# Hydrodynamic attractor of noisy plasmas

Zenan Chen<EMAIL_ADDRESS>Institute of Modern Physics, Fudan
University, Shanghai 200433, China  Derek Teaney<EMAIL_ADDRESS>Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794, USA  Li Yan<EMAIL_ADDRESS>Institute of Modern Physics,
Fudan University, Shanghai 200433, China  Key Laboratory of Nuclear Physics
and Ion-beam Application (MOE), Fudan University, Shanghai 200433, China

###### Abstract

We provide a generalized formulation of fluctuating hydrodynamics for the far-
from-equilibrium noisy medium. As an example, we consider a noisy plasma
experiencing Bjorken expansion, for which the leading order evolution is
captured by hydrodynamic attractor of classical hydrodynamics, while the
quadratic couplings of fluctuations are solved effectively via a generalized
version of the hydrodynamic kinetic equation. In the far-from-equilibrium
plasma, backreaction of hydrodynamic fluctuations results in renormalization
of transport properties, as well as long time tails, of high orders. In
particular, corresponding to a renormalized hydrodynamic attractor, evolution
in a noisy plasma towards equilibrium becomes non-monotonic.

Introduction.—Hydrodynamics is an effective theory that by construction
applies to thermal systems close to local equilibiurm. In hydrodynamics,
departures from ideal fluids are captured by gradients of hydrodynamic fields
as well as hydrodynamic fluctuations. Hydrodynamic fluctuations are in general
suppressed in systems with large amount of constituents, it is therefore not
surprising theoretical formulations without hydrodynamic fluctuations
(frameworks sometimes referred to as the classical hydrodynamics Kovtun
(2012)) have achieved remarkable successes. Such examples include in
particular the hydrodynamic modeling of quark-gluon plasma (QGP) in high-
energy nuclear physics Shen and Yan (2020).

Nonetheless, hydrodynamic fluctuations cannot be neglected when they are
substantial to system dynamical evolution. For instance, when a thermal system
evolves close to a critical point, correlations among fluctuations of order
parameters diverge, resulting in novel hydrodynamic modes Stephanov and Yin
(2018). Hydrodynamic fluctuations are amplified in small systems, such as the
QGP droplet created in high-energy proton-lead collisions Khachatryan  _et
al._ (2015), owing to the fact that correlations among thermal fluctuations
are inversely proportional to the system volume. More importantly, the non-
linear nature of hydrodynamics allows for corrections from couplings of
fluctuations Pomeau and Résibois (1975). Backreaction of the coupled modes
renormalizes transport properties Kovtun  _et al._ (2011a); Akamatsu  _et al._
(2017, 2018); Martinez and Schäfer (2017); Kovtun  _et al._ (2011b), generates
non-analytical long time tail structures Kovtun and Yaffe (2003), and even
influences evolution history in a fluid Floerchinger  _et al._ (2015).

The equation of motion of fluctuating hydrodynamics follows the conservation
of energy and momentum

$\partial_{\mu}T^{\mu\nu}=0\,,\qquad T^{\mu\nu}=T_{\rm cl}^{\mu\nu}+\delta
T^{\mu\nu}+S^{\mu\nu}\,,$ (1)

where the classical energy-momentum tensor, $T_{\rm cl}^{\mu\nu}$, consists of
energy density $e$, pressure $P$, fluid four-velocity $u^{\mu}$, and expansion
in terms of their gradients ff: ,

$\displaystyle T_{\rm
cl}^{\mu\nu}=eu^{\mu}u^{\nu}+P\Delta^{\mu\nu}+\pi^{\mu\nu}\,.$ (2)

The expansion can be characterized by the Knudsen number Kn. Up to second
order in gradients, the constitutive equation is often formulated via the
stress tensor $\pi^{\mu\nu}$ relaxing to its Navier-Stokes correspondence,
i.e., the Israel-Stewart hydrodynamics Israel and Stewart (1979). In the
classical constitutive equation Eq. (2), variables are thermal averaged
quantities without corrections from thermal fluctuations, namely, they are
bare variables to be distinguished later from the renormalized ones.
Fluctuations of energy-momentum tensor $\delta T^{\mu\nu}$ are constructed
accordingly in terms of thermal fluctuations of hydrodynamic variables, which
are further induced through the random noise $S^{\mu\nu}$, subject to the
condition $\langle S^{\mu\nu}(x)\rangle=0$ and the fluctuation-dissipation
relation, $\langle
S^{\mu\nu}(x)S^{\alpha\beta}(y)\rangle=2T\eta\Delta^{\mu\nu\alpha\beta}\delta^{4}(x-y)$.
Here, the brackets indicate an average over ensemble of thermal fluctuations.
With an equation of state: $P=P(e)$, Eqs. (1) and (2) are closed.

Eq. (1) is stochastic hence the resulted system evolution fluctuates in space
and time. However, the averaged evolution is deterministic, which can be
obtained, for instance, through an ensemble average over numerical simulations
of the stochastic processes. Alternatively, with respect to an effective field
theory approach of fluctuating hydrodynamics Kovtun  _et al._ (2014); Liu and
Glorioso (2018); Crossley  _et al._ (2017), by treating thermal fluctuations
as perturbations, averaged quantities can be solved order by order. This is a
strategy analogous to the loop expansion in quantum field theory. The tree-
level analysis corresponds to solving the classical hydrodynamics,
$\partial_{\mu}T_{\rm cl}^{\mu\nu}=0$. Effect of thermal fluctuations then
arises when hydrodynamic fluctuations are included and constrained by
$\partial_{\mu}\delta T^{\mu\nu}=-\partial_{\mu}S^{\mu\nu}$, which accordingly
determines multi-point correlations Akamatsu  _et al._ (2017, 2018); An  _et
al._ (2019, 2021). The two-point correlations, $\langle\delta T^{\mu\nu}\delta
T^{\alpha\beta}\rangle$, in particular, contain already the information of
quadratic couplings of modes that contributes to $\langle T^{\mu\nu}\rangle$,
and the renormalization of transport properties and the long time tails.

Figure 1:  Isotropization of medium corresponding to attractors in classical
hydrodynamics (black solid line) and fluctuating hydrodynamics (colored band).

Recently, extensive studies have been devoted to the generalization of
classical hydrodynamics to far-from-equilibrium systems. These works are
motivated in part by exploring the applicability condition of hydrodynamics
through the convergence of gradient expansion Heller  _et al._ (2013); Heller
and Spalinski (2015), and in part by the experimental observations of
collectivity in QGP created from colliding nuclei of small sizes (cf. Ref.
Khachatryan  _et al._ (2015)). From either aspect, it was acknowledged that
classical hydrodynamics admits the so-called attractor solutions in some
comoving flows Romatschke (2018); Blaizot and Yan (2018); Kurkela  _et al._
(2020); Behtash  _et al._ (2018); Denicol and Noronha (2019); Jaiswal  _et
al._ (2019); Chattopadhyay and Heinz (2020); Du  _et al._ (2021); Strickland
(2018); Brewer  _et al._ (2019); Behtash  _et al._ (2019), owing to the
expected hydrodynamic fixed points in these systems Blaizot and Yan (2020).

Bjorken flow, for instance, applies approximately to the very early stages of
high-energy heavy-ion collisions, where system expands dominantly along the
beam axis ($z$-axis). In the Milne coordinates, $\tau=\sqrt{t^{2}-z^{2}}$ and
$\zeta=\tanh^{-1}(z/t)$, with respect to the Israel-Stewart formulation,
classical hydrodynamics reduces to coupled equations,

$\displaystyle\frac{de}{d\tau}$ $\displaystyle=-\frac{1}{\tau}(e+P+\pi)\,,$
(3a) $\displaystyle\pi$
$\displaystyle=-\frac{4}{3}\frac{\eta}{\tau}-\tau_{\pi}\left[\frac{d\pi}{d\tau}+\frac{4}{3}\frac{\pi}{\tau}\right]\,,$
(3b)

where $\pi=\pi^{\zeta}_{\;\zeta}$ is the $\zeta\zeta$-component of the stress
tensor. For later convenience, we introduce dimensionless constants,

$\eta=C_{\eta}s\,,\quad\tau_{\pi}=C_{\tau}C_{\eta}/T\,,\quad e=C_{e}T^{4}\,,$
(4)

to parameterize shear viscosity $\eta$, shear relaxation time $\tau_{\pi}$,
and energy density. We also consider the system conformal, which has
$P=c_{s}^{2}e$.

Defining the inverse Knudsen number Kn${}^{-1}=w\equiv\tau/\tau_{\pi}$,
Bjorken expansion of QGP is fully captured by the relative decay rate of
energy density: $g(w)\equiv d\ln e/d\ln\tau$. Especially, isotropization of
the system is related to $g(w)$ through

$\frac{P_{L}}{e}=\frac{\tau^{2}T_{\rm cl}^{\zeta\zeta}}{T_{\rm
cl}^{\tau\tau}}=-1-g(w)\,.$ (5)

As will be clear later, this relation gets renormalized by hydrodynamic
fluctuations. In terms of $g(w)$, hydrodynamic attractor behaves as a smooth
and monotonic connection between the free streaming fixed point at early
times, $g(w)\approx-1$, and ideal hydrodynamic fixed point at late times,
$g(w)=-4/3$, while evolution with arbitrary initial conditions merge swiftly
towards the attractor. Hydrodynamic attractor can be solved numerically, as
well as analytically upon approximations Blaizot and Yan (2021a). In the
leading order adiabatic approximation Blaizot and Yan (2021b) (or the leading
order slow-roll approximation Liddle  _et al._ (1994)), hydrodynamic attractor
can be written as,

$g(w)=-\frac{1}{2}\left[\frac{22}{7}+w-\sqrt{\left(\frac{10}{21}+w\right)^{2}+\frac{64}{45}}\right]\,.$
(6)

In the region $w\gtrsim 1$, Eq. (6) is not sensitive to second order transport
coefficients Blaizot and Yan (2021b).

In Fig. 1, the istropization of medium corresponding to the classical
hydrodynamic attractor, Eq. (6), is shown as the black solid line, which
evolves monotonically from free streaming towards the ideal hydrodynamic fixed
point, $1/3$, at late times. Note in particular, deviations from 1/3, at late
times, are proportional to the bare shear viscosity $\eta$.

Hydrodynamic attractor conceptually extends the applicability of classical
hydrodynamics to systems with large Kn. It is not only because attractor
universally describes system evolution irrespective of initial conditions, but
also a consequence that attractor accounts for a systematic resummation of
gradients in terms of trans-series, including non-analytical transient modes
$\propto w^{-C_{\tau}/2C_{\eta}}e^{-3/2w}$, in the far-from-equilibrium medium
Basar and Dunne (2015); Blaizot and Yan (2021a). Therefore, following the
strategy of effective field theory approach, fluctuating hydrodynamics on top
of classical hydrodynamic attractor applies to noisy fluid out of equilibrium.
It is our purpose of the Letter, to investigate the effects of hydrodynamic
fluctuations in a plasma out of equilibrium. Without loss of generality, we
take Eq. (6) as the decay rate of the bare energy density, as the input for
the next leading order analysis of fluctuating hydrodynamics.

Hydrodynamic kinetic equation in the far-from-equilibrium regime.—Bjorken
symmetry is broken by fluctuations. In terms of Fourier modes of fluctuations
of energy and momentum densities,

$\displaystyle\delta e(\tau,{\bf k})=\int d\zeta
d^{2}\vec{x}_{\perp}e^{i\vec{k}_{\perp}\cdot\vec{x}_{\perp}+i\tau
k_{\zeta}\zeta}\delta T^{\tau\tau}(\tau,\vec{x}_{\perp},\zeta)\,,$ (7a)
$\displaystyle g^{i}(\tau,{\bf k})=\int d\zeta
d^{2}\vec{x}_{\perp}e^{i\vec{k}_{\perp}\cdot\vec{x}_{\perp}+i\tau
k_{\zeta}\zeta}\delta T^{\tau i}(\tau,\vec{x}_{\perp},\zeta)\,,$ (7b)

the equation $\partial_{\mu}\delta T^{\mu\nu}=-\partial_{\mu}S^{\mu\nu}$ leads
to coupled stochastic differential equations for $\phi_{a}=(c_{s}\delta
e,g^{x},g^{y},\tau g^{\eta})$. Here $k_{\zeta}$ is dimensionful and conjugate
to $\tau\zeta$. These differential equations are equivalent to a hierarchy of
equations for multi-point correlators An  _et al._ (2021). Especially, the
equal-time two-point correlators, for which we define as $N_{ab}$ through

$\langle\phi_{a}(\tau,\vec{k})\phi_{b}(\tau,-\vec{k}^{\prime})\rangle\equiv
N_{ab}(\tau,\vec{k})(2\pi)^{3}\delta^{3}(\vec{k}-\vec{k}^{\prime})\,,$ (8)

satisfy effectively hydrodynamic kinetic equations Akamatsu  _et al._ (2017).
Of course, when applies to out-of-equilibrium systems with large Kn, the
background fluid evolution should be accounted by the hydrodynamic attractor.

To facilitate analyses, following Ref. Akamatsu  _et al._ (2017), it is
convenient to rotate in the $\vec{k}$-space, which accordingly defines two
longitudinal and two transverse modes $\phi_{A}$ and $A=(\pm,T_{1},T_{2})$.
After the rotation, the hydrodynamic kinetic equation is dominated by the
diagonal components. We therefore find formally

$\left(1+\frac{g(w)}{4}\right)\frac{\partial R_{A}}{\partial\ln
w}=-\alpha_{A}w\tilde{k}^{2}(R_{A}-1)-\beta_{A}(w)R_{A},$ (9)

where $R_{A}\equiv\tau N_{AA}/T(e+P)$ is the normalized correlator and
$\tilde{k}^{2}=\tau_{\pi}^{2}(|\vec{k}_{\perp}|^{2}+k_{\zeta}^{2})$. The
coefficients are

$\displaystyle\alpha_{\pm}=\frac{4}{3C_{\tau}}\,,\quad\alpha_{T_{1}}=\alpha_{T_{2}}=\frac{2}{C_{\tau}}\,,$
(10)

and

$\displaystyle\beta_{\pm}$
$\displaystyle=1+\frac{5}{4}g(w)+c_{s}^{2}+\cos^{2}\theta_{k}\,,$ (11)
$\displaystyle\beta_{T_{1}}$
$\displaystyle=1+\frac{5}{4}g(w)\,,\quad\beta_{T_{2}}=1+\frac{5}{4}g(w)+2\sin^{2}\theta_{k}\,,$
(12)

where $\cos\theta_{k}=k_{\zeta}/|\vec{k}|$. The left hand side of Eq. (9)
represents time derivative on top of the background attractor. The first term
($\propto\alpha_{A}$) and the second term ($\propto\beta_{A}$) on the right
hand side play the role of collision and longitudinal expansion, respectively.
System evolution towards equilibrium relies then on these two competing
effects. Once Eq. (9) is solved, the equal-time two-point correlators $N_{aa}$
in the original basis can be obtained respectively via an inverse rotation.

Figure 2:  Evolution of $R_{+}$ with respect to various initial conditions.
Dashed blue and dashed red lines are obtained with $\tilde{k}=1,3$, and
$\cos\theta_{k}=0.1,0.5$, respectively. Solid lines are the slow-roll
approximation of the corresponding attractor, $R_{+}^{\rm slow-
roll}=\alpha_{+}w\tilde{k}^{2}/(\alpha_{+}w\tilde{k}^{2}+\beta_{+})$.

Hydrodynamic kinetic equation applies between separated scales:
$c_{s}^{-1}\nabla\sim(c_{s}\tau)^{-1}\ll k\ll\lambda_{\rm mfp}^{-1}$ Akamatsu
_et al._ (2017). In the large $k$ limit, or more precisely when
$w\tilde{k}^{2}\gg 1$ according to Eq. (9), the two-point correlators approach
$T(e+P)/\tau$, which can be understood as the “equilibrium” expectation in an
out-of-equilibrium system defined according to background attractor. One
thereby introduces a critical scale $k_{*}=\sqrt{w}/\tau$, above which the
two-point correlators are well captured by the background attractor. In the
out-of-equilibrium system, the separated scales, as well as $k_{*}$, are time
dependent. In particular, because these scales merge around $w\approx 1$, a
reliable description from Eq. (9) for the out of equilibrium plasma should
only apply when $w\in[1,+\infty)$.

As a consequence of fixed points, Eq. (9) itself possesses attractor
solutions. To see this, we first notice that $R_{A}=0$ is a fixed point
solution in the small $w\tilde{k}^{2}$ extreme, which is stable only when
$\beta_{A}>0$. Nevertheless, the stability of this fixed point does not affect
the two-point correlators at late times. In the large $w\tilde{k}^{2}$
extreme, Eq. (9) allows for solution in terms of a double expansion

$R_{A}(w,w\tilde{k}^{2})=1-\frac{\beta_{A}(w)}{\alpha_{A}w\tilde{k}^{2}}+\ldots=\sum_{n,m}\frac{F_{n,m}^{(A)}}{w^{n}(w\tilde{k}^{2})^{m}}\,.$
(13)

Note that the correlator depends explicitly on $w\tilde{k}^{2}$, as being
dictated by Eq. (9). The expansion in $1/w$ is the hydrodynamic gradient
expansion of the two-point correlators, which is asymptotic. The expansion in
$1/\tilde{k}^{2}$ is asymptotic as well, which, however, differs from the
usual hydrodynamic gradient expansion in wavenumbers Grozdanov  _et al._
(2019). In the large-$w\tilde{k}^{2}$ extreme, the solution is well captured
by the first several terms in the expansion. In analogy to the hydrodynamic
fixed point in classical hydrodynamics at large $w$, the
large-$w\tilde{k}^{2}$ behavior represents a stable hydrodynamic fixed point
in the two-point correlators.

In Fig. 2, for illustrative purposes, the evolution of $R_{+}$ is shown with
two sets of $\tilde{k}$ and $\cos\theta_{k}$ values. Irrespective of initial
conditions, $R_{+}$ tends to follow universal curves at late times (dashed
lines), which is exactly the feature that one expects in an attractor
solution. The universal curves stand for the attractors, which with slow-roll
approximation ($\partial_{w}R_{A}=0$), can be approximated as $R_{A}^{\rm
slow-roll}=\alpha_{A}w\tilde{k}^{2}/(\alpha_{A}w\tilde{k}^{2}+\beta_{A})$
(solid lines in Fig. 2). Similar behavior can be found in other modes as well.

Renormalization in far-from-equilibrium noisy fluids.—The resulted equal-time
two-point correlators $N_{aa}$ suffice to determine thermal corrections to the
averaged energy-momentum tensor out of equilibrium. With respect to an
integral in $\vec{k}$-space, the thermal corrections can be classified as a
cut-off dependent correction $T^{\mu\nu}_{\Lambda}$ and a long-time tail
correction $\Delta T^{\mu\nu}$. For instance, the $\tau\tau$-component,

$\displaystyle\langle T^{\tau\tau}\rangle-T^{\tau\tau}_{\rm cl}$
$\displaystyle=\frac{1}{e+P}\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\sum_{a=\vec{x}_{\perp},\zeta}N_{aa}(w,w\tilde{k}^{2})$
$\displaystyle=T^{\tau\tau}_{\Lambda}+\Delta T^{\tau\tau}\,,$ (14)

where the explicit dependence on $w\tilde{k}^{2}$ is rooted in Eq. (9). As
indicated in Eq. (13), the integral contains a cubic and a linear divergent
pieces, which can be regulated by introducing a cut-off scale $\Lambda$. These
regulated integrals then give rise to the cut-off dependent corrections. For
$T^{\tau\tau}_{\Lambda}$ and $\tau^{2}T^{\zeta\zeta}_{\Lambda}$, one finds,

$\displaystyle T^{\tau\tau}_{\Lambda}$
$\displaystyle=\frac{T\Lambda^{3}}{2\pi^{2}}-\frac{\Lambda
T^{3}}{4\pi^{2}}\frac{C_{\tau}}{(C_{\tau}C_{\eta})^{2}}\frac{35}{8w}\left(\frac{4}{3}+g(w)\right)\,,$
(15a) $\displaystyle\tau^{2}T^{\zeta\zeta}_{\Lambda}$
$\displaystyle=\frac{T\Lambda^{3}}{6\pi^{2}}-\frac{\Lambda
T^{3}}{4\pi^{2}}\frac{C_{\tau}}{(C_{\tau}C_{\eta})^{2}}\frac{1}{w}\left(\frac{27}{10}+\frac{35}{24}g(w)\right)\,.$
(15b)

In the limit $w\gg 1$, these cut-off dependent corrections can be absorbed
into the energy-momentum tensor, so that energy density, pressure ($O(w^{0})$)
Akamatsu  _et al._ (2017) and shear viscosity ($O(w^{-1})$) Kovtun  _et al._
(2011b) get renormalized, respectively. Note that since the cut-off dependent
correction at $1/w$ is negative ($\propto-\Lambda T^{3}$), the resulted
renormalized shear viscosity is actually enhanced. In the far from equilibrium
regime, with respect to the trans-series expansion in $g(w)$, higher order
transport coefficients ($O(w^{-2}$) and beyond) are renormalized as well.

Renormalization in fluctuating hydrodynamics reflects the fact that
hydrodynamic fluctuations stay in equilibrium above the critical scale. In the
out-of-equilibrium medium with large Kn, the cut-off scale can be taken
according to $k_{*}\ll\Lambda\ll\lambda_{\rm mfp}^{-1}$. In practice, provided
the information of the physically measured quantities at a certain scale in
the expanding system, $\Lambda$ is $w$ dependent and well constrained.

Out-of-equilibrium long time tails.—After renormalization, a finite piece in
the thermal corrections remains. As shown in Eq. (Hydrodynamic attractor of
noisy plasmas), the explicit dependence on $w\tilde{k}^{2}$ implies an overall
factor $w^{-3/2}$ in the finite integral, which leads to the non-analytical
structure in the well-known long-time tails,

$\displaystyle\frac{\Delta T^{\tau\tau}}{e}$
$\displaystyle=\frac{w^{-3/2}}{C_{e}(C_{\tau}C_{\eta})^{3}}\sum_{n=0}\frac{f_{n}^{\tau\tau}}{w^{n}}\,,$
(16a) $\displaystyle\frac{\tau^{2}\Delta T^{\zeta\zeta}}{e}$
$\displaystyle=\frac{w^{-3/2}}{C_{e}(C_{\tau}C_{\eta})^{3}}\sum_{n=0}\frac{f_{n}^{\zeta\zeta}}{w^{n}}\,.$
(16b)

Note that the long-time tails are cut-off independent. The coefficients
$f_{n}$ can be solved in principle by a summation of $F_{n,m}$ in Eq. (13).
Via a polynomial fit with respect to the numerical solutions of Eq. (9), we
are allowed to identify $f_{0}^{\tau\tau}=0.45\pm 0.1$, while
$f_{0}^{\zeta\zeta}=0.15032\pm 0.00002$ and $f_{1}^{\zeta\zeta}=-0.53\pm
0.05$.

Renormalized attractor. — In fluctuating hydrodynamics, the effective out-of-
equilibrium system evolution should be monitored by the thermal averaged
components in the energy-momentum tensor. In particular, the renormalized
ratio,

$\displaystyle\frac{\langle\tau^{2}T^{\zeta\zeta}\rangle}{\langle
T^{\tau\tau}\rangle}$ $\displaystyle=\frac{\tau^{2}T_{\rm
cl}^{\zeta\zeta}+\tau^{2}T^{\zeta\zeta}_{\Lambda}+\tau^{2}\Delta
T^{\zeta\zeta}}{T^{\tau\tau}_{\rm cl}+T^{\tau\tau}_{\Lambda}+\Delta
T^{\tau\tau}}$
$\displaystyle=[-1-g(w)]\Big{(}1+\frac{3\tau^{2}T^{\zeta\zeta}_{\Lambda}}{e}+\frac{3\tau^{2}\Delta
T^{\zeta\zeta}}{e}$
$\displaystyle\qquad\qquad\quad\qquad-\frac{T^{\tau}_{\Lambda}}{e}-\frac{\Delta
T^{\tau\tau}}{e}+\ldots\Big{)}$ $\displaystyle\equiv[-1-g(w)]Z^{-1}_{\rm
att}(w)\,,$ (17)

captures the observed system isotropization in the presence of hydrodynamic
fluctuations. In Eq. (Hydrodynamic attractor of noisy plasmas), a
multiplicative renormalization factor $Z_{\rm att}^{-1}$ is introduced, which
contains expansion in $1/w$ from the cut-off dependent corrections, and non-
analytical corrections starting from $w^{-3/2}$ from the long time tails.

With respect to a noisy gluonic plasma Heller  _et al._ (2013), with
$C_{\eta}=1/4\pi$, $C_{\tau}=2(2-\ln 2)$ and $C_{e}=16\pi^{2}/30$, the
renormalized attractor is solved and shown as the colored band in Fig. 1. The
upper and lower boundaries are determined according to the two extreme cut-off
scales, $\Lambda\sim k_{*}$ and $\Lambda\sim\lambda_{\rm mfp}^{-1}$,
respectively. In both cases, when $\Lambda$ is explicitly taken into account,
both the cut-off dependent corrections and the long-time tails are constrained
by an overall factor $1/C_{e}(C_{\tau}C_{\eta})^{3}$. This factor is roughly
the inverse of the number degrees of freedom in an unit volume, in consistency
to the physical expectation of quadratic couplings of hydrodynamic
fluctuations Akamatsu  _et al._ (2017).

When $w\gg 1$, the system is close to an ideal fluid, hence effects of
hydrodynamic fluctuations are expected suppressed. Indeed, at large $w$, the
renormalized attractor follows the trend of classical hydrodynamics,
approaching 1/3 irrespective of hydrodynamic fluctuations. However, a closer
look reveals that the renormalized attractor is actually below the classical
result. This feature is expected, since the effective shear viscosity, which
quantifies the reduction from 1/3, is enhanced by the renormalization due to
hydrodynamic fluctuations Akamatsu  _et al._ (2017); Kovtun  _et al._ (2011b).
Moreover, taking into account the fact that the renormalized correction to
shear viscosity is proportional to $\Lambda$, a larger reduction in the
renormalized attractor is expected with respect to a larger cut-off scale, as
manifested in Fig. 1.

Unlike the classical attractor, which increases monotonically from far from
equilibrium towards ideal fluids, the renormalized attractor becomes non-
monotonic in the far-from-equilibrium region, as shown in Fig. 1. This non-
monotonic behavior qualitatively reflects the long time tail contribution to
$Z^{-1}_{\rm att}$. More precisely, the leading order long time tails give
rise to positive corrections in $Z_{\rm att}^{-1}$, which compensates the
reduction from the renormalized shear viscosity at large $w$ Kovtun  _et al._
(2011b), but dominates when the system is far from equilibrium.
Parametrically, by comparing the leading order long-time tails and the cut-off
corrections in $Z^{-1}_{\rm att}$ in the $\zeta\zeta$ sector, the minimum
point of the renormalized attractor can be found around $\sqrt{w}\sim
4\pi^{2}f_{0}^{\zeta\zeta}/C_{\tau}$.

Summary and discussion.—With the help of hydrodynamic attractor, fluctuating
hydrodynamics can be applied to far-from-equilibrium noisy systems. As an
example, the hydrodynamic kinetic equation, which was developed previously for
a noisy fluid close to equilibrium, can be generalized to far-from-equilibrium
noisy plasmas. In the far-from-equilibrium medium, although hydrodynamic
fluctuations lead to qualitatively similar contributions from the coupled
modes, higher order contributions, such as terms of order $O(1/w^{2})$ and
$O(w^{-5/2})$, and even some more complicated trans-series structures in
$g(w)$, are involved. Moreover, backreaction of the hydrodynamic fluctuations
are significant in the far-from-equilibrium region, which modifies the system
evolution towards equilibrium. In particular, due to long time tails, in the
noisy plasma the effective isotropization is non-monotonic.

The current analysis could be improved systematically by including more
ingredients in a noisy fluid system, such as the nonlinear couplings of modes
beyond the quadratic order and non-conformal corrections. With respect to
realistic high-energy nuclear collisions, initial state fluctuations, which
differ conceptually from hydrodynamic fluctuations, should be taken into
account as well.

Acknowledgments.— L. Y. is supported by the National Science Foundation of
China through grant No. 11975079. L. Y. thanks Yi Yin for helpful discussions.

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