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\begin{center}
\textbf{Introduction}
\end{center}
Nearly perfect fluid behavior observed in strongly interacting
quantum systems reveals the emergence of collective phenomena across extreme physical regimes,
from the ultra-hot quark-gluon plasma (QGP) at $T \sim 10^{12}$ K to ultra-cold atomic gases
at $T \sim 10^{-8}$ K. Despite their distinct degrees of freedom and symmetries, both systems
exhibit a universal description, where the short-lived microscopic interactions are integrated
out, leaving the long-range dynamics to dominate. This long-range dynamics is captured within
the hydrodynamic framework, whose applicability to far-from-equilibrium systems is explained
by the existence of hydrodynamic attractor. The hydrodynamic attractor can be defined as a
curve/surface characterised by hydrodynamic degrees of freedom, towards which the
non-equilibrium dynamics of the system evolves, irrespective of any initial conditions. The
feature of the attractor ---``memory loss of the initial condition''--- makes it a strong
phenomenological candidate for identifying universal behavior in both systems.

\begin{center}
\textbf{Observation of collective flow:}
\end{center}
In heavy-ion collisions at Relativistic
Heavy-Ion Colliders (RHIC) and the Large Hadron Collider (LHC), highly Lorentz-contracted
nuclei of Pb or Au collide at ultra-relativistic energies, resulting in the production of
QGP--- a deconfined state of quarks and gluons~\cite{Adams2005,Back2005,Arsene2005,Adcox2005}.
The initial almond-shaped overlap of the colliding nuclei generates anisotropic pressure
gradients, leading to the emission of particles boosted in the direction of the large pressure
gradient. This converts the initial space asymmetry to measurable momentum space asymmetry,
characterized by harmonic flow coefficient $\bar{v}_n$~\cite{Busza2018}. Experimental
observations reveal a significant elliptic flow in semi-central
collisions~\cite{Romatschke2007}, indicating strong collective expansion. Further analyses of
the flow demands non-zero viscosity to describe the transverse momentum dependence and
centrality of elliptic flow, as well as the emergence of higher spherical harmonics. Bayesian
studies estimate the shear-viscosity-to-entropy ratio to be $\eta/s \approx 0.15 \pm 0.07$
near the deconfinement transition~\cite{Everett2021,Heffernan2024,Virta2025}, close to the
conjectured lower bound $\eta/s = 1/4\pi$ from holographic theory~\cite{Kovtun2005}.

Analogous collective dynamics can be observed in strongly interacting cold Fermi
gases~\cite{OHara2002}. When a cloud of trapped atoms is released from a cigar-shaped optical
trap, the gas expands faster along its shorter axis, resulting in inversion of its shape due
to pressure gradients. This evolution is captured by the aspect ratio
$AR = \sqrt{\langle x_T^2 \rangle/\langle z^2 \rangle}$, which changes from $AR < 1$ to
$AR > 1$, indicating elliptic flow---similar to that in QGP. Analysis with viscous
hydrodynamics shows that near unitarity---where scale invariance eliminates bulk
viscosity---$\eta/s \approx 0.5 \pm 0.1$~\cite{Bluhm2017}, the lowest measured for any
non-relativistic fluid and close to the holographic bound~\cite{Kovtun2005}. These
observations motivate a deeper understanding of hydrodynamics as a universal description
through the concept of hydrodynamic attractors.

\begin{center}
\textbf{Hydrodynamics}
\end{center}
Hydrodynamics provides a macroscopic description of matter based on the separation
of scales, where the system is probed over spacetime intervals large enough for microscopic
dynamics to become irrelevant. At this scale, the only relevant degrees of freedom are the
conserved quantities associated with the global symmetries of the system. In a hydrodynamic
framework, a system is governed by conservation equations of energy and momentum,\footnote{Here we only focus on energy-momentum conservation. Particle number conservation is given by $\nabla_\mu J^\mu = 0$, where $J^\mu = \mathcal{N} u^\mu$, $\mathcal{N}$ is the particle number.} which in relativistic language is given by $\nabla_\mu T^{\mu\nu} = 0$. The
energy-momentum tensor is expressed as $T^{\mu\nu} = (E+P)u^\mu u^\nu + P g^{\mu\nu} +
\Pi^{\mu\nu}$, where $g^{\mu\nu}$ is the background spacetime metric. Here, $P$ and $E$
denote the equilibrium pressure and energy density, related via the equation of state. For a
conformal (or scale invariant) theory, $P(\mathcal{E}(T)) = \mathcal{E}(T)/(d-1)$ where
temperature $T \propto \mathcal{E}^{1/d}$, $d$ is the dimension of the system. The stress
tensor $\Pi^{\mu\nu}$ represents the deviation from the equilibrium, i.e., the dissipative
correction, via gradients of $E$ and fluid velocity $u^\mu$. At first-order in gradient,
$\Pi^{\mu\nu}$ reproduces the Navier-Stokes theory, which leads to acausality in the limit of
large momentum~\cite{Romatschke2010,Kostadt2000}. Various causal hydrodynamic models such as
M\"{u}ller-Israel-Stewart (MIS) formalism~\cite{Muller1967,Israel1979}, BRSSS~\cite{Baier2008},
etc., have been formulated where $\Pi^{\mu\nu}$ is treated as an independent degree of
freedom. These formalisms allow to write a relaxation-type equation of $\Pi^{\mu\nu}$ such
that when the velocity gradient is zero, $\Pi^{\mu\nu}$ vanishes on a finite microscopic
timescale referred to as the relaxation time. The MIS formalism is the first phenomenological
model to formulate an understanding of the emergence of hydrodynamic behavior in QGP via the
decay of non-hydro modes to a hydrodynamic attractor~\cite{HellerSpalinski2015}.


\begin{center}
\textbf{Attractors in heavy-ion collisions}
\end{center}
Attractors in heavy-ion collisions have been studied extensively in the
one-dimensional Bjorken model, which mimics the boost-invariant expansion of QCD matter along
the axis of collision~\cite{Soloviev2022,Florkowski2018}. The evolution is studied in the
rest frame where the proper time $\tau$ and the rapidity coordinate $\xi$ are related to the
Minkowski coordinate $(t, x_\perp, z)$ as $\tau = \sqrt{t^2 - z^2}$ and
$\xi = \arctan(z/t)$. Existence of an attractor in a system undergoing Bjorken expansion has
been identified in various frameworks, such as MIS/BRSSS~\cite{Heller2018}, kinetic
theory~\cite{Heller2018,Strickland2018}, holographic description of $\mathcal{N} = 4$ Super
Yang-Mills (SYM) theory~\cite{Spalinski2018}, Gauss-Bonnet holography~\cite{Casalderrey2018},
and also in a hybrid viscous fluid model~\cite{Mitra2020,Mitra2024}. Besides the anisotropic
Bjorken expansion, attractors have been explored in the context of isotropic Hubble expansion
in cosmology~\cite{Du2021}, and also in systems undergoing both radial and longitudinal
expansion---Gubser flow~\cite{Banerjee2024,Mitra2024b,Behtash2018,Denicol2019}.

Most commonly, attractors are quantified in terms of a dimensionless quantity $\mathcal{A}(\tau T)$,
which measures how far the system is from the local equilibrium. Fig.~\ref{fig:attractor}
illustrates the evolution of a system for arbitrary initial conditions towards the attractor
curve, characterised by hydrodynamics at late time, irrespective of significant pressure
anisotropy. The energy density of such a system is governed by the late-time behavior,
$\mathcal{E} = \epsilon_0 (\tau_0/\tau)^{4/3} + \frac{3}{4} C_\eta \epsilon_0^{3/4} \tau_0
\tau^{-2} + \mathcal{O}(\tau^{-8/3})$ where $\epsilon_0$ depends on the initial condition at
proper time $\tau_0$. $C_\eta$ is the first-order transport coefficient associated with the
shear viscosity $\eta$. The leading term provides an exact solution of the Euler equation,
representing the expansion of a conformal perfect fluid, and the sub-leading terms are the
dissipative corrections given in terms of $\nabla_\mu u^\mu \sim 1/\tau$. The late-time
behaviour of the pressure anisotropy reads,
$\mathcal{A}(\tau T) = 8 C_\eta / (\tau T) + \mathcal{O}(\tau T^{-2})$,
which is independent of any initial parameter, indicating a universal solution and hence a
natural measure for characterising hydrodynamic attractors.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.49\textwidth]{attractor_left}\hfill
  \includegraphics[width=0.49\textwidth]{attractor_right}
  \caption{The figure shows the evolution of $\mathcal{A}$ as a function of dimensionless
    time $\tau T$ in Bjorken model~\cite{Florkowski2018}. The left plot shows the analytical
    attractor (red line) in $\mathcal{N} = 4$ SYM plasma obtained by resummation, and the
    dashed magenta corresponds to the first-order hydro. The right plot shows the numerically
    determined attractor (red line) in BRSSS theory. The dashed magenta and dotted green
    curves correspond to the first-order and second-order hydro. In both, the blue curves are
    the numerical solutions obtained using AdS/CFT and BRSSS theory.}
  \label{fig:attractor}
\end{figure}

Further, the approach of the non-hydro modes towards the attractor surface has been observed
to be exponential, which, when computed from the MIS formalism, reads
$\delta\mathcal{A} \sim e^{-\frac{3(\tau T)}{2 C_{\tau_\pi} (\tau T)^{C_\eta/C_{\tau_\pi}}}}$.
Here, $C_{\tau_\pi} = \tau_\pi T$ is the transport coefficient, $\tau_\pi$ being the
relaxation time. Besides numerical understanding, the hydrodynamic attractor has a rich
mathematical structure. The analytical attractor is built out of resummation of the divergent
series of the late-time hydrodynamic expansion~\cite{Spalinski2018} via the Borel resummation
technique~\cite{Aniceto2015}, which modifies the gradient expansion to a
transseries~\cite{HellerSpalinski2015,Basar2015,Aniceto2019,Aniceto2018} that systematically
encodes the details of the non-hydro modes.

\begin{center}
\textbf{Attractors and spontaneous symmetry breaking}
\end{center}
In the Bjorken model, the system remains on the hydrodynamic attractor and hence in the hydrodynamic regime for infinite
proper time. However, the QGP evolves from quarks and gluons to hadrons at finite proper time
$\tau \approx 10\,\text{fm/c}$ around $T_c \approx 155\,\text{MeV}$, thus indicating a
chiral phase transition. This leads to an interesting question of interplay between attractors
and other important features of QGP, such as spontaneous symmetry breaking. In this
direction, the first step has been taken~\cite{Mitra2021} by studying an expanding superfluid
in the context of boost-invariant Bjorken flow~\cite{Bjorken1983}, followed by Gubser
flow~\cite{Gubser2010,Gubser2011}, and Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW)
background of the early universe~\cite{Wald1984}. The essential idea is to study an expanding
superfluid associated with spontaneous symmetry breaking. In~\cite{Mitra2021,Buza2025}, one
develops a dissipative superfluid formalism in an expanding background by coupling fluid
degrees of freedom with additional non-hydrodynamic degrees of freedom, namely the $U(1)$
scalar field $\Sigma$ in the long wavelength limit. The mass $m$ of the scalar field is
promoted to a function of temperature $T$ of the normal fluid, $m = m(T - T_c)$, where $T_c$
is the critical temperature of the system below which the system enters the spontaneously
broken phase. The primary observation of the model suggests that the evolution of the
superfluid in all three expanding backgrounds is dictated by a hydrodynamic attractor with
unbroken symmetry at intermediate times, while at late times the system approaches one of the
symmetry-breaking fixed points associated with the ground state of the broken phase. This
behaviour defines a characteristic timescale referred to as \emph{attractor time} --- the time
over which the system evolves out of hydrodynamic behaviour. Fig.~\ref{fig:superfluid}
illustrates this in the Bjorken expansion, showing the dependence of $\Delta\tau$ on the
initial conditions.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.6\textwidth]{superfluid}
  \caption{The figure shows expansion of a $U(1)$ superfluid in Bjorken
    background~\cite{Buza2025}. $T_0$ is the initial temperature of the system, and $T_c$ is
    set to unity. The system evolves to the hydrodynamic attractor as defined by the red line,
    around the same time for different initial conditions. However, it evolves to the
    symmetry-breaking fixed points at different times depending on the initial conditions.
    Here $\Delta\tau_3 > \Delta\tau_2 > \Delta\tau_1$ depending on large $T_0$.}
  \label{fig:superfluid}
\end{figure}

\begin{center}
\textbf{Cold atomic gases}
\end{center}
In cold atomic gases, hydrodynamic attractors have been explored in a two-component Fermi gas at arbitrary temperature in three dimensions with short-range interaction characterized by the two-body $s$-wave scattering length $a(t)$, which is temporally driven at fixed volume~\cite{Fujii2024}. This temporal variation of $a(t)$ drives the system out of equilibrium, equivalent to isotropic fluid expansion, while the fluid remains uniform and at rest. From a hydrodynamic point of view, the amount of energy density produced due to the time variation in $a(t)$ is given by the dynamic sweep
theorem~\cite{Tan2008}, $\dot{\mathcal{E}}(t) = \mathcal{C}(t)\dot{a}(t)/(4\pi m a(t)^2)$
where the expectation value of the contact density $\mathcal{C}(t)$ incorporates both the
thermal equilibrium value $\mathcal{C}_{eq}[a(t)]$ and the dissipative correction $\pi(t)$ to
ideal hydrodynamics. This dissipative correction can be interpreted as the deviation of the
expectation value of pressure $p(t)$ from equilibrium.

A hydrodynamic attractor has been first observed in the relaxation dynamics of cold
atoms~\cite{Fujii2024}, where the scattering length $a(t)$ is asymptotically driven to the
unitarity limit by power-law variation with positive exponent $\alpha$ as,
\begin{equation}
a_k(t)^{-1} =
\begin{cases}
  a_k^{-1} & t < t_k, \\
  a_k^{-1}(t/t_k)^{-\alpha} & t > t_k
\end{cases}
\label{eq:protocol}
\end{equation}
where $a_k$ is a constant scattering length up to time $t_k$, for $k = 1, 2, \cdots$. Here,
$k$ differentiates the drives for different initial conditions. To ensure relaxation of the
out-of-equilibrium fluid towards the equilibrium at long times, the scattering length is
defined at the relaxation time $\tau_\zeta$ as $a_k(\tau_\zeta/t_k)^\alpha := \tilde{a}$, so
that all the driving protocols at long times converge onto a single curve that asymptotically
approaches the unitarity limit $a^{-1} \to 0$ with power $\alpha$. Near unitarity, the bulk
viscosity $\zeta[a]$ vanishes as $a^{-2}$, allowing the approximation $\zeta[a] \simeq
\zeta^{(2)} a^{-2}$, with $\zeta^{(2)}$ being a constant. At unitarity, following this
protocol, the attractor solution in the long time limit, times larger than $\tau_\zeta$ reads,
$\pi_{att}(t) = 3\zeta[\tilde{a}] \frac{\alpha\tau_\zeta^{2\alpha}}{t^{2\alpha+1}} \left[1 +
(2\alpha+1) \frac{\tau_\zeta}{t} + \mathcal{O}((\tau_\zeta/t)^2)\right]$,
where the leading term corresponds to the Navier-Stokes result and the next term gives the
second-order hydro correction. Moreover, the non-hydro modes decay exponentially to the
attractor, following $e^{-(t-t_k)/\tau_\zeta}$, a behavior analogous to what is observed in
heavy-ion collisions.

\begin{figure}[h]
  \centering
  \includegraphics[width=0.55\textwidth]{coldatoms_left}\hfill
  \includegraphics[width=0.42\textwidth]{coldatoms_right}
  \caption{The left figure shows the deviation of the dimensionless contact density from the
    equilibrium for $\alpha = 1/2$. The right figure shows the cyclic attractor for the MIS
    model. Here, the full dynamical bulk pressure $\Pi(t)$ is plotted against the Navier-Stokes
    bulk pressure. Figures from~\cite{Fujii2024,Mazeliauskas2025}.}
  \label{fig:coldatoms}
\end{figure}

Beyond the regime of monotonic expansion, a new kind of cyclic hydrodynamic attractors has
been observed in the cold atoms when subjected to periodically driven scattering length,
$a(t)^{-1} = a_0^{-1}(1 + A\sin\omega t)$~\cite{Mazeliauskas2025}. The periodic expansion
and contraction of the system never allows the system to relax towards the Navier-Stokes
limit, driving it to a new cyclic attractor as shown in Fig.~\ref{fig:coldatoms}. The
existence of the cyclic attractors opens a new avenue in the direction of experimental
observation of the attractors, where the closed cycle deviation from equilibrium measures
provides clear evidence of attractor behavior. Following this, an ongoing research is
exploring the possibility of cyclic attractors arising from anisotropic oscillation in the
context of heavy-ion collisions~\cite{InProgress}.

\begin{center}
\textbf{Discussion}
\end{center}
Both quark-gluon plasma and ultracold atomic gases provide an
excellent platform to test the universality of hydrodynamics in strongly interacting systems
at extreme conditions. The existence of hydrodynamic attractors in far-from-equilibrium
systems, with a typical size of $N \approx 10^4$--$10^5$, has been instrumental in
understanding the evolution of the system to hydrodynamic behavior. Primarily, in both cases,
the attractors have been discussed in the context of highly symmetric systems, encompassing
both space-time and internal symmetries. Despite significant success, hydrodynamics and
attractors remains an active field to explore. Some of the open questions are:
\begin{itemize}
\item \textbf{Experimental realisation of the attractors:} Cold atomic gases offer a
  controlled laboratory environment to probe hydrodynamic attractors. Recent
  work~\cite{Fujii2024} outlines a possible experimental strategy to detect attractors and
  highlights the contact density as a possible observable for this purpose. However,
  confirming whether the contact density is a suitable observable that characterizes the
  attractor remains to be established.

\item \textbf{Attractors and symmetries:} In heavy-ion collisions, QGP evolves to a hadronic
  phase associated with broken chiral symmetry. Likewise, the ultracold atom exhibits a
  BCS-BEC crossover with distinct phases at different scattering lengths. Recent
  studies~\cite{Mitra2021,Buza2025} have explored $U(1)$ superfluid as a simplified framework
  to examine the interplay between attractors and symmetries. Extending such models to
  realistic systems, such as QCD viewed as a non-abelian superfluid, and to ultracold atomic
  gases, remains an interesting direction to explore.

\item \textbf{Hydrodynamic behaviour in small systems:} Collective phenomena have also been
  observed in small systems, such as proton-nucleus~\cite{Khachatryan2015},
  proton-proton collisions~\cite{Adam2017,Khachatryan2017}, and recently in light-ions (O-O,
  Ne-Ne) collisions~\cite{Park2025,Loizides2025}---and even in a mesoscopic
  system~\cite{Brandstetter2025} of size $N \approx 10$ atoms. These observations challenge
  the traditional concept of scale separation in hydrodynamics and raise the question of the
  existence of attractors in such systems~\cite{Berges2025}.

\item \textbf{Nuclear structure and interactions:} Furthermore, the emergence of
  hydrodynamics in small systems prompts investigation into the role of initial geometry, the
  nuclear structure, and many- or few-body correlations to gain deeper insight into the
  universality of hydrodynamics. In this context, the various discussions at this workshop on
  bound states and scaling symmetries provide an opportunity to explore these effects on
  hydrodynamics.
\end{itemize}

\begin{acknowledgments}
We thank Aleksas Mazaliauskas, Alexander Soloviev, and Tilman Enss for the helpful comments
on the manuscript. We also thank the organisers for hosting the workshop --- ``Universality in
strongly interacting systems: from QCD to atoms'' at ECT$^*$. TM has been supported by the
DFG through the Emmy Noether Programme (project number 496831614), through CRC~1225 ISOQUANT
(project number 27381115).
\end{acknowledgments}

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