#

Reghukrishnan Gangadharan<EMAIL_ADDRESS>Ankit Kumar Panda
<EMAIL_ADDRESS>Victor Roy<EMAIL_ADDRESS>School of Physical
Sciences, National Institute of Science Education and Research, An OCC of Homi
Bhabha National Institute, Jatni-752050, India

(August 28, 2024)

# Study of Early time attractor with Longitudinal Forces with Bjorken Symmetry

Reghukrishnan Gangadharan<EMAIL_ADDRESS>Ankit Kumar Panda
<EMAIL_ADDRESS>Victor Roy<EMAIL_ADDRESS>School of Physical
Sciences, National Institute of Science Education and Research, An OCC of Homi
Bhabha National Institute, Jatni-752050, India

(August 28, 2024)

###### Abstract

We study the effect of external proper time-dependent longitudinal forces on
the evolution of the distribution function using the Boltzmann Equation with a
relaxation time collision kernel under Bjorken flow. We derive an exact
solution and study the early time attractor behavior of different components
of energy-momentum tensor under Bjorken symmetry. We show that the arbitrary
initial data approaches the attractor solution but is influenced by the
external force with slightly broken Bjorken symmetry.

††preprint: APS/123-QED

##  I Introduction

In the realm of high-energy heavy-ion collisions, two massive nuclei collide
at relativistic speeds and create quark-gluon plasma (QGP)[1]. This novel
state of matter is composed of deconfined quarks and gluons, almost freely
traversing the nuclear volume. Owing to the initial velocity of the two
colliding nuclei, initially, the hot and dense fireball expands primarily
along the beam direction; the transverse expansion due to the hydrodynamic
response of the system takes some time to catch up with the longitudinal
expansion[2, 3, 4, 5]. Due to this initial asymmetric flow, the momentum
anisotropy can be substantial. This may give rise to a situation where the
system is far away from local thermal equilibrium. The momentum anisotropy is
also supposed to be larger for smaller systems such as p-p, p-Au, d-Au, etc.
However, relativistic hydrodynamics seems to work well even for these small
systems [6, 7].

Hydrodynamics is a valid description of dynamics when a system under scrutiny
has attained a state of near local thermal equilibrium (i.e., the ratio of the
off-equilibrium part of the energy-momentum tensor to the equilibrium one is
much smaller than one). This raises the question of how the hydrodynamics can
be a valid description for the QGP evolution at small time scales of $\sim$
$0.1$ to $0.2$ fm/c (at LHC) or for smaller systems where it is supposed to be
highly anisotropic. Hydrodynamic attractors were proposed [8] as a possible
solution to why hydrodynamics effectively describe QGP evolution even for
systems far from local thermal equilibrium. For systems with certain
symmetries- conformal and Bjorken/Gubser, it was observed that the non-
hydrodynamic modes decay exponentially, and the system relaxes to an attractor
solution regardless of the initial conditions (near or far away from local
thermal equilibrium). It was speculated then that the validity of
hydrodynamics could be expanded beyond the limit of small gradients as
previously held. Several subsequent studies [9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21], showed that attractor solutions could be observed in less
symmetric systems albeit in a more general sense of an attractor. However,
more recent non-conformal studies [11, 12, 22] using exact solutions to the
Boltzmann equation pointed out that attractor behavior is not as general as
previously thought as it was absent in several hydrodynamic variables when
conformal symmetry was broken and non-conformal systems only show an early-
time attractor for the case of scaled longitudinal pressure $P_{L}/P$.

Given that a universal attractor is absent in non-conformal systems, one could
ask whether attractors still exist when other symmetries are broken. One
method to explore this is to introduce an external force. Due to the
relativistic speed of the charged protons inside the colliding nuclei, there
exists a strong transient electromagnetic field in the initial stage of heavy-
ion collisions. These fields are estimated to possess strengths on the order
of $10^{18}$ to $10^{19}$ G at the apex of RHIC or LHC energies [23, 24, 25,
26, 27]. Advancements have recently been made across various fronts,
encompassing theoretical [28, 29, 30, 31, 32], phenomenological [33, 34], and
experimental domains [35, 36], all in pursuit of detecting the telltale
signals of these potent electromagnetic fields. Given that early-time
attractors explore the journey toward equilibrium from highly non-equilibrated
states, it is both logical and imperative to investigate their behavior in the
presence of electromagnetic fields. In this study, we intend to explore the
evolution of a far-from-equilibrium system under an external force arising due
to the interaction of electrically conducting QGP fluid and the initial strong
EM fields.

We use the RTA-Boltzmann equation to study the effects of external forces in a
$0+1$ dimensional setting under approximate Bjorken symmetry. Specifically, we
subject the system to a longitudinal force and explore how the evolution of
the system varies for varying field strengths and initial anisotropies. We
check for the presence of attractor behavior in hydrodynamic variables for the
non-conformal systems and if they persist in the conformal limit. For our
calculations, we will be using natural units,
$\bar{h}=k_{B}=c=\epsilon_{0}=\mu_{0}=1$, and the metric signature is mostly
negative $g_{\mu\nu}=diag(+,-,-,-)$.

Our paper is structured as follows. In Sec.(II), we discuss the relativistic
Boltzmann equation with the RTA collision kernel. Then, in the next section,
Sec.(III), we discuss symmetries and the Bjorken flow. In the next section,
Sec.(IV), we talk about the formal solution of the Boltzmann equation with
external forces. We also cover the discussions on the initial distribution
function and how to describe these external forces in Sec.(V). Finally, we
present our findings for different situations involving external forces and
summarize our findings in Sec.(VI),(VII) respectively.

##  II The Boltzmann Equation

The Boltzmann equation describes the evolution of the single-particle phase
space distribution function $f(x,p)$ of a statistical system. The expression
of the Boltzmann equation in the presence of external forces can be written as
[37]

$\displaystyle
p^{\mu}\partialderivative{f}{x^{\mu}}+mK^{\mu}\partialderivative{f}{p^{\mu}}-\Gamma^{\sigma}_{\mu\nu}p^{\mu}p^{\nu}\partialderivative{f}{p^{\sigma}}=C(f)\,,$
(1)

here $K^{\mu}$ is a 4-force and $\Gamma^{\rho}_{\mu\nu}$ is the Christoffel
symbol. The momenta $p^{\mu}$ satisfy the onshell relation
$g_{\mu\nu}p^{\mu}p^{\nu}=m^{2}$ and the 4-force satisfy
$g_{\mu\nu}p^{\mu}K^{\mu}/p^{0}=0$. As the general collision kernel is quite
complex, we use Anderson-Witting relaxation time approximation(RTA) [37]

$\displaystyle C(f)=-u\cdot p\frac{(f-f_{eq})}{\tau_{R}}.$ (2)

Here, $f_{eq}\sim e^{-(u\cdot p)/T}$ represents the local equilibrium
distribution function. $u\cdot p=g_{\mu\nu}u^{\nu}p^{\mu}$ represents the
inner product. $T$ is the local temperature and $u^{\mu}$ is the fluid
4-velocity ($g_{\mu\nu}u^{\mu}u^{\nu}=1$). We consider the relaxation time
$\tau_{R}$ is a function of temperature but not of momentum.

The energy-momentum tensor $T^{\mu\nu}(x)$ is obtained from the distribution
function by taking its second moment

$\displaystyle T^{\mu\nu}=\int d{\Xi}p^{\mu}p^{\nu}f,$ (3)

where the integration measure is given by
$d\Xi=\frac{\sqrt{-g}d^{3}p}{(2\pi)^{3}p^{0}}$. The energy-momentum
conservation $\nabla_{\mu}T^{\mu\nu}=0$ is ensured for the RTA kernel by
imposing the Landau matching condition

$\displaystyle\int d{\Xi}(u_{\mu}p^{\mu})^{2}f=\int
d{\Xi}(u_{\mu}p^{\mu})^{2}f_{eq},$ (4)

where $\nabla_{\mu}$ is the covariant derivative. Here the fluid four-velocity
$u^{\mu}$ is defined as the Eigenvector of $T^{\mu\nu}$ with energy density
$\epsilon$ as the Eigenvalue

$\displaystyle T^{\mu\nu}u_{\nu}=\epsilon u^{\mu}.$ (5)

##  III Symmetries and Bjorken Flow

The symmetries of the system set restrictions on the flow profile of the
fluid. A Bjorken system is characterized by translation, rotational symmetry
in the transverse plane ($x-y$), boost invariance along the longitudinal
direction $z$, and a reflection symmetry($z\rightarrow-z$). The symmetries
completely determine the Bjorken flow profile,

$\displaystyle
u^{\mu}=\left(\frac{t}{\sqrt{t^{2}-z^{2}}},0,0,\frac{z}{\sqrt{t^{2}-z^{2}}}\right)\,.$
(6)

We use Milne coordinate system $(\tau,x,y,\eta)$, where

$\displaystyle\tau=\sqrt{t^{2}-z^{2}},$ (7)
$\displaystyle\eta=\tanh^{-1}{\frac{z}{t}},$ (8)

where $\tau$ is the longitudinal proper time, $\eta$ the space-time rapidity.
The Milne metric is given by $g_{\mu\nu}=diag(1,-1,-1,-\tau^{2})$ with the non
zero Christoffels symbols

$\displaystyle\Gamma^{\tau}_{\eta\eta}=\frac{1}{\tau}$ ,
$\displaystyle\Gamma^{\eta}_{\tau\eta}=\tau.$ (9)

The flow profile in the Milne coordinate with Bjorken symmetry ($\eta$
invariance) becomes $u^{\mu}=(1,0,0,0)$. Rotational and translational
symmetries in the transverse plane imply that the distribution function cannot
depend on $x$ and $y$. The spatial dependence then reduces to only that of the
proper time $\tau$ due to boost invariance and reflection symmetry. We use
transverse momentum $p_{T}=\sqrt{p_{x}^{2}+p_{y}^{2}}$ and $w\equiv
p_{\eta}=\tau p^{z}=tp^{z}-zp^{t}$ in the longitudinal direction. The
distribution function is then only dependent on three variables, one space,
and two momenta and the Boltzmann Equation in Milne coordinates takes the
simple form

$\displaystyle\partialderivative{f(\tau,p_{T},w)}{\tau}=-\frac{(f-f_{eq})}{\tau_{R}}\,.$
(10)

A formal solution can be found for this equation in [38, 39, 40, 41] which is
given as:

$\displaystyle f(\tau,w,p_{T})=$ $\displaystyle
D(\tau,\tau_{0})f_{0}(w,p_{T})$ (11)
$\displaystyle+\int_{\tau_{0}}^{\tau}\frac{d\tau^{\prime}}{\tau_{R}(\tau^{\prime})}D(\tau,\tau^{\prime})f_{eq}(\tau^{\prime},w,p_{T})\,,$
(12)

where $D(\tau_{2},\tau_{1})$ the damping function defined as

$D(\tau_{2},\tau_{1})=\exp\left[-\int_{\tau_{1}}^{\tau_{2}}\frac{d\tau^{\prime\prime}}{\tau_{R}(\tau^{\prime\prime})}\right]\,,$
(13)

and $f_{0}=f(\tau=\tau_{0},w,p_{T})$.

###  III.1 RTA Boltzmann Equation under Longitudinal Force

The introduction of an external force is bound to break one of the four
symmetries. This means that Bjorken flow could be modified in the presence of
force. In order to approximately retain Bjorken flow, the simplest force one
can then introduce is one that only depends on the proper time $\tau$ and is
along the $z$ direction. The introduction of this force will break reflection
symmetry about $\eta=0$. Considering the force to be weak enough so that the
change in fluid velocity ($\delta u^{\mu}$) is small and the new velocity is
written as

$\displaystyle u^{\mu}=u^{\mu}_{B}+\delta u^{\mu}.$ (14)

This approximation is valid as long as the correction to the fluid variables
due to the force is smaller than the equilibrium values. Assuming Bjorken
flow, the Boltzmann equation with force along $z$ direction can then be
written as (see App.(A) for details)

$\displaystyle\left(\mathcal{F}(\tau)\tau\partialderivative{w}+\partialderivative{\tau}\right)f=-\frac{\left(f-f_{eq}\right)}{\tau_{R}}\,.$
(15)

##  IV Solution under external force

Here we assume $\mathcal{F}(\tau)\neq 0\,$ for any value of
$\tau_{i}<\tau<\tau_{f}$ (where $\tau_{i}$ and $\tau_{f}$ are initial and
final time).For convenience we introduce a dimensionless function F($\tau$)
and express $\mathcal{F}=\frac{\alpha}{\tau_{F}}F(\tau)$. After a change of
variables (see App.(A.1)), Eq.(15) takes the following form,

$\displaystyle\partialderivative{f(r,s,p_{T})}{s}=-\frac{\tau_{F}}{\tau_{R}}\frac{\left(f-f_{eq}\right)}{F(\tau(s))\,\tau(s)\,}\,.$
(16)

Where we have defined a new variable $s$.

$\displaystyle s(\tau)$
$\displaystyle=\int_{0}^{\tau}F\left(\tau^{\prime}\right)\frac{\tau^{\prime}}{\tau_{F}}d\tau^{\prime}\,,$
(17)

and used the coordinate transformations $(w,\tau)\to(r,s)$,

$\displaystyle r$ $\displaystyle=w-\alpha s\,,$ (18) $\displaystyle s$
$\displaystyle=s\,.$ (19)

Here $\alpha$ and $\tau_{F}$ are constants with dimensions of momentum and
time respectively so that $\alpha/\tau_{F}$ has dimensions of force. Then we
can write a solution

$\displaystyle f(s,r,p_{T})=D(s,s_{0})$ $\displaystyle f_{0}(r,p_{T})$
$\displaystyle+\int_{s_{0}}^{s}\frac{ds^{\prime}}{\tau_{R}(s^{\prime})}D(s,s^{\prime})f_{eq}(s^{\prime},r,p_{T}),$
(20)

where

$\displaystyle
D(s_{1},s_{2})=\exp\left[-\int_{s_{1}}^{s_{2}}\frac{ds^{\prime}}{F(\tau(s^{\prime}))\,(\tau(s^{\prime})/\tau_{F})\,\tau_{R}(s^{\prime})}\right].$
(21)

The above expression for $D(s_{1},s_{2})$ gives the impression that it depends
explicitly on the force. However, if one were to change the coordinates back
from $s\to\tau$, we can see that $D(s_{1},s_{2})$ takes the form

$\displaystyle
D(\tau_{1},\tau_{2})=\exp\left[-\int_{\tau_{1}}^{\tau_{2}}\frac{d\tau^{\prime}}{\tau_{R}(\tau^{\prime})}\right]\,.$
(22)

Therefore dependence of $D(\tau_{1},\tau_{2})$ on force is only implicitly
through $\tau_{R}(\tau^{\prime})$.

##  V Initial Conditions

The initial anisotropic distribution is chosen to be that of Romatschke-
Strickland and form [42],

$\displaystyle
f_{0}=\frac{2}{(2\pi)^{3}N_{0}}\exp{-\frac{\sqrt{p_{T}^{2}+(1+\xi_{0})p_{z}^{2}+m^{2}}}{\Lambda_{0}}}.$
(23)

The three initial free parameters $N_{0}$, $\xi_{0}$ and $\Lambda_{0}$ allow
us to specify the initial fluid parameters $T$,$P_{L}$ and $P_{T}$. Here
$\xi_{0}$ corresponds to the longitudinal anisotropic parameter in momentum
space, and $\Lambda_{0}$ is an energy scale, and in the limit it reduces to
local temperature. The value of $\Lambda_{0}$ and $N_{0}$ is set such that the
energy density matches the energy density of an equilibrium distribution with
the initial temperature $T_{0}$. To study the effect of varying
$\frac{m}{\Lambda_{0}}$ we can change $N_{0}$ while ensuring that the initial
energy density and $\xi_{0}$ remain fixed. For the non-conformal case
($m/\Lambda_{0}\neq 0$) we use the same set of initial parameters
(Tab.(2),App.(3)) as [43] for ease of comparison.

The hydrodynamic variables like energy density ($\mathcal{E}$), longitudinal
$P_{L}$ and transverse pressure $P_{T}$ in the presence of external forces can
be obtained from Eq.(3) and Eq.(IV)

$\displaystyle{\mathcal{E}}(s)$
$\displaystyle=D(s,s_{0})\frac{\Lambda_{0}^{4}}{4\pi
N_{0}}{H}^{F}_{\epsilon}\left[s_{0},s,\xi_{0},\frac{\alpha}{\Lambda_{0}},\frac{m}{\Lambda_{0}}\right]$
$\displaystyle+\int_{s_{0}}^{s}\frac{ds^{\prime}}{\tau_{R}(s^{\prime})}D(s^{\prime},s)T^{4}(s^{\prime}){H}^{F}_{\epsilon}\left[s^{\prime},s,\frac{\alpha}{T},\frac{m}{T}\right]\,,$
(24) $\displaystyle P_{L}(s)$
$\displaystyle=D(s,s_{0})\frac{\Lambda_{0}^{4}}{4\pi
N_{0}}{H}^{F}_{L}\left[s_{0},s,\xi_{0},\frac{\alpha}{\Lambda_{0}},\frac{m}{\Lambda_{0}}\right]$
$\displaystyle+\int_{s_{0}}^{s}\frac{ds^{\prime}}{\tau_{R}(s^{\prime})}D(s^{\prime},s)T^{4}(s^{\prime}){H}^{F}_{L}\left[s^{\prime},s,\frac{\alpha}{T},\frac{m}{T}\right]\,,$
(25) $\displaystyle P_{T}(s)$
$\displaystyle=D(s,s_{0})\frac{\Lambda_{0}^{4}}{4\pi
N_{0}}{H}^{F}_{T}\left[s_{0},s,\xi_{0},\frac{\alpha}{\Lambda_{0}},\frac{m}{\Lambda_{0}}\right]$
$\displaystyle+\int_{s_{0}}^{s}\frac{ds^{\prime}}{\tau_{R}(s^{\prime})}D(s^{\prime},s)T^{4}(\tau^{\prime}){H}^{F}_{T}\left[s^{\prime},s,\frac{\alpha}{T},\frac{m}{T}\right]\,,$
(26)

by evaluating the corresponding integrals App.[B]. Note that due to the
breaking of reflection symmetry, the energy-momentum tensor is no longer
diagonal. The non-diagonal term $T^{03}$ can be evaluated as

$\displaystyle
T^{03}=\int_{s_{0}}^{s}\frac{ds^{\prime}}{\tau_{R}(s^{\prime})}D(s,s^{\prime})T^{4}(s^{\prime}){H}^{F}_{03}\left[s^{\prime},s,\frac{m}{T}\right].$
(27)

We can use it to calculate the modified flow velocity $u^{\mu}$ using the
Landau frame definition

$\displaystyle T^{\mu\nu}u_{\nu}=\epsilon u^{\mu}\,.$ (28)

$T^{03}/P_{eq}$ contains the directionality of the force as its sign changes
with the force direction. Therefore we can use $T^{03}/P_{eq}$ and $\delta
u/u<<1$ as a measure of self consistency. Here $P_{eq}$ is the equilibrium
isotropic pressure defined by

$P_{eq}=\int d{\Xi}\frac{{\bm{p}^{2}}}{3}\,f_{eq}\,.$ (29)

For the rest of our calculation, we use the conformal relation
$\tau_{R}(\tau)=5c/T(\tau)$, where $c\propto\frac{\eta_{0}}{s_{0}}$ is a
dimensionless number which sets the initial viscosity of the system. In this
study, we set the initial time to $\tau_{0}=0.1$fm and the initial temperature
to $T_{0}=500$ MeV. For the non-conformal computations, we set the mass to
$m=200$ MeV.

The Eq.(24) can be solved using iterative techniques (A faster algorithm
compared to the iterative technique used in previous studies is given in
App.(C)) and using the energy density-temperature relation,

$\mathcal{E}(T)=\frac{3T^{4}}{\pi^{2}}\left(\frac{z^{2}}{2}K_{2}(z)+\frac{z^{3}}{6}K_{1}(z)\right)\,,$
(30)

where $z=m/T$ and $K_{1},K_{2}$ are modified Bessel functions of the second
kind. The relations for pressure Eq.(25),Eq.(26) can be evaluated once the
temperature is obtained using the above procedure.

###  V.1 Parametrised Force

####  V.1.1 Force Type 1

Motivated by the exponentially decaying fields in high-energy heavy-ion
collisions we use the following parametrisation of the force

$\displaystyle F(\tau)=\frac{\tau}{\tau_{F}}\exp{1-\frac{\tau}{\tau_{F}}},$
(31)

where $\tau_{F}$ is a decay time scale. The parameter $\tau_{F}$ is the proper
time at which $\mathcal{F}$ takes its maximum value $\frac{\alpha}{\tau_{F}}$.
Here $\alpha$ has the dimension of momentum, $\frac{\alpha}{\tau_{F}}$ has
dimensions of $p^{2}$(Force). By varying the value of $\alpha$ and $\tau_{F}$
we can control the strength and the duration of the force. A plot of the
normalized force for various values of decay time is given in Fig.(1).

Figure 1: Normalised force (max($F$) = 1) for different timescales. $\tau_{F}$
= 1 fm/c (dot), $\tau_{F}$ = 2.5 fm/c (dashed), $\tau_{F}$ = 5 fm/c (dot-
dashed)

####  V.1.2 Force Type 2

For exploratory purposes, we also try a constant force

$\displaystyle F(\tau)=1,$ (32)

where $\mathcal{F}$ takes the maximum value $\mathcal{F}_{0}$. For ease of
comparison, we rewrite $\mathcal{F}_{0}=\frac{\alpha}{\tau_{F}}$.

##  VI Effect of force on fluid variables

###  VI.1 Force Type 1

The application of an external force should change the particle momenta and
therefore could have observable effects on pressure. We explore these effects
in Fig.[2-5] for various values of $m/\Lambda_{0}$. The figures [2-5] show the
proper time evolution of temperature (top left panel), transverse pressure
(top right panel), longitudinal pressure (bottom right panel) and the scaled
off-diagonal component $T^{03}/P_{eq}$ (bottom left panel) of the energy-
momentum tensor for various values of initial anisotropy. Different lines in
these figures correspond to various values of $\alpha$. Fig [2] shows the time
evolution of temperature for negative $\xi_{0}$ values whereas Fig[3] is for a
sharply peaked $p_{z}$ distribution. Fig.(4),(5) is for isotropic initial
distribution but for varying $m/\Lambda_{0}$ values. In each figure showing
the temperature evolution, the red line corresponds to ideal hydro evolution,
and green, orange, and blue correspond to Boltzmann solution with
$\alpha=200,\,100,\,50,\,0$ respectively. The same color code is used in
$\bar{P}_{L},\bar{P}_{T}$ , and $\bar{T}^{\tau z}-\tau$ evolution.

Figure 2: Evolution of $T$, $\bar{P}_{L}$, $\bar{P}_{T}$, and $\bar{T}^{03}$
for $\alpha=[200,100,50,0]$, $\tau_{m}=1$, and for $\bar{\Pi}_{0}=0$,
$\bar{\pi}_{0}=-1$.

Figure 3: Evolution of $T$, $\bar{P}_{L}$, $\bar{P}_{T}$, and $\bar{T}^{03}$
for $\alpha=[200,100,50,0]$, $\tau_{m}=1$, and for $\bar{\Pi}_{0}=0$,
$\bar{\pi}_{0}=0.99$.

Figure 4: Evolution of $T$, $\bar{P}_{L}$, $\bar{P}_{T}$, and $\bar{T}^{03}$
for $\alpha=[200,100,50,0]$, $\tau_{m}=1$ and for
$\bar{\Pi}_{0}=0.85,\bar{\pi}_{0}=0$.

Figure 5: Evolution of $T$, $\bar{P}_{L}$, $\bar{P}_{T}$, and $\bar{T}^{03}$
for $\alpha=[200,100,50,0]$, $\tau_{m}=1$ and for
$\bar{\Pi}_{0}=0.25,\bar{\pi}_{0}=0$.

It is seen that for the magnitude of the force considered in this study, there
is a marginal impact on the evolution of energy density (temperature). A
possible explanation is the following: the expansion cools down the system
whereas force, in the presence of dissipation, should increase the temperature
of the system. The effect is small because we are assuming the force and the
correction to the fluid velocity is small.

It is also seen that $\bar{P}_{L}$ increases while $\bar{P}_{T}$
correspondingly decreases during the time period in which the force acts.
$P_{L}$ increases as it is driven by an external force, which in turn drives
the transverse pressure down due to the constraint
$\mathcal{E}-2P_{T}-P_{L}=\langle m^{2}\rangle$. We see that the long-term
behavior of the evolution is such that the system relaxes back to the zero-
force curve, and loses memory of the variations due to force.

From Fig.(2) and Fig.(3) we see that the effect of longitudinal force is
diminished when there is a large initial shear anisotropy. That means that the
predominant factor driving the evolution of the system is the shear force. On
the other hand in Fig.(4) and Fig.(5) when the shear anisotropy is low, we see
that the effect of force is pronounced and the effect of bulk anisotropy is
subdominant.

###  VI.2 Force Type 2

In the case of constant force, the force-to-temperature ratio cannot be kept
constant throughout the evolution of the system. So, we use smaller
$\alpha=2,5,10$ (MeV) values to make the corrections small. We see again that
the variation in temperature is marginal except for late times when the force
is comparatively larger than the effective temperature. The variations in the
bulk observables are also only noticeable at late times. The qualitative
behavior of the bulk observables remains the same as that for the case of the
decaying force. We show the results for a constant force for
$\bar{\Pi}_{0}=-0.37$,$\bar{\pi}_{0}=-1$ in Fig.(6).

Figure 6: Evolution of $T$,$\bar{P}_{L}$,$\bar{P}_{T}$ and $\bar{T}^{03}$ for
$\alpha=[10,5,2,0]$,$\tau_{m}=1$ and for
$\bar{\Pi}_{0}=-0.37,\bar{\pi}_{0}=-1$

###  VI.3 Attractor Behaviour: Non-conformal

It was reported that for non-conformal systems a universal attractor was not
obtained for fluid variables other than for scaled longitudinal pressure
$\bar{P}_{L}=P_{L}/P$ [43]. The authors also computed the bounds for $\Pi/P$
(scaled bulk pressure) and $\pi/P$ (scaled sheer pressure) so that the
positivity of the distribution function, the transverse and the longitudinal
pressure is maintained. Following their findings we use the same set of
initial parameters given in Table (2). The corresponding values for
$m/\Lambda_{0}$, $N_{0}$ and $\xi_{0}$ are reported in Table.(3). With this,
we check how the attractor behavior is modified by an external force. In
Fig.(7) the scaled longitudinal pressure $\bar{P}_{L}$ is plotted for three
different force $\alpha=0$ (top panel), $\alpha=50$ (middle panel), and
$\alpha=100$ (bottom panel). In each subplot, the different colored lines
represent different anisotropies as given in Table.(3). For zero force we
reproduce the results in the the previous study [43]. However, for finite
forces, only a late-time attractor is present.

Figure 7: Evolution of $\bar{P}_{L}$ for $\alpha=[0,50,100]$,$\tau_{m}=1$ and
for various initial conditions

###  VI.4 Attractor Behaviour: Massless Limit

Systems with conformal symmetry were shown to exhibit universal early time
attractor behavior for a variety of system configurations and flow profiles
[8, 44, 45, 46, 9, 47]. Here we check if early-time attractors can be
reproduced for systems under external forces. In Fig.(8) the transverse and
longitudinal pressure is plotted for varying initial conditions (given in
Table.(1)) in the presence of an external force in the massless limit. We
observe that even when the system is subjected to an external force it shows
an attractor in $\bar{P}_{L}$ and $\bar{P}_{T}$ much earlier than the massive
case (as compared to Fig.(7)). We note however that this is subject to the
condition that the force has died out much before equilibration is achieved.

The equilibration rate is mostly determined by the exponentially decaying
damping factor $D(\tau,\tau^{\prime})$ given in Eq.(22). One can see from
Eq.(24), Eq.(25) and Eq.(26) that the decay rate is determined by the
interplay between the damping factor and the $H_{\mathcal{E}/P_{L}/P_{T}}$
functions. It is observed from the $\tau-T$ graphs that the force has
negligible impact on the temperature evolution and, therefore, doesn’t modify
the damping function(as $\tau_{R}\sim 1/T$). However, it significantly
modifies $H_{P_{L}/P_{T}}$ functions. Therefore, the relative change in
behavior is due to the higher sensitivity of these functions on mass and
force. Within the approximation scheme employed, in the massless limit, the
system regains conformal symmetry. A combination of these factors gives rise
to the observed behaviour.

Figure 8:  Scaled Transverse and longitudinal pressure plotted for varying
initial conditions (given in Table.(1)) in the presence of an external force
$\alpha=100$ with m=0.

##  VII Summary and Conclusion

We solve the RTA Boltzmann equation incorporating the external force and study
the dynamics of the system far away from local equilibrium under Bjorken flow.
We also study the attractor behavior of a system undergoing Bjorken expansion
in the presence of external longitudinal force. For this exploratory study, we
consider a constant force and time-parameterized force which initially peaks
and then exponentially decays afterward. We treat this force as a small
perturbation so that the overall flow profile remains unchanged. For a
constant force the temperature evolution was observed to vary at late times
while for the time-varying force considered here, no significant change was
observed for temperature evolution. The longitudinal pressure was observed to
increase as it is driven by the external longitudinal force which in turn
drove the transverse pressure down due to the constraint
$\mathcal{E}-2P_{T}-P_{L}=\langle m^{2}\rangle$. The force was also observed
to have a larger effect for smaller-scaled mass values.

We explored the attractor behavior by considering the time evolution of scaled
longitudinal and transverse pressure for various initial anisotropies. For
conformal systems an early time attractor was observed for various bulk scaled
observables [8]. Whereas in the non-conformal case [43] an early time
attractor was observed only for scaled longitudinal pressure. The addition of
force however modifies this behaviour and an attractor is only found at late
times. In the conformal limit, we observed that, within our approximation
scheme, the attractor for scaled longitudinal pressure reappears even in the
presence of an external force. This behavior is due to the non-trivial
interplay between the damping function $D(\tau,\tau^{\prime})$ and
$H_{P_{L}/P_{T}}$. The results in this work were obtained by assuming that the
system obeys approximate Bjorken symmetry. However, the non-diagonal terms
$T^{03}$ become comparable to $\mathcal{E}$ and $\mathcal{P}$ for large values
of force. Therefore, a proper quantitative estimate of the evolution of the
bulk properties cannot be obtained using the techniques used in this paper.

For future work, one can explore the effect of force on non-Bjorken flows like
Gubser and even arbitrary flow profiles. One can consider more realistic
electromagnetic forces and a self-consistent solution can be obtained by
considering a $1+1$ dimensional expansion. One can also solve for systems
under a magnetic field which would necessarily include transverse dynamics and
momentum dependant forces, Further, one can explore the effects of Chromo-
Electric and Chromo-magnetic fields which would require modifications in the
collision kernel.

##  VIII Acknowledgements

AP acknowledges the CSIR-HRDG financial support. RG and VR acknowledge support
from the DAE, Govt. of India. RG also acknowledges fruitful discussions with
Sunil Jaiswal during the early stages of this work.

##  Appendix A RTA Boltzmann Equation with Force

Boltzmann equation would then read

$\displaystyle\mathcal{F}$
$\displaystyle\tau\partialderivative{f}{w}+\partialderivative{f}{\tau}+\frac{\delta
u_{z}p^{z}}{u_{\tau}p^{\tau}}\partialderivative{f}{z}$ (33)
$\displaystyle=-\frac{p^{\tau}}{u_{\tau}p^{\tau}}\frac{\left(f-f_{eq}\right)}{\tau_{R}}-\frac{\delta
u\cdot p}{u_{\tau}p^{\tau}}\frac{\left(f-f_{eq}\right)}{\tau_{R}}\,.$ (34)

If we now assume that $\delta u/u_{\tau}<<1$, we can ignore the $\delta u$
terms from the above equation and get

$\displaystyle\left(\mathcal{F}(\tau)\tau\partialderivative{w}+\partialderivative{\tau}\right)f=-\frac{\left(f-f_{eq}\right)}{\tau_{R}}\,.$
(35)

For the consistency of the above approximation, we need to keep the strength
of the force sufficiently small so that the background flow is still close to
Bjorken.

###  A.1 Change of Variables

For computational convenience we write the force $\mathcal{F}$ as

$\mathcal{F}=\frac{\alpha}{\tau_{F}}F(\tau)\,.$ (36)

We then rewrite the Boltzmann equation as

$\displaystyle\left(\frac{\alpha}{\tau_{F}}F(\tau)\tau\partialderivative{w}+\partialderivative{\tau}\right)f=-\frac{\left(f-f_{eq}\right)}{\tau_{R}}\,.$
(37)

To convert Eq.(35) to a simpler form, we divide both sides by
$F(\tau)\tau/\tau_{F}$ getting

$\displaystyle\left(\alpha\partialderivative{w}+\frac{\tau_{F}}{F(\tau)\tau}\partialderivative{\tau}\right)f=-\frac{\tau_{F}}{\tau_{R}}\frac{\left(f-f_{eq}\right)}{F(\tau)\tau}\,.$
(38)

We define the variable $s$

$\displaystyle s(\tau)$
$\displaystyle=\int_{0}^{\tau}F\left(\tau^{\prime}\right)\frac{\tau^{\prime}}{\tau_{F}}d\tau^{\prime},$
(39)

and rewrite the proper time derivative to get Eq.(15),

$\displaystyle\left(\alpha\partialderivative{w}+\partialderivative{s}\right)f=-\frac{\tau_{F}}{\tau_{R}}\frac{\left(f-f_{eq}\right)}{F(\tau)\tau}\,.$
(40)

We now use the standard technique of characteristics for solving partial
differential equations and define new variables $(w,s)\rightarrow(r,s)$

$\displaystyle r$ $\displaystyle=w-\alpha s,$ (41) $\displaystyle s$
$\displaystyle=s,$ (42)

and obtain the equation

$\displaystyle\partialderivative{f(r,s,p_{T})}{s}=-\frac{\tau_{F}}{\tau_{R}}\frac{\left(f-f_{eq}\right)}{F(\tau(s))\tau(s)}\,.$
(43)

##  Appendix B Integrals

In this section we give the explicit form of the $H$ functions appearing in
the equation for $\mathcal{E}$, $P$ etc given in Eq.(V) We define the
integrals,

$\displaystyle\tilde{H}^{F}_{\epsilon}\left[s^{\prime},s,m,\alpha\right]=\int
du\,u^{3}\,e^{-\sqrt{u^{2}+z^{2}}}H^{F}_{\epsilon}(u,z,s,s^{\prime},\alpha),$
(44) $\displaystyle\tilde{H}^{F}_{L}\left[s^{\prime},s,m,\alpha\right]=\int
du\,u^{3}\,e^{-\sqrt{u^{2}+z^{2}}}H^{F}_{L}(u,z,s,s^{\prime},\alpha),$ (45)
$\displaystyle\tilde{H}^{F}_{T}\left[s^{\prime},s,m,\alpha\right]=\int
du\,u^{3}\,e^{-\sqrt{u^{2}+z^{2}}}H^{F}_{T}(u,z,s,s^{\prime},\alpha),$ (46)

where $u=p/T$ and $z=m/T$.

The functions inside the integrals can be obtained using analytical
techniques. Define

$\displaystyle h(s)\equiv\tau(s)\,.$ (47)

For notational simplicity, we define the following variables

$\displaystyle\frac{h(s^{\prime})}{h(s)}$
$\displaystyle\equiv\mathbf{\bar{\text{$h$}}}(s,s^{\prime})\equiv\mathbf{\bar{\text{$h$}}}\,,$
(48) $\displaystyle\frac{\alpha\times(s-s^{\prime})}{h(s^{\prime})}$
$\displaystyle\equiv g(s,s^{\prime})\equiv g,$ (49)

and

$\displaystyle N^{2}$
$\displaystyle\equiv\frac{1}{(1-\mathbf{\bar{\text{$h$}}}^{2})}\left(\frac{u^{2}+\mathbf{\bar{\text{$h$}}}^{2}g^{2}+z^{2}}{u^{2}}+\frac{g^{2}\mathbf{\bar{\text{$h$}}}^{4}}{u^{2}(1-\mathbf{\bar{\text{$h$}}}^{2})}\right)\,,$
(50) $\displaystyle G$
$\displaystyle\equiv\frac{g\mathbf{\bar{\text{$h$}}}^{2}}{u(1-\mathbf{\bar{\text{$h$}}}^{2})}\,.$
(51)

Using the above definitions we can write

$\displaystyle H^{F}(u,z,s,s^{\prime},\alpha)$
$\displaystyle=-\mathbf{\bar{\text{$h$}}}\sqrt{(1-\mathbf{\bar{\text{$h$}}}^{2})}N^{2}\left\\{\frac{1}{2}\left[\sin^{-1}{\frac{G-1}{N}}-\sin^{-1}{\frac{\left(G+1\right)}{N}}\right]\right.$
$\displaystyle\left.+\frac{1}{2}\left[-\frac{1+G}{N}\sqrt{1-\left(\frac{1+G}{N}\right)^{2}}+\frac{G-1}{N}\sqrt{1-\left(\frac{G-1}{N}\right)^{2}}\right]\right\\},$

$\displaystyle H_{L}(u,z,s,s^{\prime},\alpha)$
$\displaystyle=-\frac{\mathbf{\bar{\text{$h$}}}^{3}}{\sqrt{1-\mathbf{\bar{\text{$h$}}}^{2}}}\left\\{\left(\frac{N^{2}}{2}+(g+G)^{2}\right)\left[\sin^{-1}{\frac{G-1}{N}}-\sin^{-1}{\frac{1+G}{N}}\right]\right.$
$\displaystyle+\frac{N^{2}}{2}\left[\frac{1+G}{N}\sqrt{1-\left(\frac{1+G}{N}\right)^{2}}-\frac{G-1}{N}\sqrt{1-\left(\frac{G-1}{N}\right)^{2}}\right]$
$\displaystyle-\left.2N(g+G)\left[\sqrt{1-\left(\frac{1+G}{N}\right)^{2}}-\sqrt{1-\left(\frac{1-G}{N}\right)^{2}}\right]\right\\},$

$\displaystyle H_{T}(u,z,s,s^{\prime},\alpha)$
$\displaystyle=-\frac{\mathbf{\bar{\text{$h$}}}}{\sqrt{(1-\mathbf{\bar{\text{$h$}}}^{2})}}\left\\{\left(1-G^{2}-\frac{N^{2}}{2}\right)\left[\sin^{-1}{\frac{G-1}{N}}-\sin^{-1}{\frac{1+G}{N}}\right]\right.$
$\displaystyle\left.-\frac{N^{2}}{2}\left[\frac{1+G}{N}\sqrt{1-\left(\frac{1+G}{N}\right)^{2}}-\frac{G-1}{N}\sqrt{1-\left(\frac{G-1}{N}\right)^{2}}\right]\right.$
$\displaystyle\left.+2NG\left[\sqrt{1-\left(\frac{1+G}{N}\right)^{2}}-\sqrt{1-\left(\frac{1-G}{N}\right)^{2}}\right]\right\\}.$
(52)

##  Appendix C Iterative Solution Algorithm

Consider the integral equation

$\displaystyle y(t)$ $\displaystyle=$ $\displaystyle K(t,t_{0},y)v(t,t_{0})$
(54)
$\displaystyle+\int_{t_{0}}^{t}dt^{\prime}K(t,t^{\prime},y)\frac{H(t,t^{\prime},y(t^{\prime}))}{\tau_{R}(y(t^{\prime}))}f(y(t^{\prime}))$

where,

$\displaystyle K(t,t^{\prime},y)$ $\displaystyle=$
$\displaystyle\exp{\int_{t^{\prime}}^{t}g(y(t^{\prime\prime}))dt^{\prime\prime}}$
(55)

and we are given the initial condition

$y(t_{0})=y_{0}.$ (56)

This implies

$y(t_{0})=v(t_{0},t_{0})$ (57)

as

$K(t,t,y)=1$ (58)

The usual method for solving this equation is to consider an arbitrary
functional form for the solution $y(t)$ and substitute it in the RHS and get a
new solution. This process is continued until the iteration converges to some
desired level of accuracy. The drawback of this method is that the speed of
convergence depends on the initial guess $y(t)$. Here we take an alternate
approach. If the function $y(t)$ is continuous then we approximate at each
step $t_{n}=t_{n-1}+\Delta t_{n}$,

$y(t_{n-1}+\Delta t_{n})=y(t_{n-1})+\mathcal{O}(\Delta t_{n})\,.$ (59)

Therefore with the initial condition $y(t_{0})=v(t_{0},t_{0})$, we start with

$y(t_{0}+\Delta t)\sim y(t_{0})$ (60)

and continue this procedure in order for each $n$ successively. We show
elsewhere that after one iteration this produces an approximation for $y(t)$
accurate up to $\mathcal{O}(\Delta t^{2})$. This process gets rid of the need
for a good initial guess. It not only converges faster but also allows us to
estimate order or error for our computation.

##  Appendix D Inital Conditions

For the conformal case, we fix the anisotropy and normalization the same as
the non-conformal case but vary the value of $\Lambda_{0}$ to match the
initial energy density. The initial conditions for the distribution function
were chosen to match [43] for comparison and are reproduced in the table (1).

No. | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
$\Lambda_{0}$ | 321.74 | 314 | 275 | 1089 | 198 | 500 | 500
$\xi_{0}$ | -0.832 | -0.908 | -0.949 | 1208.05 | -0.987 | 0 | 0
Table 1: Various values of $\Lambda_{0}$,$N_{0}$ and $\xi_{0}$ corresponding to different initial values of $(\pi/P)_{0}$ for m=0. No. | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
$(\Pi/P)_{0}$ | 0 | -0.25 | -0.37 | 0 | 0 | 0.25 | -0.85
$(\pi/P)_{0}$ | -1 | -1 | -1 | 0.99 | -1.8 | 0 | 0
Table 2: Various initial values of $(\Pi/P)_{0}$,$(\pi/P)_{0}$ and their number code. No. | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
$m/\Lambda_{0}$ | 0.016 | 4.808 | 10.89 | 0.294 | 1.818 | 2.023 | 20
$N_{0}$ | 0.655 | $4\times 10^{-5}$ | $2.5\times 10^{-8}$ | 0.078 | 0.0632 | $1.06\times 10^{-3}$ | $1.48\times 10^{-13}$
$\xi_{0}$ | -0.832 | -0.908 | -0.949 | 1208.05 | -0.987 | 0 | 0
Table 3: Various values of $m/\Lambda_{0}$,$N_{0}$ and $\xi_{0}$ corresponding
to different initial values of $(\Pi/P)_{0}$,$(\pi/P)_{0}$ given in (2).

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