# Early time behavior of spatial and momentum anisotropies in a kinetic
approach to nuclear collisions

Marc Borrell<EMAIL_ADDRESS>Nicolas Borghini
<EMAIL_ADDRESS>Fakultät für Physik, Universität Bielefeld,
D-33615 Bielefeld, Germany

###### Abstract

We derive a general formula for the early time dependence of a phase space
distribution evolving according to the kinetic Boltzmann equation. Assuming
that the early evolution of the system created in high-energy nuclear
collisions can be described by kinetic theory, we calculate the scaling
behaviors for the onset of various characteristics of the transverse dynamics.
In particular, we show that the scaling behavior of the anisotropic flow
coefficients $v_{n}$ at early times does not depend on the details of the
collision kernel or the system composition, while at the same time it differs
from the prediction of fluid dynamics.

##  I Introduction

Ultrarelativistic nuclear collisions at the Relativistic Heavy-Ion Collider
(RHIC) and the Large Hadron Collider (LHC) produce a large number of
particles, whose bulk, consisting of soft hadrons, exhibits many signals of
collective dynamical behavior. The latter, referred to as collective flow, is
almost universally interpreted as the genuine space-time evolution of the
created system. One of the key observables of bulk collectivity involved is
anisotropic flow, i.e. the asymmetry in the transverse emission pattern of
particles [1, 2].

The collective dynamical evolution of the bulk is commonly modeled as the
dissipative expansion of a relativistic fluid, at least for a significant part
of the system history. Such modeling, possibly supplemented with a dynamical
“pre-equilibrium” stage (see Refs. [3, 4] for recent reviews) and a hadronic
afterburner following the fluid-dynamical evolution, yields a very good
description of experimentally measured soft hadron production in collisions of
heavy nuclei. More surprisingly, it can also satisfactory describe the bulk in
collisions of smaller systems with a large final-state multiplicity [5].

The use of fluid dynamics is however more questionable in such systems with
few degrees of freedom [6]. Moreover, since the overall system lifetime is
shorter, the pre- and post-equilibrium stages become comparatively more
important in the evolution. This has led to a renewed interest in alternative
descriptions of collective flow, in particular in kinetic-theory models. In
this framework, a number of recent studies investigated anisotropic flow for
setups with a simplified initial geometry [7, 8, 9, 10, 11, 12, 13] — to put
aside the uncertainties in the initial state of actual small systems —,
revisiting and extending earlier approaches [14, 15, 16].

In this spirit, we consider in this paper a system of degrees of freedom
described by a single-particle phase space distribution
$f(t,\vec{x},\\!\vec{\,p})$ obeying the kinetic Boltzmann equation [17]

$p^{\mu}\partial_{\mu}f(t,\vec{x},\\!\vec{\,p})={\cal C}[f],$ (1)

with ${\cal C}[f]$ the collision term modeling the effect of rescatterings,
whose dependence on the position $\vec{x}$ and momentum $\\!\vec{\,p}$ we do
not denote. As is customary in the study of nuclear collisions at
ultrarelativistic energies, we do not include the possible influence of a mean
field on the left hand side of this equation. We focus on the development of
anisotropic transverse flow and the system geometry at early times,
investigating their respective scaling behaviors as a function of time and of
the average number of rescatterings per particle, characterized by an inverse
Knudsen number.

The time evolution of anisotropic flow and spatial characteristics of the
system created in heavy-ion collisions has already been studied before, either
in fluid-dynamical [18, 19, 20, 21] or transport [15, 13] calculations,
extended over the whole system lifetime, yet always for specific initial
conditions. In contrast, it was found in Ref. [22] that irrespective of the
initial state, the early-time development of (the azimuthally symmetric
component of) transverse flow seems to be universal, with a growth linear in
time for various classes of models. Here we do not investigate several classes
of models but “only” kinetic theory — yet at a quite general level — and we
also do not specify the initial state of the system but leave it arbitrary.

We begin in Sect. II by discussing the time development of the quantities we
are interested in in a collisionless system. In Sect. III, we introduce the
early-time expansion of the phase space distribution, following from the
Boltzmann equation, which we later use to derive the behavior of anisotropies
in an interacting system. We then investigate in Sect. IV the special case of
a two-dimensional system of massless particles with elastic binary
rescatterings, before generalizing in Sect. V to the case of a generic kinetic
theory, before we discuss our results (Sect. VI). A few calculations relevant
to Sect. III–V are included in appendices. Throughout the paper we set
$c=\hbar=1$ and we use a metric with negative signature.

##  II Free streaming system

Let us first investigate a non-interacting system, for which the collision
term on the right hand side of Eq. (1) vanishes. How anisotropic flow and the
spatial eccentricities behave in such a system is well known. Since there are
no rescatterings, the momentum distribution of the particles cannot change, in
particular the flow harmonics $v_{n}$. In turn, in the absence of initial
anisotropic flow the spatial eccentricities monotonously decrease in absolute
value, tending towards 0 at large times. We shall nevertheless discuss the
case of such a system in some detail, first, to introduce a few notations. And
secondly, because our results in this section will prove to be useful when we
consider the early-time evolution of an interacting system, in which the
calculations will be performed in the vicinity of the non-interacting case.

As is well known, the solutions of the collisionless Boltzmann equation are
free-streaming solutions obeying the functional relation

$f_{\rm f.s.}(t,\vec{x},\\!\vec{\,p})=f_{\rm
f.s.}\\!\bigg{(}t_{0},\vec{x}-\frac{\\!\vec{\,p}}{E}(t-t_{0}),\\!\vec{\,p}\bigg{)},$
(2)

where $t_{0}$ denotes a reference time, in particular the initial time of the
evolution. Throughout the paper we shall often denote with a subscript 0 the
value at $t_{0}$ of a function of time (and possibly other variables), as e.g.
$f_{0}(\vec{x},\\!\vec{\,p})=f(t_{0},\vec{x},\\!\vec{\,p})$. Without loss of
generality, we shall take $t_{0}=0$ in our calculations.

Let $g(\vec{x},\\!\vec{\,p})$ be a function of the phase space coordinates. We
denote by $\left\langle\cdots\right\rangle_{t}^{\rm f.s.}$ an average with a
free-streaming distribution $f_{\rm f.s.}(t,\vec{x},\\!\vec{\,p})$ taken at
time $t$:

$\left\langle g(\vec{x},\\!\vec{\,p})\right\rangle_{t}^{\rm
f.s.}\equiv\int\\!g(\vec{x},\\!\vec{\,p})f_{\rm
f.s.}(t,\vec{x},\\!\vec{\,p})\,{\rm d}^{3}\vec{x}\,{\rm d}^{3}\\!\vec{\,p},$
(3)

where the integral runs over the whole phase space. More generally, all
averages throughout the paper are performed over phase space and use a
particle-number density as weight.

Using the characteristic relation (2) in the integrand, a straightforward
change of variables allows one to express $\left\langle
g(\vec{x},\\!\vec{\,p})\right\rangle_{t}^{\rm f.s.}$ in terms of an average at
$t_{0}$:

$\left\langle g(\vec{x},\\!\vec{\,p})\right\rangle_{t}^{\rm f.s.}=\left\langle
g\big{(}\vec{x}+\vec{v}\,t,\\!\vec{\,p}\big{)}\right\rangle_{0},$ (4)

where $\vec{v}\equiv\\!\vec{\,p}/E$. Note that we drop the the superscript
f.s. when denoting the average in the initial state. With the help of this
identity one can readily derive the time evolution in a non-interacting system
of the spatial “eccentricities” [21, 23]

$\epsilon^{\bf x}_{n\,}{\rm e}^{{\rm i}n\Phi_{n}}\equiv-\frac{\left\langle
r_{\perp}^{n}{\rm e}^{{\rm i}n\theta}\right\rangle}{\left\langle
r_{\perp}^{n}\right\rangle},$ (5)

where $(r_{\perp},\theta)$ are polar coordinates in the transverse plane
evaluated in a centered frame, so that the definition is only interesting for
$n\geq 2$. For simplicity, we assume that the initial phase space distribution
is isotropic in momentum space at each point $\vec{x}$.

Setting $\Phi_{2}=0$ for the moment, we being with the “ellipticity”

$\epsilon_{2}^{\bf x}\equiv\frac{\left\langle
y^{2}-x^{2}\right\rangle}{\left\langle x^{2}+y^{2}\right\rangle},$ (6)

whose behavior in a free-streaming system has already been investigated [24].
If each average in Eq. (6) is computed with a free-streaming solution,
relation (4) yields

$\left\langle x^{2}\right\rangle_{t}^{\rm
f.s.}=\left\langle(x+v_{x}t)^{2}\right\rangle_{0}=\left\langle
x^{2}\right\rangle_{0}+t^{2}\left\langle v_{x}^{2}\right\rangle_{0},$

where we used $\left\langle v_{x}\right\rangle_{0}=0$ as follows from the
assumed isotropy in momentum space. Similarly, one has

$\left\langle y^{2}\right\rangle_{t}^{\rm f.s.}=\left\langle
y^{2}\right\rangle_{0}+t^{2}\left\langle v_{y}^{2}\right\rangle_{0}.$

Momentum-space isotropy also yields

$\left\langle v_{x}^{2}\right\rangle_{0}=\left\langle
v_{y}^{2}\right\rangle_{0}=\frac{1}{2}\left\langle
v_{\perp}^{2}\right\rangle_{0},$

where $v_{\perp}=|\bm{v}_{\perp}|$ is the modulus of the transverse velocity,
so that one quickly finds [24]

$\epsilon_{2}^{\bf x}(t)=\frac{\epsilon_{2}^{\bf x}(t_{0})}{1+\left\langle
v_{\perp}^{2}\right\rangle_{0}t^{2}/\left\langle
r_{\perp}^{2}\right\rangle_{0}},$ (7)

where $\left\langle r_{\perp}^{2}\right\rangle_{0}\equiv\left\langle
x^{2}+y^{2}\right\rangle_{0}$. This result is unchanged if the initial
participant-plane angle $\Phi_{2}$ is not aligned with the $x$-axis: in a non-
interacting system without initial flow, $\Phi_{2}$ does not change with time.
As anticipated, (the modulus of) $\epsilon_{2}^{\bf x}(t)$ decreases with
time, with an early-time departure from its initial value that is quadratic in
time:

$\epsilon_{2}^{\bf x}(t)\simeq\epsilon_{2}^{\bf
x}(t_{0})\bigg{(}1-\frac{\left\langle
v_{\perp}^{2}\right\rangle_{0}}{\left\langle
r_{\perp}^{2}\right\rangle_{0}}t^{2}\bigg{)}\ \text{for
}t\ll\Bigg{(}\\!\frac{\left\langle
r_{\perp}^{2}\right\rangle_{0}}{\left\langle
v_{\perp}^{2}\right\rangle_{0}}\\!\Bigg{)}^{\\!\\!1/2}.$ (8)

As we shall now show, the spatial eccentricity $\epsilon_{n}^{\bf x}$ in the
$n$-th harmonic also departs quadratically with time from its initial value in
a free streaming system without initial anisotropic flow.111To the best of our
knowledge, this behavior has not been reported in the literature before.
Consider thus

$\epsilon^{\bf x}_{n\,}{\rm e}^{{\rm i}n\Phi_{n}}=-\frac{\left\langle
r_{\perp}^{n}{\rm e}^{{\rm i}n\theta}\right\rangle}{\left\langle
r_{\perp}^{n}\right\rangle}=-\frac{\left\langle(x+{\rm
i}y)^{n}\right\rangle}{\left\langle(x^{2}+y^{2})^{n/2}\right\rangle}.$ (9)

In the case of a non-interacting system, invoking Eq. (4) for the numerator
yields

$\displaystyle\left\langle(x+{\rm i}y)^{n}\right\rangle_{t}^{\rm f.s.}$
$\displaystyle=\left\langle\big{[}(x+v_{x}t)+{\rm
i}(y+v_{y}t)\big{]}^{n}\right\rangle_{0}$
$\displaystyle=\left\langle\big{[}(x+{\rm i}y)+(v_{x}+{\rm
i}v_{y})t\big{]}^{n}\right\rangle_{0}.$

Introducing the azimuthal angle $\varphi$ of the velocity (or equivalently the
momentum), we can switch back to polar coordinates: $x+{\rm i}y=r_{\perp}{\rm
e}^{{\rm i}\theta}$ and $v_{x}+{\rm i}v_{y}=v_{\perp}{\rm e}^{{\rm
i}\varphi}$. This yields

$\left\langle r_{\perp}^{n}{\rm e}^{{\rm i}n\theta}\right\rangle_{t}^{\rm
f.s.}=\sum_{k=0}^{n}\binom{n}{k}\\!\left\langle r_{\perp}^{n-k}{\rm e}^{{\rm
i}(n-k)\theta\,}v_{\perp}^{k}{\rm e}^{{\rm
i}k\varphi}\right\rangle_{\\!0}t^{k}.$

Due to the local isotropy in momentum space, every average of ${\rm e}^{{\rm
i}k\varphi}$ with $k\neq 0$ vanishes, so that the above average is actually
time-independent:

$\left\langle r_{\perp}^{n}{\rm e}^{{\rm i}n\theta}\right\rangle_{t}^{\rm
f.s.}=\left\langle r_{\perp}^{n}{\rm e}^{{\rm i}n\theta}\right\rangle_{0}.$
(10)

In particular the phase of $\left\langle r_{\perp}^{n}{\rm e}^{{\rm
i}n\theta}\right\rangle_{t}^{\rm f.s.}$ remains constant, i.e. the
participant-plane angle $\Phi_{n}$ does not rotate in the absence of
interactions, which is quite intuitive.

For the denominator of Eq. (9), we similarly write

$\displaystyle\left\langle r_{\perp}^{n}\right\rangle_{t}^{\rm
f.s.}=\left\langle\big{[}(x+v_{x}t)^{2}+(y+v_{y}t)^{2}\big{]}^{n/2}\right\rangle_{0}.$
(11)

If $n$ is even, the quantity to be averaged can be exactly computed for any
$t$ with the binomial theorem. Invoking the isotropy in momentum space, only
the terms with even powers of both $v_{x}$ and $v_{y}$ remain after averaging.
If we only look at the early-time behavior, a Taylor expansion valid for both
even and odd values of $n$ yields

$\displaystyle\left\langle\big{[}(x+v_{x}t)^{2}+(y+v_{y}t)^{2}\big{]}^{n/2}\right\rangle_{\\!0}$
$\displaystyle\simeq$
$\displaystyle\left\langle(x^{2}+y^{2})^{n/2}\right\rangle_{\\!0}\bigg{[}1\,+\,$
$\displaystyle{\cal
O}\bigg{(}\frac{\left\langle\bm{v}_{\perp}^{2}\right\rangle_{0}}{\left\langle
r_{\perp}^{2}\right\rangle_{0}}t^{2}\bigg{)}\bigg{]},$

where the factor multiplying the term in $t^{2}$ depends on $n$ and is
positive. All in all, one finds

$\epsilon_{n}^{\bf x}(t)\simeq\epsilon_{n}^{\bf x}(t_{0})\bigg{[}1-{\cal
O}\bigg{(}\frac{\left\langle v_{\perp}^{2}\right\rangle_{0}}{\left\langle
r_{\perp}^{2}\right\rangle_{0}}t^{2}\bigg{)}\bigg{]}\ \text{for
}t\ll\\!\Bigg{(}\\!\frac{\left\langle
r_{\perp}^{2}\right\rangle_{0}}{\left\langle
v_{\perp}^{2}\right\rangle_{0}}\\!\Bigg{)}^{\\!\\!1/2},$ (12)

similar to Eq. (8).

The results of this section can be extended in a straightforward way to the
generalized spatial eccentricities [21, 23]

$\epsilon^{\bf x}_{n,m\,}{\rm e}^{{\rm i}n\Phi_{m,n}}\equiv-\frac{\left\langle
r_{\perp}^{m}{\rm e}^{{\rm i}n\theta}\right\rangle}{\left\langle
r_{\perp}^{m}\right\rangle}.$ (13)

In a free-streaming system without initial anisotropic flow, the numerator of
the ratio on the right hand side is actually independent of time. In turn, the
denominator increases with $t^{2}$, so that eventually one finds the same
behavior as in Eq. (12), with a different factor in front of the term in
$t^{2}$.

##  III Interacting system: general idea

In this section we introduce a general early-time expansion of the phase space
distribution of an interacting system obeying the Boltzmann equation, by
carefully exploiting the latter. This expansion will form the basis of our
calculations of the development of anisotropic flow and spatial eccentricities
in the following two sections.

Consider an arbitrary single-particle phase space distribution
$f(t,\vec{x},\\!\vec{\,p})$. (Strictly speaking, the distribution needs to
vanish quickly enough as $|\vec{x}|$ or $|\\!\vec{\,p}|$ go to infinity to be
normalized to the number of particles in the system. In addition, we assume
that it is sufficiently continuously differentiable for our equations to make
sense.) If one is interested in the early-time evolution starting from a known
initial condition at $t=t_{0}$, one can begin with the Taylor expansion

$\displaystyle f(t,\vec{x},\\!\vec{\,p})=f_{0}(\vec{x},\\!\vec{\,p})$
$\displaystyle+t\,\partial_{t}f(\vec{x},\\!\vec{\,p})\big{|}_{0}$ (14)
$\displaystyle+\frac{t^{2}}{2}\partial_{t}^{2}f(\vec{x},\\!\vec{\,p})\big{|}_{0}+\cdots$
(15)

where $f$ and its successive time derivatives are evaluated at the initial
time $t_{0}=0$. The early-time behavior is thus governed by these time
derivatives.

Making use of the relativistic Boltzmann equation (1), the first time
derivative in Eq. (15) trivially reads

$\partial_{t}f(\vec{x},\\!\vec{\,p})\big{|}_{0}=-\frac{\\!\vec{\,p}}{E}\cdot\vec{\nabla}_{\\!x}f(\vec{x},\\!\vec{\,p})\big{|}_{0}+\frac{1}{E}{\cal
C}[f]\big{|}_{0}.$ (16)

If we now differentiate the Boltzmann equation with respect to time, we obtain
an expression for $\partial_{t}^{2}f$ as a function of
$\partial_{t}\vec{\nabla}_{\\!x}f$ and the time derivative $\partial_{t}{\cal
C}[f]$ of the collision term. Exchanging the order of time derivative and
gradient in the term
$\partial_{t}\vec{\nabla}_{\\!x}f=\vec{\nabla}_{\\!x}\partial_{t}f$ and
replacing $\partial_{t}f$ by its expression as given by Eq. (1), one finds

$\displaystyle\partial_{t}^{2}f(t,\vec{x},\\!\vec{\,p})=$
$\displaystyle\,\bigg{(}\frac{\\!\vec{\,p}}{E}\cdot\vec{\nabla}_{\\!x}\bigg{)}^{\\!\\!2}f(t,\vec{x},\\!\vec{\,p})$
(17) $\displaystyle-\frac{\\!\vec{\,p}}{E^{2}}\cdot\vec{\nabla}_{\\!x}{\cal
C}[f]+\frac{1}{E}\partial_{t}{\cal C}[f],$ (18)

which can then be evaluated at $t_{0}$. Given an explicit expression for the
collision term ${\cal C}[f]$, the time derivative $\partial_{t}{\cal
C}[f]|_{0}$ will generically involve $\partial_{t}f|_{0}$, which can again be
replaced by the right hand side of Eq. (16). We can thus eliminate time
derivatives at $t_{0}$ in a systematic manner, replacing them by expressions
that only involve the spatial dependence of the initial distribution
$f_{0}(\vec{x},\\!\vec{\,p})$.

Iterating this approach, one finds

$\displaystyle f(t,\vec{x},\\!\vec{\,p})=f_{0}(\vec{x},\\!\vec{\,p})$
$\displaystyle+t\bigg{(}\\!\\!-\\!\frac{\\!\vec{\,p}}{E}\cdot\vec{\nabla}_{\\!x}f(\vec{x},\\!\vec{\,p})+\frac{1}{E}{\cal
C}[f]\bigg{)}_{\\!0}+\frac{t^{2}}{2}\bigg{(}\frac{\big{(}\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{\\!2}}{E^{2}}f(\vec{x},\\!\vec{\,p})-\frac{\\!\vec{\,p}}{E^{2}}\cdot\vec{\nabla}_{\\!x}{\cal
C}[f]+\frac{1}{E}\partial_{t}{\cal C}[f]\bigg{)}_{\\!0}$ (19)
$\displaystyle+\frac{t^{3}}{3!}\bigg{(}\\!\\!-\\!\frac{\big{(}\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{\\!3}}{E^{3}}f(\vec{x},\\!\vec{\,p})+\frac{\big{(}\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{\\!2}}{E^{3}}{\cal
C}[f]-\frac{\\!\vec{\,p}}{E^{2}}\cdot\vec{\nabla}_{\\!x}\partial_{t}{\cal
C}[f]+\frac{1}{E}\partial_{t}^{2}{\cal C}[f]\bigg{)}_{\\!0}$ (20)
$\displaystyle+\frac{t^{4}}{4!}\bigg{(}\frac{\big{(}\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{\\!4}}{E^{4}}f(\vec{x},\\!\vec{\,p})-\frac{\big{(}\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{\\!3}}{E^{4}}{\cal
C}[f]+\frac{\big{(}\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{\\!2}}{E^{3}}\partial_{t}{\cal
C}[f]-\frac{\\!\vec{\,p}}{E^{2}}\cdot\vec{\nabla}_{\\!x}\partial_{t}^{2}{\cal
C}[f]+\frac{1}{E}\partial_{t}^{3}{\cal C}[f]\bigg{)}_{\\!0}$ (21)
$\displaystyle+{\cal O}(t^{5}).$ (22)

This lengthy expression can be shortened if one realizes that the terms that
do not involve the collision kernel ${\cal C}[f]$ or its derivatives are
actually the successive time derivatives of the free-streaming distribution
$f_{\rm f.s.}(t,\vec{x},\\!\vec{\,p})$ that coincides with
$f_{0}(\vec{x},\\!\vec{\,p})$ at the initial time $t_{0}$, see Eq. (2). Thus
we may rewrite Eq. (22) as

$\displaystyle f(t,\vec{x},\\!\vec{\,p})=f_{\rm f.s.}(t,\vec{x},\\!\vec{\,p})$
$\displaystyle+t\,\frac{{\cal
C}[f]\big{|}_{0}}{E}+\frac{t^{2}}{2}\bigg{(}\\!\\!-\\!\frac{\\!\vec{\,p}}{E^{2}}\cdot\vec{\nabla}_{\\!x}{\cal
C}[f]+\frac{1}{E}\partial_{t}{\cal C}[f]\bigg{)}_{\\!0}$ (23)
$\displaystyle+\frac{t^{3}}{3!}\bigg{(}\frac{\big{(}\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{\\!2}}{E^{3}}{\cal
C}[f]-\frac{\\!\vec{\,p}}{E^{2}}\cdot\vec{\nabla}_{\\!x}\partial_{t}{\cal
C}[f]+\frac{1}{E}\partial_{t}^{2}{\cal C}[f]\bigg{)}_{\\!0}$ (24)
$\displaystyle+\frac{t^{4}}{4!}\bigg{(}\\!\\!-\\!\frac{\big{(}\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{\\!3}}{E^{4}}{\cal
C}[f]+\frac{\big{(}\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{\\!2}}{E^{3}}\partial_{t}{\cal
C}[f]-\frac{\\!\vec{\,p}}{E^{2}}\cdot\vec{\nabla}_{\\!x}\partial_{t}^{2}{\cal
C}[f]+\frac{1}{E}\partial_{t}^{3}{\cal C}[f]\bigg{)}_{\\!0}+{\cal O}(t^{5}).\
\quad$ (25)

Note that the first term on the right hand side actually resums all orders in
$t$, i.e. is automatically valid at any order in $t$ in the absence of
interactions. In the following sections we shall exploit Eq. (25) to
investigate how the early time evolution of geometric eccentricities or
anisotropic flow coefficients departs in the presence of rescatterings from
their free-streaming behavior.

Before that, let us further discuss expansion (25). The coefficient of the
linear term in $t$ and the first terms in the factors within parentheses
multiplying the higher-order powers of $t$ are of the generic form

$\frac{\big{(}\\!\\!-\\!\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{k}}{E^{k+1}}{\cal
C}[f]\big{|}_{0},$ (26)

with $k\geq 0$. These contributions only involve the initial phase space
distribution $f_{0}$ and its spatial derivatives, which can be signaled by
writing the collision term ${\cal C}[f_{0}]$. If $\sigma$ is a typical cross
section for the rescatterings modeled by the collision term, the terms (26)
are of order ${\cal O}(\sigma)$. Equivalently, these terms are of order ${\cal
O}(\mathrm{Kn}^{-1})$, where $\mathrm{Kn}$ denotes a characteristic Knudsen
number built from the mean free path and a typical length scale of the initial
state distribution. Although the definition of $\mathrm{Kn}$ is arbitrary, we
shall from now on systematically use ${\cal O}(\mathrm{Kn}^{-1})$ instead of
${\cal O}(\sigma)$, since the Knudsen number is dimensionless while the cross
section is not. Since $\mathrm{Kn}^{-1}$ is roughly speaking a measure of the
average number of rescatterings undergone by each particle in the system, it
is similar to the opacity used in a number of related studies [8, 10, 11, 13].

Starting from order $t^{2}$ the expansion (25) contains terms of the form

$\frac{\big{(}\\!\\!-\\!\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x}\big{)}^{k}}{E^{k+1}}\partial_{t}{\cal
C}[f]\big{|}_{0}\frac{t^{k+2}}{(k+2)!},$ (27)

with $k\geq 0$. As was already mentioned, the time derivative
$\partial_{t}{\cal C}[f]$ can be computed when the collision kernel is known,
by replacing every $\partial_{t}f$ as in Eq. (16). Accordingly, the term (27)
for a given $k$ will yield two types of contributions: on the one hand, terms
involving $(k+1)$-th spatial derivatives of $f_{0}$ while being of order
${\cal O}(\mathrm{Kn}^{-1})$, like the contributions of the form (26). On the
other hand, there also come terms of order ${\cal O}(\mathrm{Kn}^{-2})$, with
spatial derivatives of $f_{0}$ of order $k$ only. As an example, we present in
Appendix A the calculation of $\partial_{t}{\cal C}[f]$ for a specific choice
of collision kernel.

More generally, transforming the factor in front of the term of order $t^{k}$
with $k\geq 1$ in Eq. (25) so as to express every time derivative at $t_{0}$
in terms of spatial gradients of $f_{0}$ — possibly entering (iterated)
collision kernels —, one finds that the factor involves contributions at order
$\mathrm{Kn}^{-1}$, $\mathrm{Kn}^{-2}$, …$(\mathrm{Kn}^{-1})^{k}$. Equation
(25) thus implicitly contains a double expansion in powers of both $t$ and
$\mathrm{Kn}^{-1}$, where for consistency only the powers
$(\mathrm{Kn}^{-1})^{j}$ with $j\leq k$ should appear at order $t^{k}$.
Conversely, corrections to the free-streaming distribution of order
$(\mathrm{Kn}^{-1})^{j}$ only come up at order $t^{j}$ and higher in Eq. (25).
Thus, we are in principle able to go beyond the linear order in
$\mathrm{Kn}^{-1}$ to which the (semi-)analytical results existing in the
literature [14, 16, 9, 25, 13] are usually restricted.

##  IV Early-time behavior of a two-dimensional system of massless particles
with elastic rescatterings

To illustrate how the ideas introduced in the previous section can be used to
derive the early time behavior of quantities characterizing the expanding
system, we shall now choose a specific ansatz for the collision kernel ${\cal
C}[f]$ of the Boltzmann equation. Since it is possibly the simplest possible
case, we consider a system of massless particles without spin, undergoing
elastic binary collisions, and propagating in two dimensions222Two-dimensional
vectors will be denoted in boldface: ${\bf x}$, ${\bf p}$. only, namely the
transverse plane of a nucleus-nucleus collision. As we discuss below, the
latter assumption — which will be relaxed in Sect. V.1 — together with that of
massless particles significantly simplifies the form of the Møller
velocity333We use the notation $v_{\rm rel.}$, although the Møller velocity
does not coincide with the relative velocity.

$v_{\rm rel.}=\sqrt{(\vec{v}-\vec{v}_{1})^{2}-|\vec{v}\times\vec{v}_{1}|^{2}}$
(28)

in the collision integral [17]

${\cal C}[f]=\frac{E}{2}\\!\int\\!(f^{\prime}f^{\prime}_{1}-ff_{1})v_{\rm
rel.}\frac{{\rm d}\sigma}{{\rm d}\Theta}\,{\rm d}\Theta\,{\rm d}^{2}{\bf
p}_{1},$ (29)

where we did not denote the arguments of the phase space distribution before
(unprimed) or after (primed) a collision, while $\Theta$ is the scattering
angle and ${\rm d}\sigma/{\rm d}\Theta$ the corresponding differential cross
section. To remain as general as possible, we do not specify the latter —
which in the two-dimensional case has the dimension of a length —, nor the
form of the initial phase space distribution. Instead of Eq. (29), we shall
also use the equivalent form [17]

$\displaystyle{\cal C}[f]=\frac{1}{2}$
$\displaystyle\\!\int\\!(f^{\prime}f^{\prime}_{1}-ff_{1})W({\bf p},{\bf
p}_{1}\to{\bf p}^{\prime},{\bf p}^{\prime}_{1})$ (30) $\displaystyle\hskip
56.9055pt\times\frac{{\rm d}^{2}{\bf p}_{1}}{(2\pi)^{2}E_{1}}\frac{{\rm
d}^{2}{\bf p}^{\prime}}{(2\pi)^{2}E^{\prime}}\frac{{\rm d}^{2}{\bf
p}^{\prime}_{1}}{(2\pi)^{2}E^{\prime}_{1}},\qquad$ (31)

with the transition rate $W({\bf p},{\bf p}_{1}\to{\bf p}^{\prime},{\bf
p}^{\prime}_{1})$, which has the advantage to be more easily generalized to
other types of rescatterings.

We shall assume that the initial distribution $f_{0}({\bf x},{\bf p})$ is
isotropic in momentum space at each point ${\bf x}$, so that there is no
initial anisotropic flow in the system. In contrast, the spatial distribution
in the initial state is asymmetric and in particular depends on the polar
angle $\theta$. To represent this variation with $\theta$, we symbolically
write

$f_{0}=\bar{f}_{0}+\sum_{n\neq 0}\delta_{n}f_{0}^{(n)}{\rm e}^{-{\rm
i}n\theta},$ (32)

where the real-valued functions $\bar{f}_{0}$ and $f_{0}^{(n)}$ are
independent of $\theta$, while $\delta_{n}$ is a dimensionless complex number
characterizing the modulation of period $2\pi/n$ in $\theta$. To ensure that
$f_{0}$ is real-valued, both $\delta_{-n}=\delta_{n}^{*}$ and
$f_{0}^{(-n)}=f_{0}^{(n)}$ should hold for every $n$. Clearly, the
eccentricity $\epsilon_{n}^{\bf x}$ and its generalizations (13) will be
proportional to $\delta_{n}$:

$\epsilon_{n,m}^{\bf x}\propto\delta_{n}.$ (33)

Equation (32) is meant to be schematic and to stand for the polar dependence
of any systematic expansion relying on a true complete basis of functions on
the transverse plane, like the cumulant [21] or the Bessel–Fourier [26, 27]
expansions.

###  IV.1 Anisotropic flow

In a kinetic description, anisotropic flow results from the rescatterings in
the system, which convert the asymmetry of its initial geometry into an an
anisotropy in momentum space. The Fourier coefficients quantifying the
momentum-distribution anisotropies can be obtained from the phase space
distribution:

$v_{n}(t)\equiv\big{|}v_{n}(t)\big{|}{\rm e}^{{\rm
i}n\Psi_{n}(t)}=\frac{\displaystyle\int\\!f(t,{\bf x},{\bf p})\,{\rm e}^{{\rm
i}n\varphi_{\bf p}}\,{\rm d}^{2}{\bf x}\,{\rm d}^{2}{\bf
p}}{\displaystyle\int\\!f(t,{\bf x},{\bf p})\,{\rm d}^{2}{\bf x}\,{\rm
d}^{2}{\bf p}},$ (34)

where $\varphi_{\bf p}$ denotes the azimuth of ${\bf p}$. While the integral
over ${\bf x}$ runs over the whole position space, that over ${\bf p}$ can
either be over the whole momentum space or restricted to an interval in $|{\bf
p}|$ (while still running over the whole range for $\varphi_{\bf p}$). In the
following we solely discuss the first possibility (“integrated flow”), since
the second one depends more crucially on the details of the microscopic
interaction rate and of the initial phase space distribution. Accordingly, we
do not consider directed flow $v_{1}$, since its integrated value is fixed to
its initial value (here zero) by global momentum conservation.

When the integration runs over the whole phase space, the denominator of Eq.
(34) is simply the total number of particles in the system. Since we assume in
this section that the latter only has elastic scatterings, this number remains
constant in time.

####  IV.1.1 Leading contribution at early times

Since the denominator of Eq. (34) is constant, the whole time dependence of
$v_{n}(t)$ comes from the numerator, in which we can now substitute the early-
time expansion (25) for $f(t,{\bf x},{\bf p})$. The first term from the free-
streaming distribution $f_{\rm f.s.}$ does not contribute to $v_{n}(t)$, since
we assumed that there is no initial anisotropic flow. The next term linear in
$t$ does not contribute either: it consists of the collision kernel ${\cal
C}[f_{0}]$ in the initial state, and every phase space distribution entering
it is isotropic in momentum space.444To be more precise, a dependence on
$\varphi_{\bf p}$ appears in ${\cal C}[f_{0}]$ via the relative velocity. But
it appears in the form $\varphi_{\bf p}-\varphi_{1}$ relative to the azimuth
$\varphi_{1}$ of the “collision partner” labeled with 1, and thus disappears
in the integration over ${\bf p}_{1}$ with a momentum-isotropic distribution
$f_{1}\big{|}_{0}$.

The term quadratic in time in expansion (25) is the first one involving a
contribution of the form (26) with $k\geq 1$, namely $-({\bf
p}/E^{2})\cdot\bm{\nabla}_{\\!x}{\cal C}[f]\big{|}_{0}$. This term does not
contribute to anisotropic flow at early times: performing first the
integration over the transverse plane in Eq. (34), one encounters the integral

$\int\\!\bm{\nabla}_{\\!x}{\cal C}[f]\big{|}_{0}\,{\rm d}^{2}{\bf x}=0,$ (35)

since the collision kernel vanishes at infinity. More generally, at every
order in expansion (25) the terms involving (powers of) the spatial gradient
of ${\cal C}[f]\big{|}_{0}$ or of its time derivatives will not contribute to
the early-time behavior of anisotropic flow, thanks to a similar argument: the
integration over the transverse plane yields zero at once. Thus the only terms
in Eq. (25) that can eventually contribute to $v_{n}(t)$ at early times are
those of the form $(1/E)\partial_{t}^{k}{\cal C}[f]\big{|}_{0}$, without
spatial gradient in front, where the $k$-th time derivative of the collision
kernel appears at order $t^{k+1}$.

Although $(1/E)\partial_{t}^{k}{\cal C}[f]\big{|}_{0}$ is not a total spatial
gradient, still it contains (powers of) $\bm{\nabla}_{\\!x}$ — multiplied by a
momentum — when one computes explicitly the time derivative, as exemplified in
Eqs. (53)–(55). Such terms thus couple the spatial and momentum “components”
of the phase space distribution, via the inner products ${\bf
p}\cdot\bm{\nabla}_{\\!x}$, under the influence of the particle rescatterings
encoded in the transition rate, which is why they are crucial for the
development of anisotropic flow. Let us sketch how a contribution to the
$n$-th harmonic $v_{n}$ can arise in the explicit calculation of Eq. (34).

First, note that ${\rm e}^{{\rm i}n\varphi_{\bf p}}$, or equivalently its real
and imaginary parts, can be expressed in terms of powers of the components of
${\bf p}$ and in fact involves the $n$-th power.555This is of course obvious
since ${\rm e}^{{\rm i}n\varphi_{\bf p}}={\bf p}^{n}/|{\bf p}|^{n}$, where we
identify the two-dimensional momentum ${\bf p}$ with a complex number
$p_{x}+{\rm i}p_{y}$. Accordingly, any non-zero contribution to $v_{n}(t)$
must involve a term in ${\bf p}^{n}$ from the early-time expansion of
$f(t,{\bf x},{\bf p})$. In the time derivative $\partial_{t}^{k}{\cal
C}[f]\big{|}_{0}$, such terms trivially appear if $k=n$, i.e. at order
$t^{n+1}$. Closer investigation reveals that there are already terms in ${\bf
p}^{n}$ at order $t^{n}$ — but they do not contribute to $v_{n}(t)$ at early
times. Keeping the discussion to first order in the inverse Knudsen number,
these terms come from combining an “obvious” $({\bf
p}\cdot\bm{\nabla}_{\\!x})^{n-1}$ in $\partial_{t}^{n-1}{\cal
C}[f]\big{|}_{0}$ with the factor ${\bf p}$ hidden in the Møller velocity
(28), which for massless particles reads

$v_{\rm rel.}=1-\frac{{\bf p}\cdot{\bf p}_{1}}{E\,E_{1}}=1-\cos(\varphi_{\bf
p}-\varphi_{1}),$ (36)

where the second identity only holds in two dimensions (or if the two
particles have the same polar angle along the third direction). As this
expression shows, the term that could contribute to $v_{n}(t)$ at order
$t^{n}$ also involves a multiplicative factor ${\bf p}_{1}$, i.e. will yield
an odd function of ${\bf p}_{1}$ in the integrand of the collision kernel:
this vanishes in the integration over ${\bf p}_{1}$, thus giving no
contribution to $v_{n}(t)$.666To be thorough, the term in ${\bf p}\cdot{\bf
p}_{1}$ from the Møller velocity can contribute to the early time development
of $v_{n}(t)$, but it has to multiply not only a term in $({\bf
p}\cdot\bm{\nabla}_{\\!x})^{n-1}$, but also a term ${\bf
p}_{1}\cdot\bm{\nabla}_{\\!x}$ (or an odd power thereof): such a contribution
can only appear at order $t^{n+1}$ or higher.

We mentioned above that the terms from expansion (25) of the form
$(1/E)\partial_{t}^{k}{\cal C}[f]\big{|}_{0}$ involve contributions of higher
order in the inverse Knudsen number $\mathrm{Kn}^{-1}$. Careful accounting
based on iterating Eq. (16) shows that at order $t^{n+1}$ a term in
$\big{(}\mathrm{Kn}^{-1}\big{)}^{j}$ is accompanied by $n+1-j$ powers of
$\bm{\nabla}_{\\!x}$ (see Appendix A): if $j>1$, this cannot contribute to
$v_{n}(t)$. Thus the higher orders in $\mathrm{Kn}^{-1}$ affect the early time
development of $v_{n}$ at a higher order in $t$, as we shall discuss in next
subsection.

From the above discussion, any non-zero contribution to $v_{n}(t)$ at early
times actually comes from a term in expansion (25) that contains the $n$-th
power of the spatial gradient $\bm{\nabla}_{\\!x}$. By investigating explicit
examples, or more formally by following a similar approach to that developed
in Ref. [21], one finds that this operator $(\bm{\nabla}_{\\!x})^{n}$ together
with the subsequent integration over ${\bf x}$ isolates the component with
periodicity $2\pi/n$ in the polar angle $\theta$ of the function it acts upon.
In the term in $\partial_{t}^{n}{\cal C}[f]\big{|}_{0}$, these gradients
appear in the form of products [see Eqs. (53)–(55)]

$\bigg{[}\frac{{\bf p}\cdot\bm{\nabla}_{\\!x}f_{0}({\bf x},{\bf
p})}{E}\bigg{]}^{k}\bigg{[}\frac{{\bf p}_{1}\cdot\bm{\nabla}_{\\!x}f_{0}({\bf
x},{\bf p}_{1})}{E_{1}}\bigg{]}^{n-k},$

with $0\leq k\leq n$. Replacing each $f_{0}$ by its schematic expansion (32),
one finds that two kinds of terms contribute to the $n$-th polar mode of
$\partial_{t}^{n}{\cal C}[f]\big{|}_{0}$: First, terms involving the symmetric
part $\bar{f}_{0}$ of one factor with the term in $\delta_{n}f_{0}^{(n)}$ from
the other factor, yielding a contribution proportional to $\delta_{n}$. And
secondly, terms of the form $\delta_{k}\delta_{n-k}f_{0}^{(k)}f_{0}^{(n-k)}$
with $|k|\geq 1$. Since every $\delta_{k}$ is proportional to the eccentricity
$\epsilon_{k}^{\bf x}$, we recover the known fact [28, 29, 30, 31] that
$v_{n}$ gets a linear contribution in $\epsilon_{n}^{\bf x}$ together with
nonlinear contributions $\epsilon_{n-k}^{\bf x}\epsilon_{k}^{\bf x}$. What we
show here for the first time is that in a kinetic-transport approach all
contributions to $v_{n}(t)$ grow with the same power of $t$ at early times,
namely $t^{n+1}$:

$v_{n}(t)\\!\underset{\text{early }t}{\sim}\\!\mathrm{Kn}^{-1}\bigg{(}\\!{\cal
K}^{(1)}_{n,n}\epsilon_{n}^{\bf x}+\sum_{k\geq 1}\\!{\cal
K}^{(1)}_{n,n-k,k}\epsilon_{n-k}^{\bf x}\epsilon_{k}^{\bf x}\bigg{)}t^{n+1},$
(37)

where the scale defining the notion of an early time is given by the typical
transverse size of the system. The overall growth $v_{n}(t)\propto t^{n+1}$ in
transport computations was already observed in Ref. [15, 16] (and in Refs.
[14, 32] for $v_{2}$), where it was noted that it differs from fluid-dynamical
calculations, which yield the faster increase $v_{n}(t)\propto t^{n}$.

Anticipating on the following subsection, let us list which expected
contributions to $v_{n}$ are missing from the scaling behavior (37) but appear
at higher orders in $t$. First, we have already mentioned the terms of higher
order in the inverse Knudsen number. Secondly, fluid-dynamical simulations
[34, 33], numerical solution of the Boltzmann equation [11] or transport
calculations [35] have revealed cubic contributions in the eccentricities to
the flow harmonics, as e.g. contributions in $(\epsilon_{2}^{\bf x})^{3}$ to
$v_{2}$ or $v_{6}$. Such terms are also absent from Eq. (37), because in a
scenario with only binary rescatterings and no quantum effects they appear at
a higher order in $\mathrm{Kn}^{-1}$ [9].

####  IV.1.2 Higher order contributions at early times

From the previous subsection we know that at early times anisotropic flow
harmonics are “created” by the terms of the type $(1/E)\partial_{t}^{k}{\cal
C}[f]\big{|}_{0}$ in expansion (25). For $v_{n}(t)$, this gives non-zero
contributions starting at $k=n$, resulting in the scaling behavior (37) in
$t^{n+1}$. We have also emphasized that a necessary ingredient is the presence
of (at least) $n$ powers of the spatial gradient $\bm{\nabla}_{\\!x}$ in the
expression of $\partial_{t}^{k}{\cal C}[f]\big{|}_{0}$.

What happens if we push expansion (25) to order $t^{n+2}$, thus including the
term in $\partial_{t}^{n+1}{\cal C}[f]\big{|}_{0}$? First, the terms of order
$\mathrm{Kn}^{-1}$ contain $n+1$ powers of $\bm{\nabla}_{\\!x}$, which
necessarily multiply $n+1$ momentum variables (${\bf p}$, ${\bf p}_{1}$…). In
addition, the contribution may include extra momentum factors — for example
from the ${\bf p}\cdot{\bf p}_{1}$ term in the Møller velocity (36) —, which
only come in pairs. Because the momentum variables come in a number of parity
opposite to that of $n$, such terms cannot contribute to $v_{n}(t)$: when
integrating over all momenta to obtain the integrated $v_{n}$, the integrand
will turn out to be odd in at least one of the momentum variables, and thus
yield zero. On the other hand, there can be non-vanishing contributions to
$v_{n}(t)$ of order $\mathrm{Kn}^{-1}$ at order $t^{n+3}$, $t^{n+5}$, and so
on.

The derivative $\partial_{t}^{n+1}{\cal C}[f]\big{|}_{0}$ also includes terms
of order $\mathrm{Kn}^{-2}$, see for instance the second line of Eq. (53) in
the case $n=0$ or Eq. (57) for $n=1$. These terms involve $n$ powers of
$\bm{\nabla}_{\\!x}$, multiplied with their respective momentum variables.
Thus these terms can yield non-zero contributions to $v_{n}(t)$ at early
times, scaling as $t^{n+2}$. Moreover, these terms include products of (powers
of gradients of) $f$ with (powers of gradients of) the collision kernel ${\cal
C}[f]$, and thus are “cubic” in the phase space distribution for the Boltzmann
kernel (29)–(30). Accordingly, one quickly sees that the contributions to
$v_{n}(t)$ at order $\mathrm{Kn}^{-2}$ will not only include linear and
quadratic terms in the initial anisotropies $\epsilon_{n}^{\bf x}$, similar to
those in the parentheses of Eq. (37), but also cubic terms of the type

$\sum_{k,l\geq 1}{\cal K}^{(2)}_{n,n-k-l,k,l}\epsilon_{n-k-l}^{\bf
x}\epsilon_{k}^{\bf x}\epsilon_{l}^{\bf x},$ (38)

as for instance contributions in $(\epsilon_{2}^{\bf x})^{3}$ to $v_{2}$ or
$v_{6}$ [9]. However, contributions to $v_{n}(t)$ that are quartic in the
eccentricities — for instance a contribution in $(\epsilon_{2}^{\bf x})^{4}$
to $v_{8}$ — cannot appear at order $\mathrm{Kn}^{-2}$ (and thus scales as
${\cal O}(t^{n+2})$), but only at order $\mathrm{Kn}^{-3}$ (resp. ${\cal
O}(t^{n+3})$).

Summarizing, we have found the following scaling behavior of $v_{n}(t)$ at
early times, starting from an initial state without anisotropic flow:

$v_{n}(t)\\!\underset{\text{early }t}{\sim}\sum_{j\geq
1}\big{(}\mathrm{Kn}^{-1}\big{)}^{\\!j}\sum_{k\geq 0}a^{(j)}_{n,k}t^{n+2k+j}.$
(39)

In the case of a system of particles undergoing elastic binary rescatterings
and leaving aside quantum effects from the collision kernel, the coefficient
$a^{(j)}_{n,k}$, which differ from one harmonic to the other, may contain
products of one, two, till at most $j+1$ spatial eccentricities
$\epsilon_{l}^{\bf x}$, see e.g. Eqs. (37) (for $j=1$) and (38) (for $j=2$).

###  IV.2 Spatial characteristics

The expansion (25) also allows one to determine to the early-time development
of quantities that characterize the spatial geometry of the system. In
contrast to the anisotropic flow harmonics, these quantities are usually non-
zero in the initial state, and in addition they already evolve in a free-
streaming system, i.e. under the influence of the first term $f_{\rm f.s.}$ in
Eq. (25). The subsequent terms, involving the collision kernel, thus describe
the change in the early-time development due to rescatterings. As examples, we
shall now discuss the behaviors of the root-mean-square transverse radius
$\left\langle r_{\perp}^{2}\right\rangle$ or the eccentricities
$\epsilon_{n}^{\bf x}$ defined by Eq. (5).

In this subsection, we shall make extensive use of the property (valid at any
fixed ${\bf x}$) [17]

$\int\\!\big{[}a({\bf x})+p_{\mu}b^{\mu}({\bf x})\big{]}{\cal
C}[f]\,\frac{{\rm d}^{2}\bf p}{E}=0,$ (40)

for arbitrary functions $a$ and $b^{\mu}$ (with $\mu\in\\{0,1,2\\}$) of
position. Physically, the identity encodes the conservation of particle
number, energy and momentum in the binary collisions.

####  IV.2.1 Root-mean-square radius

Let us first study how the typical size of the system, which expands into the
vacuum, increases. For that purpose, we can for instance characterize the
system size by the mean-square (transverse) radius

$\left\langle
r_{\perp}^{2}\right\rangle_{t}\equiv\frac{\displaystyle\int\\!(x^{2}+y^{2})f(t,{\bf
x},{\bf p})\,{\rm d}^{2}{\bf x}\,{\rm d}^{2}{\bf
p}}{\displaystyle\int\\!f(t,{\bf x},{\bf p})\,{\rm d}^{2}{\bf x}\,{\rm
d}^{2}{\bf p}}.$ (41)

The idea of the following calculations is readily extended to other powers of
$r_{\perp}$. From Sect. II we know that the mean-square radius already
increases for a free-streaming system, namely as

$\left\langle r_{\perp}^{2}\right\rangle_{t}^{\rm f.s.}=\left\langle
r_{\perp}^{2}\right\rangle_{0}+\left\langle\bm{v}_{\perp}^{2}\right\rangle_{0}t^{2}=\left\langle
r_{\perp}^{2}\right\rangle_{0}+t^{2},$ (42)

where the second identity follows from assuming massless particles propagating
in two dimensions only. Our task is to determine how rescatterings modify this
behavior at early times.

The denominator in Eq. (41) is the total particle number of the system and
remains constant if only elastic rescatterings are present in the system. In
the following, we thus focus on the numerator of Eq. (41). Inserting expansion
(25) in this numerator, the linear term in $t$ from the rescatterings involves
an integral of the form

$\int\\!g({\bf x})\,{\cal C}[f]\,{\rm d}^{2}{\bf x}\,\frac{{\rm d}^{2}{\bf
p}}{E},$ (43)

with $g({\bf x})=x^{2}+y^{2}$, independent of momentum, in the present case.
The integral over ${\bf p}$ at a fixed position ${\bf x}$ is of the form (40)
— with only $a({\bf x})=g({\bf x})$ being non-zero — and hence vanishes. Thus
there is no contribution linear in time to the early time development of
$\left\langle r_{\perp}^{2}\right\rangle_{t}$ in the classical Boltzmann
scenario, but we can already note that this is not necessarily true for
collision kernels that do not implement particle-number conservation.

Turning next to the contribution from the term due to collisions in $t^{2}$ in
expansion (25), it also vanishes for a collision kernel ${\cal C}[f]$
conserving particle number and if the initial distribution is isotropic in
momentum space. First, one can argue that an integral of the form (43) with
any time derivative $\partial_{t}^{k}{\cal C}[f]$ instead of ${\cal C}[f]$ is
also zero when the collision kernel conserves particle number. Indeed, taking
the time derivative out of the integral over the phase space variables yields

$\int\\!g({\bf x})\,\partial_{t}^{k}{\cal C}[f]\,{\rm d}^{2}{\bf
x}\,\frac{{\rm d}^{2}{\bf p}}{E}=\partial_{t}^{k}\\!\\!\int\\!g({\bf
x})\,{\cal C}[f]\,{\rm d}^{2}{\bf x}\,\frac{{\rm d}^{2}{\bf p}}{E}=0,$ (44)

since one eventually computes the total time derivative of a function which is
identically zero at any time. This ensures that the term in $\partial_{t}{\cal
C}[f]\big{|}_{0\,}t^{2}$ does not contribute to $\left\langle
r_{\perp}^{2}\right\rangle_{t}$ at early times.

In turn, we argue in Appendix B.1 that the other collision-induced term at
order $t^{2}$ in expansion (25), that involving $({\bf
p}/E^{2})\cdot\bm{\nabla}_{\\!x}{\cal C}[f]\big{|}_{0}$, does not contribute
either to the early-time development of the mean-square radius if the initial
distribution is isotropic in momentum.

Thus the elastic binary rescatterings described by the Boltzmann collision
kernel do not affect the leading contribution to the increase of the root-
mean-square transverse radius $\left\langle r_{\perp}^{2}\right\rangle_{t}$ of
the system, which remains dominated by the free-streaming expansion at early
times. The first modification from rescatterings to that behavior can
generally occur at order $t^{3}$ in the expansion (25), and thus are
subleading compared to the ballistic motion of the particles, at least at
early times.

$\left\langle r_{\perp}^{2}\right\rangle_{t}=\left\langle
r_{\perp}^{2}\right\rangle_{t}^{\rm f.s.}+{\cal O}(t^{3}).$ (45)

Let us sketch how things behave at order $t^{3}$ and higher, and in particular
which terms from expansion (25) can yield a non-zero contribution to
$\left\langle r_{\perp}^{2}\right\rangle_{t}$. First, the term in
$\partial_{t}^{k-1}{\cal C}[f]\big{|}_{0}$ at order $t^{k}$ consistently gives
zero for a particle-number conserving collision kernel, thanks to Eq. (44).
The terms in $({\bf p}\cdot\bm{\nabla}_{\\!x})^{k}$ with $k\geq 3$ (thus
starting at order $t^{4}$) also yield zero, irrespective of any assumption on
the collision kernel apart from its being zero at infinity. Indeed, the
contributions of such terms can be integrated by parts over $x$ or $y$ twice,
to get rid of $r_{\perp}^{2}$, leaving the integral over position space of the
derivative of a function that vanishes at infinity: for instance

$\int\\!x^{2}\frac{(p_{x}\partial_{x})^{k}}{E^{k+1}}{\cal
C}[f]\big{|}_{0}\,{\rm
d}x=2\\!\int\\!\frac{p_{x}^{k}\partial_{x}^{k-2}}{E^{k+1}}{\cal
C}[f]\big{|}_{0}\,{\rm d}x=0.$

Interestingly, the terms involving the second spatial derivatives of either
${\cal C}[f]\big{|}_{0}$ or its time derivatives — e.g. the first term of
order $t^{3}$ or the second of order $t^{4}$ in Eq. (25) — also do not
contribute to $\left\langle r_{\perp}^{2}\right\rangle_{t}$ in the special
case of massless particles propagating in two dimensions (or at least such
that $p_{z}=0$), as shown in Appendix B.2. But this need not be true more
generally.

Thus the only collision-induced terms that can affect the early-time increase
of a two-dimensional gas of massless particles undergoing elastic binary
collisions are those in expansion (25) involving a single power of $({\bf
p}/E)\cdot\bm{\nabla}_{\\!x}$ applied to the time derivatives of ${\cal C}[f]$
at $t=0$. For instance, the second term of order $t^{3}$ or the third term of
order $t^{4}$ in Eq. (25).

All in all, if the phase-space distribution is initially isotropic in momentum
at each point in the transverse plane and if its subsequent evolution
preserves particle number, then rescatterings only affect the growth of the
mean square transverse radius $\left\langle r_{\perp}^{2}\right\rangle_{t}$ at
order $t^{3}$ or higher, i.e. subleadingly compared to the effect of the free-
streaming expansion. One can readily check that this behavior also holds for
any moment $\left\langle r_{\perp}^{n}\right\rangle_{t}$ with arbitrary
integer $n$, using the same arguments.

####  IV.2.2 Eccentricities

Consider now the spatial eccentricities (5). We have seen in Sect. II that
they decrease in an collisionless system without initial anisotropic flow,
symbolically in $1/[1+{\cal O}(t^{2})]$ at early times. In the previous
subsection, we saw that rescatterings change the behavior of the term in the
denominator of Eq. (5) only at order $t^{3}$ or higher. Accordingly, we shall
now only investigate the behavior of $\left\langle r_{\perp}^{n}{\rm e}^{{\rm
i}n\theta}\right\rangle_{t}$, which remains constant in time for a free-
streaming system, see Eq. (10).

Repeating the steps used in computing $\left\langle
r_{\perp}^{2}\right\rangle_{t}$, one quickly finds that the contributions to
$\left\langle r_{\perp}^{n}{\rm e}^{{\rm i}n\theta}\right\rangle_{t}$ in $t$
and $t^{2}$ vanish. Again, the term in $t$ and one of the terms in $t^{2}$
[that with the time derivative of the collision kernel in expansion (25)] are
zero thanks to particle-number conservation — technically, invoking Eqs.
(43)–(44) with $g({\bf x})=r_{\perp}^{n}{\rm e}^{{\rm i}n\theta}$. The
remaining term in $t^{2}$ is also zero if the initial phase space distribution
is isotropic in momentum space, thanks to Eq. (60) – which plays here the role
played by relation (58) in the calculation of $\left\langle
r_{\perp}^{2}\right\rangle_{t}$.

Therefore any change of $\left\langle r_{\perp}^{n}{\rm e}^{{\rm
i}n\theta}\right\rangle_{t}$ from rescatterings comes at order $t^{3}$ or
higher. Since this is also the order at which rescatterings affect the
evolution of $\left\langle r_{\perp}^{n}\right\rangle_{t}$, it will also hold
for their ratio, namely the spatial eccentricities (5) [and their generalized
version (13)]:

$\epsilon_{n}^{\bf x}(t)\\!\underset{\text{early }t}{\sim}\epsilon_{n}^{\bf
x}(t)\big{|}^{\rm f.s.}+{\cal O}(t^{3}),$ (46)

where $\epsilon_{n}^{\bf x}(t)\big{|}^{\rm f.s.}$ denotes the time dependence
of the eccentricity $\epsilon_{n}^{\bf x}$ in a free-streaming system with the
same initial phase space distribution, as computed in Sect. II. This means
that at early times the effect of rescatterings on $\epsilon_{n}^{\bf x}(t)$
is subleading compared to that of the free-streaming expansion.

As a final remark, one would intuitively expect a connection between the early
time behaviors of $\epsilon_{n}^{\bf x}(t)$ and $v_{n}(t)$: a non-zero
anisotropic flow coefficient $v_{n}$ clearly affects the spatial eccentricity
$\epsilon_{n}^{\bf x}$, in particular the azimuthally asymmetric numerator
$\left\langle r_{\perp}^{n}{\rm e}^{{\rm i}n\theta}\right\rangle_{t}$.
However, we argued in the previous section and the present one that different
terms in Eq. (25) are responsible for the respective onsets of $v_{n}(t)$ and
$\epsilon_{n}^{\bf x}(t)$. From that observation we would tentatively conclude
that in the absence of initial anisotropic flow, the early time behaviors of
spatial and momentum anisotropies are not related, or at least not obviously.

##  V Generalizations

In this section we discuss how the early-time behavior of the anisotropic flow
harmonics [Eq. (39)] and the spatial eccentricities [Eq. (46)] change when one
departs from the two-dimensionless system of massless “classical” particles
with elastic binary rescatterings assumed in Sect. IV.

###  V.1 3-dimensional expansion

What happens if the system expands in 3 dimensions instead of 2 in the
previous section? Clearly, this should slow down the transverse expansion,
since $\left\langle v_{\perp}^{2}\right\rangle$ is now in general smaller than
1, so that the characteristic time scale $\sqrt{\left\langle
r_{\perp}^{2}\right\rangle/\left\langle v_{\perp}^{2}\right\rangle}$ increases
(by a factor $\sqrt{3/2}$, in case the velocity distribution is also isotropic
along the third direction). But the actual question is, whether the scaling
behaviors found in Sect. IV are modified or not.

Looking back at the reason why $v_{n}(t)$ grows in $t^{n+1}$ at early times in
the absence of initial anisotropic flow, we see that the arguments that were
used never involved the dimensionality of the space into which the system is
expanding. Instead, the proof rather relied on the necessity to have a term
that contains $n$ powers of the transverse momentum ${\bf p}$ and that is not
odd in any of the other momenta appearing in the collision kernel.777In
particular, the longitudinal components of the momenta play no role in the
reasoning. Accordingly, whether the longitudinal motion of the system is boost
invariant or not does not affect the scaling behaviors. That these
requirements are not fulfilled at any order $t^{k}$ with $k\leq n$ remains
true in the case of a three-dimensional expansion, and thus the scaling
behavior (39), including the dependence on the inverse Knudsen number, still
holds. Of course the coefficients $a^{(j)}_{n,k}$ do depend on whether the
system is expanding in two or three dimensions. But their generic dependence
on the initial spatial eccentricities is unchanged: for instance, the
$a^{(1)}_{n,k}$, which are the relevant coefficients at order
$\mathrm{Kn}^{-1}$, only depend linearly or quadratically on the initial
$\\{\epsilon_{l}^{\bf x}\\}$.

Similarly, the space dimension did not play any role in our reasoning at
orders ${\cal O}(t)$ and ${\cal O}(t^{2})$ in the calculation of the mean
square radius (or more generally $\left\langle
r_{\perp}^{n}\right\rangle_{t}$) and the spatial eccentricities
$\epsilon_{n}^{\bf x}$. We actually encountered a term that vanishes only for
the special case of massless particles propagating in two dimensions (or with
$p_{z}=0$), but only at order ${\cal O}(t^{3})$, i.e. subleading with respect
to the free-streaming behavior. So here again, the scaling law (46) determined
in Sect. IV.2 is still valid in the three-dimensional case.

###  V.2 Massive particles

Another assumption of Sect. IV was that of considering massless degrees of
freedom. This hypothesis leads to a simpler expression of the Møller velocity
(28) [see Eq. (36)], which is helpful when dealing with explicit semi-
analytical examples [36]. However this feature is actually irrelevant for our
derivations of the scaling behavior of anisotropic flow and the
eccentricities.

Indeed, the Møller velocity of massive particles with momenta $\\!\vec{\,p}$,
$\\!\vec{\,p}_{1}$ can only depend on $\\!\vec{\,p}$ via the inner product
$\\!\vec{\,p}\cdot\\!\vec{\,p}_{1}$,888The square modulus $\\!\vec{\,p}^{2}$
“does not know” about the azimuth $\varphi_{\bf p}$, and thus cannot
contribute to anisotropic flow. just like in the massless case. To contribute
to the early-time development of a flow harmonic $v_{n}(t)$, a factor
$(\\!\vec{\,p}\cdot\\!\vec{\,p}_{1})^{k}$ from expanding $v_{\rm rel.}$ has to
multiply a contribution $(\\!\vec{\,p}_{1}\cdot\vec{\nabla}_{\\!x})^{k}$ to
match the $k$ powers of $\\!\vec{\,p}_{1}$ from the Møller velocity, and
another contribution $(\\!\vec{\,p}\cdot\vec{\nabla}_{\\!x})^{n-k}$, to obtain
the necessary $n$ powers of $\\!\vec{\,p}$. That is, $n$ powers of the spatial
gradient $\vec{\nabla}_{\\!x}$ should appear in the relevant term in expansion
(25), and this is only possible at order ${\cal O}(t^{n+1})$ or higher.
Accordingly, the scaling behavior (39) remains valid in the case of massive
particles.

It is also clear that the early-time behavior (46) of the spatial
eccentricities remains valid too if the particles are massive. Indeed, the
arguments to establish that the possible contributions from scatterings at
orders ${\cal O}(t)$ and ${\cal O}(t^{2})$ are zero were very general ones
(particle number conservation, momentum isotropy of the initial distribution)
and totally irrespective of any detail of the collision kernel — apart from
its conserving particle number.

Let us also mention in this subsection another possible modification of the
composition of the system that does not affect the scaling behaviors (39) and
(46). Until now we only considered systems consisting of a single species of
particles. In heavy ion collisions, it is certainly more realistic to consider
a mixture of several species that can not only self-interact but also interact
with each other. One should then introduce several phase space distributions,
which obey coupled Boltzmann equations [17]. If the number of particles of
each species remains constant, i.e. if all possible scattering processes are
elastic, then one can check that the scaling behavior (39) of the anisotropic-
flow coefficients is unchanged, since the arguments used to derive it still
hold. One then finds that the flow coefficients $v_{n}(t)$ of a given species
depend not only on the initial eccentricities of that species itself, but also
on those of the other species, which may be interesting if different particle
species have different initial geometrical profiles due to different
production mechanisms [25].

###  V.3 Inclusion of quantum statistics in the collision kernel

In this subsection and the the next we investigate how modifying the form of
the collision kernel ${\cal C}[f]$ affects the scaling behaviors of the flow
harmonics $v_{n}(t)$ and the eccentricities $\epsilon_{n}^{\bf x}$ of Sect.
IV.

The first modification we consider consists in incorporating quantum effects —
Pauli blocking or Bose enhancement — into the binary collision kernel
(29)–(30), via the usual substitution

$f^{\prime}f^{\prime}_{1}-ff_{1}\to
f^{\prime}f^{\prime}_{1}FF_{1}-ff_{1}F^{\prime}F^{\prime}_{1}$ (47)

in the integrand, where $F(t,\vec{x},\\!\vec{\,p})\equiv 1\pm
f(t,\vec{x},\\!\vec{\,p})$.

Now, the extra terms introduced in the collision kernel by the substitution
(47) do not introduce any extra power of $\\!\vec{\,p}$ or the spatial
gradient $\vec{\nabla}_{\\!x}$, and thus they cannot modify the scaling
behavior (39) of the flow harmonics. Similarly, the modified collision term
still conserves particle number, and also leaves the early-time behavior (46)
qualitatively unchanged.

However, as was already noted in Ref. [9] the longer form of the collision
kernel due to the change (47) does induce a modification, namely in the
dependence of the coefficients $a^{(j)}_{n,k}$ in Eq. (39) on the initial
spatial eccentricities $\\{\epsilon_{l}^{\bf x}\\}$. Indeed, since products of
three or four distribution functions now appear in the integrand of ${\cal
C}[f]$, the coefficients $a^{(1)}_{n,k}$ can now include terms depending
cubically or quartically on the initial eccentricities. For instance, the
early-time expansion $v_{2}(t)$ or $v_{6}(t)$ can now have a term in
$(\epsilon_{2}^{\bf x})^{3}$ at leading order $\mathrm{Kn}^{-1}$, or
$v_{4}(t)$ a contribution in $(\epsilon_{2}^{\bf x})^{4}$. In practice, this
will only be the case if the initial state is dense enough for quantum
corrections to become relevant.

###  V.4 Alternative collision kernel

Another type of modification of the collision kernel ${\cal C}[f]$ has a
higher impact on the scaling behaviors of Sect. IV, namely if we drop the
assumption that the rescatterings in the system are elastic. Instead, one can
include inelastic two-to-two ($2\leftrightarrow 2$) processes (if the system
consists of several species), as well as particle-number changing processes:
for instance $1\leftrightarrow 2$ gluon splitting or fusion processes [37, 38]
or $2\leftrightarrow 3$ parton processes [39].

Indeed, in Sect. IV.2 we invoked several times particle-number conservation to
cancel terms when deriving the scaling behaviors of the mean square radius
$\left\langle r_{\perp}^{2}\right\rangle_{t}$ [Eq. (45)] or of $\left\langle
r_{\perp}^{n}{\rm e}^{{\rm i}n\theta}\right\rangle_{t}$, and as a result of
the spatial eccentricities, Eq. (46). If particle number is not conserved,
then the early-time evolution of every average $\left\langle
r_{\perp}^{n}\right\rangle_{t}$ or $\left\langle r_{\perp}^{n}{\rm e}^{{\rm
i}n\theta}\right\rangle_{t}$ and therefore of $\epsilon_{n}^{\bf x}(t)$ will
generally include linear and quadratic terms in $t$, both at order
$\mathrm{Kn}^{-1}$, which represents a significant modification of the free-
streaming evolution.

Turning to the anisotropic flow coefficients $v_{n}(t)$, their early-time
behavior is also affected if the collision kernel does not conserve particle
number, although much less than the spatial eccentricities. As a matter of
fact, particle-number conservation plays no role in the time dependence of the
numerator of Eq. (34), which determined the overall scaling of $v_{n}(t)$ in
Sect. IV.1. On the other hand, the denominator of Eq. (34) is obviously no
longer constant if particle number is not conserved, which has to be taken
into account. By integrating expansion (25) first over $\vec{x}$ (so that
total-gradient terms vanish) and then over $\\!\vec{\,p}$ (where only terms
even in all momenta can contribute), one finds using the same arguments as in
Sect. IV.1 that the total particle number behaves at early times like

$N(t)\\!\underset{\text{early }t}{\sim}N_{0}+\sum_{j\geq
1}\big{(}\mathrm{Kn}^{-1}\big{)}^{\\!j}\sum_{k\geq 0}b_{k}^{(j)}t^{2k+j},$
(48)

where $N_{0}$ denotes its initial value. Combining this behavior with that of
the numerator and reorganizing the double expansion in powers of
$\mathrm{Kn}^{-1}$ and $t$, one finds that $v_{n}(t)$ still obeys a scaling
law of the form (39), with modified coefficients $a^{(j)}_{n,k}$ — apart from
the leading coefficient $a^{(1)}_{n,0}$ of the term in
$\mathrm{Kn}^{-1}t^{n+1}$, which is the same as if $N(t)$ stayed constant.

###  V.5 Initial anisotropic flow

Eventually, a last key ingredient in Sect. IV was the hypothesis of vanishing
initial anisotropic flow. This assumption may naturally be released, not only
out of mathematical curiosity, but also for several physically motivated
reasons: If $t_{0}$ still represents the initial time of the fireball
expansion, one may consider that anisotropic flow is already present due to
initial-state correlations, as investigated for instance in AMPT in Ref. [40],
or that the finite particle multiplicity unavoidably leads to (small) flow
harmonics. Or $t_{0}$ may instead represent a later time in the system
evolution, say the time at which the description changes from fluid dynamics
to a transport approach, in which case the presence of anisotropic flow
results from the preceding evolution.

In any case, if the assumption of initial momentum isotropy is released, then
one should beware that the free-streaming behaviors are modified. Indeed, in
the absence of rescatterings each coefficient $v_{n}(t)$ remains at its
initial value, whether it vanishes or not. In turn, the spatial eccentricities
no longer decrease as found in Sect. II. For instance, one can see than an
initial $v_{n}$ will spoil relation (10) and lead to a time-dependent average
$\left\langle r_{\perp}{\rm e}^{{\rm i}n\theta}\right\rangle_{t}^{\rm f.s.}$
instead.

That being told, the scaling behaviors (39) and (46), now viewed as describing
the departure from the free-streaming behaviors, will generally no longer
hold. Thus we have seen in Sect. IV.1 that the assumption of vanishing initial
anisotropic flow was necessary to cancel the influence on the flow harmonics
of the term linear in $t$ in expansion (25). If there is some initial
anisotropic flow $v_{n}(t_{0})$, then $v_{n}(t)$ may depart linearly from that
value at (shortly) later times $t$, as was found on a toy example in Ref.
[41]. Similarly, in Sect. IV.2 initial momentum isotropy was a necessary
ingredient to cancel one of the terms in $t^{2}$ contributing to the mean
square radius $\left\langle r_{\perp}^{2}\right\rangle_{t}$ or the asymmetry
$\left\langle r_{\perp}^{2}{\rm e}^{{\rm i}n\theta}\right\rangle_{t}$, so that
departure from momentum isotropy will generally lead to terms at order ${\cal
O}(t^{2})$ in the scaling behavior (46).

##  VI Discussion

The main result of this paper is that in a system described by the kinetic
Boltzmann equation with a particle-number conserving collision kernel, and
with transverse momentum isotropy in the initial state, the anisotropic flow
coefficients $v_{n}(t)$ resp. the spatial eccentricities $\epsilon_{n}^{\bf
x}(t)$ scale according to Eq. (39) resp. Eq. (46) at early times. In the case
of the harmonics $v_{n}$, we also detailed the dependence on the average
number of rescatterings per particle, characterized here by the inverse
Knudsen number $\mathrm{Kn}^{-1}$.

These behaviors are quite generic, since they follow from writing the early-
time expansion of the phase space distribution as a Taylor series (25) and
invoking general principles. They thus hold whether the system is expanding in
two or three dimensions, and whether it consists of massless or massive
classical or quantum particles. Simplifying the collision kernel, for instance
using the popular relaxation time approximation [8, 11, 42, 13], should also
not spoil the scaling behaviors (39), (46), at least as long as conservation
laws are properly implemented. The only exceptions we encountered are twofold,
namely if there is already anisotropic flow in the initial state or if the
particles can undergo inelastic collisions. Actually, the latter possibility
only affects the scaling of eccentricities, not that of the flow harmonics,
whose behavior is thus more robust.

The early-time scaling $v_{n}(t)\propto t^{n+1}$ of momentum anisotropies in
transport calculations was already observed in special cases before: in
numerical simulations with a given initial phase distribution in Ref. [15],
and in analytical calculations restricted to leading order in
$\mathrm{Kn}^{-1}$ with another specific initial profile [16]. Here we have
shown that this scaling behavior is very general if there is no initial flow.

The most studied anisotropic-flow harmonic at ultrarelativistic energies, both
experimentally and theoretically, is elliptic flow, for which Eq. (39) yields

$v_{2}(t)\\!\underset{\text{early }t}{\propto}\\!\mathrm{Kn}^{-1\,}t^{3},$
(49)

at lowest order in $t$. This behavior should be contrasted with the findings
in fluid dynamics, either in simulations [18, 19, 20, 21] or using general
arguments [22], which give $v_{2}(t)\propto t^{2}$ at early times.999Strictly
speaking, the measure of the momentum anisotropy is sometimes $v_{2}$ itself,
and sometimes related to the asymmetry $T^{xx}-T^{yy}$ of the energy-momentum
tensor, which necessitates no particlization of the fluid. The latter also
scales as $t^{3}$ for the system investigated in the present paper [36]. The
mismatch between the scaling exponents predicted by kinetic theory and fluid
dynamics was already noted in Ref. [15] but only for two-dimensional
expansions. In this paper we showed that the difference remains for kinetic
theory in three dimensions. This implies that there is no “universal behavior”
for the development of momentum anisotropies — in contrast to that found for
radial transverse flow [22] — but rather that different classes of theories
(kinetic theory or fluid dynamics) may lead to different behaviors. If there
exist “late-time attractor solutions” for truly three-dimensional dynamical
scenarios without cylindrical symmetry, as was found empirically for one-
dimensional motion in both strong and weak coupling regimes [43, 44, 45, 46],
then it would be interesting to see how this could be reconciled with the
different behaviors of anisotropic flow at early times.

The results reported in the present paper are admittedly rather formal,
because we deliberately aimed at staying as general as possible, with minimal
assumptions on the initial phase-space distribution and the collision kernel.
This is what allows us to derive generic results. In a companion paper [36] we
shall present comparisons with numerical simulations, including a few
straightforward generalizations — for instance, looking at energy-weighted
anisotropies. Accordingly, we shall be able to illustrate the effect of
including higher-order terms in $t$ and/or $\mathrm{Kn}^{-1}$ from expansion
(25). The price to pay is that we use a single, simple setup (two-dimensional
system of massless particles, toy initial profile) and thus lose the
generality of the present paper.

Eventually, one should naturally ask whether the different scaling behaviors
of say $v_{2}(t)$ in kinetic theory vs. fluid dynamics are relevant for heavy-
ion phenomenology. The main issue is clearly that experimentally only the
final values of the flow harmonics, and to a lesser extent the spatial
eccentricities, are directly accessible. In collisions of heavy nuclei, one
may hope that accessing some characteristics of the early-time dynamics of the
bulk could become feasible by studying particles that decouple early from the
system, as e.g. photons or dileptons pairs in an appropriate invariant-mass
interval [47]. The onset of “pre-flow” anisotropies of the bulk in large
systems may also be relevant for determining the initial condition to the
subsequent fluid-dynamical evolution. Accessing the early-time behavior may be
more feasible in “small systems”, since in that case the evolution lasts less
long, so that the final flow values are possibly more influenced by the early
stage. One should however remember that the characteristic time scale of the
early-time evolution also becomes smaller in such systems.

###### Acknowledgements.

We thank Nina Kersting and Hendrik Roch for discussions. We acknowledge
support by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) through the CRC-TR 211 “Strong-interaction matter under extreme
conditions” – project number 315477589 – TRR 211.

##  Appendix A Time derivative of the collision kernel: an example

To illustrate the procedure for eliminating the time derivatives of the
collision term ${\cal C}[f]$, we take as an example the kernel (30), which we
recast in the shorter form

${\cal C}[f]=\frac{1}{2}\\!\int_{{\bf p}_{1},{\bf p}^{\prime},{\bf
p}^{\prime}_{1}}\\!(f^{\prime}f^{\prime}_{1}-ff_{1})W({\bf p},{\bf
p}_{1}\to{\bf p}^{\prime},{\bf p}^{\prime}_{1}),$ (50)

in which the precise form of the integration measure in momentum space is no
longer written. Differentiating this kernel with respect to time yields at
once

$\partial_{t}{\cal C}[f]=\frac{1}{2}\\!\int_{{\bf p}_{1},{\bf p}^{\prime},{\bf
p}^{\prime}_{1}}\\!\big{[}(\partial_{t}f^{\prime})f^{\prime}_{1}+(\partial_{t}f^{\prime}_{1})f^{\prime}-(\partial_{t}f)f_{1}-(\partial_{t}f_{1})f\big{]}W({\bf
p},{\bf p}_{1}\to{\bf p}^{\prime},{\bf p}^{\prime}_{1}).$ (51)

If we now invoke the Boltzmann equation (1), we can re-express every time
derivative of the phase space distribution in the integrand, which leads to

$\displaystyle\partial_{t}{\cal C}[f]=$ $\displaystyle\
\frac{1}{2}\\!\int_{{\bf p}_{1},{\bf p}^{\prime},{\bf
p}^{\prime}_{1}}\\!\bigg{(}\\!\\!-\\!\frac{{\bf
p}^{\prime}\cdot\bm{\nabla}_{\\!x}f^{\prime}}{E^{\prime}}f^{\prime}_{1}-\frac{{\bf
p}^{\prime}_{1}\cdot\bm{\nabla}_{\\!x}f^{\prime}_{1}}{E^{\prime}_{1}}f^{\prime}+\frac{{\bf
p}\cdot\bm{\nabla}_{\\!x}f}{E}f_{1}+\frac{{\bf
p}_{1}\cdot\bm{\nabla}_{\\!x}f_{1}}{E_{1}}f\bigg{)}W({\bf p},{\bf
p}_{1}\to{\bf p}^{\prime},{\bf p}^{\prime}_{1})$ (52)
$\displaystyle+\frac{1}{2}\\!\int_{{\bf p}_{1},{\bf p}^{\prime},{\bf
p}^{\prime}_{1}}\\!\bigg{(}\frac{{\cal
C}[f^{\prime}]}{E^{\prime}}f^{\prime}_{1}+\frac{{\cal
C}[f^{\prime}_{1}]}{E^{\prime}_{1}}f^{\prime}-\frac{{\cal
C}[f]}{E}f_{1}+\frac{{\cal C}[f_{1}]}{E_{1}}f\bigg{)}W({\bf p},{\bf
p}_{1}\to{\bf p}^{\prime},{\bf p}^{\prime}_{1}).$ (53)

The term on the right hand side of the first line is of order
$\mathrm{Kn}^{-1}$, while that in the second line is of order
$\mathrm{Kn}^{-2}$, as mentioned in Sec. III.

Iterating the procedure, one finds that the $k$-th time derivative
$\partial_{t}^{k}{\cal C}[f]$ will involve the $k$-th power of the spatial
gradient, $(\bm{\nabla}_{\\!x})^{k}$, at order $\mathrm{Kn}^{-1}$, together
with lower powers $(\bm{\nabla}_{\\!x})^{k-j}$ at order
$\big{(}\mathrm{Kn}^{-1}\big{)}^{j+1}$ for all $1\leq j\leq k$. For instance,
the second time derivative reads

$\displaystyle\partial_{t}^{2}{\cal C}[f]=$ $\displaystyle\
\frac{1}{2}\int_{{\bf p}_{1},{\bf p}^{\prime},{\bf
p}^{\prime}_{1}}\\!\bigg{[}\frac{\big{(}{\bf
p}^{\prime}\cdot\bm{\nabla}_{\\!x}\big{)}^{\\!2\\!}f^{\prime}}{E^{\prime
2}}f^{\prime}_{1}+\frac{\big{(}{\bf
p}^{\prime}_{1}\cdot\bm{\nabla}_{\\!x}\big{)}^{\\!2\\!}f^{\prime}_{1}}{E^{\prime
2}_{1}}f^{\prime}-\frac{\big{(}{\bf
p}\cdot\bm{\nabla}_{\\!x}\big{)}^{\\!2\\!}f}{E^{2}}f_{1}-\frac{\big{(}{\bf
p}_{1}\cdot\bm{\nabla}_{\\!x}\big{)}^{\\!2\\!}f_{1}}{E_{1}^{2}}f\bigg{]}W({\bf
p},{\bf p}_{1}\to{\bf p}^{\prime},{\bf p}^{\prime}_{1})\quad$ (54)
$\displaystyle+\\!\int_{{\bf p}_{1},{\bf p}^{\prime},{\bf
p}^{\prime}_{1}}\\!\bigg{(}\frac{{\bf
p}^{\prime}\cdot\bm{\nabla}_{\\!x}f^{\prime}}{E^{\prime}}\frac{{\bf
p}^{\prime}_{1}\cdot\bm{\nabla}_{\\!x}f^{\prime}_{1}}{E^{\prime}_{1}}-\frac{{\bf
p}\cdot\bm{\nabla}_{\\!x}f}{E}\frac{{\bf
p}_{1}\cdot\bm{\nabla}_{\\!x}f_{1}}{E_{1}}\bigg{)}W({\bf p},{\bf p}_{1}\to{\bf
p}^{\prime},{\bf p}^{\prime}_{1})+{\cal
O}(\mathrm{Kn}^{-2},\mathrm{Kn}^{-3}),$ (55)

where we only wrote the term of (leading) order $\mathrm{Kn}^{-1}$. The term
of order $\mathrm{Kn}^{-2}$ is

$\displaystyle\int_{{\bf p}_{1},{\bf p}^{\prime},{\bf
p}^{\prime}_{1}}\\!\bigg{(}\\!\\!-\\!\frac{{\bf
p}^{\prime}\cdot\bm{\nabla}_{\\!x}f^{\prime}}{E^{\prime}}\frac{{\cal
C}[f^{\prime}_{1}]}{E^{\prime}_{1}}-\frac{{\bf
p}^{\prime}_{1}\cdot\bm{\nabla}_{\\!x}f^{\prime}_{1}}{E^{\prime}_{1}}\frac{{\cal
C}[f^{\prime}]}{E^{\prime}}+\frac{{\bf
p}\cdot\bm{\nabla}_{\\!x}f}{E}\frac{{\cal C}[f_{1}]}{E_{1}}+\frac{{\bf
p}_{1}\cdot\bm{\nabla}_{\\!x}f_{1}}{E_{1}}\frac{{\cal C}[f]}{E}\bigg{)}W({\bf
p},{\bf p}_{1}\to{\bf p}^{\prime},{\bf p}^{\prime}_{1})$ (56) $\displaystyle\
+\int_{{\bf p}_{1},{\bf p}^{\prime},{\bf
p}^{\prime}_{1}}\\!\bigg{(}\\!\\!-\\!\frac{{\bf
p}^{\prime}\cdot\bm{\nabla}_{\\!x}{\cal C}[f^{\prime}]}{E^{\prime
2}}f^{\prime}_{1}-\frac{{\bf p}^{\prime}_{1}\cdot\bm{\nabla}_{\\!x}{\cal
C}[f^{\prime}_{1}]}{E^{\prime 2}_{1}}f^{\prime}+\frac{{\bf
p}\cdot\bm{\nabla}_{\\!x}{\cal C}[f]}{E^{2}}f_{1}+\frac{{\bf
p}_{1}\cdot\bm{\nabla}_{\\!x}{\cal C}[f_{1}]}{E_{1}^{2}}f\bigg{)}W({\bf
p},{\bf p}_{1}\to{\bf p}^{\prime},{\bf p}^{\prime}_{1}),$ (57)

while that of order $\mathrm{Kn}^{-3}$ includes in its integrand contributions
of the type ${\cal C}[f]{\cal C}[f_{1}]$ and ${\cal C}[{\cal C}[f]]$, both
multiplying the transition rate $W$.

##  Appendix B Calculation of contributions to the early-time development of
spatial characteristics

###  B.1 Contributions at order ${\cal O}(t^{2})$

Let us show that the integral

$\int\\!(x^{2}+y^{2})\frac{{\bf p}\cdot\bm{\nabla}_{\\!x}}{E^{2}}{\cal
C}[f]\big{|}_{0}\,{\rm d}^{2}{\bf p}\,{\rm d}^{2}{\bf x},$ (58)

which enters the early-time behavior of the mean square radius $\left\langle
r_{\perp}^{2}\right\rangle_{t}$ at order $t^{2}$, vanishes when the initial
distribution is isotropic in momentum.

Performing integration by parts over the spatial variables using the fact that
${\cal C}[f]\big{|}_{0}$ vanishes at large distances, the integral becomes

$-2\\!\int\\!\frac{xp_{x}+yp_{y}}{E^{2}}{\cal C}[f]\big{|}_{0}\,{\rm
d}^{2}{\bf p}\,{\rm d}^{2}{\bf x}.$

The integral over momentum is then vanishing at every ${\bf x}$: Replacing
${\cal C}[f]$ by its explicit expression, one obtains an integral over all
momenta of an integrand which is always odd in ${\bf p}$ or ${\bf p}_{1}$
(when the Møller velocity is involved), since $f_{0}$ is isotropic in momentum
space.

The same reasoning gives

$\int\\!(x^{2}+y^{2})^{n/2\,}\frac{{\bf p}\cdot\bm{\nabla}_{\\!x}}{E^{2}}{\cal
C}[f]\big{|}_{0}\,{\rm d}^{2}{\bf p}\,{\rm d}^{2}{\bf x}=0,$ (59)

resp.

$\int\\!r_{\perp}^{n}{\rm e}^{{\rm i}n\theta\,}\frac{{\bf
p}\cdot\bm{\nabla}_{\\!x}}{E^{2}}{\cal C}[f]\big{|}_{0}\,{\rm d}^{2}{\bf
p}\,{\rm d}^{2}{\bf x}=0,$ (60)

which is relevant for the average $\left\langle
r_{\perp}^{n}\right\rangle_{t}$ resp. $\left\langle r_{\perp}^{n}{\rm e}^{{\rm
i}n\theta}\right\rangle_{t}$ at order $t^{2}$.

###  B.2 Influence of mass on the early-time development of $\left\langle
r_{\perp}^{2}\right\rangle_{t}$

In this appendix we detail how the particle mass affects the integral

$\int\\!(x^{2}+y^{2})\frac{\big{(}{\bf
p}\cdot\bm{\nabla}_{\\!x}\big{)}^{2}}{E^{3}}{\cal C}[f]\big{|}_{0}\,{\rm
d}^{2}{\bf p}\,{\rm d}^{2}{\bf x},$ (61)

which appears when one investigates the behavior of the mean-square radius
$\left\langle r_{\perp}^{2}\right\rangle_{t}$ at order $t^{3}$ (see Sec.
IV.2).

Writing $({\bf
p}\cdot\bm{\nabla}_{\\!x})^{2}=p_{x}^{2}\partial_{x}^{2}+2p_{x}p_{y}\partial_{x}\partial_{y}+p_{y}^{2}\partial_{y}^{2}$,
one can first handle the term in $p_{x}^{2}\partial_{x}^{2}$ by performing two
successive integrations by parts over $x$. Since ${\cal C}[f]\big{|}_{0}$ or
its derivative $\partial_{x}{\cal C}[f]\big{|}_{0}$ vanish as $|x|\to\infty$ —
the system has a finite initial size —, one finds

$\int\\!(x^{2}+y^{2})\frac{p_{x}^{2}}{E^{3}}\partial_{x}^{2}{\cal
C}[f]\big{|}_{0}\,{\rm d}x=2\frac{p_{x}^{2}}{E^{3}}\\!\int\\!{\cal
C}[f]\big{|}_{0}\,{\rm d}x.$

With the help of the similar result for the term in
$p_{y}^{2}\partial_{y}^{2}$, the contribution from the two corresponding terms
to the integral (61) becomes

$2\\!\int\\!\frac{p_{x}^{2}+p_{y}^{2}}{E^{2}}{\cal
C}[f]\big{|}_{0}\,\frac{{\rm d}^{2}{\bf p}}{E}\,{\rm d}^{2}{\bf x}.$ (62)

If the particles are massless, the ratio in the integrand equals 1, and the
integral over ${\bf p}$ vanishes for a particle-number-conserving collision
kernel, thanks to property (40).

Eventually, the mixed term in $\partial_{x}\partial_{y}$ does not contribute
to the integral (61) either. Indeed, the integral over the spatial variables
are readily performed and yield ${\cal C}[f]\big{|}_{0}$ (or one of its
spatial derivatives) at infinity, where it vanishes.

Altogether, we have thus showed that the integral (61) vanishes for a two-
dimensional system of massless particles. This is no longer necessarily true
for massive particles, or if they propagate in three dimensions.

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