put row wise to form a determinant as

$\displaystyle\begin{vmatrix}&&a_{11}&&a_{12}&&a_{13}~{}~{}~{}~{}\\\
&&a_{21}&&a_{22}&&a_{23}~{}~{}~{}~{}\\\
&&a_{31}&&a_{32}&&a_{33}~{}~{}~{}~{}\end{vmatrix}$ (5.16)

and upon the expansion of the determinant above, yield

$\displaystyle a\omega^{3}+b\omega^{2}+c\omega=0\hskip
14.22636pt\rightarrow\hskip 14.22636pt\omega(a\omega^{2}+b\omega+c)$
$\displaystyle=$ $\displaystyle 0\,.$ (5.17)

The coefficients $a,b$ and $c$ are determined by solving Eqs. (5.8), (5.9) and
(5.10), simultaneously. The solutions of this equation which provide a
relation between $\omega$ and $k$ which is called the dispersion relation. The
equation, (5.17) has three roots, one real which is $\omega=0$ and two complex
roots with real ($\omega_{{\mathcal{R}}e}$) and imaginary ($\omega_{im}$)
parts given below by Eqs. (5.18) and (5.21) respectively. The real part of
$\omega$ can be expressed as:

$\omega_{{\mathcal{R}}e}=\sqrt{\frac{a_{0}k^{2}-a_{1}k^{3}+a_{2}k^{4}}{b_{0}-b_{1}k^{2}}}\,.$
(5.18)

where,

$\displaystyle a_{0}=9h\Big{[}\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n}+\alpha_{1}n\Big{\\{}\Big{(}\frac{\partial\epsilon}{\partial
n}\Big{)}_{T}\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n}-\Big{(}\frac{\partial\epsilon}{\partial
T}\Big{)}_{n}\Big{(}\frac{\partial P}{\partial
n}\Big{)}_{T}\Big{\\}}\Big{]}\,,$ $\displaystyle
a_{1}=\frac{9\alpha\beta_{1}n^{2}T^{2}\kappa^{2}}{h}+12\alpha_{1}\eta\kappa
nT\Big{[}\alpha+\frac{\alpha n}{h}\Big{(}\frac{\partial\epsilon}{\partial
n}\Big{)}_{T}-\frac{T}{h}\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n}+\frac{T}{h}\Big{(}\frac{\partial\epsilon}{\partial
T}\Big{)}_{n}\Big{]}\,,$ $\displaystyle
a_{2}=\frac{9\beta_{1}\kappa^{2}n^{2}T}{h}\Big{[}\Big{(}\frac{\partial
n}{\partial T}\Big{)}_{\mu}\Big{(}\frac{\partial\mu}{\partial
n}\Big{)}_{T}+\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n}\Big{(}\frac{\partial P}{\partial n}\Big{)}_{T}\Big{]}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{12\alpha_{1}\eta\kappa
nT}{h}\Big{[}\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n}+n\Big{(}\frac{\partial n}{\partial
T}\Big{)}_{\mu}\Big{(}\frac{\partial\mu}{\partial
n}\Big{)}_{T}+\frac{n}{h}\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n}\Big{(}\frac{\partial\epsilon}{\partial n}\Big{)}_{T}\Big{]}\,,$
$\displaystyle b_{0}=9h\Big{(}\frac{\partial\epsilon}{\partial
T}\Big{)}_{n}\,,$ $\displaystyle
b_{1}=24\beta_{2}\eta^{2}\Big{(}\frac{\partial\epsilon}{\partial
T}\Big{)}_{n}+\frac{9\beta_{1}\kappa^{2}n^{2}}{h}\Big{[}T\Big{(}\frac{\partial\mu}{\partial
n}\Big{)}_{T}\Big{(}\frac{\partial n}{\partial T}\Big{)}_{\mu}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,-T^{2}\Big{(}\frac{\partial\epsilon}{\partial
n}\Big{)}_{T}\Big{(}\frac{\partial\mu}{\partial
n}\Big{)}_{n}+\alpha\Big{(}\frac{\partial\epsilon}{\partial
n}\Big{)}_{T}\Big{]}\,.$ (5.19)

and, $h=\epsilon+P$ is the enthalpy density. We have kept terms up to
quadratic power of transport coefficients in Eq. (5.18). We have also
neglected the higher order terms in
$\alpha_{0},\alpha_{1},\beta_{0},\beta_{1},\beta_{2}$. Expanding
$\omega_{{\mathcal{R}}e}$ in powers of $k$ and keeping terms up to
$\cal{O}$$(k^{4})$ we obtain,

$\omega_{{\mathcal{R}}e}=\sqrt{\frac{a_{0}}{b_{0}}}\left[k-\frac{1}{2}\frac{a_{1}}{a_{0}}k^{2}+(\frac{1}{2}\frac{a_{2}}{a_{0}}-\frac{1}{8}{a_{1}}{a_{0}^{2}}+\frac{b_{1}}{b_{0}})k^{3}+(\frac{1}{4}\frac{a_{1}a_{2}}{a_{0}^{2}}+\frac{1}{16}{a_{1}^{2}}{a_{0}^{3}}-\frac{1}{2}\frac{a_{1}b_{1}}{a_{0}b_{0}})k^{4}\right]\,.$
(5.20)

Similarly, the expression for the imaginary part of $\omega$ reads as:

$\omega_{Im}=\frac{-c_{0}k^{2}+c_{1}k^{3}+c_{2}k^{4}}{d_{0}+d_{1}k^{2}}\,.$
(5.21)

where,

$\displaystyle c_{0}=2\eta h^{2}\Big{(}\frac{\partial\epsilon}{\partial
T}\Big{)}_{n}-3hn^{2}\kappa
T\beta_{1}\Big{[}\alpha\kappa\Big{(}\frac{\partial\epsilon}{\partial
n}\Big{)}_{T}+h\Big{(}\frac{\partial n}{\partial
T}\Big{)}_{\mu}\Big{(}\frac{\partial\mu}{\partial n}\Big{)}_{T}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,+\frac{\alpha_{1}}{\beta_{1}}\Big{(}\frac{\partial
P}{\partial T}\Big{)}_{n}\Big{(}\frac{\partial\epsilon}{\partial
n}\Big{)}_{T}\Big{]}\,,$ $\displaystyle c_{1}=2\alpha\beta_{1}\eta
n^{2}T\Big{(}T\kappa^{2}+4h\eta\frac{\beta-2}{\beta_{1}}\Big{)}\,,$
$\displaystyle c_{2}=8\beta_{2}\eta\kappa
n^{2}T\Big{[}\Big{(}\frac{\partial\epsilon}{\partial
T}\Big{)}_{n}-\Big{(}\frac{\partial P}{\partial
n}\Big{)}_{T}^{2}\Big{(}\frac{\partial\epsilon}{\partial
n}\Big{)}_{T}\Big{]}\,,$ $\displaystyle
d_{0}=3h^{3}\Big{(}\frac{\partial\epsilon}{\partial T}\Big{)}_{n}\,,$
$\displaystyle
d_{1}=3h\beta_{1}n^{2}\kappa\Big{[}\alpha\kappa\Big{(}\frac{\partial\epsilon}{\partial
n}\Big{)}_{T}+4T^{2}\kappa\Big{(}\frac{\partial\epsilon}{\partial
T}\Big{)}_{n}\Big{(}\frac{\partial\mu}{\partial
n}\Big{)}_{T}^{2}-T\Big{(}\frac{\partial n}{\partial
T}\Big{)}_{n}\Big{(}\frac{\partial\epsilon}{\partial n}\Big{)}_{T}\Big{]}\,.$
(5.22)

The imaginary part of $\omega$ up to $\cal{O}$$(k^{4})$ is given by,

$\omega_{Im}=-\frac{c_{0}}{d_{0}}\left[k^{2}-\frac{c_{1}}{c_{0}}k^{3}-(\frac{d_{1}}{d_{0}}+\frac{c_{2}}{c_{0}})k^{4}\right]\,.$
(5.23)

The dispersion relation for first order hydrodynamics can be obtained by
setting the relaxation coefficients ($\beta_{0},\beta_{1},\beta_{2}$) and the
coupling coefficients ($\alpha_{0}$, $\alpha_{1}$) to zero which allow only
$a_{0}$, $b_{0}$, $c_{0}$ and $d_{0}$ to be non-zero. Therefore, keeping terms
up to $\cal{O}$$(k^{2})$ in Eqs. (5.20) and (5.23) we get (see also [183]),

$\omega(k)=c_{s}k-\frac{i}{2}k^{2}\frac{\eta}{s}s\frac{4/3}{h}\,,$ (5.24)

where, $c_{s}=\sqrt{\big{(}\frac{\partial P}{\partial\epsilon}\big{)}_{s/n}}$
is the speed of sound and $\eta/s$ is shear viscosity ($\eta$) to the entropy
density ($s$) ratio. The Eq.(5.24) is the dispersion relation for NS
hydrodynamics, which is obtained from the limiting case of MIS theory.

####  5.4.1 Fluidity near the critical region

The imaginary and real parts of $\omega(k)$ provide the information
respectively on the attenuation and the propagation of the sound wave in the
dissipative fluid. Thus, if magnitude of the imaginary part is greater than
the real part, the wave will dissipate quickly. Therefore, the dispersion
relations in Eqs.(5.18) and (5.21) can be used to determine the upper limit of
$k$ of the sound wave for its survivability in the medium. The threshold value
of $k$, i.e. $k_{th}$ can be evaluated by using the following condition [184]

$\displaystyle\Big{|}\frac{\omega_{Im}(k)}{\omega_{{\mathcal{R}}e}(k)}\Big{|}_{k=k_{th}}=1\,.$
(5.25)

i.e. any wave with wave vector higher than $k_{th}$ will get dissipated in the
fluid. Solving the above equation, we get,

$k_{th}=\sqrt{\frac{\mathcal{P}}{\mathcal{Q}}}~{}.$ (5.26)

where,

$\displaystyle\mathcal{P}$ $\displaystyle=$
$\displaystyle\frac{a_{0}}{c_{0}^{2}}-\frac{a_{0}d_{1}c^{2}_{2}}{b_{0}c_{0}^{2}d_{0}^{2}}-\frac{a_{1}d_{0}}{c_{0}^{3}}+\frac{a_{1}c_{2}^{2}d_{1}}{b_{0}c_{0}^{3}d_{0}}-\frac{b_{1}^{2}c_{2}^{3}d_{1}}{b_{0}c_{0}^{2}d_{0}^{3}}+\frac{a_{1}d_{0}}{a_{0}b_{0}c_{0}^{2}}~{},$
(5.27) $\displaystyle\mathcal{Q}$ $\displaystyle=$
$\displaystyle\frac{b_{0}}{d_{0}^{2}}-\frac{b_{1}}{c_{0}^{2}}-\frac{b_{0}c_{2}}{c_{0}^{3}}+\frac{a_{1}}{a_{0}b_{0}d_{0}}-\frac{a_{1}b_{1}d_{0}}{a_{0}b_{0}^{2}c_{0}^{2}}+\frac{a_{1}c_{2}d_{0}}{b_{0}^{2}c_{0}^{3}}~{}.$
(5.28)

Expanding $\mathcal{P}$ and $\mathcal{Q}$ and keeping the leading terms of the
series we get,

$\displaystyle k_{th}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{a_{0}d_{0}^{2}}{b_{0}c_{0}^{2}}}\Bigg{[}1-\frac{1}{2}\Bigg{(}\frac{d_{1}c_{2}^{2}}{b_{0}d_{0}^{2}}+\frac{a_{1}d_{0}}{a_{0}c_{0}}-\frac{a_{1}c_{2}^{2}d_{1}}{a_{0}b_{0}c_{0}d_{0}}+\frac{b_{1}^{2}c_{2}^{3}d_{1}}{a_{0}b_{0}d_{0}^{3}}-\frac{a_{1}d_{0}}{a_{0}^{2}b_{0}}\Bigg{)}\Bigg{]}$
(5.29)
$\displaystyle\Bigg{[}1+\frac{1}{2}\Bigg{(}\frac{b_{1}d_{0}^{2}}{b_{0}c_{0}^{2}}+\frac{c_{2}d_{0}^{2}}{c_{0}^{3}}-\frac{a_{1}d_{0}}{a_{0}b_{0}^{2}}+\frac{a_{1}b_{1}d_{0}^{3}}{a_{0}b_{0}^{3}c_{0}^{2}}-\frac{a_{1}c_{2}d_{0}^{3}}{b_{0}^{3}c_{0}^{3}}\Bigg{)}\Bigg{]}$

The first term of the above expression gives rise to the value of $k_{th}$ in
the NS limit,
$k_{th}=\sqrt{\frac{a_{0}d_{0}^{2}}{b_{0}c_{0}^{2}}}=\frac{3}{2}c_{s}\frac{h}{s}\frac{1}{\eta/s}$
[184]. The subsequent terms are arising from the second-order hydrodynamical
effects as indicated by the presence of coupling and relaxation coefficients
appearing through the quantities defined in Eqs. (5.3).

The threshold wavelength ($\lambda_{th}$) corresponding to $k_{th}$ is given
by $\lambda_{th}=2\pi/k_{th}$. Sound waves with wavelength,
$\lambda<\lambda_{th}$ will dissipate in the medium. However, sound waves with
$\lambda\geq\lambda_{th}$ will survive to propagate in the fluid without much
dissipative effects. The quantity $\lambda_{th}$ can also be used to define
the fluidity ($\mathcal{F}$) of the system with widely varying particle
density and temperature by selecting a length scale (inter-particle
separation), $l\sim\rho^{-1/3}$ [184] of the system as:

$\mathcal{F}\sim\frac{\lambda_{th}}{l}.$ (5.30)

where, $\rho$ is the particle number density of the fluid (for a relativistic
fluid $l$ can be chosen as $l\sim s^{-1/3}$). The length scale, $R_{v}\sim
1/k_{th}$, called viscous horizon [185], which fixes the limit for the sound
wave with $\lambda$ smaller than $R_{v}$ will be dissipated due to highly
viscous and thermal conduction effects. $R_{v}$ can also be used to estimate
the value of order of the highest harmonics $n_{v}=2\pi R/R_{v}$
($n=2,3,4,...$) which will survive the dissipation i.e. any harmonics of order
higher than $n_{v}$ will not survive against dissipation. We find that
$k_{th}$ is directly proportional to the $c_{s}$, which approaches zero at the
critical point, that is the sound wave loses its strength and get damped
strongly near critical point. Therefore, we can argue that $k_{th}$ also
vanishes or in other words $\lambda_{th}$ diverges at the critical point.

###  5.5 Results and discussion

Figure 5.1:  a) Dissipation of sound modes with time for different k values at
$(\mu_{c},T_{c})=$(367 MeV,154 MeV) and ($\eta/s=\kappa T/s=1/4\pi).$ b)
Damping of the sound waves with different sets of transport coefficients for
$k=250MeV$.

Now we discuss the dissipation of the perturbation in the fluid when the
system hits the CEP in the QCD phase diagram. The damping caused by the
imaginary part of the frequency of hydrodynamic modes of perturbation at the
critical point $(T_{c},\,\mu_{c})=(154,\,367)\,MeV$ is depicted in Fig.5.1. It
is seen that the waves with larger (smaller) values of wavenumber ($k$) damp
faster (slower). The waves in fluid damp highly for larger values of transport
coefficients (right panel). The results displayed in Fig.5.2 (left panel) show
that when the system is away from the critical point then the waves damp
slowly for fluid with higher temperatures and densities. The waves in a medium
with higher viscosity and thermal conductivity damp faster (Fig. 5.2, right
panel) for obvious reasons.

Figure 5.2:  a) Dissipation of sound waves with k for different sets of values
of $(\mu,T)$ at $t=0.6$ fm/c and $\eta/s=\kappa T/s=1/4\pi$. b) Damping of the
sound waves with different sets of value of transport coefficients with
$(\mu_{c},T_{c})=$ (367 MeV,154 MeV).

Figure 5.3:  a) Left panel: The variation of $\lambda_{th}$ (fm) with
$T/T_{c}$ for different values of $\mu$. The red line shows the diverging
nature of $\lambda_{th}$ at $T/T_{c}=1$ for $\mu_{c}=367$ MeV. b)Right panel
displays $\lambda_{th}$ (fm) as a function of $T/T_{c}$ at $\mu_{c}=367$ MeV
for different set of values of the transport coefficients.

The variation of $\lambda_{th}$ with temperature ($T/T_{c}$) is shown in
Fig.5.3. It is evident that $\lambda_{th}(=2\pi/{k_{th}})$ depends on the
transport coefficients ($\eta,\kappa$) of the system, as well as on the
various response functions appearing through the thermodynamic derivatives,
$\Big{(}\frac{\partial\epsilon}{\partial
T}\Big{)}_{n},\Big{(}\frac{\partial\epsilon}{\partial n}\Big{)}_{T},$
$\Big{(}\frac{\partial P}{\partial T}\Big{)}_{n},\Big{(}\frac{\partial
P}{\partial n}\Big{)}_{T},\Big{(}\frac{\partial
n}{\partial\mu}\Big{)}_{T},\Big{(}\frac{\partial n}{\partial T}\Big{)}_{\mu}$,
coupling constant ($\alpha_{1}$), and the relaxation coefficients
($\beta_{1},\beta_{2}$). To observe the behaviour of $k_{th}$ through the EoS
only, we have not considered the scaling behaviour (near the CEP) of the
transport coefficients but we have taken as $\eta/s=\kappa T/s=1/4\pi$. The
values of $\Big{(}\frac{\partial\epsilon}{\partial
T}\Big{)}_{n},\Big{(}\frac{\partial\epsilon}{\partial
n}\Big{)}_{T},\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n},\Big{(}\frac{\partial P}{\partial
n}\Big{)}_{T},\Big{(}\frac{\partial
n}{\partial\mu}\Big{)}_{T},\Big{(}\frac{\partial n}{\partial T}\Big{)}_{\mu}$
are calculated in terms of different response function by using relevant
thermodynamic relations (Appendix B). In the left panel of Fig.5.3, the
variation of $\lambda_{th}$ with $T/T_{c}$ is presented. It is observed that
at the CEP ($T_{c}=154\text{MeV}$, $\mu_{c}=367\text{MeV}$) the $\lambda_{th}$
diverges. As mentioned above, the $\lambda_{th}$ is defined as the wavelength
such that waves with wavelengths, $\lambda\geq\lambda_{th}$ are only allowed
to propagate and others get dissipated. It is observed that when we consider
$\mu=347$ MeV and 387 MeV (away from the critical point) a finite value of
$\lambda_{th}$ is obtained, that is wave with $\lambda\geq\lambda_{th}$ will
propagate in the medium without substantial damping in such cases. At the
critical point, however, $\lambda_{th}$ diverges which infer that waves with
all wavelength will dissipate in the fluid. We also observe that for the lower
value of $T$ the value of $\lambda_{th}$ is smaller. At lower temperature
region, the magnitude of $\lambda_{th}$ for $\mu=387$ MeV is larger compared
to $\mu=347$ MeV. This indicates that for higher values of temperature and
chemical potential $\lambda_{th}$ is larger. The fluidity defined in Eq.(5.30)
is directly proportional to $\lambda_{th}$ which diverges at CEP, implies that
fluidity also diverges at the CEP. Away from CEP, the fluidity decreases. The
fluidity is larger for $\mu=387$ MeV compared to $\mu=347$ MeV.

The viscous damping of a perturbation can be understood from the relation:

$\displaystyle T^{\mu\nu}_{1}(t)=T^{\mu\nu}_{1}(0)exp({-\omega_{Im}t})~{},$
(5.31)

where, $T^{\mu\nu}_{1}(0)$ is the perturbation in EMT at $t=0$ and
$T^{\mu\nu}_{1}(t)$ is at some later time $t$ which gets dissipated as
indicated by the exponential term. The spectrum of the initial $(t=0)$
perturbations can be associated with the harmonics of the shape deformations
and density fluctuations [186]. The dispersion relation for $\omega$ provides
the value, $k_{th}$, which can be used to define a length scale, $R_{v}\sim
1/k_{th}$. For system of size $R$, $R_{v}$ can be used to define
$n_{v}=\frac{2\pi R}{R_{v}}$ which is linked to the value of the highest
harmonic $n_{v}$ (eccentricity-driven) that will effectively survive damping.
We have seen that the nature of the plot, $\lambda_{th}$ vs $T$ does not
change considerably with the change in shear viscosity ($\eta/s$) but changes
significantly with the variation of thermal conductivity ($\kappa T/s$). The
right panel of Fig.5.3 displays the variation of $\lambda_{th}$ with
temperature for higher values of $\kappa T/s$ and $\eta/s$. As the magnitude
of $\kappa T/s$ increases the gap between the divergences in the two phases
gets narrower. It is well-known [162] that the thermal conductivity diverges
at the critical point. Therefore, the nature of the variation of
$\lambda_{th}$ with $T/T_{c}$ near the CEP will be essentially governed by the
thermal conductivity. We have found that with increasing thermal conductivity
the width of the divergence gets narrower.

## CHAPTER 6 Dynamic structure factor near the QCD critical point

###  6.1 Introduction

This chapter is based on the publication as shown in Ref.[73]. Here, we
calculate the dynamic structure factor, which is the time-dependent correlator
of density fluctuations. This is evaluated from the linear analysis of the
hydrodynamic equations, and we will see its behaviour near the QCD critical
point. The importance of the spectral structure, more specifically the
structure factor, denoted by,
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$, is an accessible
quantity in laboratory, in condensed matter physics by light scattering, X-ray
diffraction, and neutron scattering. The spectrum of light scattered by a
fluid is proportional to
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$, where
$k=|\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}}|$ is to be identified with the
so-called scattering wave vector, which is related to the geometry of the
scattering experiment and the wavelength of the incident light. The spectrum
of scattered light contains separately identifiable Lorentzians, called
Rayleigh-line (peaks) named after Lord Rayleigh [187] and Brillouin-line
(peaks), measured by Fleury and Boon [188] in 1969. Rayleigh line (R-line),
arises from entropy or, equivalently, temperature fluctuations at constant
pressure, whereas B-lines arise from thermally excited propagating sound waves
(modes) associated with adiabatic pressure fluctuations [60, 189]. The
Rayleigh line and the Brillouin lines are usually investigated separately,
both in theory and in experiments.

It is well-known that the correlation length ($\xi$) diverges at the critical
point. The effects of the divergence on the particle correlations, baryon
number fluctuations, and correlations of density fluctuations will have a
better chance to get detected provided the fluctuations to survives the
evolution of the hadronic phase. It is important to understand the
correlations of those fluctuations theoretically by identifying signatures of
the CEP in hadronic spectra. The effects of CEP on the evolution of QGP go as
input to the hydrodynamic equations through the equation of state (EoS) and
via the criticality of the transport coefficients and the thermodynamic
response functions of the medium [162, 145, 146].

The dynamical structure factor,
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ has been estimated
earlier in Ref. [62, 164] without taking into account the effect of EoS. It is
shown in this work that the EoS plays a vital role in determining the
behaviour of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$, especially its
strength at the R-peak, which is modified by several orders of magnitude when
the effects of the CEP in EoS is incorporated. The EoS has a strong effects on
B-peaks too because it determines $c_{s}$ (speed of sound) and hence the
location of the B-peaks. It is also very crucial to understand whether all the
hydrodynamic modes ($k$-modes) travel at the same speed or not. The R-peak and
the B-peaks will be closer for slower modes even at points away from the CEP.
Therefore, the structure of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ will shed light on the
speed of the perturbation propagating as a sound wave.

###  6.2 Linearized Hydrodynamic equations

In this chapter, we use the Eckart’s frame of reference as discussed in
Sec.3.3 in Ch.3, where it is considered that the heat flux is non-zero but the
particle current is zero. Therefore, energy-momentum tensor (EMT) and the the
particle current ($N^{\mu}$) are given by Eqs.(3.30) and (3.31). The
conservations of EMT and the net baryon number follows the Eq.(3.50).

The hydrodynamic Eqs. (3.50) are partial differential equations, which are
non-linear, could not be solved analytically in general. Presently we aim to
obtain the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ of the dynamical
density fluctuations in $\omega-k$ space. The hydrodynamical equations
presented above can be linearized to express small perturbations (in
magnitude) in the thermodynamical variables (small deviations from the
equilibrium values of the variable). These linearized equations can be solved
to obtain the dynamical density fluctuations. Let $Q_{0}$ ($Q$) represent a
thermodynamic quantity in (away from) equilibrium. A small perturbation
$\delta Q$ can be written as: $Q=Q_{0}+\delta Q$ where $Q$ can be any of the
thermodynamic quantities such as
$n,\epsilon,u^{\alpha},q^{\alpha},s,\Pi,\pi^{\alpha\beta}$ (baryon number
density, energy density, fluid four-velocity, heat flow vector, entropy
density, bulk pressure, shear stress tensor respectively), etc. The linearized
hydrodynamic equations around the equilibrium [62, 190, 72, 191, 183] thus
become:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\partial\delta
n}{\partial
t}+n_{0}\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}.\delta\mathchoice{\mbox{\boldmath$v$}}{\mbox{\boldmath$v$}}{\mbox{\boldmath$\scriptstyle
v$}}{\mbox{\boldmath$\scriptscriptstyle v$}}~{},$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle h_{0}\frac{\partial\delta v}{\partial
t}+\nabla(\delta P+\delta\Pi)+\frac{\partial\delta q}{\partial
t}+\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}.\delta\mathchoice{\mbox{\boldmath$\pi$}}{\mbox{\boldmath$\pi$}}{\mbox{\boldmath$\scriptstyle\pi$}}{\mbox{\boldmath$\scriptscriptstyle\pi$}}~{},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\delta\Pi+\zeta[\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}.\delta\mathchoice{\mbox{\boldmath$v$}}{\mbox{\boldmath$v$}}{\mbox{\boldmath$\scriptstyle
v$}}{\mbox{\boldmath$\scriptscriptstyle
v$}}+\beta_{0}\frac{\partial\delta\Pi}{\partial
t}-\tilde{\alpha_{0}}\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}.\delta\mathchoice{\mbox{\boldmath$q$}}{\mbox{\boldmath$q$}}{\mbox{\boldmath$\scriptstyle
q$}}{\mbox{\boldmath$\scriptscriptstyle q$}}]~{},$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle\delta\pi^{ij}+\eta[\partial^{i}\delta
v^{j}+\partial^{j}\delta
v^{i}-\frac{2}{3}g^{ij}\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}.\delta\mathchoice{\mbox{\boldmath$v$}}{\mbox{\boldmath$v$}}{\mbox{\boldmath$\scriptstyle
v$}}{\mbox{\boldmath$\scriptscriptstyle
v$}}+2\beta_{2}\frac{\partial\delta\pi^{ij}}{\partial t}$
$\displaystyle-\tilde{\alpha_{1}}(\partial^{i}\delta q^{j}+\partial^{j}\delta
q^{i}-\frac{2}{3}g^{ij}\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}.\delta\mathchoice{\mbox{\boldmath$q$}}{\mbox{\boldmath$q$}}{\mbox{\boldmath$\scriptstyle
q$}}{\mbox{\boldmath$\scriptscriptstyle q$}})~{},$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle\delta q+\kappa T_{0}[\frac{\nabla\delta
T}{T_{0}}+\frac{\partial\delta v}{\partial
t}+\tilde{\beta_{1}}\frac{\partial\delta q}{\partial
t}-\alpha_{0}\nabla\delta\Pi-\tilde{\alpha_{1}}\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}.\delta\mathchoice{\mbox{\boldmath$\pi$}}{\mbox{\boldmath$\pi$}}{\mbox{\boldmath$\scriptstyle\pi$}}{\mbox{\boldmath$\scriptscriptstyle\pi$}}]~{},$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle n_{0}\frac{\partial\delta
s}{\partial
t}+\frac{1}{T_{0}}\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}.\delta\mathchoice{\mbox{\boldmath$q$}}{\mbox{\boldmath$q$}}{\mbox{\boldmath$\scriptstyle
q$}}{\mbox{\boldmath$\scriptscriptstyle q$}}~{},$ (6.2)

where, $h_{0}=\epsilon_{0}+P_{0}$ is the enthalpy density in equilibrium and
$v$ is the space dependent velocity of fluid four-velocity. Now we decompose
the fluid velocity along the directions parallel and perpendicular to the
direction of wavevector, $k$ and denote them as
$\delta\mathchoice{\mbox{\boldmath$v_{||}$}}{\mbox{\boldmath$v_{||}$}}{\mbox{\boldmath$\scriptstyle
v_{||}$}}{\mbox{\boldmath$\scriptscriptstyle v_{||}$}}$ and
$\delta\mathchoice{\mbox{\boldmath$v_{\perp}$}}{\mbox{\boldmath$v_{\perp}$}}{\mbox{\boldmath$\scriptstyle
v_{\perp}$}}{\mbox{\boldmath$\scriptscriptstyle v_{\perp}$}}$ respectively.
Here, we have considered the longitudinal components only by setting
$\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}.\delta\mathchoice{\mbox{\boldmath$v_{\perp}$}}{\mbox{\boldmath$v_{\perp}$}}{\mbox{\boldmath$\scriptstyle
v_{\perp}$}}{\mbox{\boldmath$\scriptscriptstyle v_{\perp}$}}=0$. The
hydrodynamic equations can be solved for a given set of initial condition,
$n(0),\,v_{||}(0),\,T(0),\,q(0),\,\Pi(0)$ and $\pi(0)$, by using the Fourier-
Laplace transformation as the following:

$\displaystyle\delta
Q(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=\int^{\infty}_{-\infty}d^{3}{r}\,\,\int^{\infty}_{0}dte^{-i(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}.\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\boldmath$\scriptstyle
r$}}{\mbox{\boldmath$\scriptscriptstyle r$}}-\omega
t)}\delta\tilde{Q}(\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\boldmath$\scriptstyle
r$}}{\mbox{\boldmath$\scriptscriptstyle r$}},t)~{}.$ (6.3)

And,

$\displaystyle\delta
Q(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,0)=\delta
Q(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},t=0)=\int^{\infty}_{-\infty}d^{3}{r}\,\,e^{-i(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}.\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\boldmath$\scriptstyle
r$}}{\mbox{\boldmath$\scriptscriptstyle
r$}})}\delta\tilde{Q}(\mathchoice{\mbox{\boldmath$r$}}{\mbox{\boldmath$r$}}{\mbox{\boldmath$\scriptstyle
r$}}{\mbox{\boldmath$\scriptscriptstyle r$}},t=0)~{}.$ (6.4)

The $\delta P$ and $\delta s$ can be expressed in terms of the independent
variables $n$ and $T$ as follows by using the thermodynamic relations:

$\displaystyle\delta P$ $\displaystyle=$ $\displaystyle\Big{(}\frac{\partial
P}{\partial n}\Big{)}_{T}\delta n+\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n}\delta T~{},$ $\displaystyle\delta s$ $\displaystyle=$
$\displaystyle\Big{(}\frac{\partial s}{\partial n}\Big{)}_{T}\delta
n+\Big{(}\frac{\partial s}{\partial T}\Big{)}_{n}\delta T~{}.$ (6.5)

We use Eqs. (6.3), (6.4) and (6.5) to write down the longitudinal linearized
hydrodynamic equation as:

$\displaystyle\delta
Q(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)=\mathcal{M}^{-1}\delta
Q(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)~{},$ (6.6)

where,

$\displaystyle\mathcal{M}=\begin{bmatrix}i\omega&ikn_{0}&0&0&0&0\\\
\frac{ik}{h_{0}}\Big{(}\frac{\partial P}{\partial
n}\Big{)}_{T}&i\omega&\frac{ik}{h_{0}}\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n}&\frac{i\omega}{h_{0}}&\frac{ik}{h_{0}}&\frac{ik}{h_{0}}\\\
0&ik\zeta&0&-ik\tilde{\alpha_{0}}\zeta&1+i\omega\beta_{0}\zeta&0\\\
0&-i\frac{4}{3}k\eta&0&i\frac{4}{3}\tilde{\alpha_{1}}k\eta&0&1+2i\omega\beta_{2}\eta\\\
0&i\omega\kappa T_{0}&ik\kappa&1+i\omega\tilde{\beta_{1}}\kappa
T_{0}&ik\alpha_{0}\kappa T_{0}&ik\tilde{\alpha_{1}}\kappa T_{0}\\\ -i\omega
n_{0}\Big{(}\frac{\partial s}{\partial n}\Big{)}_{T}&0&i\omega
n_{0}\Big{(}\frac{\partial s}{\partial T}\Big{)}_{n}&\frac{ik}{T_{0}}&0&0\\\
\end{bmatrix}$ , (6.7)

and

$\displaystyle\delta
Q(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)=\begin{bmatrix}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)\\\ \delta
v_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)\\\
\delta\Pi(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)\\\
\delta\pi_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)\\\ \delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)\\\ \delta
T(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)\end{bmatrix};~{}\delta
Q(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,0)=\begin{bmatrix}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\\\ \delta
v_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)+\frac{1}{h_{0}}\delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\\\
i\omega\beta_{0}\zeta\delta\Pi(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\\\
-2\beta_{2}\eta\delta\pi_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\\\ -\kappa T_{0}\delta
v_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)+\kappa
T_{0}\tilde{\beta_{1}}\delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\\\ -n_{0}\Big{(}\frac{\partial
s}{\partial n}\Big{)}_{T}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)+n_{0}\Big{(}\frac{\partial
s}{\partial T}\Big{)}_{n}\delta
T(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\\\ \end{bmatrix}$ (6.8)

We are concerned here about the density fluctuation which is given by,

$\displaystyle\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ $\displaystyle=$
$\displaystyle\Big{[}\mathcal{M}^{-1}_{11}-n_{0}\Big{(}\frac{\partial
s}{\partial n}\Big{)}_{T}\mathcal{M}^{-1}_{16}\Big{]}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)+\Big{[}\mathcal{M}^{-1}_{12}-\kappa
T_{0}\mathcal{M}^{-1}_{15}\Big{]}\mathchoice{\mbox{\boldmath$\delta
v_{||}$}}{\mbox{\boldmath$\delta v_{||}$}}{\mbox{\boldmath$\scriptstyle\delta
v_{||}$}}{\mbox{\boldmath$\scriptscriptstyle\delta
v_{||}$}}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)$ (6.9)
$\displaystyle+\mathcal{M}^{-1}_{13}\Big{[}i\omega\beta_{0}\zeta\Big{]}\delta\Pi(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)-\mathcal{M}^{-1}_{14}\Big{[}2\beta_{2}\eta\Big{]}\delta\pi_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)$
$\displaystyle\Big{[}\frac{1}{h_{0}}\mathcal{M}^{-1}_{12}+\kappa
T_{0}\tilde{\beta_{1}}\mathcal{M}^{-1}_{15}\Big{]}\delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)+\mathcal{M}^{-1}_{16}\Big{[}n_{0}\Big{(}\frac{\partial s}{\partial
T}\Big{)}_{n}\Big{]}\delta
T(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)~{}.$

We define
$\mathcal{S^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ by the following
expression:

$\displaystyle\mathcal{S^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)=\Big{<}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\Big{>}~{}.$ (6.10)

The correlation between two independent thermodynamic variables, say, $Q_{i}$
and $Q_{j}$ vanishes i.e.

$\displaystyle\Big{<}\delta
Q_{i}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)\delta
Q_{j}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\Big{>}=0,\,\,\,\,i\neq j~{}.$
(6.11)

Using Eq.(6.11) into Eq.(6.9), the required correlator,
$\mathcal{S^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ is obtained as:

$\displaystyle\mathcal{S^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ $\displaystyle=$
$\displaystyle\Big{[}\mathcal{M}^{-1}_{11}-n_{0}\Big{(}\frac{\partial
s}{\partial n}\Big{)}_{T}\mathcal{M}^{-1}_{16}\Big{]}\Big{<}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\Big{>}~{}.$ (6.12)

Finally,
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ is defined as:

$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=\frac{\mathcal{S^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)}{\Big{<}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\Big{>}}~{}.$ (6.13)

The variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $k$ and $\omega$
for given values of transport coefficients, thermodynamic variables and the
response functions are studied below. In the small $k$ limit with
$\tilde{\alpha_{0}}\rightarrow 0,\tilde{\alpha_{1}}\rightarrow
0,\beta_{0}\rightarrow 0,\tilde{\beta_{1}}\rightarrow 0,\beta_{2}\rightarrow
0$ the results for Navier-Stokes hydrodynamics can be retrieved from the full
expression given in the Appendix C. We will see below that the variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$
possesses with three peaks positioned at $\omega=0$ and
$\omega=\pm\omega_{B}$. The $\omega_{B}$ is a function of $k$, $c_{s}$ and
other thermodynamic variables. The peak at $\omega=0$ is called the Rayleigh-
peak and the doublet symmetrically situated at $\pm\omega_{B}$ are called
Brillouin-peaks. The quantities $\Big{(}\frac{\partial P}{\partial
n}\Big{)}_{T},\Big{(}\frac{\partial P}{\partial
T}\Big{)}_{n},\Big{(}\frac{\partial s}{\partial
n}\Big{)}_{T},\Big{(}\frac{\partial s}{\partial T}\Big{)}_{n}$ appearing in
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ can be evaluated in
terms of relevant thermodynamic variables (see Appendix C).

The width of this R-line is proportional to $\kappa/nC_{P}$ [60], where
$\kappa$ is the thermal conductivity, $C_{P}$ is the isobaric specific heat.
Therefore, if the order of divergence for $\kappa$ is less compared to
$C_{P}$, then we see a narrow R-line. Two Brillouin lines are positioned at
$\omega_{B}\pm c_{s}k$ with respect to the frequency of the incident light
source, which may give rise to the speed of sound. The finite width of the
B-line can provide information about the finite value of transport
coefficients such as shear, and bulk viscosities. Also, the ratio of
intensities of R-line and B-line is expressed as
$I_{R}/2I_{B}=C_{P}/C_{V}-1=\kappa_{T}/\kappa_{S}-1$ (where, $\kappa_{T}$ and
$\kappa_{S}$ are the isothermal and adiabatic compressibilities respectively),
called the Landau-Placzek ratio. Therefore, studying the structure factor
could be a useful theoretical as well as experimental aspects to investigate
the behaviour of the medium near the CEP.

In condensed matter physics, the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ is measured by light
scattering, utilizing the relation between intensity of light with density-
density correlation [60]. However, any such direct measurement of the
corresponding critical opalescence in QCD is still not achieved. The
possibility of such measurement of QCD opalescence has been proposed by
measuring jet quenching [78].

###  6.3 Behaviour of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ near the CEP

The dynamical structure factor,
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ defined in Eq. (6.13)
directly depends on the transport coefficients, such as $\eta,\zeta$ and
$\kappa$ as well as on the four thermodynamic partial derivatives,
$\Big{(}\frac{\partial P}{\partial n}\Big{)}_{T},\Big{(}\frac{\partial
P}{\partial T}\Big{)}_{n}$, $\Big{(}\frac{\partial s}{\partial n}\Big{)}_{T}$
and $\Big{(}\frac{\partial s}{\partial T}\Big{)}_{n}$. It depends on the
coupling coefficients $(\tilde{\alpha_{0}},\tilde{\alpha_{1}})$ and also on
the relaxation coefficients $(\beta_{0},\tilde{\beta_{1}},\beta_{2})$ which
pervade into
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ through the Müller-
Israel-Stewart (MIS) hydrodynamics. In
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$, the effects of the
CEP (through the EoS) is carried by the thermodynamic derivatives. It is well
known that the behaviour of the transport coefficients and the thermodynamic
response functions near the CEP are characterized by the critical exponents.
As the CEP is approached, some of those quantities start to diverge. A
thermodynamic variable, $f(t)$ near the CEP can be written as [189]:

$\displaystyle f(t)=Ae^{\lambda}(1+Be^{y}+...)~{},$ (6.14)

where, $y>0$, $t=(T-T_{c})/T_{c}$ is the reduced temperature. The critical
exponent $\lambda$ can be defined as:

$\displaystyle\lambda=\lim_{t\to 0}\frac{lnf(t)}{ln(t)}~{}.$ (6.15)

$\lambda$ can be either positive or negative correspondingly $f(t)$ will
vanish or diverge at the CEP.

Now the partial derivatives appearing in the expression for
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ can be expressed as
(see Appendix C):

$\displaystyle\Big{(}\frac{\partial P}{\partial n}\Big{)}_{T}$
$\displaystyle=$
$\displaystyle\frac{1}{n_{0}\kappa_{T}},\,\,\,\,\Big{(}\frac{\partial
P}{\partial T}\Big{)}_{n}=\mu n_{0}c^{2}_{s}\alpha_{p}\frac{C_{V}}{C_{P}}~{},$
$\displaystyle\Big{(}\frac{\partial s}{\partial n}\Big{)}_{T}$
$\displaystyle=$
$\displaystyle\frac{h_{0}c^{2}_{s}\alpha_{p}}{n_{0}\gamma},\,\,\,\,\Big{(}\frac{\partial
s}{\partial T}\Big{)}_{n}=\frac{C_{V}}{T}~{},$ (6.16)

where, $\kappa_{T}=\frac{1}{n_{0}}\Big{(}\frac{\partial n}{\partial
P}\Big{)}_{T}$, is the isothermal compressibility,
$\alpha_{P}=-\frac{1}{n_{0}}\Big{(}\frac{\partial n}{\partial T}\Big{)}_{P}$
is the volume expansivity coefficient. The symbol $C_{P}$ and $C_{V}$ are the
specific heats at constant pressure and volume respectively. The critical
behaviour of various transport coefficients and thermodynamic response
functions plays important roles to determine the strength of the signal of the
presence of the CEP. We take the following $t$ dependence near CEP for the
present purpose [162, 145, 146].

$\displaystyle\kappa_{T}=\kappa_{T}^{0}|t|^{-\gamma^{\prime}},C_{V}=C_{0}|t|^{-\alpha},C_{P}=\frac{\kappa_{0}T_{0}}{n_{0}}\Big{(}\frac{\partial
P}{\partial T}\Big{)}^{2}_{n}|t|^{-\gamma^{\prime}},$ $\displaystyle
c^{2}_{s}=\frac{T_{0}}{n_{0}h_{0}C_{0}}\Big{(}\frac{\partial P}{\partial
T}\Big{)}^{2}_{n}|t|^{\alpha},\alpha_{P}=\kappa_{0}\Big{(}\frac{\partial
P}{\partial T}\Big{)}_{n}|t|^{-\gamma^{\prime}}~{},$
$\displaystyle\eta=\eta_{0}|t|^{1+a_{\kappa}/2-\gamma^{\prime}},\zeta=\zeta_{0}|t|^{-\alpha_{\zeta}},\kappa=\kappa_{0}|t|^{-a_{\kappa}}~{},$
(6.17)

where, $\alpha,\gamma^{\prime},a_{\zeta},a_{\kappa}$ are the critical
exponents. Here $\alpha=0.11$ and $\gamma^{\prime}=1.2$. For a liquid-gas
critical point, $a_{\zeta}=\nu d-\alpha=1.78$ (here, $d=3,\nu=0.63$), and
$a_{\kappa}=0.63$. For consistency the partial derivative,
$\left(\frac{\partial P}{\partial T}\right)_{n}$ has been calculated by using
the the EoS. The critical behaviours of the second-order coupling as well as
the relaxation coefficients enter into the calculation through the
thermodynamic variables which contain the effects of the CEP (see Ch.4).

###  6.4 Results and discussion

Now we discuss the effects of the CEP through the EoS and subsequently on the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$. We have evaluated the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ by including the
effects of the CEP through: (i) the EoS and (ii) the critical behaviour of
various transport coefficients as discussed above. As mentioned in Ch.4, we
assume that the CEP is located at $(T_{c},\,\mu_{c})=(154,\,367)\,MeV$ in the
QCD phase diagram.

Figure 6.1:  a) Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ for
($r=0.2$) and $k=0.1$ fm-1 are plotted when the system is away from CEP for
first-order (NS) and second-order (MIS) hydrodynamics. b) Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ for
($r=0.2$) and $k=0.1$ fm-1 in second-order hydrodynamics. The results obtained
with $\eta/s=\zeta/s=\kappa T/s=1/4\pi$ is shown by red line. The green line
is obtained when parametric form of transport coefficients and response
functions are used.

Fig.6.1(a) shows a comparison of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ for first-order and
second-order hydrodynamics. The results show a difference in the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ estimated by using the
Naiver-Stokes (NS) and the Müller-Israel-Stewart (MIS) hydrodynamics. Fig.
6.1(b) depicts the variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ when the
system is away from CEP, which is represented by $r=0.2$. To examine the
effects of the EoS, we have kept the transport coefficients at their lower
bounds, $\eta/s,\zeta/s,\kappa T/s=1/4\pi$. The coupling and relaxation
coefficients are evaluated from Appendix A. We notice that there are three
distinct peaks (red line). The central one is larger in magnitude and is
called the R-peak which originates from the thermal fluctuations. The
symmetric doublet about the R-peak is identified as the B-peaks. The B-peaks
originate from pressure fluctuation at constant entropy, which is related to
the propagation of the sound waves. We find that the B-peaks are smaller in
magnitude than the R-peaks. We exclude the effects of the EoS to investigate
the effects of transport coefficients and the thermodynamic response functions
only near the CEP via their critical exponents in evaluating
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ (depicted by the green
line).

Figure 6.2:
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ near the critical
point for ($r=0.01$) and $k=0.1$ fm-1. The red curve represents the effects of
EoS and $\eta/s=\zeta/s=\kappa T/s=1/4\pi$. The green line is obtained by
using the scaling hypothesis of thermodynamic variables near CEP and both
effects are shown in blue curve. Inset plot is for the broader range in
$\omega$ ($-0.2\leq\omega\leq 0.2$).

Closer to the CEP, however, the scenario changes drastically. The
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ near the CEP
($r=0.01$) reveals only the R-peak with the magnitude enhanced by more than an
order of magnitude. The B-peaks do not appear due to the absorption of sound
waves near the CEP (Fig.6.2). The vanishing of the B-peaks can be understood
from the fact that in the leading order of Brillouin frequencies, it depends
as $\omega_{B}\sim\pm c_{s}k$. The effects of EoS are seen to reduce the
magnitude of the R-peak (red line). The speed of sound is determined by EoS,
therefore, the position, as well as the height of the B-peaks will depend on
the EoS.

Figure 6.3:  Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ for
$r=0.01$ (red-line) and 0.2 (blue-line) corresponding to the situation of the
system near and away from the CEP respectively. The value of $k$ is $0.02$
fm-1 and $\eta/s=\zeta/s=\kappa T/s=1/4\pi$.

The
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ is plotted as a
function of $\omega$ for a smaller $k$ ($k=0.02$ fm-1) in Fig.6.3 when the
effects of the EoS only is considered. The red (blue) line corresponds to
results closer to the CEP with $r=0.01$ (away from CEP with $r=0.2$). We
observe that for smaller $k$ value, the R-peak gets bigger as well as sharper
(in frequency axis). A comparison with results shown in Fig.6.4 reveals that
near the CEP, the B-peaks vanish and the R-peak gets sharper and bigger.

Figure 6.4:  Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ and $k$
near the CEP ($r=0.01$) for $\eta/s=\zeta/s=\kappa T/s=1/4\pi$ Figure 6.5:
Same as Fig. 6.4 for $r=0.2$ i.e. the system is away from CEP.

Motivated by the results shown in Figs. 6.2 and 6.3 i.e. by looking into the
sensitivity of the results on the $k$ values we investigate the behaviour of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ for different
hydrodynamic modes ($k-$modes). The variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $k$ and $\omega$
near the CEP is displayed in Fig.6.4. We see that for all values of $k$, the
B-peaks vanish and the R-peak remains. The magnitude of the R-peak is maximum
in the neighbourhood of $k\rightarrow 0$. The
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ is plotted with
$\omega$ and $k$ in Fig.6.5 when the system is away from the CEP. We observe
that the B-peaks shift away from R-peak with the increase in $k$. This is
better manifested in Fig.6.6, where the Brillouin frequency $\omega_{B}$ is
plotted against $k$. A linear variation is obtained for given values of $T$
and $\mu$. The slope ($c_{s}$) of the line is found to be $c_{s}=0.532$ (i.e.
$c^{2}_{s}=0.283$) close to the speed of sound $c^{2}_{s}=0.25$ obtained from
Eq. C.9 at the same value of temperature and chemical potential.

Figure 6.6:  Variation of the Brillouin frequency, $\omega_{B}$ with $k$ when
the system is away from CEP ($r=0.2$) with $\eta/s=\zeta/s=\kappa T/s=1/4\pi$.

Therefore, we observe that the B-peaks move toward the R-peak when the system
approaches the CEP and ultimately merges with R-peak at the CEP. This
indicates that the speed of propagation of all hydrodynamic modes disappears
at the CEP. This is quite consistent with the finding that the speed of sound
reaches its minimum at the QCD critical point. The location of the B-peaks
depend on $k$ when the system is away from the CEP, indicating that the
different k-modes travel with different speeds in the fluid.

In non-relativistic limit, it is evident that the width of the R-peak is
determined by $\kappa$, the widths of the $B$-peaks can be calculated in terms
of $\kappa$, $\eta$, $\zeta$ and the ratio $C_{P}/C_{V}$ [60]. The integrated
intensity of the R and B-peaks are determined by the ratio $C_{V}/C_{P}$, is
connected with the ratio of isothermal to adiabatic compressibilities.
Therefore, the spectral function contains useful information about the
thermodynamic state of the system. The spectral function in QGP is calculated
as a function of $\omega$ and $k$ which is not directly measurable through
external probe in experiments due to transient nature of the extremely small
system. In such a situation, one should depend on the indirect measurement of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$. Since the width and
the integrated intensities of the R and B peaks are related to the various
transport coefficients and thermodynamic response functions, thus, the
determination of these coefficients by some other experimental measurements
will help in constructing the spectral functions.

Some kind of mapping between $k$ and some observables is required to gain
insight on
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$. The spectral function
can be measured by using the bin by bin density fluctuations. Then $k$’s can
be associated to the inverse angular separation of the bins as
$\delta\phi\sim\frac{1}{kR}$, where $R$ is the radius of the freeze-out
surface (which analogous to the analysis of temperature fluctuation of CMB
radiation [192]). Therefore, the analysis of the bin by bin density
correlation in $\delta\phi\sim(kR)^{-1}$ in the transverse plane for different
beam energies will carry the effects of the CEP. However, for $k\sim
0.1fm^{-1}$, $\delta\phi$ is large for typical freeze-out radius in
relativistic heavy-ion collider experiments (RHIC-E), ($\sim 5$ fm). This
implies that the correlation can be observed for large angular separation.
That is the pattern corresponding to these $k$ values will show up over the
angular size $\delta\phi$.

In the condensed matter physics, the effects of the critical point are usually
investigated by measuring the intensity of the light scattering from the
system. In contrast to this, such external probes are unavailable for
detecting the CEP in QCD matter. However, numerous possibilities for the
detection of the CEP in QCD have been discussed in the literature. The
phenomenon of critical opalescence is considered as a signal of large density
fluctuations in a condensed matter system [60]. The possibility of detecting
such phenomenon of QCD opalescence can in principle be done by measuring the
suppression of hadronic spectra in heavy-ion collisions ($R_{AA}$) as
demonstrated in Ref. [78]. The $R_{AA}$ can guide to estimate the opacity
factor, ${\mathcal{K}}$ as:

${\mathcal{K}}=-\frac{ln(R_{AA})}{R_{HBT}}~{},$ (6.18)

where, $R_{HBT}$ is the Hanbury-Brown-Twiss(HBT) radius of the system.

The Fourier coefficients of the azimuthal distributions of hadrons can be used
to understand various properties of the matter produced in RHIC-E. The
coefficient of $cos3\phi$ (triangular flow, $v_{3}$) sheds light on the
quantum fluctuations of the initial state. Similarly, the coefficients of
$cos2\phi$ (elliptic flow, $v_{2}$) is concerned with the EoS of the system.
Near the CEP, the highest order of the harmonics to survive vary as $\sim
1/\lambda_{th}$, where $\lambda_{th}$ is some wavelength of the pressure
perturbation (sound) which blows up at the CEP and hence all the flow
harmonics will vanish [72] (discussed in Ch.5). However, the hadronic spectra
contain contribution from all the space-time points i.e. from creation of the
fireball to the stage of freeze out with all possible values of $T$ and $\mu$.
Therefore, instead of vanishing, the suppression of flow harmonics may be
observed.

The physics of the hadronic matter under extreme conditions of temperatures or
densities and the QCD phase transition has been dubbed as the condensed matter
physics of elementary particles [193] where the microscopic interaction is
controlled by the non-abelian gauge theory whereas, the condensed matter
physics is governed by the abelian gauge theory. Identification of some
appropriate probes analogous to light in the abelian system will go a long way
to unfold the physics of QCD matter near the CEP. It is shown [166] that the
dynamical universality class of the QCD critical point belong to the model H
[165] with a combination (linear) of the chiral condensate and the baryon
density as hydrodynamic mode. The evaluation of baryon number fluctuation by
using the correlation length prevailed in [166] will be an interesting problem
to address, however, this is beyond the scope of the present work.

## CHAPTER 7 Understanding the parametric slow mode through the dynamic
structure factor

###  7.1 Introduction

This chapter is based on the arXiv preprint [194]. The background fluid QGP is
described by the hydrodynamic equations of second-order. Without considering
the effect of critical point, hydrodynamic modelling of such collisions have
been developed to a certain degree of sophistication [195, 196, 197, 5]. The
hydrodynamics is used to describe the slowly evolving modes of the macroscopic
system while the faster non-hydrodynamic modes are driven by the collision
dynamics at the microscopic level. The time required to achieve local
equilibrium is much shorter than the time required to attain global
equilibrium, and applicability of hydrodynamics depends on the separation of
this time scale. The hydrodynamic modelling has been used to study the
evolution of the fireball created in relativistic nuclear collisions with the
inclusion of the CEP [195, 196, 197, 5], although strictly speaking it breaks
down near the CEP [71]

The underlying reason of the failure of validity of hydrodynamics near the CEP
[71, 62] happens due to long range correlations and enhanced fluctuations [89,
90, 91]. In other words, if the system experience the CEP, the correlation
length ($\xi$) diverges and the relaxation time which evolves as
$\sim\xi^{3}$, also diverges leading to critical slowing down [70, 60, 61].
Consequently, for the system stays away from local thermal equilibrium, the
first-order or the second-order hydrodynamical model may become inapplicable.
However, hydrodynamics can still be applied for systems far away from
equilibrium by including the higher order gradients of hydrodynamic fields.
This is a very active field of contemporary research but beyond the scope of
present work (we refer to [198] and references therein for details).

It has been shown in Ref. [71] that the validity of the hydrodynamics can be
extended near CEP by introducing a slowly evolving scalar non-hydrodynamic
field or the slow out-of-equilibrium mode (OEM), $\phi$, which is subsequently
incorporated in the definition of the entropy along with other hydrodynamical
variables. The slow OEM are not treated separately [71, 199] as they are
coupled with hydrodynamic modes.

It is important to point out here that the description of far-from-equilibrium
conformal systems may converge to hydrodynamic evolution [200, 201, 202, 203,
204]. However, this may not possible for general non-conformal systems [205,
206] as the gradient expansion is not always convergent [207]. The above
mentioned gradient expansion of the hydrodynamic field may not be suitable for
systems with large fluctuations near the CEP where the OEM relax slowly.
Instead, a separate treatment of these modes will allow to avoid the problem
of convergence of gradient expansion of hydrodynamic fields. The presence of
slow modes and its coupling with different hydrodynamic fields allow the study
of the fluctuations of large magnitudes near the CEP. In the present work, the
role of $\phi$ on
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ will be studied using
relativistic causal dissipative hydrodynamics proposed by Müller-Israel-
Stewart (MIS) [108]. The dispersion relations for the hydrodynamic modes are
governed by the set of hydrodynamic equations and the extensivity of
thermodynamics restricts the coupling of slow modes with gradients of other
hydrodynamic fields.

In condensed matter system, the fluctuations has been extensively studied,
specifically, the spectral structure of the correlation of static as well as
dynamical density fluctuations, also called the structure factor
($\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$). It has been shown
both theoretically and experimentally that
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ depends on the
transport coefficients of the system [60, 61, 189]. The thermal fluctuations
are controlled by the the transport properties of the medium following the
Onsager’s hypothesis [208]. The properties of the fluctuation are studied by
the light scattering and neutron scatterings experiments. The scattered light
spectrum contains separately identifiable Lorentzian peaks, called Rayleigh
(R) peak [187] situated at angular frequency, $\omega=0$, and two Brillouin
(B) peaks symmetrically located about the R-peak, was first experimentally
detected by Fleury and Boon [188]. The R-line is originates from the
temperature fluctuations at constant pressure, whereas the B-lines originate
from the pressure fluctuations at constant entropy. The width and the
integrated intensities of the R-peak and B-peaks provide information about
various thermodynamic susceptibilities and transport coefficients as discussed
in Ch.6. As some of the transport coefficients and response functions change
drastically, the study of the structure factor is very useful to understand
the behaviour of the fluid near the CEP where the B-peaks tend to vanish,
giving rise to severe modification in
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$. Since the width of a
distribution represents the decay rate of the fluctuation a distribution of
narrow width indicate the slower decay of the fluctuation.

The CEP in QCD belongs to the same universality class [51, 54],
$\mathcal{O}(4)$, as that of 3D Ising model and the liquid-gas critical point.
The universality hypothesis is understood from the long ranged behaviour of
the system and is independent on the microscopic characteristic. Therefore,
understanding the slow modes in one system can be a useful guidance for the
theoretical modelling relevant to the CEP of the QCD phase diagram. Near the
CEP, the dynamic structure factor without any slow mode has been quite
adequately studied earlier [62, 73]. The static and dynamic structure factors
have also been studied for QCD matter near CEP with stochastic diffusion
dynamics of net baryon number [209, 210, 211]. In this chapter, specifically,
we will study the role of the slow modes on the Rayleigh and Brillouin peaks
of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$.

By definition of the critical point the quantity, $\left(\partial P/\partial
V\right)_{\text{critical point}}$ approaches zero, leading to the divergence
of the isothermal compressibility ($\kappa_{T}\sim\left(\partial V/\partial
P\right)_{T}$) and hence reulting in large fluctuation in density, as
$\overline{\left(\delta n\right)^{2}}\propto\kappa_{T}$). This large
fluctuation may reflect on experimental observables. Thermal fluctuations can
cause entropy production leading to fluctuation in multiplicity, however,
their ensemble average is unaffected [212]. These fluctuations may not
influence the lower flow harmonics but affect their correlation. The CMS
collaborations have measured higher flow harmonics which again carry a strong
signature of thermal fluctuations [213], indicating that the measurement of
fluctuations in multiplicity and higher harmonics as a function of collision
energy at various rapidity bins may provide signal of the CEP (see Ref. [214]
for a recent review). An enhancement of event-by-event fluctuation in net
proton due to the CEP has been observed in Ref. [215]. The explicit dependence
of the correlation of the multiplicity fluctuation on the hydrodynamic
properties (speed of sound and shear viscosity) of the fluid in rapidity space
has been shown [216] within the scope of longitudinally expanding system with
boost invariance [217]. It will be interesting to study the same correlation
with the inclusion of the CEP (as both the viscosity and speed of sound get
affected by the CEP) by using full (3+1) dimensional expansion within the
ambit of relativistic second-order viscous hydrodynamics. The effects of out
of equilibrium dynamics and space-time inhomogeneity on the kurtosis of net
baryon and $\sigma$ field containing the signal of the CEP have been studied
in Refs. [218].

###  7.2 Hydrodynamic framework

In this chapter, we use the Eckart’s frame of reference as discussed in
Sec.3.3 in Ch.3, where it is considered that the heat flux is non-zero but the
particle current is zero. Therefore, energy-momentum tensor (EMT) and the the
particle current ($N^{\mu}$) are given by Eqs.(3.30) and (3.31). The
conservations of EMT and the net baryon number follows the Eq.(3.50).

The equation for relaxation including the additional scalar soft mode $\phi$
is introduced as [71]:

$D\phi=-F_{\phi}+A_{\phi}\theta\,,$ (7.1)

where, $\theta=\partial_{\mu}u^{\mu}$ and $D=u^{\mu}\partial_{\mu}$. Forms of
$F_{\phi}$ and $A_{\phi}$ are obtained by the second law of thermodynamics:

$\displaystyle\partial_{\mu}s^{\mu}\geq 0\,,$ (7.2)

where, $s^{\mu}$ is identified as the entropy four current and is defined as

$\displaystyle s^{\mu}=su^{\mu}+\Delta s^{\mu}\,.$ (7.3)

In the hydro+ formalism [71], the partially equilibrated entropy gets an extra
contribution from the soft mode $\phi$, and the change in entropy density can
be written as

$ds_{+}=\beta_{+}d\epsilon-\alpha_{+}dn-\pi d\phi\,,$ (7.4)

where, $\epsilon$ is the energy density, $n$ is the baryon number density, and
$\pi$ is the cost in energy for including $\phi$ modes in the system, called
the corresponding chemical potential. Here, $\beta_{+}=1/T$ and
$\alpha_{+}=\mu_{+}/T$, where, $T$ is the temperature $\mu_{+}$ is the
chemical potential.

In the present work it is found that, imposition of the extensivity condition
plays an important role to determine the form of $A_{\phi}$. From here onward,
for simplicity we drop the subscript “$+$” and write the following relations
as obtained from the extensivity condition discussed in Appendix D.1, in the
presence of $\phi$ as:

$\displaystyle\beta\,dP$ $\displaystyle=$
$\displaystyle-(\epsilon+P)d\beta+nd\alpha+\phi d\pi\,,$ (7.5) $\displaystyle
s$ $\displaystyle=$ $\displaystyle\beta(\epsilon+P-\mu n)-\pi\phi\,.$ (7.6)

The equation for the slow mode is a form of relaxation type, therefore,
keeping up to first order of slowly evolving soft mode is adequate to maintain
causality. Therefore,

$\partial_{\mu}s^{\mu}=(F_{\phi}-b\partial_{\mu}q^{\mu})\pi-q^{\mu}\partial_{\mu}(\beta+b\pi)-\beta(\partial_{\mu}u_{\nu})\Delta
T^{\mu\nu}+\partial_{\mu}(\Delta s^{\mu}+\beta q^{\mu}+b\pi
q^{\mu})+S_{n}\theta\,,$ (7.7)

where, $b$ represents the coupling strength and

$S_{n}=s-\beta(\epsilon+P)+\mu\beta n+\pi A_{\phi}\,.$ (7.8)

The soft mode couples with the heat flux satisfying $\partial_{\mu}s^{\mu}\geq
0$. Now, with $S_{n}=0$ and redefining $\pi$, $q^{\mu}$, $\Delta T^{\mu\nu}$
and $s^{\mu}$, we have

$\displaystyle\Delta s^{\mu}=-\beta q^{\mu}-b\pi q^{\mu}\,,$ (7.9)

and

$\displaystyle q^{\mu}$ $\displaystyle=$ $\displaystyle-\kappa
T[Du^{\mu}-\frac{1}{\beta}\Delta^{\mu\nu}\partial_{\nu}(\beta+b\pi)]\,,$
(7.10) $\displaystyle F_{\phi}$ $\displaystyle=$
$\displaystyle\gamma\pi-b\partial_{\mu}[\kappa
TDu^{\mu}-\frac{\kappa}{\beta}\Delta^{\mu\nu}\partial_{\nu}(\beta+b\pi)]\,,$
(7.11)

where, the proportionality constants $\kappa\geq 0$, $\gamma\geq 0$ are
respectively thermal conductivity and relaxation rate of slow modes. Along
with the above expression of fluxes one also needs, $S_{n}=c\,\theta$ with
$c\,\geq 0$ being a constant to satisfy the relation
$\partial_{\mu}s^{\mu}\geq 0$. However, since, $S_{n}$ contains local
thermodynamic quantities at zeroth order in derivative, it can not be a
function of the four divergence of fluid velocity, $\theta$. In other words,
the relation $S_{n}=c\,\theta$ with non-zero $c$ will make the local
equilibrium entropy density in fluid rest frame a function of the dissipative
gradient of the fluid velocity which is frame dependent. In that case, the
local equilibrium entropy will be frame dependent which is contradictory to
its definition as a frame independent local thermodynamic quantity, so $c$
should be zero, implies $S_{n}=0$. Therefore, the dissipative fluxes (in Eqs.
(3.43)-(3.45)) get modified with the coupling to soft modes in the MIS theory
as:

$\displaystyle\Pi=-\frac{1}{3}\zeta\Big{[}\partial_{\mu}u^{\mu}+\beta_{0}D\Pi-\tilde{\alpha}_{0}\partial_{\mu}q^{\mu}\Big{]}\,,$
(7.12a) $\displaystyle q^{\mu}=-\kappa
T\Delta^{\mu\nu}\Big{[}\beta\partial_{\nu}(T+b\pi)+Du_{\nu}+\beta_{1}Dq_{\nu}-\tilde{\alpha}_{0}\partial_{\nu}\Pi-\tilde{\alpha}_{1}\partial_{\lambda}\pi^{\lambda}_{\nu}\Big{]}\,,$
(7.12b)
$\displaystyle\pi^{\mu\nu}=-2\eta\Big{[}\Delta^{\mu\nu\rho\lambda}\partial_{\rho}u_{\lambda}+\beta_{2}D\pi^{\mu\nu}-\tilde{\alpha}_{1}\Delta^{\mu\nu\rho\lambda}\partial_{\rho}q_{\lambda}\Big{]}\,,$
(7.12c)

with proportionality constants $\eta\geq 0$, $\zeta\geq 0$, where $\eta$ and
$\zeta$ are respectively the shear and bulk viscous coefficients. The
coefficients $\tilde{\alpha_{0}},\tilde{\alpha_{1}},\tilde{\beta_{1}}$ are the
relaxation and coupling coefficients, and the expressions for those
coefficients are given in Appendix-A, which are evaluated through the
thermodynamic integral from kinetic theory prescription. [110].

####  7.2.1 Linearized equations and the dynamic structure factor

To linearize the hydrodynamic equations for small deviations from equilibrium
field quantities, we assume $\mathcal{Q}=\mathcal{Q}_{0}+\mathcal{\delta Q}$,
where $\mathcal{Q}$, $\mathcal{Q}_{0}$ and $\delta\mathcal{Q}$ represent
general hydrodynamic variables, their average values and fluctuations
respectively. $\mathcal{Q}_{0}=0$ is its average value for dissipative degrees
of freedom and $u_{0}^{\mu}=(-1,\,0,\,0,\,0)$ and $\delta
u^{\mu}=(0,\,\delta\mathchoice{\mbox{\boldmath$u$}}{\mbox{\boldmath$u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle u$}})$. The fluxes given in Eq. (7.8)
are substited in Eq. (3.30) and solved in the linearized domain in frequency-
wave vector space to obtain
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$. In the linearized
domain, the equations of motion for different space-time components of the
energy-momentum tensor and baryonic charge current obtained from Eqs. (3.50)
read as:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\frac{\partial\delta\epsilon}{\partial
t}-(\epsilon_{0}+P_{0})+\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}\cdot\delta\mathchoice{\mbox{\boldmath$u$}}{\mbox{\boldmath$u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}-\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}\cdot\delta\mathchoice{\mbox{\boldmath$q$}}{\mbox{\boldmath$q$}}{\mbox{\boldmath$\scriptstyle
q$}}{\mbox{\boldmath$\scriptscriptstyle q$}}\,,$ (7.13a) $\displaystyle 0$
$\displaystyle=$ $\displaystyle-(\epsilon_{0}+P_{0})\frac{\partial}{\partial
t}\delta u^{i}-\partial^{i}(\delta P+\delta\Pi)+\frac{\partial}{\partial
t}\delta q^{i}-\partial_{j}\Pi^{ij}\,,$ (7.13b) $\displaystyle 0$
$\displaystyle=$ $\displaystyle-\frac{\partial}{\partial t}\delta
n-n_{0}\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}\cdot\delta\mathchoice{\mbox{\boldmath$u$}}{\mbox{\boldmath$u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle u$}}\,,$ (7.13c) $\displaystyle 0$
$\displaystyle=$
$\displaystyle\delta\Pi+\frac{1}{3}\zeta[\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}\cdot\delta\mathchoice{\mbox{\boldmath$u$}}{\mbox{\boldmath$u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle u$}}+\beta_{0}\frac{\partial}{\partial
t}\delta\Pi-\tilde{\alpha_{0}}\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}\cdot\delta\mathchoice{\mbox{\boldmath$q$}}{\mbox{\boldmath$q$}}{\mbox{\boldmath$\scriptstyle
q$}}{\mbox{\boldmath$\scriptscriptstyle q$}}]\,,$ (7.13d) $\displaystyle 0$
$\displaystyle=$ $\displaystyle\delta q^{i}+\kappa T_{0}\nabla^{i}\delta
T+\kappa T_{0}\frac{\partial}{\partial t}\delta u^{i}+\kappa
T_{0}\beta_{1}\frac{\partial}{\partial t}\delta q^{i}-\kappa
T_{0}\tilde{\alpha}_{0}\nabla^{i}\delta\Pi-\kappa
T_{0}\tilde{\alpha}_{1}\nabla_{j}\pi^{ij}\,,$ (7.13e) $\displaystyle 0$
$\displaystyle=$
$\displaystyle\delta\pi^{ij}+2\eta\delta^{ijlm}(\partial_{l}\delta
u_{m}-\tilde{\alpha}_{1}\partial_{l}\delta
q_{m})+2\eta\beta_{2}\frac{\partial}{\partial t}\delta\pi^{ij}\,,$ (7.13f)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle-\frac{\partial}{\partial
t}\delta\phi-(\gamma+T_{0}^{2}\frac{K_{q\pi}^{2}}{\kappa}\nabla^{2})C_{\phi\pi}\delta\phi-\big{[}\gamma+(T_{0}^{2}\frac{K_{q\pi}^{2}}{\kappa}-\frac{K_{q\pi}}{C_{T\pi}})\nabla^{2}\big{]}C_{T\pi}\delta
T$ (7.13g)
$\displaystyle-(\gamma+T_{0}^{2}\frac{K_{q\pi}}{\kappa}\nabla^{2})C_{\pi
n}\delta n-T_{0}K_{q\pi}\frac{\partial}{\partial t}(\nabla_{i}\delta
u^{i})+\tilde{\phi}\nabla_{i}\delta u^{i},$

where $K_{q\pi}=b\kappa$ and

$\displaystyle C_{A\pi}=\frac{\partial\pi}{\partial
A},\,\,\,\,\text{with}\,\,\,A\equiv(T,n,\phi)\,.$ (7.14)

We get a set of linear equations in $\omega-k$ space (Appendix E.1) by taking
the Fourier-Laplace transformation of the above equations. The set of
equations given in Eqs. (E.3)-(E.13) can be written in matrix form as:

$\displaystyle\mathcal{M}\delta\mathcal{Q}=\mathcal{A}\,,$ (7.15)

where $\mathcal{M}$, is an $11\times 11$ matrix representing the coefficients
of the column vector comprinsing the quantities, $\delta n,\delta T,\delta
u_{||},\delta u_{\perp},\delta\Pi,\delta q_{||}$, $\delta
q_{\perp},\delta\pi_{||\,||},\delta\pi_{||\,\perp},\delta\pi_{\perp\,\perp}$,
and $\delta\phi$. The solution of the set of linear equations can be written
as:

$\delta\mathcal{Q}=\mathcal{M}^{-1}\mathcal{A}\,,$ (7.16)

In the present work we are interested in evaluating the two point correlation
of density fluctuation. The solution of the set of equations represented by
Eq. (7.16) leads to the following expression for density fluctuation ($\delta
n$),

$\displaystyle\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ $\displaystyle=$
$\displaystyle\big{[}\epsilon_{n}\mathcal{M}^{-1}_{12}+\mathcal{M}^{-1}_{11}\big{]}\delta{n}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)-\epsilon_{T}\big{[}\mathcal{M}^{-1}_{12}\big{]}\delta{T}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)$ (7.17) $\displaystyle+$
$\displaystyle\big{[}\epsilon_{\phi}\mathcal{M}^{-1}_{12}+\mathcal{M}^{-1}_{13}\big{]}\delta{\phi}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)$ $\displaystyle+$
$\displaystyle\big{[}\epsilon_{0}\mathcal{M}^{-1}_{14}-ikT_{0}\mathcal{M}^{-1}_{13}\kappa_{\text{q$\pi$}}+\mathcal{M}^{-1}_{14}P_{0}-T_{0}\chi\mathcal{M}^{-1}_{15}\big{]}\delta{u_{||}}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)$ $\displaystyle-$
$\displaystyle\big{[}\beta_{1}T_{0}\chi\mathcal{M}^{-1}_{15}+\mathcal{M}^{-1}_{14}\big{]}\delta{q_{||}}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)+\frac{1}{3}\zeta\mathcal{M}^{-1}_{16}\beta_{0}\delta{\pi}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)$ $\displaystyle-$
$\displaystyle
2\eta\mathcal{M}^{-1}_{17}\beta_{2}\delta{\pi}_{||||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)-2\eta\mathcal{M}^{-1}_{18}\beta_{2}{\pi}_{\perp\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)$ $\displaystyle-$
$\displaystyle\big{[}\beta_{1}T_{0}\chi\mathcal{M}^{-1}_{110}+\mathcal{M}^{-1}_{19}\big{]}\delta{q}_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)-2\eta\mathcal{M}^{-1}_{111}\beta_{2}\delta{\pi}_{||\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0),$ $\displaystyle+$
$\displaystyle\big{[}\epsilon_{0}\mathcal{M}^{-1}_{19}+\mathcal{M}^{-1}_{19}P_{0}-T_{0}\chi\mathcal{M}^{-1}_{110}\big{]}\delta{u}_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\,,$

where, $\epsilon_{x}=\frac{\partial\epsilon}{\partial x}$ and $x=n$,$T$,
$\phi$.

Now we define the correlation of density fluctuations, as
$\mathcal{S^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ by the following
expression [60]:

$\displaystyle\mathcal{S^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)=\Big{<}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)\delta{n}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,0)\Big{>}\,.$ (7.18)

Since the correlation between two independent thermodynamic variables, say,
$\mathcal{Q}_{i}$ and $\mathcal{Q}_{j}$ vanishes, i.e. we have,

$\displaystyle\Big{<}\delta\mathcal{Q}_{i}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)\delta{\mathcal{Q}}_{j}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,0)\Big{>}=0,\,\,\,\,i\neq j\,.$
(7.19)

Imposing Eq.(7.19) into Eq.(7.17), the $\mathcal{S}^{{}^{\prime}}_{nn}$ can
only contain two elements of the matrix $\mathcal{M}^{-1}$ as:

$\displaystyle\mathcal{S}^{{}^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=-\Big{[}\big{(}\frac{\partial{\epsilon}}{\partial
n}\big{)}\mathcal{M}^{-1}_{12}-\mathcal{M}^{-1}_{11}\Big{]}\Big{<}{\delta{n}}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\,0){\delta{n}}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,0)\Big{>}\,,$ (7.20)

The final expression for the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ can be obtained as:

$\displaystyle\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=\frac{\mathcal{S}^{{}^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)}{\Big{<}{\delta{n}}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\,0){\delta{n}}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\,0)\Big{>}}=-\Big{[}\big{(}\frac{\partial{\epsilon}}{\partial
n}\big{)}\mathcal{M}^{-1}_{12}-\mathcal{M}^{-1}_{11}\Big{]}\,.$ (7.21)

This is the correlation in density fluctuation or the power spectrum with the
inclusion of the extra degree of freedom, $\phi$.

###  7.3 Results and discussion

Figure 7.1:  (a) Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $\eta/s=\zeta/s=\kappa T/s=1/4\pi$, when the system is
away from CEP ($r=0.2$). The peaks are appearing as very narrow due to the
scale chosen along x-axis, change in x-axis scale makes the width visible (see
the inset for the R-peak). (b) The system is close to the CEP ($r=0.01$). The
results are obtained with the EoS containing the CEP and the transport
coefficients, thermodynamic response function and the relaxation coefficients
are estimated from the scaling behaviour.

We aim to study the dynamic structure factor near the CEP by introducing a
slow scalar field. The effect of the CEP is taken through an EoS, containing
the critical point. The various transport coefficients and the thermodynamic
response functions near the CEP play important roles determining the structure
factor are taken from the Eq.(6.17).

Figure 7.2:  Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ and $k$
for $r=0.2$, i.e. when the system is away from CEP. Figure 7.3:  Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $r=0.2$, i.e. the system is away from CEP for
$K_{q\pi}=0.$ Figure 7.4:  Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $r=0.2$, i.e. the system is away from CEP but with higher
value of $(\frac{\partial P}{\partial\phi})$. The asymmetry in the B-peaks is
clearly visible. Figure 7.5:  Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $r=0.2$, i.e. the system is away from CEP with increased
$\phi$ modes in the system which increases the thermal fluctuation.

The variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ is shown
in Fig.7.1(a) for $k=0.1$ fm-1, when the system is away from the CEP. Without
the presence of $\phi$, the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ has three peaks (see
Fig.6.1(b)). These are identified as the R-peak located at $\omega=0$ and the
B-peaks symmetrically situated on either sides of the R-peak. While with the
presence of the slow mode $\phi$, the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ admits four peaks. The
locations of the four peaks of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ for different values
of $\omega$ are obtained from the dispersion relation provided in the Appendix
F. The Brillouin peaks (B-peaks) are identified as the away side peaks of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$. The Stokes (left
side) and the anti-Stokes (right side) components are located asymmetrically
(unlike the figure shown in Fig.6.1) on either side of the origin with uneven
magnitudes. The other two peaks (closer to $\omega=0$) are present even with
vanishing speed of sound in the neighbourhood of CEP (as shown in Fig.7.1(b)),
arising due to the coupling of $\phi$ with $q^{\mu}$. The coupling of $\phi$
with the hydrodynamical fields results in one extra peak, and shifted the
$R$-peak from $\omega=0$. The peak appears at $\omega=0$ represents static
thermal fluctuations (without time variation) on the magnitude of the
fluctuations. But their appearance at non-zero $\omega$ represents that there
is variation in the magnitude of the fluctuation with time. So the appearance
of these peaks near $\omega\neq 0$ is due to the time varying thermal
fluctuations which is resulted from the coupling of the heat flux to slow OEM.
The OEM have much slower dynamic evolution rate, so that thermal fluctuations
become time dependent instead of being time independent. A non-relativistic
fluid system placed in a stationary temperature gradient showed asymmetry
between Stokes and anti-Stokes components, the asymmetry is maximum when
$k||\nabla T$ and vanishes with $k\perp\nabla T$. From Fig.7.1(a), it is seen
that the asymmetry in Rayleigh component is exactly opposite to the Brillouin
components. In the vicinity of the CEP (Fig.7.1(b)), the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ shows two peaks closer
to $\omega=0$ as the B-peaks disappear due to the absorption of sound at the
CEP.

In Fig. 7.2 the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ is plotted in
$\omega-k$ plane when the system is away from CEP. The
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with non-zero $\phi$
is quite similar (as shown in Fig.6.4) in a sense that the B-peaks are moving
away from the R-peak for larger $k$-values, and their magnitudes gets smaller
and smaller. However, in presence of $\phi$, these peaks may reappear at non-
zero $k$ values too due to the coupling coupling of $\phi$ with the
hydrodynamic fields. The symmetrically positioned B-peaks of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ is reattained when the
coupling $K_{q\pi}$ is set to zero (shown in Fig. 7.3).

Figure 7.6:  Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $r=0.2$. The results for second-order and first-order
hydrodynamics are compared here. Figure 7.7:  Variation of
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $r=0.2$ with (dotted line) and without (solid) $\phi$
mode, when the longitudinal modes are only considered.

The asymmetry in the position in
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ of the B-peaks
increases with increase in $\big{(}\partial P/\partial\phi\big{)}$, represents
the effect of slow modes on the local pressure, which accounts for the effect
of slow modes in the speed of sound in the system (Appendix F). The asymmetry
in
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with respect to
$\omega$ increases with increase in $\big{(}\partial P/\partial\phi\big{)}$.
This is distinctly visible in Fig. 7.4 in comparison to results displayed in
Fig. 7.1(a).

Increasing $\phi$ modes in a system induces fluctuations in the system. In
Fig.7.5, it is observed that the magnitudes of the peaks get enhanced due to
the enhancement in $\phi$ value. The introduction of slow modes is intended to
account for the higher order gradients (required for system far-away from
equilibrium) which become relevant at hydrodynamic scales.

In Fig. 7.6, the structure factor for second-order hydrodynamics has been
compared with first-order hydrodynamics. Interestingly, the structure factor
for first-order hydrodynamics admits a R-peak at the origin and two symmetric
B-peaks located on the opposite sides of the R-peak. The effects of $\phi$
seems to be inconsequential in-terms of the arising of extra peaks in the
first-order theory because of the vanishing of various coupling and relaxation
coefficients in first order approximation. However, the effect of slow modes
in first-order theory is there in the asymmetry of B-peak magnitudes.

In Eq.(7.12b) shows that the coupling of $\phi$ enters the system through
$q^{\mu}$. We know that first-order theory can be obtained from second-order
theory by setting the coupling coefficients ($\tilde{\alpha_{0}}$ and
$\tilde{\alpha_{1}}$) and relaxation coefficients
($\beta_{0},\tilde{\beta_{1}},\beta_{2}$) to zero in Eqs.(7.12a)-(7.12c).
Therefore, the only term through which the coupling of $\phi$ enters the
first-order hydrodynamics is given by the first term of $q^{\mu}$ in
Eq.(7.12c) that is, $q^{\mu}=-\kappa
T\Delta^{\mu\nu}\Big{[}\beta\partial_{\nu}(T+b\pi)\Big{]}$. It is interesting
to note that the incorporation of $\phi$ has just shifted the temperature from
$T$ to $T+b\pi$ through its chemical potential $\pi$. This term does not
introduce any $\omega$ dependence, therefore, the presence of $\phi$ in first-
order hydrodynamics will not introduce any extra peak.

Till now, it is evident that the extra-peaks are originating from coupling of
heat flux with the scalar field in second-order theory. However, it is not
clear whether, the coupling with the longitudinal or transverse modes or both
are responsible for these peaks in second order theory. To verify, the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ derived from the
longitudinal (by neglecting the transverse modes) dispersion relation with and
without $\phi$ has been depicted in Fig. 7.7. We find that the width of the
peaks are narrowed down due to the inclusion of the $\phi$ field (blue dashed
line). The decay rate of the thermal fluctuation is reflected through the
width of the R-peak, which becomes smaller due to the introduction of slow
mode $\phi$. Therefore, the decay of the fluctuation becomes relaxed in the
presence of the $\phi$. In comparison of Figs.7.1(a), 7.3, 7.6 and 7.7, we can
say that the extra peak appeared in Fig. 7.1(a) is due to the coupling of the
transverse modes with $\phi$. It is also iimportant to mention that the
presence of $\phi$ modes can result in the extra peak in the dynamic structure
factor when the transverse modes in the causal theory of hydrodynamics is
concerned.

However, it may create confusion whether the absence of extra peaks at
$\phi\neq 0$ is due to the longitudinal modes only or because of first order
hydrodynamics. This can be resolved from the observation that the height
asymmetry is different for these two scenarios.

In summary, we can argue that the dynamic structure factor with presence of a
slow out-of-equilibrium mode changes significantly. We found four peaks of
Lorentzian distribution, whereas the dynamic structure factor without any out-
of-equilibrium mode shows three peaks. The peaks in
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\phi$ are
narrower in width compared to the peaks appeared in the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ without any $\phi$.
This narrower width of the distribution indicates the slow relaxation of the
fluctuation in the system, causing critical slowing down. The extra peak
appears in the
$\mathcal{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\,\omega)$ with $\phi$ is due to
the coupling of the $\phi$ mode with the transverse modes when the second-
order theory of hydrodynamics is concerned.

## CHAPTER 8 Propagation of Nonlinear waves near the QCD critical point

###  8.1 Introduction

In this chapter, we discuss about our publication, presented in Ref. [219]. So
far in Ch.5, Ch.6 and Ch.7, we have discussed about the possible fate of small
perturbations (under linear approximation) near the QCD critical point. But in
this chapter, we discuss about the perturbations which are comparatively
larger in magnitudes. In relativistic heavy-ion collider experiments (RHIC-E),
the partons are produced with a wide range of transverse momentum ($p_{T}$).
The partons with relatively smaller $p_{T}$, on subsequent scattering,
contribute to produce a locally thermalized hot medium QGP, whereas the
partons with high $p_{T}$ do not contribute in the medium formation, but they
traverse the medium as jets with associated radiated partons, and produce
disturbances inside the medium by depositing energy into it through
interactions. The supersonic partons, can produce larger perturbations in the
medium leading to a nonlinear wave. The quantum fluctuations may also lead to
inhomogeneity in the medium, which may serve as perturbations on the
hydrodynamically evolving medium. The hydrodynamic response of the medium to
such perturbations, will be imprinted into the medium and those imprint are
translated into the hadrons spectra at the freeze-out hyper-surface. This
imprint will also have effects on other penetrating particles like photons and
lepton pairs emitted throughout the evolution history. Specifically, the
emergence of two maximas at $\Delta\phi=\pi\pm 1.2$ radian in the quenched
away side jet [220, 221]. This double-hump structure in the correlation
function of jet is elucidated as the effect of the produced Mach cone in
hydrodynamic response to the perturbation created by jets [222]. The flow
harmonics (created due to the momentum anisotropy) of produced particle is
attributed to the hydrodynamic response of the QGP to the initial geometry.
During expansion and cooling of the QGP, on transition to hadronic phase,
these anisotropies gets transmitted into hadronic momentum spectra for the
conservation of momentum.

The damping of the Mach cone is related to the hydrodynamic response of the
medium. Any change in the behaviour of QGP and in the nature of transition
from QGP to hadron gas e.g. presence of the CEP will affect the response. In
general, the hydrodynamic response can also be treated as linear as well as
nonlinear, depending on the magnitude of the perturbations. A small
disturbance is treated as a linear perturbation [223, 224, 225, 226, 72, 73,
62] whereas a relatively large perturbations (comparable to unperturbed value)
should be treated as nonlinear perturbation [227, 228, 229, 230, 131, 231].
The momentum anisotropy can be accounted mostly as a linear response to the
initial geometry (eccentricity) with a small contribution from nonlinear
effects [232, 233]. Moreover, the produced Mach front has been found to travel
as a shock front [234]. The propagation of this shock front is controlled by
the hydrodynamic response which may be linear or nonlinear. Due to the large
amplitude of this shock front one expects the perturbation to be nonlinear in
nature [235]. In Refs. [236, 237, 238] the presence of Mach cone effect on the
away side jet in two particle and three particle correlations have been
explained by the non-dissipative property of the nonlinear waves.

The fate of nonlinear waves in QGP has been studied by Fogaça et al. [131,
231] with the ambit of Naiver-Stokes (NS) as well as Müller-Israel-Stewart
(MIS) hydrodynamics with the incorporation of the shear viscosity only. They
have argued that the nonlinear waves survive, despite the presence of the
shear viscosity. The propagation of nonlinear perturbation due to energy
deposition by the jets into the medium have been argued to be responsible for
broadening of away the side jet [131]. The nonlinear radial spreading of
localized waves (solitons) of large magnitude, which are created from the
thermalization of the energy deposited by jets considered to be the cause of
broadening of the peak.

The hydrodynamic response will rely on the state of the fluid phase and also
on the nature of the perturbations (linear or nonlinear). The effects of
linear response has already been analysed and found to be suppressed near the
CEP [62, 72, 73]. In this regard, however, it will be relevant to investigate
the possible fate of the nonlinear perturbations in presence of the CEP. This
is particularly crucial because physical process like the energy deposition by
jets can be large enough to ignite nonlinear effects. In such cases, the
nonlinear wave should play dominant role in the formation of Mach-cone.
Therefore, the study of the effects of nonlinear perturbations in QGP under
the influence of the CEP is crucially important.

Near the CEP where fluctuations are expected to grow, the study of the
nonlinear waves is important for understanding the broadening of the away side
jet and the fate of double-hump structure in the correlation function. This
could be a very good guide for the critical point search in QCD phase diagram.
As a matter of fact to the best of the knowledge of the author of this
dissertation, the fate of the nonlinear disturbances due to the presence of
the CEP has not been reported till date. In this work, we address the
propagation of nonlinear waves with the presence of the CEP, where we have
considered the relevant dissipative coefficients. The inclusion of all those
coefficients are crucial because some of these are known to diverge near the
CEP [239, 240, 241, 162, 169]. However, the equations which govern the
evolution of the nonlinear waves in the presence of all the relevant transport
coefficients i.e. shear viscosity ($\eta$), bulk viscosity ($\zeta$) and
thermal conductivity ($\kappa$) are not readily available within the scope of
the MIS hydrodynamics, and hence we deduce those equations for the present
work.

###  8.2 1-D flow equations

In this section, the relevant equations are derived to study the propagation
of nonlinear perturbation in a fluid in $(1+1)$ dimension. We have used the
MIS theory for the evolution of the QGP and adopted the Eckart’s frame of
reference (see Ch.3, Sec.3.3) to define fluid velocity, where it is considered
that the heat flux is non-zero but the particle current is zero. Therefore,
energy-momentum tensor (EMT) and the the particle current ($N^{\mu}$) are
given by Eqs.(3.30) and (3.31). The conservations of EMT and the net baryon
number are written as follows

$\displaystyle\partial_{\mu}T^{\mu\nu}=0~{},\,\,\,\,\,\partial_{\mu}N^{\mu}=0~{}.$
(8.1)

The evolution equations of energy density and fluid velocity are obtained from
the Eqs.(8.1) by taking projection along the directions parallel and
perpendicular to $u^{\mu}$ as:

$\displaystyle D\epsilon=-(\epsilon+P)\theta+(\partial_{\mu}u_{\nu})\Delta
T^{\nu\mu}-(\partial_{\mu}q^{\mu})~{},$ (8.2)
$\displaystyle(\epsilon+P)Du^{\alpha}=\Delta^{\alpha\mu}\partial_{\mu}-\Delta^{\alpha}_{\nu}\partial_{\mu}\Delta
T^{\nu\mu}~{},$ (8.3)

where, $u^{\mu}u_{\mu}=-1,$ and

$\displaystyle\theta$ $\displaystyle=$
$\displaystyle\partial_{\mu}u^{\mu}=v\gamma^{3}\frac{\partial v}{\partial
t}+\gamma^{3}\frac{\partial v}{\partial x}~{},$ (8.4)
$\displaystyle\partial_{\mu}q^{\mu}$ $\displaystyle=$ $\displaystyle
q^{x}\frac{\partial v}{\partial t}+v\frac{\partial q^{x}}{\partial
t}+\frac{\partial q^{x}}{\partial x}~{}.$ (8.5)

Here,

$\displaystyle
u_{\mu}q^{\nu}=0,\,\,\,u_{\mu}\pi^{\mu\nu}=0,\,\,\,g_{\mu\nu}\pi^{\mu\nu}=0,\,\,\,\pi^{xy}=\pi^{zx}=\pi^{yz}=0,\,\,\,\pi^{yy}=\pi^{zz}=\pi^{T}~{},$
(8.6)

which leads to,

$\displaystyle\pi^{00}=v^{2}\pi^{xx},\,\,\,\pi^{xt}=\pi^{tx}=v^{2}\pi^{xx},\,\,\,\pi^{T}=\pi^{tx}=\frac{(v^{2}-1)}{2}\pi^{xx}~{},$
(8.7)

and

$\displaystyle q^{t}=vq^{x},\,\,\,q^{y}=q^{z}=q^{T}~{}.$ (8.8)

Now from the energy-momentum conservation equations i.e.  from Eqs. (8.2) and
(8.3), we get,

$\displaystyle\gamma v\Big{(}\frac{\partial\epsilon}{\partial
t}+\frac{\partial\epsilon}{\partial x}\Big{)}=$
$\displaystyle\Big{[}(h+\Pi)v\gamma^{3}+v\gamma\pi^{xx}+q^{x}\\}\frac{\partial
v}{\partial t}\Big{]}$ (8.9)
$\displaystyle+\Big{[}(h+\Pi)\gamma^{3}+\gamma\pi^{xx}\Big{]}\frac{\partial
v}{\partial x}+v\frac{\partial q^{x}}{\partial t}+\frac{\partial
q^{x}}{\partial x},$
$\displaystyle\Big{[}(h+\Pi)v\gamma^{4}+v\gamma^{2}(1-2v^{2})v\pi^{xx}+\gamma^{3}(1+v^{2})vq^{x}\Big{]}\frac{\partial
v}{\partial t}$ (8.10) $\displaystyle+$
$\displaystyle\Big{[}(h+\Pi)v\gamma^{4}-\gamma^{2}v^{2}\pi^{xx}+\gamma
v(1+2\gamma^{2})q^{x}\Big{]}\frac{\partial v}{\partial x}$ $\displaystyle=$
$\displaystyle-v^{2}\Big{[}\gamma^{2}\frac{\partial P_{\zeta}}{\partial
t}+\frac{\gamma^{2}}{v}\frac{\partial P_{\zeta}}{\partial
x}+\frac{\partial\pi^{xx}}{\partial
t}+\frac{1}{v}\frac{\partial\pi^{xx}}{\partial
x}+\frac{\gamma}{v}\frac{\partial q^{x}}{\partial t}-\gamma\frac{\partial
q^{x}}{\partial x}\Big{]}$

We define $P+\Pi=P_{\zeta}$. The equation governing the net baryon number
conservation reads,

$\displaystyle\frac{\partial n}{\partial t}+v\frac{\partial n}{\partial
x}=-n\gamma^{2}\Big{[}v\frac{\partial v}{\partial t}+\frac{\partial
v}{\partial x}\Big{]}~{},$ (8.11)

The other three equations originating from dissipative fluxes are given by the
Eqs.(3.43), (3.44), and (3.45), which can be written as:

$\displaystyle\Pi+\zeta\beta_{0}\Big{[}\gamma\frac{\partial\Pi}{\partial
t}+v\frac{\partial\Pi}{\partial x}\Big{]}$ $\displaystyle=$
$\displaystyle-\zeta\Big{[}(v\gamma^{3}-\tilde{\alpha_{0}}q^{x})\frac{\partial
v}{\partial t}+\gamma^{3}\frac{\partial v}{\partial
x}-\tilde{\alpha_{0}}(v\frac{\partial q^{x}}{\partial t}-\frac{\partial
q^{x}}{\partial x})\Big{]}~{},$ (8.12) $\displaystyle q^{x}+\kappa
T\tilde{\beta_{1}}\Big{[}\gamma\frac{\partial q^{x}}{\partial
t}+v\frac{\partial q^{x}}{\partial x}\Big{]}=\kappa
T\Big{[}(-\gamma^{4}+\tilde{\beta_{1}}\gamma^{3}vq^{x}+\tilde{\alpha_{1}}(2-\gamma^{2})\pi^{xx})\frac{\partial
v}{\partial t}$
$\displaystyle+(-v\gamma^{4}+\tilde{\beta_{1}}\gamma^{3}v^{2}q^{x}-\tilde{\alpha_{1}}\gamma^{2}v\pi^{xx})\frac{\partial
v}{\partial x}+\tilde{\alpha_{1}}v\frac{\partial\pi^{xx}}{\partial
t}+\tilde{\alpha_{1}}\frac{\partial\pi^{xx}}{\partial x}$
$\displaystyle-\tilde{\alpha_{0}}v\gamma^{2}\frac{\partial\Pi}{\partial
t}-\tilde{\alpha_{0}}\gamma^{2}\frac{\partial\Pi}{\partial
x}+\frac{1}{T}v\gamma^{2}\frac{\partial T}{\partial
t}+\frac{1}{T}\gamma^{2}\frac{\partial T}{\partial x}\Big{]}~{},$ (8.13)
$\displaystyle\pi^{xx}+\frac{4}{3}\eta\beta_{2}\Big{[}\gamma\frac{\partial\pi^{xx}}{\partial
t}+v\frac{\partial\pi^{xx}}{\partial
x}\Big{]}=\frac{4}{3}\eta\Big{[}(\gamma^{5}+\tilde{\alpha_{1}}\gamma^{4}vq^{x}+2\beta_{2}\gamma^{3}\pi^{xx})\frac{\partial
v}{\partial t}$
$\displaystyle+(\gamma^{5}+\tilde{\alpha_{1}}\gamma^{4}vq^{x}+2\beta_{2}\gamma^{3}v^{2}\pi^{xx})\frac{\partial
v}{\partial x}-\tilde{\alpha_{1}}\gamma^{2}v\frac{\partial q^{x}}{\partial
t}-\tilde{\alpha_{1}}\gamma^{2}\frac{\partial q^{x}}{\partial x}\Big{]}~{}.$
(8.14)

####  8.2.1 Derivation of nonlinear wave equations

The above hydrodynamic equations from MIS theory is used to derive the
equations governing the motion of the nonlinear perturbations in the fluid. We
have adopted Reductive Perturbative Method (RPM) technique [242, 243, 244,
245], where we need to define ‘stretched co-ordinates’ as,

$\displaystyle X=\frac{\sigma^{1/2}}{L}(x-c_{s}t),\text{and}\hskip
11.38092ptY=\frac{\sigma^{3/2}}{L}(c_{s}t)~{},$ (8.15)

where, $L$ is some characteristic length and $c_{s}$ is the speed of sound.
Therefore, from the above equations we get

$\displaystyle\frac{\partial}{\partial
x}=\frac{\sigma^{1/2}}{L}\frac{\partial}{\partial X},\text{and}\hskip
11.38092pt\frac{\partial}{\partial
t}=-c_{s}\frac{\sigma^{1/2}}{L}\frac{\partial}{\partial
X}+c_{s}\frac{\sigma^{3/2}}{L}\frac{\partial}{\partial Y}~{},$ (8.16)

where, $\sigma$ is the expansion parameter. The coordinate $X$ is measured
from the frame of propagating sound waves, whereas the $Y$ represents fast
moving coordinate. The RPM technique is designed to preserve the structural
form of the parent equation. Now we perform the simultaneous series expansion
of hydrodynamic quantities in powers of $\sigma$ to get,

$\displaystyle\hat{\epsilon}$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{\epsilon_{0}}=1+\sigma\epsilon_{1}+\sigma^{2}\epsilon_{2}+\sigma^{3}\epsilon_{3}+...,\,\hat{p}=\frac{P}{P_{0}}=1+\sigma
p_{1}+\sigma^{2}p_{2}+\sigma^{3}p_{3}+...,$ $\displaystyle\hat{v}$
$\displaystyle=$ $\displaystyle\frac{v}{c_{s}}=\sigma
v_{1}+\sigma^{2}v_{2}+\sigma^{3}v_{3}+...,\,\,\,\,\,\,\hat{\Pi}=\frac{\Pi}{P_{0}}=\sigma\Pi_{1}+\sigma^{2}\Pi_{2}+\sigma^{3}\Pi_{3}+...,$
$\displaystyle\hat{q}^{x}$ $\displaystyle=$
$\displaystyle\frac{q^{x}}{\epsilon_{0}}=\sigma
q^{x}_{1}+\sigma^{2}q^{x}_{2}+\sigma^{3}q^{x}_{3}+...,\,\hat{\pi}^{xx}=\frac{\pi^{xx}}{P_{0}}=\sigma\pi^{xx}_{1}+\sigma^{2}\pi^{xx}_{2}+\sigma^{3}\pi^{xx}_{3}+...,$
(8.17)

where $\epsilon_{0}$ and $P_{0}$ are the background equilibrium energy density
and pressure respectively on which the perturbations propagate. By collecting
terms with different order of $\sigma$ from the above series, different types
of equation like Breaking wave equation, Burgers equation or KdV (Korteweg-De
Vries) like equation, etc can be obtained [131, 246, 247].

We rewrite the equations of motion i.e. Eq. (8.9)-Eq.(8.2), using Eqs. (8.16)
and (8.17) and collect terms with different orders in $\sigma$ and then
finally revert from $(X,Y)\to(t,x)$ to obtain the evolution equations of
different order of perturbations. It is found in Ref. [231] that for the
conformal background, the evolution equation of the first-order perturbation
does not include any relaxation and coupling coefficients arising from the MIS
theory. Therefore, to get the effect of second-order, it is required to go to
the second-order in perturbation ($\hat{\epsilon_{2}}$). To obtain the time
evolution of second order perturbation, we need to collect terms up to third
order in $\sigma$ in the expansion, because the equations in $n$-th order in
$\sigma$ contains time derivatives of $\hat{\epsilon}_{n-1}$. Therefore, to
get equations containing time derivative of $\hat{\epsilon}_{2}$ we go up to
3rd order in $\sigma$. This leads to the following equations for the
perturbation in the energy density ($\hat{\epsilon}$) as:

$\displaystyle\frac{\partial\hat{\epsilon_{1}}}{\partial
t}+\left[1+(1-c_{s}^{2})\frac{\epsilon_{0}}{\epsilon_{0}+P_{0}}\hat{\epsilon_{1}}\right]c_{s}\frac{\partial\hat{\epsilon_{1}}}{\partial
x}-\left[\frac{1}{2(\epsilon_{0}+P_{0})}(\zeta+\frac{4}{3}\eta)\right]\frac{\partial^{2}\hat{\epsilon_{1}}}{\partial
x^{2}}=0,$ (8.18)

and

$\displaystyle\frac{\partial\hat{\epsilon_{2}}}{\partial
t}+\mathcal{S}_{1}\frac{\partial\hat{\epsilon_{2}}}{\partial
x}+\mathcal{S}_{2}\frac{\partial\hat{\epsilon_{1}}}{\partial
x}+\mathcal{S}_{3}\frac{\partial^{2}\hat{\epsilon_{1}}}{\partial
x^{2}}+\mathcal{S}_{4}\frac{\partial^{3}\hat{\epsilon_{1}}}{\partial
x^{3}}+\mathcal{S}_{5}\frac{\partial^{2}\hat{\epsilon_{2}}}{\partial
x^{2}}=0~{}.$ (8.19)

where $\hat{\epsilon_{i}}=\sigma^{i}\epsilon_{i}$ for $i=1,2,.....$. The
coefficients $\mathcal{S}_{i}$’s for $i=1$ to $5$ are:

$\displaystyle\mathcal{S}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon_{0}+P_{0}}\Big{[}c_{s}\\{\epsilon_{0}\left(1-\hat{\epsilon}_{1}\left(c_{s}^{2}-1\right)\right)+P_{0}\\}\Big{]},$
$\displaystyle\mathcal{S}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon_{0}+P_{0}}\Big{[}\epsilon_{0}c_{s}\\{c_{s}^{2}-1\\}\\{\epsilon_{0}\left(\left(2c_{s}^{2}+1\right)\hat{\epsilon}_{1}{}^{2}-\hat{\epsilon}_{2}\right)-P_{0}\hat{\epsilon}_{2}\\}\Big{]},$
$\displaystyle\mathcal{S}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{12(\epsilon_{0}+P_{0})^{2}}\Big{[}\epsilon_{0}\hat{\epsilon}_{1}\\{3c_{s}^{2}(7\zeta+8\eta)+3\zeta+4\eta\\}\Big{]},$
$\displaystyle\mathcal{S}_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{72c_{s}c_{V}(\epsilon_{0}+P_{0})^{2}}\Big{[}4c_{s}^{2}\\{3\kappa
T\epsilon_{0}(3\alpha_{0}\zeta+4\alpha_{1}\eta)$
$\displaystyle+3\zeta+4\eta+P_{0}(3\alpha_{0}\zeta+4\alpha_{1}\eta)\\}+(\epsilon_{0}+P_{0})(9\beta_{0}\zeta^{2}+16\beta_{2}\eta^{2})-c_{V}(3\zeta+4\eta)^{2}$
$\displaystyle+12\kappa\\{\epsilon_{0}+P_{0}\\}\\{\epsilon_{0}(3\alpha_{0}\zeta+4\alpha_{1}\eta)\\}+3\zeta+4\eta+P_{0}(3\alpha_{0}\zeta+4\alpha_{1}\eta)\Big{]},$
$\displaystyle\mathcal{S}_{5}$ $\displaystyle=$
$\displaystyle-\frac{3\zeta+4\eta}{6\left(\epsilon_{0}+P_{0}\right)}\,.$
(8.20)

Eqs. (8.18) and (8.19) have been solved to analyse the fate of the nonlinear
waves propagating through a relativistic viscous fluid. The relevant EoS
(i.e., relation between $\epsilon_{0}$ and $P_{0}$) and the initial conditions
required to solve these equations.

We observe that the governing equation Eq.(8.18) does not have any effect of
second order theory. This equation can be derived from the NS theory as well.
However, the second equation contains the second order effects via the
coupling and relaxation coefficients of the MIS theory. Here, we have
considered all the relevant transport coefficients to provide a general
equations for the propagation of nonlinear waves. The dispersive term in
Eq.(8.19) (third order space derivative term) indicate that the combined
effects of shear and bulk viscosities in the diffusive term act against the
effect of thermal conductivity. This might weaken the diffusion of nonlinear
waves in a dissimilar way.

###  8.3 Results and discussion

We present here the results on the fate of the nonlinear perturbations in the
QGP when it passes near the region of the CEP. The equations we obtained look
like KdV equation (ideal background) whose general analytic solution is a sec-
hyperbolic function with a shifted argument. To better understand the
deviation of propagation from that of the ideal background, the initial
profile of perturbations of both the orders are taken as of the same form as
the ‘sech’ function. We solve the equations describing the evolution of
$\hat{\epsilon_{i}}$ for the following initial profile of the perturbations:

$\displaystyle\hat{\epsilon}_{1}$ $\displaystyle=$ $\displaystyle
A_{1}\big{[}\mathrm{sech}\frac{(x-x_{0})}{B_{1}}\big{]}^{2}$
$\displaystyle\hat{\epsilon}_{2}$ $\displaystyle=$ $\displaystyle
A_{2}\big{[}\mathrm{sech}\frac{(x-x_{0})}{B_{2}}\big{]}^{2},$ (8.21)

where, $x_{0}$ is the initial position of the peak of the perturbations.
Since, here we are dealing with 1D propagation in $+x$ direction in an
extended and static background, any choice of value of $x_{0}$ should provide
the same salient features of the propagation. Here we have taken $x_{0}=10$ fm
to demonstrate the propagation of the perturbation however, one may choose any
other value of $x_{0}$ as well. The solutions of the Eqs.(8.18) and (8.19) are
solitonic in nature (with sech2 dependence), which is similar to the solutions
of KdV equations for small dissipations. This motivates us to choose initial
profile given by Eq. (8.21). Also to note that for small spatial gradient the
solitonic behaviour of the solution has also been demonstrated in Ref. [231]
for Gaussian initial profiles in a conformally invariant hydrodynamic
background. In the initial profile, the $A_{i}$ and $B_{i}$ respectively are
determining the height and width of the initial profile. The effects of the
CEP has been taken into consideration through the EoS (Ch.4) and the scaling
behaviour of transport coefficients and thermodynamic response functions in
Eq.(6.17).

Figure 8.1:  The spatial variation of perturbations in energy density when the
system is away from CEP at $\mu=400$ MeV and $T=185$ MeV. The left panel shows
$x$ dependence of $\hat{\epsilon_{1}}=\sigma\epsilon_{1}$. The middle and
right panels indicate the variation of
$\hat{\epsilon_{2}}=\sigma^{2}\epsilon_{2}$ and
$[1+\hat{\epsilon_{1}}+\hat{\epsilon_{2}}]$ (see Eq. (8.17)) respectively with
$x$. It is clear that when the system is away from CEP, the perturbations
survive despite the dissipative effects. The results are obtained here for
$A_{1}=0.6,B_{1}=1,A_{2}=0.4$ and $B_{2}=1$.

Fig.8.1 depicts the propagation of a nonlinear wave in the medium when the
system is passing away from the CEP. It is observed that, though the amplitude
of the wave is reduced, the nonlinear wave survives the dissipation despite
the presence of the non-zero values of transport coefficients of the medium.
However, the nonlinear wave is substantially dissipated (almost died) near the
CEP as evident from the results shown in Fig. 8.2. It has been observed that
the dissipative feature remains unaltered with the variation of the location
of the CEP along the transition line in the QCD phase diagram. This clearly
indicates that the nonlinear perturbations will exhibit unique detectable
effects of the CEP. It also shows that the speed of the attenuated wave is
non-vanishing in contrast to linear waves [72]. This is due to the amplitude
dependent propagation speed of the nonlinear waves, shown in Fig.8.3. The
attenuation of the perturbation is in comparison smaller for the second-order
correction ($\hat{\epsilon_{2}}$). Furthermore, the second-order perturbation
travels a bit faster than the first-order one. This is clear from the results
displayed in Figs.8.1 and Figs.8.2, which can also be understood from
Eq.(8.19), where the second-order correction contains the third-order
derivatives of the first-order perturbation. This is equivalent to the
dispersive term in the KdV equation, which is responsible for height
preserving solitonic behaviour [246]. Therefore, this dispersive terms compete
with the diffusive terms, results in weakening the damping effect. The
relaxation terms in the dissipative fluxes incorporated in MIS theory
(Eq.(8.19)) makes the dissipation lesser in comparison to the NS theory
(Eq.(8.18)).

Figure 8.2:  The spatial variation of perturbations in energy density when the
system is close to CEP at $\mu=367$ MeV and $T=158$ MeV. The left panel shows
$x$ dependence of $\hat{\epsilon_{1}}=\sigma\epsilon_{1}$. The middle and
right panels indicate the variation of
$\hat{\epsilon_{2}}=\sigma^{2}\epsilon_{2}$ and
$[1+\hat{\epsilon_{1}}+\hat{\epsilon_{2}}]$ (see Eq. (8.17)) respectively with
$x$. It is clear that when the system is near the CEP, the nonlinear
perturbations is strongly dissipated. The results are obtained here for
$A_{1}=0.6,B_{1}=1,A_{2}=0.4$ and $B_{2}=1$.

Therefore, near the CEP, the disturbances with larger magnitudes, which are
created in the medium due to energy deposition by energetic particles or jets,
will be highly dissipated. In earlier studies [72], it was shown that linear
waves will be completely stopped and dissipated near the CEP. Similar
suppression for nonlinear waves too are found in the presence of the CEP.
Although the speed of propagation of the nonlinear waves are amplitude
dependent, in presence of the CEP, irrespective of their amplitude all the
waves are highly suppressed. This may have important consequences in detecting
the CEP from the analysis of the hadron spectra.

In has been speculated earlier [62, 72, 73] that the formation of Mach cone is
forbidden near the CEP for the propagation of linear perturbations. The
remaining question to be reported if the Mach cone formation will be there for
nonlinear waves or not. This is important because nonlinear waves are found to
maintain the solitonic nature in comparison to linear perturbations.
Nevertheless, we find that even the nonlinear perturbations will not be able
to preserve the Mach cone effects if it hits the CEP. So the propagation of
both types of the perturbations (linear and nonlinear) are stopped due to the
presence of the CEP. These results can be used to detect the CEP by looking
into the suppression of the Mach cone effect in two particle correlation [237,
236]. Due to the high dissipation of the nonlinear waves, the broadening
effect of localized waves will also vanish [131] along with the Mach cone
effect.

Figure 8.3:  The spatial variation of perturbations in energy density when the
system is away from the CEP. Same plot as Fig.8.1(c), but with larger
amplitude.

The flow harmonics play important role in characterizing the medium formed in
RHIC-E. It was shown in Refs. [248, 249] that some of the flow harmonics will
collapse at the CEP (along phase boundary of the QCD phase diagram). Based on
linear analysis it has been argued in [62, 72, 73] that $v_{2}$ or higher
harmonics will be reduced near the CEP. The similar conclusion can be drawn
for the nonlinear perturbations too near the CEP. If the initial spatial shape
of the system formed in HICs is highly distorted azimuthally, the fluid
dynamical response to the initial eccentricity can be nonlinear. The
suppression of the nonlinear waves also suggests that $v_{2}$ and higher order
flow harmonics will be suppressed near the CEP due to the absorption of sound
wave.

In a recent study [250], it is argued that the path to the critical point is
greatly influenced by far-from equilibrium initial conditions, where viscous
effects may lead to dramatically different $(T,\mu)$ trajectories. The sound
wave propagating through the system will be ceased near the CEP. Therefore,
the pressure gradient produced (due to EoS) due to initial spatial anisotropy
will not be converted effectively to momentum anisotropy through hydrodynamic
expansion in an event of isentropic trajectory passing through the CEP. In
such events the flow harmonics get suppressed. In another event, due to the
different initial conditions, the system will follow a trajectory away from
the CEP (see Ref.[5]). In such events, sound will propagate accordingly and
thus no suppression in flow harmonics should be observed. This difference in
flow harmonics (between two events one nearby and the other away from the CEP)
will lead to large event-by-event fluctuations in flow harmonics. Therefore,
large event-by-event fluctuation in flow harmonics could be a signature of the
CEP.

So far it is evident that the nonlinear waves survive dissipation even in the
presence of $\eta,\zeta$ and $\kappa$ without the critical effect, and the
suppression of nonlinear wave only happens almost exclusively because of the
presence of the CEP. This hints that the formation of Mach cone will be
forbidden in the presence of the CEP. Therefore, the present theoretical
investigation indicates that the Mach cones may disappear in the presence of
the CEP. The Mach cone effect manifests as a double hump in the two-particle
correlation in the low momentum domain of associated particles, and the CEP
plays a unique role in suppressing the double hump in the two-particle
correlation contrary to the other mechanisms: e.g. (i) deflection of away side
jets, (ii) Cherenkov radiation and (iii) radiation of gluons which produce the
double hump. These mechanisms have the ability to obscure the suppression due
to the CEP by creating double hump.

The dip in the azimuthal distribution of two particle-correlation at
$\Delta\phi(=\phi-\phi_{trig}=\pi)$ accompanied by two local maxima on the
both side of $\Delta\phi=\pi$ for the transverse momentum range
$0.15<p_{assoc}^{T}<4$ GeV [221, 251], is attributed to the processes pointed
above. Therefore, we contrast the effect of the CEP to the following
mechanism.