  *

# Field-Theory of Active Chiral Hard Disks: A First-Principles Approach to
Steric Interactions

Erik Kalz1, Abhinav Sharma2,3, Ralf Metzler1,4 1Institute of Physics and
Astronomy, University of Potsdam, D-14476 Potsdam, Germany  2Institute of
Physics, University of Augsburg, D-86159 Augsburg, Germany  3Institute Theory
of Polymers, Leibniz-Institute for Polymer Research, D-01069 Dresden, Germany
4Asia Pacific Centre for Theoretical Physics, KR-37673 Pohang, Republic of
Korea<EMAIL_ADDRESS><EMAIL_ADDRESS>ralf.metzler@uni-
potsdam.de

###### Abstract

A first-principles approach for active chiral hard disks is presented, that
explicitly accounts for steric interactions on the two-body level. We derive
an effective one-body equation for the joint probability distribution of
positions and angles of the particles. By projecting on the angular modes, we
write a hierarchy for the lowest hydrodynamic modes, i.e. particle density,
polarization, and nematic tensor. By undimensionalising the equations, we
highlight the assumptions, which - though inherent - are often included
implicit in closing the hierarchy for finally arriving at an effective field-
theoretical equation for the particle density. By considering different
regimes of the Péclet number, the well-known models in active matter can be
obtained through our conisderation. Explicitly, we derive the phenomenological
Model B and by going to higher orders in the closure scheme, we show that this
first-principles approach results in the recently introduced Active Model B +,
a natural extension of the Model B for active processes. Remarkably, here we
find that chirality can change the sign of the phenomonological activity
parameters.

Keywords: first-principles approach, active chiral particles, steric
interactions, hierarchy of angular modes, Active Model B+

###### Contents

  1. 1 Introduction.
  2. 2 Theory: From Active Chiral Particles to the Active Model B +
    1. 2.1 Model
      1. 2.1.1 Setup
      2. 2.1.2 Joint Fokker-Planck Equation
    2. 2.2 Effective One-Body Description
      1. 2.2.1 The Collision Integral
      2. 2.2.2 Matched Asymptotic Expansion
      3. 2.2.3 Evaluation of the Collision Integral
      4. 2.2.4 Effective One-Body Equation
    3. 2.3 Hierarchy of Hydrodynamic Equations
      1. 2.3.1 Expansion in Harmonic Modes
      2. 2.3.2 Time-Evolution of Modes
    4. 2.4 Closure of the Hierarchy
      1. 2.4.1 Adiabatic Approximation
      2. 2.4.2 Closure of the Hierarchy via Perturbation Approach
      3. 2.4.3 Model B
    5. 2.5 Active Model B +
      1. 2.5.1 Effect of Mixed Perturbation Parameters
      2. 2.5.2 Active Model B +
      3. 2.5.3 First-Principles Expressions for Field-Theoretical Parameters
  3. 3 Conclusion
  4. A Evaluation of the Integral ()

##  1 Introduction.

The analytic model of an Active Chiral Particle (ACP) [1, 2, 3, 4] represents
an extension to the well-known Active Brownian Particle (ABP) [5, 6, 7, 8],
where additional active chirality is included. These models have proven very
useful as they come as a first step in generalising equilibrium models to
include non-equilibrium, active, contributions for microscopic agents. They
constitute a minimalist attempt to describe directed motion on the microscale
and are found to lead to emerging complex structures on the macroscale. For
that reason, they are widely used in the analytic description of non-
equilibrium, more precisely, living matter systems [9, 10, 11].

The ABP model describes the overdamped motion of a tagged particle influenced
by two contributions: the first one comes from the equilibrium fluctuations of
the surrounding medium and is given in terms of the Langevin description. The
particle is assumed to diffuse with a spatial diffusion coefficient $D_{T}$.
Secondly, the particle experiences a non-equilibrium driving force, that might
originate from an internal energy depot or external energy input [12, 13]. In
the simplest case, this scenario is modelled as a self-propulsion with a
(constant) velocity $v$ along an orientation vector
$\hat{\mathbf{e}}(\theta)$. In the model of an ABP, the self-propulsion
direction gets randomized by assuming that the orientation vector is
performing rotational diffusion on the unit-sphere with the diffusion
coefficient $D_{R}$.

Despite its simplicity, this toy model has proven tremendously successful in
analytical analysis and as a simulation model to describe active matter on a
macroscale [14, 15]. Emerging complex macroscopic phases such as motility-
induced phase separations [16, 17, 18, 19], flocking [20, 21], swarming [22,
23, 24], and even more exotic phenomena like bubbly phase transitions [25] or
active foams [26] have been reported. The ABP model has been further developed
to include active processes that explicitly break the spatial symmetry via an
effect of active chirality $\omega$ [1, 2, 3, 4]. This generalisation in the
model allows to describe even more complex emerging phenomena, like odd
viscosity [27, 28], finite-size rotating clusters [29] and edge currents at
interfaces [30, 31]. This analytical extension is inspired by experimental
observation of bacteria [32, 33, 34], sperm cells [35, 36], or syntactical
particles [37, 38], which show the archetypal chirality in their trajectories
on the microscale. Remarkably, already over a century ago, experimental
observations were made in living organisms, which showed that trajectories of
certain microorganisms such as Loxodes and Paramecium break the spatial
symmetry and need an interpretation in terms of chiral self-propulsion [39].

Another fruitful approach for describing the emerging complex structures in
active systems is based on continuum field-theoretical models of active
matter. The starting point is to address the diffusive behaviour of a
conserved order parameter, such as the mean particle probability-density
function (PDF) $\varrho$, as in the well-known Model B [40]. This model
successfully describes the equilibrium behaviour of matter, and especially the
dynamics of phase separation [41]. An inherent assumption in the derivation of
these field-theoretical models is that the underlying processes follow the
detailed-balance dynamics, that is broken in active systems. It is, therefore,
natural that field theories that aim to describe the behaviour of active
matter have to revisit the foundations and need to go beyond the detailed-
balance restrictions. Successive works culminated in the recently introduced,
so-called Active Model B+ (AMB+) [25, 26], that attracted a lot of interest
lately [42, 43, 44].

Continuum approaches, very recently, have been extend to include active
chirality and specifically account for broken-time reversal symmetries [45,
46, 47]. Modelling of interactions is crucial for these approaches. While
previous approaches are predominantly focused on interaction potentials, here
we model the interactions via a geometric approach instead. The approach was
originally introduced by Bruna and Chapman in Refs. [48, 49] and thereafter
successfully applied to ABPs [50]. Here we take that approach and extend it to
active chirality. The basic idea is to include particle interactions by
restricting the domain of definition of the time-evolution equation and
thereby creating forbidden areas, which correspond to situations with a
particle overlap. We apply this idea to the ACP model and derive an effective
one-body description of the full one-particle PDF $p(\mathbf{x},\theta,t)$ to
find a particle at position $\mathbf{x}$ with the self-propulsion vector of
angle $\theta$ at time $t$ after starting with
$\delta(\mathbf{x}-\mathbf{x}_{0})$ at $t=t_{0}$. The resulting time-evolution
equation explicitly accounts for two-particle steric interactions and
therefore its validity is restricted to the dilute limit. To arrive at a
field-theoretical description for the mean-particle PDF
$\varrho(\mathbf{x},t)$, one typically proceeds by integrating out the effect
of the angle [51, 52]. We follow this procedure and arrive at an (infinite)
hierarchy of the hydrodynamic modes of the active particle, of which the mean
particle PDF constitutes the zeroth order mode. This hierarchy is
mathematically very similar to the famous Bogoliubov–Born–Green–Kirkwood–Yvon
hierarchy in the kinetic theory [53], also known as BBGKY hierarchy. Similar
to that hierarchy, also in continuous active matter theories, one is required
to close the hierarchy to make analytic progress [54, 55, 56].

We transform the closure problem into a perturbation problem and derive two
field-theoretical descriptions for ACPs, based on the strictness of assumption
on the perturbation parameters. This gives us access to the otherwise
phenomenological coefficients in these models. We show that both Model B and
the AMB+ can be obtained within our approach, depending on the order of the
closure scheme. Our work moreover suggests that the AMB+ is a natural
generalisation of equilibrium field theories to describe the continuum
behaviour of active matter. We also show the physical regime of validity of
this model. We further have first-principles access to the coefficients in the
AMB+ and it turns out, that, surprisingly, they are altered by active
chirality in such a way, that they can even change sign as a function of
chirality.

The remainder of this work is organized as follows. In Section 2.1 we
introduce the mathematical model and describe the way to deal with inter-
particle interactions in the geometric sense. In Section 2.2 we derive in
detail the effective one-body time-evolution equation of the PDF. To our
understanding, the physics community is rather unaware of this specific method
to handle particle interactions, for which we introduce it in appropriate
detail. In Section 2.3 we thereafter derive the hierarchy of hydrodynamic
modes, and in Section 2.4 we introduce the mathematical steps to close the
hierarchy. In Section 2.5 we go one step beyond the simplest closure and find
that the time-evolution of the mean-particle PDF equals the AMB+ predicted
form. In Section 3 we conclude and give an outlook on further and related
works.

##  2 Theory: From Active Chiral Particles to the Active Model B +

###  2.1 Model

In this Section, we introduce the model of interacting ACPs and how to deal
with their excluded-volume interactions in a geometric sense. We finish by
formulating an effective one-body description.

####  2.1.1 Setup

We consider the dynamics of $N$ interacting ACPs in two dimensions. The
particles centers $\mathbf{x}_{i}(t)$ and angular coordinates $\theta_{i}(t)$,
$i\in\\{1,\ldots,N\\}$ move according to the overdamped Langevin dynamics [57,
58]

$\displaystyle\frac{\partial}{\partial t}\mathbf{x}_{i}$
$\displaystyle=v\hat{\mathbf{e}}(\theta_{i})+\sqrt{2D_{T}}\
\boldsymbol{\eta}_{i}(t),$ (1a) $\displaystyle\frac{\partial}{\partial
t}\theta_{i}$ $\displaystyle=\omega+\sqrt{2D_{R}}\ \zeta_{i}(t).$ (1b)

Here $\boldsymbol{\eta}_{i}(t)$ and $\zeta_{i}(t)$ are independent Gaussian
white noises, $v$ is the (constant) active self-propulsion velocity, $\omega$
the active torque and
$\hat{\mathbf{e}}(\theta_{i})=(\cos(\theta_{i}),\sin(\theta_{i}))^{\mathrm{T}}$
is the unit orientation vector. $(\cdot)^{\mathrm{T}}$ denotes a matrix
transpose. Note that for $\omega=0$, the model of Eqs. (1a) and (1b) reduces
to the well-known model of ABPs. $D_{T}$ and $D_{R}$ are the translational and
rotational diffusion coefficients, respectively. The diffusion coefficients
have different physical units, $[D_{T}]=m^{2}/s$ and $[D_{R}]=1/s$.

The $N$ identical disk-like particles have a diameter $d$ and are assumed to
interact hard-core with each other. For an illustration of the model system
see also Fig. 1(a). They are modelled to diffuse in a spatially bounded domain
$\mathbf{x}_{i}(t)\in\Omega\subset\mathbb{R}^{2}$ of typical size $L\times L$
and their angular coordinates are $\theta_{i}(t)\in[0,2\pi)$. By
undimensionalising space with $L$, the typical size of the domain is set to
unity, whereas the diameter of the particle becomes $\varepsilon_{d}=d/L$. We
restrict the analysis to a dilute system, i.e. we assume that
$N\varepsilon_{d}\ll 1$. In contrast to typical approaches to model the
interactions, that is via an interaction-potential term in the spatial
Langevin equation (1a), we, instead, restrict the domain of definition of the
centre-of-mass coordinates $\mathbf{x}_{i}(t)$. We, thereby, follow a
geometric approach to model particle interactions, established by Bruna and
Chapman [48, 49] that already has proven successful in various contexts [59,
60, 61]. These authors showed in a recent work [50] that within this model
they can derive additional nonlinear cross-diffusion terms for the description
of stericically interacting ABPs, compared to more classical treatments of
interacting ABPs [62, 63].

The fundamental idea in modelling steric interactions in this geometric sense
is that Eq. (1a) is defined on a restricted (spatial) domain
$\Omega^{N}_{\varepsilon_{d}}=\\{(\mathbf{x}_{1},\ldots,\mathbf{x}_{N})\in\Omega^{N};\forall
i\neq j\colon|\mathbf{x}_{i}(t)-\mathbf{x}_{j}(t)|\geq\varepsilon_{d}\\}$ due
to the excluded volume. The full model including Eq. (1b) therefore is defined
on $\Lambda^{N}=\Omega^{N}_{\varepsilon_{d}}\times[0,2\pi)^{N}$. Note that as
the angular coordinate is not affected by the steric interactions, its domain
of definition is not restricted. While for the angular coordinate we assume
periodic boundary conditions, the cost of treating the particle interactions
by restricting the domain of definition is that we have generated an inner
(moving) boundary as apparent from Fig. 1(a). At this boundary, particles
perform hard elastic collisions similar to the container walls. Both inner and
container-wall boundaries therefore are treated as those with reflective
boundaries. To simplify this problem, we observe that for a dilute system
finding configurations in the system where three particles are close, or two
particles are close at a container wall is of the order
$\mathcal{O}(\varepsilon_{d}^{2}N^{2})$. Configurations where two particles
are close or one particle is close to the wall, in contrast, are of the order
$\mathcal{O}(\varepsilon_{d}N)$ [49, 48]. It is therefore reasonable to assume
that in a dilute system, two-body collisions dominate the interactions and we
can safely ignore higher-order correlations. As we assume that all particles
are identical, it is sufficient to consider a system with $N=2$ particles. In
the end we will scale the resulting interaction contribution by the particle
number, which is justified as long as we stay within the dilute limit.

####  2.1.2 Joint Fokker-Planck Equation

The Fokker-Planck Equation (FPE) for the $2$-particle joint PDF
$P_{2}(t)=P_{2}(\mathbf{x}_{1},\theta_{1},\mathbf{x}_{2},\theta_{2},t)$,
defined on $\Lambda^{2}$, reads [64]

$\displaystyle\frac{\partial}{\partial
t}P_{2}(t)=\nabla_{1}\cdot\bigg{[}D_{T}\nabla_{1}-v\hat{\mathbf{e}}(\theta_{1})\bigg{]}P_{2}(t)+\frac{\partial}{\partial\theta_{1}}\left[D_{R}\frac{\partial}{\partial\theta_{1}}-\omega\right]P_{2}(t)$
$\displaystyle\quad+\nabla_{2}\cdot\bigg{[}D_{T}\nabla_{2}-v\hat{\mathbf{e}}(\theta_{2})\bigg{]}P_{2}(t)+\frac{\partial}{\partial\theta_{2}}\left[D_{R}\frac{\partial}{\partial\theta_{2}}-\omega\right]P_{2}(t).$
(2a) Here $\nabla_{i}$ denotes the partial differential operator with respect
to position of particle $i\in\\{1,2\\}$. The reflective boundary condition
reads
$\mathbf{n}_{1}\cdot\bigg{[}D_{T}\nabla_{1}-v\hat{\mathbf{e}}(\theta_{1})\bigg{]}P_{2}(t)+\mathbf{n}_{2}\cdot\bigg{[}D_{T}\nabla_{2}-v\hat{\mathbf{e}}(\theta_{2})\bigg{]}P_{2}(t)=0,$
(2b) valid on
$\partial\Lambda^{2}=\partial\Omega_{\varepsilon_{d}}^{2}=\partial\Omega^{2}\cup\mathcal{S}_{\mathrm{coll}}$,
where
$\mathcal{S}_{\mathrm{coll}}=\\{(\mathbf{x}_{1},\mathbf{x}_{2})\in\Omega^{2};|\mathbf{x}_{1}(t)-\mathbf{x}_{2}(t)|=\varepsilon_{d}\\}$
is the so-called (inner) collision surface. For an illustration see Fig. 1(b).
$\mathbf{n}_{i}$ in Eq. (2b) is the outward unit normal vector of disk $i$.
Note that $\mathbf{n}_{i}=\mathbf{0}$ for
$\mathbf{x}_{j}\in\partial\Omega^{2}$ for $(i,j)=(1,2)$ and $(2,1)$ due to
particle conservation, as well as $\mathbf{n}_{1}=-\mathbf{n}_{2}$ on
$\mathcal{S}_{\mathrm{coll}}$, due to elastic collisions of particles. For the
angular coordinate Eq. (2) is supplemented with the periodic boundary
condition $P_{N}(\theta_{i}=0)=P_{N}(\theta_{i}=2\pi).$ (2c)

It is convenient to use the structural similarities of the spatial and angular
coordinates and write Eq. (2) in terms of joint variables
$\chi_{i}=(\mathbf{x}_{i},\theta_{i})$. The diffusion coefficients form a
diffusion matrix $\mathbf{D}=\mathrm{diag}(D_{T},D_{T},D_{R})$ and the joint
drift reads $\mathbf{f}(\theta_{i})=(v\
\hat{\mathbf{e}}(\theta_{i}),\omega)^{\mathrm{T}}$. The FPE (2) written in the
joint variables becomes

$\displaystyle\frac{\partial}{\partial
t}P_{2}(t)=\nabla_{\chi_{1}}\cdot\left[\mathbf{D}\nabla_{\chi_{1}}-\mathbf{f}(\theta_{1})\right]P_{2}(t)+\nabla_{\chi_{2}}\cdot\left[\mathbf{D}\nabla_{\chi_{2}}-\mathbf{f}(\theta_{2})\right]P_{2}(t),$
(3)

valid on $\Lambda^{2}$. We have defined the joint partial differential
operator
$\nabla_{\chi_{i}}=(\nabla_{i},\partial/\partial\theta_{i})^{\mathrm{T}}$.

(a)

(b)

Figure 1: (a) Sketch of the model setup of the $i$th spherical ACP of diameter
$\varepsilon_{d}$. The particle is self-propelled by the active speed $v$
which direction rotates with the active frequency $\omega$. It is embedded in
a thermal bath giving rise to fluctuations in the spatial and angular
coordinates $(\mathbf{x}_{i},\theta_{i})$ with respective strengths
$\propto\sqrt{D_{T}}$ and $\propto\sqrt{D_{R}}$, see Eqs. (1a) and (1b),
respectively. (b) The particle coordinates are defined on
$\Lambda^{2}=\Omega_{\varepsilon_{d}}^{2}\times[0,2\pi)^{2}$, where
$\Omega_{\varepsilon_{d}}^{2}=\\{(\mathbf{x}_{1},\mathbf{x}_{2})\in\Omega^{2};|\mathbf{x_{1}}(t)-\mathbf{x}_{2}(t)|\geq\varepsilon_{d}\\}$
is the allowed configuration space $\Omega\subset\mathbb{R}^{2}$ without the
regions of an particle overlap due to the excluded volume (white area). This
excluded-volume creates a reflecting boundary
$\partial\Lambda^{2}=\partial\Omega^{2}\cup\mathcal{S}_{\mathrm{coll}}$, at
the container walls $\partial\Omega^{2}$ and an inner moving, so-called
collision surface
$\mathcal{S}_{\mathrm{coll}}=\\{(\mathbf{x}_{1},\mathbf{x}_{2})\in\Omega^{2};|\mathbf{x_{1}}(t)-\mathbf{x}_{2}(t)|=\varepsilon_{d}\\}$.

We are interested in analytically capturing the effects of particle collisions
on the one-body diffusive behaviour. Therefore, we want to derive an effective
description for the full one-body PDF
$p(\mathbf{x}_{1},\theta_{1},t)=p(\chi_{1},t)$, which is defined as

$\displaystyle p(\chi_{1},t)=\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
P_{2}(t)=\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\Omega\setminus\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathbf{x}_{2}\
P_{2}(t),$ (4)

where we take particle one to be the exemplary particle. The area of
integration $\Lambda(\chi_{1})$ is given by all allowed configurations for the
second particle to be placed everywhere but not in excluded volume due to the
first particle, i.e.
$\Lambda(\chi_{1})=\Omega\setminus\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})\times[0,2\pi)$,
where $\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})$ is the disk of radius
$\varepsilon_{d}$ centred at $\mathbf{x}_{1}$.

###  2.2 Effective One-Body Description

In this Section, we perform the integration to arrive at an effective one-body
description and encounter that the effect of particle collisions results in an
integral. We solve this so-called collision integral by the method of matched
asymptotic expansions and give the effective one-body description. This
Section follows Ref. [50] and adapts it to ACPs to review and introduce the
geometric method to deal with particle interactions to the reader.

####  2.2.1 The Collision Integral

Integrating Eq. (2), over the reduced configuration space $\Lambda(\chi_{1})$
results in

$\displaystyle\frac{\partial}{\partial
t}p(\chi_{1},t)=\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\nabla_{\chi_{1}}\cdot\bigg{[}\mathbf{D}\nabla_{\chi_{1}}-\mathbf{f}(\theta_{1})\bigg{]}P_{2}(t)$
$\displaystyle\quad+\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\nabla_{\chi_{2}}\cdot\bigg{[}\mathbf{D}\nabla_{\chi_{2}}-\mathbf{f}(\theta_{2})\bigg{]}P_{2}(t).$
(5)

We can easily evaluate the second integral, where integration and
differentiation are with respect to the same variable. Using the divergence
theorem and applying the boundary conditions of Eqs. (2b) and (2c), we find

$\displaystyle\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\nabla_{\chi_{2}}\cdot\bigg{[}\mathbf{D}\nabla_{\chi_{2}}-\mathbf{f}(\theta_{2})\bigg{]}P_{2}(t)$
$\displaystyle=\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\cdot\bigg{[}D_{T}\nabla_{1}-v\hat{\mathbf{e}}(\theta_{1})\bigg{]}P_{2}(t),$
(6)

where $\mathrm{d}\mathrm{S}_{2}\ \mathbf{n}_{2}$ is the outward surface
element of $\mathbf{x}_{2}$ and we used that $\mathbf{n}_{1}=-\mathbf{n}_{2}$
on $\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})$ and
$\mathbf{n}_{1}=\mathbf{0}$ for $\mathbf{x}_{2}\in\partial\Omega$.

The first integral in Eq. (2.2.1) cannot be evaluated that simple since
integration and differentiation are with respect to different particle labels.
Instead, we have to use the Reynolds transport theorem extended to spatial
variation of integrals [65, 66]. Together with an additional use of the
divergence theorem this results in

$\displaystyle\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\nabla_{\chi_{1}}\cdot\bigg{[}\mathbf{D}\nabla_{\chi_{1}}-\mathbf{f}(\theta_{1})\bigg{]}P_{2}(t)=\nabla_{\chi_{1}}\cdot\left[\mathbf{D}\nabla_{\chi_{1}}-\mathbf{f}(\theta_{1})\right]p(\chi_{1},t)$
$\displaystyle\quad-\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\cdot\bigg{[}D_{T}\left(2\nabla_{1}+\nabla_{2}\right)+v\hat{\mathbf{e}}(\theta_{1})\bigg{]}P_{2}(t).$
(7)

For details of this calculation, see A. We combine this integral with Eq.
(2.2.1) and find the effective equation for the full one-body PDF
$p(\chi_{1},t)$ according to Eq. (2.2.1) as

$\displaystyle\frac{\partial}{\partial
t}p(\chi_{1},t)=\nabla_{\chi_{1}}\cdot\left[\mathbf{D}\nabla_{\chi_{1}}-\mathbf{f}(\theta_{1})\right]p(\chi_{1},t)$
$\displaystyle\quad-
D_{T}\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\cdot\left(\nabla_{1}+\nabla_{2}\right)P_{2}(\chi_{1},\chi_{2},t).$
(8)

In analogy to kinetic theory [53], we refer to the integral as the collision
integral and denote it by $I(\chi_{1},t)$. It captures the effect of two-body
hard-disk collisions on a probabilistic level. To evaluate this integral, we
have to find an expression for the joint PDF $P_{2}(\chi_{1},\chi_{2},t)$ in
terms of the one-body PDF $p(\chi_{1},t)$, similar to the classical closure
problem in kinetic theory, known as the BBGKY hierarchy. This relation will be
specifically relevant in regions where the particles are close, and where
evaluating the collision integral will contribute to the effective
description.

####  2.2.2 Matched Asymptotic Expansion

We aim at an approximation for $P_{2}$ via the method of matched asymptotic
expansion [50, 48, 49, 59, 67]. We can suppose that when the particles are far
apart, they are independent, given the hard-disk nature of the interaction. In
this so-called outer region, we define the outer joint PDF as
$P^{\mathrm{out}}(\chi_{1},\chi_{2},t)=P_{2}(\chi_{1},\chi_{2},t)$ and due to
the independency argument we find that

$P^{\mathrm{out}}(\chi_{1},\chi_{2},t)=p(\chi_{1},t)\
p(\chi_{2},t)+\varepsilon_{d}P^{\mathrm{out}}_{(1)}(\chi_{1},\chi_{2},t)+\mathcal{O}(\varepsilon_{d}^{2}),$
(9)

where $P^{\mathrm{out}}_{(1)}$ is a function denoting the corrections at first
order $\mathcal{O}(\varepsilon_{d})$ to the independence argument [68].

(a)

(b)

Figure 2: The description of the problem changes from a particle-focused to a
fixed-particle perspective. The second particle is measured in relative units.
Technically we change from a description in (a) of
$(\mathbf{x}_{1},\theta_{1})$ for particle one and
$(\mathbf{x}_{2},\theta_{2})$ for particle two to a description in (b) of
$(\mathbf{x}_{1},\theta_{1})$ for the effectively fixed particle one and
$(\mathbf{x}_{1}+\varepsilon_{d}\mathbf{x},\theta_{1}+\theta)$ relative for
particle two. One advantage of this coordinate change is that the collision
surface in the new coordinates is given by the condition $|\mathbf{x}|=1$.

In contrast, when the two particles are close, the particles are correlated
due to interactions and we perform a variable change in this so-called inner
region. We fix particle one and measure the distance to the particle with
respect to $\varepsilon_{d}$, see Fig. 2. The coordinate change reads as
$(\chi_{2},\mathbf{x}_{2},\theta_{2})\mapsto(\chi_{1},\mathbf{x}_{1}+\varepsilon_{d}\mathbf{x},\theta_{1}+\theta)$.
The inner PDF is defined as
$P^{\mathrm{inn}}(\chi_{1},\mathbf{x},\theta,t)=P_{2}(\chi_{1},\chi_{2},t)$.
Rewriting the two-particle problem into inner coordinates, we find

$\displaystyle\varepsilon_{d}^{2}\frac{\partial}{\partial
t}P^{\mathrm{inn}}=2D_{T}\
\nabla_{\mathbf{x}}^{2}P^{\mathrm{inn}}+\varepsilon_{d}\nabla_{\mathbf{x}}\cdot\bigg{[}v\left(\hat{\mathbf{e}}(\theta_{1})-\hat{\mathbf{e}}(\theta_{1}+\theta)\right)-2D_{T}\
\nabla_{\mathbf{x}_{1}}\bigg{]}P^{\mathrm{inn}}$
$\displaystyle\quad+\varepsilon_{d}^{2}\nabla_{\mathbf{x}}\cdot\left[D_{T}\
\nabla_{\mathbf{x}}-v\ \hat{\mathbf{e}}(\theta_{1})\right]P^{\mathrm{inn}}$
$\displaystyle\quad+\varepsilon_{d}^{2}\left[D_{R}\left(\frac{\partial}{\partial\theta_{1}}-\frac{\partial}{\partial\theta}\right)^{2}-\omega\frac{\partial}{\partial\theta_{1}}\right]P^{\mathrm{inn}}.$
(10a) The no-flux boundary condition translates into inner coordinates as
$2D_{T}\
\mathbf{x}\cdot\nabla_{\mathbf{x}}P^{\mathrm{inn}}=\varepsilon_{d}\mathbf{x}\cdot\left[D_{T}\
\nabla_{\mathbf{x}_{1}}-v\left(\hat{\mathbf{e}}(\theta_{1})-\hat{\mathbf{e}}(\theta_{1}+\theta)\right)\right]P^{\mathrm{inn}},$
(10b) which is valid on the collision surface $\mathcal{S}_{\mathrm{coll}}$,
given in inner coordinates by
$\\{\mathbf{x}\in\mathbb{R}^{2};|\mathbf{x}|=1\\}$. When formulating the
problem in inner coordinates, we have an additional boundary condition, which
is replacing the otherwise implicit natural boundary condition on $P_{2}$,
i.e. $\lim_{\mathbf{x}_{i}\to\infty}P_{2}=0,i\in\\{1,2\\}$. In the framework
of matched asymptotic expansion, this condition implies that the inner PDF has
to match the outer PDF as $|\mathbf{x}|\to\infty$. An expansion of the outer
solution in inner coordinates gives $P^{\mathrm{out}}(\chi_{1},\chi_{2},t)\sim
p\ p^{+}+\varepsilon_{d}\left(p\
\mathbf{x}\cdot\nabla_{\mathbf{x}_{1}}p^{+}+P^{\mathrm{out},+}_{(1)}\right),$
(10c)

where we have introduced the shorthand notations $p=p(\chi_{1},t),\
p^{+}=p(\mathbf{x}_{1},\theta_{1}+\theta,t)$ and
$P^{\mathrm{out},+}_{(1)}=P^{\mathrm{out}}_{(1)}(\chi_{1},\mathbf{x}_{1},\theta_{1}+\theta,t)$.
Expanding the inner solution in powers of $\varepsilon_{d}$,
$P^{\mathrm{inn}}\sim
P^{\mathrm{inn}}_{(0)}+\varepsilon_{d}P^{\mathrm{inn}}_{(1)}+\mathcal{O}(\varepsilon_{d}^{2})$,
the zeroth-order inner problem becomes

$\displaystyle 0=2D_{T}\ \nabla_{\mathbf{x}}^{2}P^{\mathrm{inn}}_{(0)},$ (11a)
$\displaystyle 0=2D_{T}\
\mathbf{x}\cdot\nabla_{\mathbf{x}}P^{\mathrm{inn}}_{(0)}$
$\displaystyle\qquad\mathrm{on}\ |\mathbf{x}|=1,$ (11b) $\displaystyle
P^{\mathrm{inn}}_{(0)}\sim pp^{+}$ $\displaystyle\qquad\mathrm{as}\
|\mathbf{x}|\to\infty,$ (11c)

with the straight-forward solution

$P^{\mathrm{inn}}_{(0)}=pp^{+}.$ (12)

Note here, that $P^{\mathrm{inn}}_{(0)}$ is no function of the separation
coordinate $\mathbf{x}$.

Before we can write the first-order inner problem, we note from Eq. (10), that
at order $\mathcal{O}(\varepsilon_{d})$, $P^{\mathrm{inn}}_{(0)}$ being no
function of the separation variable $\mathbf{x}$ has the consequence of

$\displaystyle 0$ $\displaystyle=2D_{T}\
\nabla_{\mathbf{x}}^{2}P^{\mathrm{inn}}_{(1)}+\varepsilon_{d}\nabla_{\mathbf{x}}\cdot\bigg{[}v\left(\hat{\mathbf{e}}(\theta_{1})-\hat{\mathbf{e}}(\theta_{1}+\theta)\right)-2D_{T}\
\nabla_{\mathbf{x}_{1}}\bigg{]}P^{\mathrm{inn}}_{(0)}$ (13)
$\displaystyle=2D_{T}\ \nabla_{\mathbf{x}}^{2}P^{\mathrm{inn}}_{(1)}.$

Taking this into account, the first-order inner problem reads

$\displaystyle 0=2D_{T}\ \nabla_{\mathbf{x}}^{2}P^{\mathrm{inn}}_{(1)},$ (14a)
$\displaystyle\mathbf{x}\cdot\nabla_{\mathbf{x}}P^{\mathrm{inn}}_{(1)}=\mathbf{x}\cdot\mathbf{A}(\chi_{1},\theta,t)$
$\displaystyle\qquad\mathrm{on}\ |\mathbf{x}|=1,$ (14b) $\displaystyle
P^{\mathrm{inn}}_{(1)}\sim\mathbf{x}\cdot\mathbf{B}(\chi_{1},\theta,t)+P^{\mathrm{out},+}_{(1)}$
$\displaystyle\qquad\mathrm{as}\ |\mathbf{x}|\to\infty,$ (14c)

where

$\displaystyle\mathbf{A}(\chi_{1},\theta,t)$
$\displaystyle=\frac{1}{2D_{T}}\Big{(}D_{T}\nabla_{\mathbf{x}_{1}}(pp^{+})-v(\hat{\mathbf{e}}(\theta_{1})-\hat{\mathbf{e}}(\theta_{1}+\theta))pp^{+}\Big{)},$
(15) $\displaystyle\mathbf{B}(\chi_{1},\theta,t)$
$\displaystyle=p\nabla_{\mathbf{x}_{1}}p^{+}.$ (16)

The solution can be obtained straightforwardly and is given by [48, 49, 61,
65]

$P^{\mathrm{inn}}_{(1)}=a+P^{\mathrm{out},+}_{(1)}+\mathbf{x}\cdot\mathbf{B}-\frac{\mathbf{x}}{|\mathbf{x}|^{2}}\cdot\left(\mathbf{A}-\mathbf{B}\right),$
(17)

where $a$ is an arbitrary integration constant. According to the expansion-
ansatz $P^{\mathrm{inn}}\sim
P^{\mathrm{inn}}_{(0)}+\varepsilon_{d}P^{\mathrm{inn}}_{(1)}+\mathcal{O}(\varepsilon_{d}^{2})$
for the inner PDF, we have found that

$\displaystyle P^{\mathrm{inn}}=pp^{+}+\varepsilon_{d}\mathbf{x}\cdot
p\nabla_{\mathbf{x}_{1}}p^{+}-\varepsilon_{d}\frac{\mathbf{x}}{2|\mathbf{x}|^{2}}\cdot\Big{[}p^{+}\nabla_{\mathbf{x}_{1}}p-p\nabla_{\mathbf{x}_{1}}p^{+}$
$\displaystyle\quad-\frac{v}{D_{T}}\left(\hat{\mathbf{e}}(\theta_{1})-\hat{\mathbf{e}}(\theta_{1}+\theta)\right)pp^{+}\Big{]}+\varepsilon_{d}\big{(}a+P^{\mathrm{out},+}_{(1)}\big{)}+\mathcal{O}(\varepsilon_{d}^{2}).$
(18)

####  2.2.3 Evaluation of the Collision Integral

We want to use this approximate inner solution to evaluate the collision
integral. In inner coordinates, this is given by

$I(\chi_{1},t)=\varepsilon_{d}D_{T}\int_{0}^{2\pi}\mathrm{d}\theta\int_{|\mathbf{x}|=1}\mathrm{d}\mathrm{S}_{\mathbf{x}}\
\mathbf{x}\cdot\nabla_{\mathbf{x}_{1}}P^{\mathrm{inn}}(\chi_{1},\mathbf{x},\theta,t),$
(19)

where we have used that that $\mathbf{n}_{\mathbf{x}}=-\mathbf{x}$ on the
collision surface $\mathcal{S}_{\mathrm{coll}}$. Using the inner solution of
(2.2.2), the collision integral becomes

$\displaystyle
I(\chi_{1},t)=\varepsilon_{d}D_{T}\int_{0}^{2\pi}\mathrm{d}\theta\left[\nabla_{\mathbf{x}_{1}}(pp^{+})\right]_{\alpha}\
\int_{|\mathbf{x}|=1}\mathrm{d}\mathrm{S}_{\mathbf{x}}\ x_{\alpha}$
$\displaystyle\quad+\varepsilon_{d}^{2}D_{T}\int_{0}^{2\pi}\mathrm{d}\theta\bigg{[}\nabla_{\mathbf{x}_{1}}P^{\mathrm{out},+}_{(1)}\bigg{]}_{\alpha}\
\int_{|\mathbf{x}|=1}\mathrm{d}\mathrm{S}_{\mathbf{x}}\ x_{\alpha}$
$\displaystyle\quad+\varepsilon_{d}^{2}D_{T}\int_{0}^{2\pi}\mathrm{d}\theta\bigg{[}\nabla_{\mathbf{x}_{1}}(p\nabla_{\mathbf{x}_{1}}p^{+})_{\alpha}\bigg{]}_{\beta}\
\int_{|\mathbf{x}|=1}\mathrm{d}\mathrm{S}_{\mathbf{x}}\ x_{\alpha}x_{\beta}$
$\displaystyle\quad-\varepsilon_{d}^{2}\frac{D_{T}}{2}\int_{0}^{2\pi}\mathrm{d}\theta\bigg{[}\nabla_{\mathbf{x}_{1}}(p^{+}\nabla_{\mathbf{x}_{1}}p-p\nabla_{\mathbf{x}_{1}}p^{+})_{\alpha}\bigg{]}_{\beta}\
\int_{|\mathbf{x}|=1}\mathrm{d}\mathrm{S}_{\mathbf{x}}\
\frac{x_{\alpha}x_{\beta}}{|\mathbf{x}|^{2}}$
$\displaystyle\quad-\varepsilon_{d}^{2}\frac{v}{2}\int_{0}^{2\pi}\mathrm{d}\theta\bigg{[}\hat{\mathbf{e}}(\theta_{1})-\hat{\mathbf{e}}(\theta_{1}+\theta)\bigg{]}_{\alpha}\bigg{[}\nabla_{\mathbf{x}_{1}}(pp^{+})\bigg{]}_{\beta}\
\int_{|\mathbf{x}|=1}\mathrm{d}\mathrm{S}_{\mathbf{x}}\
\frac{x_{\alpha}x_{\beta}}{|\mathbf{x}|^{2}},$ (20)

where we have used the Einstein convention, i.e. that the sum over double
indices is implicit:
$x_{\alpha}x_{\alpha}=\sum_{\alpha=1}^{2}x_{\alpha}x_{\alpha}$. There are two
types of integrals appearing in (2.2.3), namely an integral of an outer unit
normal vector over the whole unit sphere;
$\int_{|\mathbf{x}|=1}\mathrm{d}\mathrm{S}_{\mathbf{x}}\ x_{\alpha}=0$, by
geometrical insight and

$\int_{|\mathbf{x}|=1}\mathrm{d}\mathrm{S}_{\mathbf{x}}\
\frac{x_{\alpha}x_{\beta}}{|\mathbf{x}|^{2}}=\int_{|\mathbf{x}|=1}\mathrm{d}\mathrm{S}_{\mathbf{x}}\
x_{\alpha}x_{\beta}=\pi\ \delta_{\alpha\beta}$ (21)

in two dimensions.

As $\theta_{1}$ is independent of $\theta$ and kept constant for the variation
of $\theta$, we can define a new variable $\tilde{\theta}=\theta_{1}+\theta$
and use it to integrate out the $\theta$-dependence in (2.2.3). Therefore, we
define the mean particle PDF

$\varrho(\mathbf{x},t)=\int_{0}^{2\pi}\mathrm{d}\tilde{\theta}\
p(\mathbf{x},\tilde{\theta},t),$ (22)

and the polarization

$\boldsymbol{\sigma}(\mathbf{x},t)=2\int_{0}^{2\pi}\mathrm{d}\tilde{\theta}\
p(\mathbf{x},\tilde{\theta},t)\ \hat{\mathbf{e}}(\tilde{\theta}),$ (23)

as the zeroth and first-order moment of the full one-body PDF, respectively.
Note the factor of 2 in the definition of polarization, which is necessary for
later consistency. Thus, the collision integral becomes

$I(\chi_{1},t)=\varepsilon_{d}^{2}\frac{\pi}{2}\nabla_{\mathbf{x}_{1}}\cdot\Big{[}3D_{T}\
p\nabla_{\mathbf{x}_{1}}\varrho-D_{T}\
\varrho\nabla_{\mathbf{x}_{1}}p+v\left(\hat{\mathbf{e}}(\theta_{1})\varrho-\frac{\boldsymbol{\sigma}}{2}\right)p\Big{]},$
(24)

where $\varrho=\varrho(\mathbf{x}_{1},t),\
\boldsymbol{\sigma}=\boldsymbol{\sigma}(\mathbf{x}_{1},t)$, and, as a reminder
$p=p(\mathbf{x}_{1},\theta_{1},t)$.

As expected, the dependence on the separation coordinates $\mathbf{x}$ and
$\theta$ vanished, since we have integrated out the effect of the second
particle on the first. Thus, we can back-transform into the original variables
and then drop the index in the notation
$(\mathbf{x}_{1},\theta_{1})\mapsto(\mathbf{x},\theta)$, similarly for the
operator $\nabla_{1}\mapsto\nabla$. We insert the evaluated collision integral
into Eq. (2.2.1), and find the effective time-evolution equation for the full
one-body PDF $p$. This equation is valid for two hard-interacting particles.
As introduced in the beginning, in the dilute system, we can assume that in a
system of in general $N$ particles, two-particle collisions dominate any
higher-order correlations. The tagged particle can have $(N-1)$inner regions
with each of the remaining particles under this assumption. It is thus
sufficient to multiply the collision integral by the factor of $(N-1)$ to
account for the interaction effect in the dilute one-body description.

####  2.2.4 Effective One-Body Equation

We introduce the dimensionless parameter $\phi=\sigma^{2}\pi(N-1)/4$, which
for large $N$ approximately equals the area fraction of the particles. We then
can write the obtained time-evolution equation for the full one-body PDF
$p=p(\mathbf{x},\theta,t)$ as

$\displaystyle\frac{\partial}{\partial t}p=-v\
\nabla\cdot\bigg{[}(1-2\phi\varrho)\
\hat{\mathbf{e}}+\phi\boldsymbol{\sigma}\bigg{]}p+\frac{\partial}{\partial\theta}\left[D_{R}\frac{\partial}{\partial\theta}-\omega\right]p$
$\displaystyle\quad+D_{T}\ \nabla\cdot\bigg{[}(1-2\phi\varrho)\nabla p+6\phi\
p\nabla\varrho\bigg{]},$ (25)

valid in the dilute limit, i.e. $\phi\ll 1$. Without the effect of chirality
($\omega=0$) this equation was recently derived in Ref. [50] for ABPs. As the
authors state there, this systematic derivation of the one-body description
results in additional cross-diffusion terms ($\propto\varrho\nabla p$,
$\propto p\nabla\varrho$) compared to more classical approaches in the
literature [62, 63]. Further, recent work shows that for ABPs, Eq.(2.2.4)
forms a well-posed problem where a stationary state exists [69].

In the context of phase-separating active matter, typically effective self-
propulsion velocities $v(\varrho)$ are introduced [16, 70], and constitute an
essential theoretical trick to achieve phase separation in purely repulsive
active systems [51]. To the lowest order in density, they are typically of the
form of $v(\varrho)=v_{0}(1-a\phi\varrho),\ v_{0}=\mathrm{const}$ which has
also been derived using linear response theory [71]. The reason is simple: in
regions where there are many particles, the effective swim speed must be
lower. It is interesting that we naturally obtain this form in the geometric
integration procedure for hard interactions, and we can constitute that $a=2$
for steric interactions.

###  2.3 Hierarchy of Hydrodynamic Equations

In this Section, we project the time-evolution equation for the full one body
PDF $p(\mathbf{x},\theta,t)$ on its angular modes, where the zeroth-order mode
is given by the mean-particle PDF $\varrho(\mathbf{x},t)$. This procedure
generates a hierarchy of coupled particle differential equations for the time
evolution of the modes. We analyse these equations to provide a footing to
systematically close the hierarchy in the next Section.

####  2.3.1 Expansion in Harmonic Modes

To overcome the angular dependence of the full one-body PDF, We expand the
full one-body PDF in eigenfunctions of the rotation operator
$\partial^{2}/\partial\theta^{2}$ in Eq. (2.2.4). This results in a Fourier
expansion, where modes of order $n\in\mathbb{N}_{0}$ have eigenvalue $-n^{2}$
in two dimensions. This expansion can be brought into the form

$\displaystyle
p(\mathbf{x},\theta,t)=\varrho(\mathbf{x},t)+\boldsymbol{\sigma}(\mathbf{x},t)\cdot\hat{\mathbf{e}}(\theta)+\mathbf{Q}(\mathbf{x},t):\hat{\mathbf{e}}(\theta)\otimes\hat{\mathbf{e}}(\theta)+\Upsilon(\theta),$
(26)

where
$\mathbf{Q}:\hat{\mathbf{e}}\otimes\hat{\mathbf{e}}=Q_{\alpha\beta}\hat{e}_{\alpha}\hat{e}_{\beta}$
denotes the full contraction with the outer product
$\hat{\mathbf{e}}\otimes\hat{\mathbf{e}}$, and $\Upsilon(\theta)$ refers to
higher order modes. Notably, higher order (outer) products of the self-
propulsion vector
$\hat{\mathbf{e}}(\theta)=\left(\cos(\theta),\sin(\theta)\right)^{\mathrm{T}}$
naturally appear as modes in this expansion. Therefore, the expansion is also
known as a Cartesian multipole expansion [72] and finds wide use in the theory
of liquid crystals [73]. Note that there exists an equivalent approach, other
works in active matter use an angular multipole expansion at this stage [51,
74, 75].

The coefficients of the expansion are given by the mean particle PDF
$\varrho(\mathbf{x},t)$ (zeroth-order mode), as defined above in Eq. (22), the
mean polarization $\boldsymbol{\sigma}(\mathbf{x},t)$ (first-order mode), as
defined above in Eq. (23), and the mean nematic order tensor (second-order
mode), which is defined as

$\mathbf{Q}(\mathbf{x},t)=4\int_{0}^{2\pi}\mathrm{d}\theta\
p(\mathbf{x},\theta,t)\
\left(\hat{\mathbf{e}}(\theta)\otimes\hat{\mathbf{e}}(\theta)-\frac{1}{2}\mathbf{1}\right),$
(27)

where $\mathbf{1}$ is the identity tensor. Note that due to the similarity
with hydrodynamic theories, the coefficients are sometimes referred to as
hydrodynamic coefficients, the hierarchy created by them as a hydrodynamic
hierarchy [10, 51].

The inner product on the unit-sphere is defined in the standard way as

$\left\langle
f(\theta),g(\theta)\right\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}\mathrm{d}\theta\
f(\theta)g(\theta),$ (28)

for two functions $f$ and $g$. The modes of the Cartesian multipole expansion
form an orthogonal basis with respect to this scalar product
$\\{\hat{\mathbf{e}}^{n};n\in\mathbb{N}_{0}\\}$. Therefore we can project the
full one-body PDF $p(\mathbf{x},\theta,t)$ on the modes and obtain the
hydrodynamic coefficients

$\displaystyle\left\langle
p(\mathbf{x},\theta,t),1\right\rangle=\varrho(\mathbf{x},t),$ (29a)
$\displaystyle\left\langle
p(\mathbf{x},\theta,t),\hat{\mathbf{e}}(\theta)\right\rangle=\frac{1}{2}\boldsymbol{\sigma}(\mathbf{x},t),$
(29b) $\displaystyle\left\langle
p(\mathbf{x},\theta,t),\hat{\mathbf{e}}(\theta)\otimes\hat{\mathbf{e}}(\theta)\right\rangle=\frac{1}{4}\
\mathbf{Q}(\mathbf{x},t).$ (29c)

####  2.3.2 Time-Evolution of Modes

Using the orthogonality of modes, we can then project the time-evolution
equation for the full one-body PDF, Eq. (2.2.4), on each mode to obtain a
time-evolution equation for the corresponding hydrodynamic coefficient.
Projecting Eq. (2.2.4) on the zeroth-order mode, we obtain a time-evolution
equation of the mean particle PDF $\varrho$, which is given by

$\displaystyle\frac{\partial}{\partial t}\varrho(\mathbf{x},t)$
$\displaystyle=D_{T}\nabla_{\alpha}\bigg{[}\left(1+4\phi\
\varrho(\mathbf{x},t)\right)\nabla_{\alpha}\varrho(\mathbf{x},t)\bigg{]}-\frac{v}{2}\nabla_{\alpha}\sigma_{\alpha}(\mathbf{x},t).$
(30)

The mean particle PDF therefore obeys a continuity equation. This is expected
on physical grounds as $\varrho$ is a conserved quantity. Specifically, this
implies that $\varrho$ is a slow variable, i.e. a density perturbation of
scale $\lambda$ relaxes on a time scale which diverges as $\lambda\to\infty$.
This observation will be confronted with the time-evolution equation for the
higher modes soon. We further observe that the equation for the mean particle
PDF needs an explicit input from the polarization $\boldsymbol{\sigma}$. This
coupling is induced by the activity $v$ in the model.

Projecting Eq. (2.2.4) on the first order mode, we obtain a time-evolution
equation for the mean polarization $\boldsymbol{\sigma}$, which is given in
component-form by

$\displaystyle\frac{\partial}{\partial
t}\sigma_{\alpha}(\mathbf{x},t)=D_{T}\nabla_{\beta}\bigg{[}\left(1-2\phi\
\varrho(\mathbf{x},t)\right)\nabla_{\beta}\sigma_{\alpha}(\mathbf{x},t)+6\phi\
\sigma_{\beta}(\mathbf{x},t)\nabla_{\alpha}\varrho(\mathbf{x},t)\bigg{]}$
$\displaystyle\quad-v\nabla_{\beta}\left[\left(1-2\phi\
\varrho(\mathbf{x},t)\right)\left(\frac{Q_{\beta\alpha}(\mathbf{x},t)}{2}+\varrho(\mathbf{x},t)\
\delta_{\beta\alpha}\right)+\phi\
\sigma_{\beta}(\mathbf{x},t)\sigma_{\alpha}(\mathbf{x},t)\right]$
$\displaystyle\quad-D_{R}\ \Gamma_{\alpha\beta}\sigma_{\beta}(\mathbf{x},t).$
(31)

Again, we observe that the time-evolution equation for the mean polarization
needs both, input from the mean-particle density $\varrho$ as well as from the
nematic order tensor $\mathbf{Q}$. This conforms with the observation, that
such hierarchies of hydrodynamic equations need information from both
neighbouring modes. Further, we observe that in contrast to the time evolution
of the mean-particle density the structure of this equation is different.
While $\varrho$ obeys a continuity equation, Eq. (2.3.2) has a sink term:
$-D_{R}\ \Gamma_{\alpha\beta}\sigma_{\beta}(\mathbf{x},t)$. The polarization,
and as we will see all higher modes, therefore, are not conserved quantities,
and their dynamics are governed by the associated time-scale of the sink term.
For the mean polarization, this is given by $\tau_{1}=1/D_{R}$ [76, 77].

This fundamental structural difference of the time-evolution equations arises
in the $\theta$-term of the parental equation (2.2.4) of the hierarchy. While
for the polarization we find that $\hat{\mathbf{e}}$ is the $n=1$st-order
eigenfunction of the rotation operator with eigenvalue $-n^{2}=-1$, for the
zeroth-order mode ($\varrho$), the eigenvalue is zero. For the chirality-
induced term of Eq. (2.2.4), we observe a similar phenomenon. In the
projection procedure, we find that

$\displaystyle\left\langle\hat{e}_{\alpha},\omega\frac{\partial}{\partial\theta}p\right\rangle=\frac{\omega}{2\pi}\int_{0}^{2\pi}\mathrm{d}\theta\
\hat{e}_{\alpha}(\theta)\
\frac{\partial}{\partial\theta}p(\mathbf{x},\theta,t)$ (32a)
$\displaystyle\quad=\frac{\omega}{2\pi}\hat{e}_{\alpha}\
p(\mathbf{x},\theta,t)\Big{|}_{0}^{2\pi}-\frac{\omega}{2\pi}\int_{0}^{2\pi}\mathrm{d}\theta\
p(\mathbf{x},\theta,t)\
\frac{\partial}{\partial\theta}\hat{e}_{\alpha}(\theta)$ (32b)
$\displaystyle\quad=\varepsilon_{\alpha\beta}\
\frac{\omega}{2\pi}\int_{0}^{2\pi}\mathrm{d}\theta\ p(\mathbf{x},\theta,t)\
\hat{e}_{\beta}(\theta)$ (32c) $\displaystyle\quad=\frac{\omega}{2}\
\varepsilon_{\alpha\beta}\ \sigma_{\beta}(\mathbf{x},t),$ (32d)

where we used that $\partial/\partial\theta\
\hat{e}_{\alpha}(\theta)=-\varepsilon_{\alpha\beta}\ \hat{e}_{\beta}(\theta)$
and $\boldsymbol{\varepsilon}$ is the Levi-Chivita symbol in two dimensions.
The boundary term in (32b) vanishes due to periodicity. When we are dealing
with the zeroth-order mode in contrast, the projection $\left\langle
1,\omega\frac{\partial}{\partial\theta}p\right\rangle$, reduces to the
boundary term, and therefore is zero due to periodicity in $\theta$.

The joint effect of rotation operator and active chirality is the origin of
the sink-term $-D_{R}\ \Gamma_{\alpha\beta}\sigma_{\beta}(\mathbf{x},t)$,
where $\boldsymbol{\Gamma}=(\mathbf{1}+\kappa\boldsymbol{\varepsilon})$ and
$\kappa=\omega/D_{R}$ is the dimensionless parameter, accounting for the
effect of chirality. In a more general context, this parameter $\kappa$ is
also referred to as the odd parameter [78, 79, 45, 80], since under a reversal
of the direction of chirality, $\omega\to-\omega$, $\kappa$ changes sign and
therefore switches the off-diagonal elements of the tensor
$\boldsymbol{\Gamma}\to\boldsymbol{\Gamma}^{\mathrm{T}}$.

We find the time-evolution equation for the nematic order tensor, from
projecting Eq. (2.2.4) on the second order mode
$\hat{\mathbf{e}}\otimes\hat{\mathbf{e}}$, to be

$\displaystyle\quad\frac{\partial}{\partial t}Q_{\alpha\beta}(\mathbf{x},t)$
$\displaystyle=D_{T}\nabla_{\gamma}\bigg{[}(1-2\phi\
\varrho(\mathbf{x},t))\nabla_{\gamma}Q_{\alpha\beta}(\mathbf{x},t)+6\phi\
Q_{\alpha\beta}(\mathbf{x},t)\nabla_{\gamma}\varrho(\mathbf{x},t))\bigg{]}$
(33) $\displaystyle\quad-v\ \nabla_{\gamma}\bigg{[}(1-2\phi\
\varrho(\mathbf{x},t))A_{\alpha\beta\gamma\delta}\
\sigma_{\delta}(\mathbf{x},t)+\phi\
\sigma_{\gamma}(\mathbf{x},t)Q_{\alpha\beta}(\mathbf{x},t)+\mathcal{O}(\Upsilon)\bigg{]}$
$\displaystyle\quad-4D_{R}\
\tilde{\Gamma}_{\alpha\gamma}Q_{\gamma\beta}(\mathbf{x},t).$

Here
$A_{\alpha\beta\gamma\varepsilon}=\frac{1}{2}\left(\delta_{\alpha\gamma}\delta_{\beta\varepsilon}+\delta_{\alpha\varepsilon}\delta_{\beta\gamma}-\delta_{\alpha\beta}\delta_{\gamma\varepsilon}\right)$
and $\tilde{\boldsymbol{\Gamma}}=\left(\mathbf{1}+\kappa/2\
\boldsymbol{\varepsilon}\right)$ again accounts for the chirality. The
structural similarities of this equation to the polarization equation are that
$(i)$ Eq. (33) couples to neighbouring modes in the hierarchy and to the
zeroth order mode $\varrho$ and $(ii)$ the relaxation dynamics of the nematic
tensor is governed by the time-scale $\tau_{2}=1/(4D_{R})$ induced by the
sink-term of Eq. (33). Again we constitute that $\mathbf{Q}$, as all higher-
order modes, is not conserved. We observe that the relaxation time scale
originates in the eigenvalues of the rotation operator, i.e. the relaxation
time scale of the $n$th mode is given by $\tau_{n}=\tau/n^{2}=1/(n^{2}D_{R})$,
where we denote $\tau=\tau_{1}=1/D_{R}$ as the fundamental time-scale and
$n\geq 1$.

In principle, the time-evolution equations for all higher-order modes can be
found by the same projection procedure as presented before. This results in an
infinite system of coupled time-evolution equations. To get an analytical
meaningful result from this hierarchy, we have to close this hierarchy based
on physical arguments about negligible contributions from higher-order modes..

###  2.4 Closure of the Hierarchy

In this Section, we introduce the physical arguments to close the hierarchy by
neglecting higher-order modes. Therefore, this problem naturally amounts to a
perturbative analysis of the hierarchy. As the result of the minimal non-
trivial closure scheme, we find an approximated time evolution of the mean-
particle density, known as Model B.

####  2.4.1 Adiabatic Approximation

It would be a formidable task, to find a general solution for the $n$th order
hierarchical equation and plug this into the $(n-1)$th order equation. By such
a procedure, one aims to identify quasi-irrelevant modes for the effect on the
time evolution of $\varrho$. But to our best knowledge, such a procedure has
never been successfully applied before for a generic order. Instead, it is
convenient [43, 52, 55, 74, 81, 82, 83], to take advantage of the time-scale
separation in the system. The dynamics of polarization and all higher-order
modes are governed by their respective time-scale $\tau=\tau_{1}=1/D_{R}$,
induced by rotational diffusion. $\tau$ then reasonably can be assumed to be
much smaller than the relaxation time scale of the density, which can be
arbitrarily large. Therefore, formally, we will investigate the limit of
$\tau\to 0$, called adiabatic or quasi-stationarity approximation, since we
adiabatically enslave the behaviour of higher order modes to the mean-particle
density and assume an instantaneous response to changes.

In some approaches, the adiabatic approximation is also handled differently.
Especially in the context of phase separations, the nematic order tensor is
assumed to be an adiabatic variable, but the polarization is treated as a
dynamical variable. This appears in active Brownian systems [55, 84, 85], but
also in active chiral systems [81, 86]. As explicitly shown there, such an
ansatz can qualitatively improve the agreement of analytical prediction with
numerical data, but can hardly be treated fully analytically when aiming for
the effect of higher-order modes on the relaxation of the mean particle PDF.
In related hierarchy-closure problems, nevertheless, analytic treatment of
more than one dynamical variable and a shift of the adiabatic assumption to
second order has recently been applied successfully in the framework of
dynamical density functional theory [87, 88]. Here many-body correlation
functions form an analogous hierarchical problem, but the approach is
analytically rather involved and hardly can be applied to our situation. We,
therefore, restrict our analysis to the basic situation and assume all higher-
order modes to be adiabatic, i.e. respond instantaneously to changes.

We will demonstrate the adiabatic approximation for the polarization equation,
but the same arguments hold true for all higher-order modes. Pointing out the
essentials, Eq. (2.3.2) can be written as

$\frac{\partial}{\partial
t}\sigma_{\alpha}+\frac{1}{\tau}\Gamma_{\alpha\beta}\cdot\sigma_{\beta}=f_{\alpha}(\varrho,\boldsymbol{\sigma},\mathbf{Q}),$
(34)

where $\mathbf{f}$ denotes the leftover gradient-structure terms of Eq.
(2.3.2). We can solve this equation formally by

$\displaystyle\sigma_{\alpha}(\mathbf{x},t)=\mathrm{e}^{-\Gamma_{\alpha\beta}t/\tau}\sigma_{\beta}(\mathbf{x},0)+\int_{0}^{t}\mathrm{d}t^{\prime}\
\mathrm{e}^{-\Gamma_{\alpha\beta}|t-t^{\prime}|/\tau}f_{\beta}(\varrho,\boldsymbol{\sigma},\mathbf{Q}),$
(35)

where $\varrho,\boldsymbol{\sigma}$ and $\mathbf{Q}$ have to be evaluated at
$t^{\prime}$ inside the integral. The integrating factor
$\mathrm{e}^{-\Gamma_{\alpha\beta}t/\tau}$ is defined as the usual matrix-
exponential and can be reformulated as

$\displaystyle\mathrm{e}^{-\Gamma_{\alpha\beta}t/\tau}$
$\displaystyle=\mathrm{e}^{-t/\tau}\ \left(\cos\left(\kappa\
\frac{t}{\tau}\right)\delta_{\alpha\beta}-\sin\left(\kappa\
\frac{t}{\tau}\right)\varepsilon_{\alpha\beta}\right),$ (36)

using the anti-symmetry property of $\boldsymbol{\Gamma}$. The active
chirality, therefore, results in oscillations, which decay exponentially on
the typical time-scale $\tau$, remember that $\kappa\propto\omega$. The
exponential factor further allows us to write $\int_{0}^{\infty}\mathrm{d}t\
\mathrm{e}^{-\Gamma_{\alpha\beta}t/\tau}=\tau\Gamma_{\alpha\beta}^{-1}$, which
together with $\lim_{\tau\to
0}\mathrm{e}^{-\Gamma_{\alpha\beta}|t|/\tau}=0_{\alpha\beta}$, for $|t|>0$,
results in $\lim_{\tau\to 0}\Gamma_{\alpha\gamma}\
\mathrm{e}^{-\Gamma_{\gamma\beta}|t|/\tau}=2\tau\
\delta_{\alpha\beta}\delta(t)$. Here the factor $2$ originates from the
integrals taken to be for positive times only. The formal solution of Eq. (35)
evaluated in the limit of $\tau\to 0$ thus reads

$\lim_{\tau\to 0}\sigma_{\alpha}(\mathbf{x},t)=\tau\Gamma^{-1}_{\alpha\beta}\
f_{\beta}(\varrho,\boldsymbol{\sigma},\mathbf{Q}),$ (37)

where now $\varrho,\boldsymbol{\sigma}$ and $\mathbf{Q}$ are functions of $t$.
The corresponding adiabatic approximation for the nematic order tensor
similarly reads $\lim_{\tau\to
0}Q_{\alpha\beta}(\mathbf{x},t)=4\tau\tilde{\Gamma}^{-1}_{\alpha\gamma}\
g_{\gamma}(\varrho,\boldsymbol{\sigma},\mathbf{Q},\Upsilon)$, where
$\mathbf{g}$ accounts for the leftover gradient-structure terms of Eq. (33)
and the modes again are evaluated at $t$.

####  2.4.2 Closure of the Hierarchy via Perturbation Approach

The adiabatic approximation, as such, does not suffice to close the hierarchy.
The coupling to higher order modes in the hierarchy for each mode takes place
in the activity-induced term $-v\nabla\cdot\left[(1-2\phi\
\varrho)\hat{\mathbf{e}}p+2\phi\ \boldsymbol{\sigma}p\right]$ from the
parental equation (2.2.4). This coupling already takes place in interaction-
free considerations ($\phi=0$), as it specifically arises due to the
appearance of the self-propulsion vector $\hat{\mathbf{e}}$. To effectively
close the hierarchy, we, therefore, have to argue for the negligibility of
higher-order modes compared to lower ones. This consequently turns the closure
of the hierarchy into a perturbation problem.

We undimensionalize time by the natural time scale of the system $\tau$ and
space by the mean-particle distance $l_{\mathrm{dist}}=L\sqrt{\phi}$. The two
natural physical length scales in the system are once the persistence length
$l_{\mathrm{pers}}=v/D_{R}$, induced by activity, and on the other hand, the
diffusion length scale $l_{\mathrm{diff}}=\sqrt{D_{T}/D_{R}}$, induced by
thermal equilibrium fluctuations [50]. We denote the parameters, which arise
when comparing the physical length scales with the undimensionalization length
scale $l_{\mathrm{dist}}$ as $\varepsilon_{p}$ and $\varepsilon_{D}$ and
observe the following relation between them:

$\displaystyle\varepsilon_{p}=\frac{l_{\mathrm{pers}}}{l_{\mathrm{dist}}},\quad\varepsilon_{D}=\frac{l_{\mathrm{diff}}}{l_{\mathrm{dist}}},\quad\mathrm{and}\quad\frac{\varepsilon_{p}}{\varepsilon_{D}}=\frac{l_{\mathrm{pers}}}{l_{\mathrm{diff}}}=\frac{v}{\sqrt{D_{T}D_{R}}}=\mathrm{Pe}.$
(38)

Here Pe stands for the Péclet number, which measures the relation of activity-
induced self-propulsion versus thermally induced displacement.

Here we take $\varepsilon_{p}$ and $\varepsilon_{D}$ to be two independent
perturbation parameters. We show that depending on their relation to the
Péclet number, different field theoretic descriptions can be obtained. The
adiabatic assumption thereby justifies independent small active and passive
length scales, since both $l_{\mathrm{pers}}\propto\tau$ and
$l_{\mathrm{diff}}\propto\tau$. This treatment is in contrast to previous
work, where this subtle point of taking both parameters small and not only
their ratio (the Péclet number) was often left implicit or was overlooked.
This work now will highlight the importance of arriving at field-theoretical
descriptions.

$\varepsilon_{p}$ and $\varepsilon_{D}$ compare the active and passive
physical length scales in the system to the undimensionalization length scale,
which we choose as the mean-particle distance. This was a careful choice,
since the other natural length scales in the system, the particle-diameter $d$
(which also equals the interaction length scale for hard systems) and the
typical box size $L$ were too small and too big, respectively, and already
form the small parameter $\phi\propto(d/L)^{2}$. With the assumption of a
dilute system, the mean-particle distance appears as the correct length scale
interpolating between a too-narrow or too-coarse-grained view of the dynamics.

The aforementioned analysis reveals that the coupling to higher-order modes
takes place in the activity-induced term at each order. To close the
hierarchy, we therefore need to decide up to which order we consider the
parameter $\varepsilon_{p}$. For this work, and referring to what is typical
in the literature [43, 52, 55, 74, 81, 82, 83], we truncate the hierarchy at
order $\mathcal{O}(\varepsilon_{p}^{3})$. We thus ignore contributions from
higher modes as $\Upsilon$ in the nematic equation. Together with the
adiabatic assumption, Eq. (33) for the nematic tensor thus becomes

$\displaystyle
Q_{\alpha\beta}(\mathbf{x},t)=\frac{\varepsilon_{D}}{4}\tilde{\Gamma}^{-1}_{\alpha\delta}\nabla_{\gamma}\bigg{[}(1-2\phi\
\varrho(\mathbf{x},t))\nabla_{\gamma}Q_{\delta\beta}(\mathbf{x},t)$
$\displaystyle\quad+6\phi\
Q_{\delta\beta}(\mathbf{x},t)\nabla_{\gamma}\varrho(\mathbf{x},t))\bigg{]}.$
(39)

Eq. (2.4.2) constitutes a fixed-point problem for $\mathbf{Q}$. We observe
that Eq. (2.4.2) has no sink-term and therefore has a definite, perturbation-
free solution given by $\mathbf{Q}=0$. This is a generic observation: closing
the hierarchy at order $\mathcal{O}(\varepsilon_{p}^{n})$ results in the
fixed-point problem for the $n-1$th order mode to only have the trivial
solution. A sink-term, and therefore a possibility to solve for non-trivial
contributions from $\mathbf{Q}$ would then result in comparing the terms to
terms of order $\mathcal{O}(\varepsilon_{D})$, and therefore inherently needs
$\varepsilon_{d}$ as a second perturbation parameter. At each order, one then
has to clarify the physical validity of the resulting theory in terms of the
relation of $\varepsilon_{D}$ to $\varepsilon_{p}$, i.e. one clarifies the
regime of the Péclet number. Note that the only sink-term in (the adiabatic
version of) Eq. (33) is $\propto vA_{\alpha\beta\gamma\delta}\
\sigma_{\delta}(\mathbf{x},t)$, but prohibited at this order due to neglecting
higher-order active terms.

####  2.4.3 Model B

In the following, we investigate the resulting time-evolution equations of the
density for different orders of truncation in the polarization equation. We
start with the first non-trivial case by allowing for terms of order
$\mathcal{O}(\varepsilon_{p}^{1})$ for the polarization. Here, the adiabatic
polarization reads

$\displaystyle\sigma_{\alpha}=-\varepsilon_{p}\
\Gamma^{-1}_{\alpha\beta}\bigg{[}\left(1-4\phi\
\varrho\right)\nabla_{\beta}\varrho\bigg{]}.$ (40)

Note that, since the diffusive parameter $\varepsilon_{D}$ originates from a
Laplace operator, it only appears at even powers. Thus, there is no term at
order $\mathcal{O}(\varepsilon_{D}^{1})$ that could be included at this
closure of the polarization. The mean-particle density from Eq. (30) becomes
at this closure

$\displaystyle\frac{\partial}{\partial
t}\varrho(\mathbf{x},t)=\varepsilon_{D}^{2}\nabla_{\alpha}\bigg{[}\left(1+4\phi\
\varrho(\mathbf{x},t)\right)\nabla_{\alpha}\varrho(\mathbf{x},t)\bigg{]}$
$\displaystyle\quad+\frac{\varepsilon_{p}^{2}}{2}\frac{1}{1+\kappa^{2}}\nabla_{\alpha}\bigg{[}\left(1-4\phi\
\varrho(\mathbf{x},t)\right)\nabla_{\alpha}\varrho(\mathbf{x},t)\bigg{]}.$
(41)

As apparent, the chosen closure scheme results in a time-evolution equation of
the mean-particle density at order $\mathcal{O}(\varepsilon_{p}^{2})$ and
$\mathcal{O}(\varepsilon_{D}^{2})$. Now, by comparing the terms under
consideration with the truncated terms, we can learn about the regime of
validity of Eq. (2.4.3). This gives us that $(i)$
$\varepsilon_{p}^{2}\gg\varepsilon_{p}^{3}$, which is consistent with our
perturbative assumption of $\varepsilon_{p}\ll 1$, but we also find that
$(ii)$ $\varepsilon_{D}^{2}\gg\varepsilon_{p}^{3}$, which tells us that Eq.
(2.4.3) is valid in the regime of $\mathrm{Pe}\ll 1/\varepsilon_{p}^{1/2}$.
Together $(i)$ and $(ii)$ do not form a precise upper bound to the Péclet
number, and therefore an analysis of the microscopic parameters is essential
when using this field-theoretical description.

Eq. (2.4.3) is written in undimensionalized form. Thus we can reintroduce the
units of space and time, i.e. $\tau$ for the time and $l_{\mathrm{dist}}$ for
the space, compare also relation (38). The time-evolution equation for the
mean density becomes

$\displaystyle\frac{\partial}{\partial
t}\varrho(\mathbf{x},t)=\nabla\cdot\bigg{[}\left(D_{T}^{\mathrm{eff}}(\varrho)+D_{A}^{\mathrm{eff}}(\varrho)\right)\nabla\varrho(\mathbf{x},t)\bigg{]}.$
(42)

The time evolution of $\varrho$ at this order thus follows the well-known
Model B dynamics for a conserved, scalar order parameter [40].
$D_{T}^{\mathrm{eff}}+D_{A}^{\mathrm{eff}}$ form the interaction-corrected
effective diffusion coefficient due to thermal and active motion,
respectively, and are given by

$\displaystyle D_{T}^{\mathrm{eff}}(\varrho)$
$\displaystyle=D_{T}\left(1+4\phi\ \varrho(\mathbf{x},t)\right),$ (43a)
$\displaystyle D_{A}^{\mathrm{eff}}(\varrho)$
$\displaystyle=D_{A}\left(1-4\phi\ \varrho(\mathbf{x},t)\right).$ (43b)

Here $D_{T}$ stands for the thermal (equilibrium) diffusion coefficient, as
introduced in the microscopic Langevin description (1a).
$D_{A}=v^{2}/(2D_{R}(1+\kappa^{2}))$ is the active diffusion coefficient,
which describes the ballistic motion of a purely active particle and relates
it to the randomization of the self-propulsion vector due to rotational
diffusion. Note that for active chirality, in accordance with the observations
in Ref. [89], the typical active diffusing coefficient in two-dimensions
$D_{A}(\kappa=0)=v^{2}/2D_{R}$ gets rescaled by chirality and can also be
written as $D_{A}=v^{2}D_{R}/(2(\omega^{2}+D_{R}^{2}))$. Remember that
$\kappa=\omega/D_{R}$.

The relations (43a) and (43b) state that hard-core interactions on one hand
enhance the thermal diffusion and on the other hand reduce the active
diffusion. For the passive motion, this is in accordance with the observation
that the collective diffusion, as what $D_{T}^{\mathrm{eff}}(\varrho)$ also
can be interpreted, gets enhanced by steric interactions [61, 65, 90]. The
collective diffusion is defined as the simultaneous motion of many Brownian
particles [90], which at first sight appears as a contradiction to our one-
particle model. But Eq. (42) can also be read as a time-evolution of the one-
body probability density field, and the derived (thermal) diffusion
coefficient as the associated interaction-corrected diffusion coefficient, the
collective diffusion coefficient. For the active motion, similarly the
interaction-reduction of the associated diffusion coefficient
$D_{A}^{\mathrm{eff}}$ is no surprise. It rather can be regarded as an
analytic necessity for the strong theory-, simulation-, and experimental-
approved existence of motility-induced phase separations [17, 18, 91, 92, 93],
i.e. the phenomenon that purely repulsive active systems can phase-separate as
a function of particle density. This phenomenon is only possible, if the
associated diffusion process becomes unstable, i.e. the effective diffusion
coefficient can formally turn negative.

###  2.5 Active Model B +

In this Section, we go one step beyond the simplest non-trivial closure
approximation by considering mixed terms of the perturbation parameters. This
amounts to a much richer field-theoretical description but for the price of a
much narrower regime of validity. We find that the dynamics of the mean-
particle PDF follow the recently introduced AMB+, to which we, therefore have
first-principles access for the parameters. Surprisingly here we find that the
characteristic AMB+ parameters all change sign as a function of chirality.

####  2.5.1 Effect of Mixed Perturbation Parameters

In this work, we treat both, the activity as well as the thermal-diffusion
induced length scales as small compared to the inter-particle distance. This
is justified by the assumption of a dilute system. We are thus formally
dealing with a two-parameter perturbation theory. It is therefore natural,
that besides arguing for the smallness of each of the parameters
$\varepsilon_{D}\ll 1$ and $\varepsilon_{p}\ll 1$, we also have to specify
their relative size. This is only possible by making a third assumption about
the Péclet number, the natural scale relating to the active and passive motion
of a tracer.

We already observed from Eq. (40) that the diffusion length-scale parameter
$\varepsilon_{D}$ only appears in even powers (due to its origin in the second
order Laplace operator). Therefore, we can go a step further in the truncation
scheme, by considering the polarization at order
$\mathcal{O}(\varepsilon_{p}^{1})$ and $\mathcal{O}(\varepsilon_{D}^{2})$. The
adiabatic polarization can self-consistently be obtained from Eq.(2.3.2) at
the desired order to be

$\displaystyle\sigma_{\alpha}=-\varepsilon_{p}\
\Gamma^{-1}_{\alpha\beta}\bigg{[}\left(1-4\phi\
\varrho\right)\nabla_{\beta}\varrho\bigg{]}$
$\displaystyle\quad-\varepsilon_{p}\varepsilon_{D}^{2}\
\left(\Gamma^{-1}\right)^{2}_{\alpha\beta}\bigg{[}\nabla_{\gamma}^{2}-2\phi\left(3\varrho\nabla_{\gamma}^{2}-(\nabla_{\gamma}^{2}\varrho)+2(\nabla_{\gamma}\varrho)\nabla_{\gamma}\right)\bigg{]}\nabla_{\beta}\varrho.$
(44)

We observe here that the higher-order gradient terms of the mean-particle
density arise together with a matrix product of the chiral matrix
$\boldsymbol{\Gamma}$. For an arbitrary vector $\mathbf{a}$, the contraction
of the relevant matrices contributes

$\displaystyle a_{\alpha}\ \Gamma^{-1}_{\alpha\beta}\
a_{\beta}=\frac{1}{1+\kappa^{2}}\ a_{\alpha}a_{\alpha},$ (45a) $\displaystyle
a_{\alpha}\ \left(\Gamma^{-1}\right)^{2}_{\alpha\beta}\
a_{\beta}=\frac{1-\kappa^{2}}{(1+\kappa^{2})^{2}}\ a_{\alpha}a_{\alpha},$
(45b)

where due to the antisymmetric structure of
$\boldsymbol{\Gamma}^{-1}=(\mathbf{1}-\kappa\boldsymbol{\varepsilon})/(1+\kappa^{2})$
only the respective diagonal elements are relevant in the full contraction.

Inserting this expression for the polarization into the time-evolution of the
mean-particle density $\varrho=\varrho(\mathbf{x},t)$, we obtain

$\displaystyle\frac{\partial}{\partial
t}\varrho(\mathbf{x},t)=\varepsilon_{D}^{2}\nabla_{\alpha}\bigg{[}\left(1+4\phi\
\varrho\right)\nabla_{\alpha}\varrho\bigg{]}+\frac{\varepsilon_{p}^{2}}{2}\frac{1}{1+\kappa^{2}}\nabla_{\alpha}\bigg{[}\left(1-4\phi\
\varrho\right)\nabla_{\alpha}\varrho\bigg{]}$
$\displaystyle\quad+\frac{\varepsilon_{p}^{2}\varepsilon_{D}^{2}}{2}\frac{1-\kappa^{2}}{(1+\kappa^{2})^{2}}\nabla_{\alpha}\bigg{[}\nabla_{\gamma}^{2}-2\phi\left(3\varrho\nabla_{\gamma}^{2}-(\nabla_{\gamma}^{2}\varrho)+2(\nabla_{\gamma}\varrho)\nabla_{\gamma}\right)\bigg{]}\nabla_{\alpha}\varrho$
(46)

It is now obvious, that to obtain this field-theoretical description for the
density compared to Eq.(2.4.3), we further have included the mixed term of
order $\mathcal{O}(\varepsilon_{p}^{2}\varepsilon_{D}^{2})$, which is only
possible if this term is assumed to be much greater then the disregarded term
of order $\mathcal{O}(\varepsilon_{p}^{3})$. An easy algebraic analysis shows
that this is only possible when $\mathrm{Pe}\ll\varepsilon_{D}\ll 1$ due to
the perturbative closure scheme.

####  2.5.2 Active Model B +

If we reintroduce physical units for space and time, i.e. $\tau$ for the time
$\l_{\mathrm{dist}}$ for the space, compare also relation (38), we can arrange
Eq. (2.5.1) in the form of

$\displaystyle\frac{\partial}{\partial
t}\varrho(\mathbf{x},t)=a\nabla^{2}\varrho+b\nabla^{2}(\varrho^{2})-k_{0}\nabla^{4}\varrho-
k_{1}\left[\nabla^{2}(\nabla\varrho)^{2}+2\nabla^{2}(\varrho\nabla^{2}\varrho)\right]$
$\displaystyle\quad+\lambda\nabla^{2}\left(\nabla\varrho\right)^{2}-\xi\nabla\cdot(\nabla\varrho)(\nabla^{2}\varrho),$
(47)

where

$\displaystyle a=D_{T}+D_{A},$ (48a) $\displaystyle
b=2\phi\left(D_{T}-D_{A}\right),$ (48b) $\displaystyle\lambda=\phi D_{A}D_{T}\
\frac{1-\kappa^{2}}{1+\kappa^{2}},$ (48c) $\displaystyle\xi=-8\phi D_{A}D_{T}\
\frac{1-\kappa^{2}}{1+\kappa^{2}},$ (48d) $\displaystyle
k(\varrho)=D_{A}D_{T}\ \frac{1-\kappa^{2}}{1+\kappa^{2}}\left(-1+6\phi\
\varrho(\mathbf{x},t)\right),$ (48e)

and $k(\varrho)=k_{0}+2k_{1}\ \varrho(\mathbf{x},t)$. Eq. (2.5.2) is known as
the (deterministic) AMB+ [25, 26, 94, 95], and can be rearranged into the
following form

$\displaystyle\frac{\partial}{\partial t}\varrho(\mathbf{x},t)$
$\displaystyle=-\nabla\cdot\left[-\nabla\left(\frac{\delta\mathcal{F}}{\delta\varrho}+\lambda(\nabla\varrho)^{2}\right)+\xi(\nabla\varrho)(\nabla^{2}\varrho)\right],$
(49a) where
$\displaystyle\mathcal{F}[\varrho]=\int\mathrm{d}\mathbf{r}\left(f_{0}(\varrho)+\frac{k(\varrho)}{2}(\nabla\varrho)^{2}\right),$
(49b)

is the free-energy functional and $f_{0}(\varrho)=a/2\ \varrho^{2}+b/3\
\varrho^{3}$ is the bulk free-energy density. Note that typically
$\varrho^{\prime}=\varrho-\varrho_{\mathrm{MF}}$ is taken to be the order
parameter, where $\varrho_{\mathrm{MF}}$ is the mean-field critical value of
the density. If one does so, a term $\propto\varrho^{3}$ is forbidden by
symmetry in $f_{0}$, but we, instead, work with the mean-field density itself.
Eq. (49) was first written down based on phenomenologically accounting for
systems with broken detailed balance to lowest order terms in Ref. [25].

####  2.5.3 First-Principles Expressions for Field-Theoretical Parameters

The parameter $k(\varrho)$ in the free energy is known as the Cahn-Hilliard
parameter [96]. When first introduced, this parameter was the minimal attempt
in an equilibrium model to extend the bulk free energy $\mathcal{F}$ to
further include density-gradient-induced inhomogeneities into the description.
Via this very successful approach, field theories could nicely capture phase-
separation dynamics in equilibrium models [41, 97]. In the general
formulation, $k(\varrho)$ is density-dependent, and a Taylor expansion to
lowest order in the density gives $k(\varrho)=k_{0}+2k_{1}\varrho$. These
coefficients $k_{0}$ and $k_{1}$ are found in our model by comparing the
first-principles time-evolution equation (2.5.1) with the phenomenological
equation of the AMB+ (2.5.2). From an equilibrium perspective, it might appear
surprising that from Eq. (48e) we find that $k_{0}<0$ for a non-chiral system
($\kappa=0$) and thus also $k(\varrho)<0$ to lowest order. In an equilibrium
field theory, this would lead to an unbounded free energy and the non-physical
possibility of a system minimizing its free energy by creating more and more
interfaces due to phase separation. But for an active field theory such as the
AMB+, the time evolution is not solely governed by a free-energy structure.
Further terms $\propto\lambda,\xi$ balance the free-energy governed evolution,
and thus equilibrium intuition can fail. Note also that in our first-
principles derivation for active systems $k(\varrho)\propto D_{A}$ and
therefore vanishes as the model is approaching an equilibrium model ($D_{A}\to
0$). Finally, the observation of $k_{0}<0$ is consistent with other works on
the AMB+ [25, 43], but (active) chirality adds a new perspective on this,
since here, $k_{0}$ and $k_{1}$ change sign as $\kappa>1$.

Figure 3: For ACPs one can define the parameter $\kappa=\omega/D_{R}$, which
is a measure of active chirality $\omega$ versus rotational diffusion $D_{R}$
of the particle. In a related context, this parameter is known as the odd
parameter [65, 75, 78, 80]. The characteristic parameters of the AMB+,
$\lambda,-\xi$ and $k(\varrho)$ all scale
$\propto(1-\kappa^{2})/(1+\kappa^{2})$, see also Eqs. (48c) - (48e). Thus,
they are positive as long as $|\omega|<D_{R}$, i.e. the circular motion is
dominated by diffusion, and are negative for $|\omega|>D_{R}$, i.e. the motion
is dominated by activity. Note that the sign of $\omega$ accounts for the
clockwise or counterclockwise direction of the active chirality.

The observation, that the coefficients which are induced by activity and
interactions ($\lambda,\xi,k(\varrho)\propto\phi D_{A}$) can change their sign
as a function of chirality is a rather surprising one. In the hard-core
interacting scenario, the angular coordinate is not altered by the excluded-
volume interaction of the spatial coordinate, and therefore, chirality does
not affect the time-evolution of the full one-body PDF beyond the (trivial)
interaction-free terms, as can be seen from Eq. (2.2.4). Any interplay of
spatial and angular is only introduced in the projection on the hydrodynamic
modes. We can, nevertheless, qualitatively justify why the change in behaviour
happens at $\kappa=1$. Similar to the rotational diffusion coefficient, which
introduces a time-scale in the system $\tau=\tau_{\mathrm{diff}}=1/D_{R}$, the
active chirality introduces a time-scale as well
$\tau_{\mathrm{act}}=1/\omega$. These time scales represent the noise and the
deterministic circular contribution to the active motion of the particle,
respectively. Hence the parameter $\
\kappa=\omega/D_{R}=\tau_{\mathrm{diff}}/\tau_{\mathrm{act}}$ measures, which
contribution is dominating the motion of a particle, similar to what was
recently reported in Ref. [98]. $\kappa<1$ corresponds to a system, in which
the diffusive motion dominates the active particle, whereas $\kappa>1$
corresponds to a deterministic-circular-motion determined motion, see also
Fig. 3. Interestingly, when both effects are of equal magnitude,
$\lambda,\xi,k(\varrho)=0$ and the system again behaves as described by the
Model B dynamics of Eq.(42).

The phenomenological parameters $\lambda$ and $\xi$ represent the active
generalisations of equilibrium Model B [40] ($\partial\varrho/\partial
t=-\nabla\mathbf{J}_{\mathrm{eq}}=\nabla(\delta\mathcal{F}[\varrho]/\delta\varrho)$,
where $\varrho$ is the conserved order parameter, $\mathbf{J}_{\mathrm{eq}}$
the equilibrium (deterministic) current and $\mathcal{F}$ is the equilibrium
free energy). Since this model was constructed on the basic principles of the
system obeying detailed balance, to describe active matter, this restriction
had to be overcome. A first step in the development of active field theories
was the so-called Active Model B (AMB) [70, 99], where $\lambda\neq 0$ but
$\xi=0$. This model still attracts interest as it represents the first step
towards an active field theory [100, 101], but it was shown, that even though
quantitatively the coexisting liquid and vapor densities are changed, the AMB
cannot report any quantitative changes in the coarsening dynamics as compared
to the known dynamics, found by phenomenological introducing activity in the
Model B [99]. Note that Eq.(42) as the lower order result of the closure
scheme, has such a Model B structure with coefficients, altered by activity.

The sought qualitative changes in the phase-separation dynamics of active
matter could thereafter be found by the further generalised AMB+ where
$\lambda\neq 0$ and additionally $\xi\neq 0$. As the AMB, the AMB+ as the most
general model at this order, goes beyond the typical free-energy structure of
Model B ($\lambda\neq 0$). For the AMB, the $\lambda$-term defines a (local)
non-equilibrium chemical potential
$\mu_{\mathrm{neq}}=\delta\mathcal{F}/\delta\varrho+\lambda(\nabla\varrho)^{2}$,
since the current is still of a gradient form. For the AMB+ instead, the
$\xi$-induced current cannot be put into a gradient structure anymore and
therefore allows for circulating real-space currents
$\nabla\wedge\mathbf{J}_{\mathrm{neq}}\propto\xi$. Further, as a result of the
non-gradient structure of the current, the non-equilibrium chemical potential
becomes non-local [25, 26] and therefore amounts to fundamental differences of
AMB and AMB+. As these studies suggest, this non-locality seems to be a
necessary ingredient to describe the behaviour of active matter from a field
theoretical perspective.

It is now interesting to note that as a result of our first principles
approach, the AMB+ appears as the most natural choice of an active field
theory which relies on an expansion in terms of density gradients. We observe
from (48c) and (48d) that $\lambda=-8\xi$. Thus it is not reasonable to
include one but leave out the other parameter in the field-theoretical
description. We further observe that as argued by phenomenology, both
$\lambda,\xi\propto D_{T}D_{A}$, such that the AMB+ reduces to the equilibrium
Model B for $D_{A}\to 0$, a purely passive system. Interestingly, it also
reduces to a non-equilibrium Model B as $D_{T}\to 0$, i.e. when the thermal
motion becomes negligible compared to the activity. Lastly, as expected,
$\lambda$ and $\xi$ are only present in an interacting system
($\lambda,\xi\propto\phi$), since they are known to alter the phase-separation
dynamics.

##  3 Conclusion

In this work, we have reviewed and extended a geometric approach [50] to deal
with particle interactions by restricting the domain of definition of their
coordinates. Forbidden overlaps of the particles correspond to forbidden areas
in the domain, which creates a configuration space with inner moving
boundaries. Based on that we derived an effective time-evolution equation for
the full one-body PDF. We proceeded by projecting the full one-body PDF on its
angular modes, which resulted in a coupled hierarchy of hydrodynamic modes. By
pointing out the underlying assumptions, we turned the closure scheme into a
perturbation problem and found effective time-evolution equations (field-
theories) for the mean-particle density.

One major result of our work is as follows. This procedure gives us first-
principles access to the otherwise phenomenological parameters of field-
theoretical descriptions of active matter. We find that the microscopically
most justified theory of a continuous model is the AMB+ [25]. Similar first-
principles derivations for the AMB+ were already found before [42, 43, 94],
but they were always restricted to the model of non-chiral ABPs. From a
technical point of view, our work unravels the theoretical necessities of the
regimes of validity to obtain such a coarse-grained model for the description
of active matter. Specifically, we find that when neglecting the time
evolution of higher modes such as polarization or nematic order, in the so-
called adiabatic limit, the AMB+ model is microscopically justified only in
the limit of low Péclet numbers, i.e. when thermal diffusion dominates active
motion. Whether the AMB+ model can be extended to the regimes of higher
activity is a subject of future research.

The main prediction of this work is, however, that active chirality has a non-
trivial influence on the dynamics of the mean-particle PDF
$\varrho(\mathbf{x},t)$. Even though, in the simplest version of the ACP
model, chirality is not altered by particle interactions and hence its effect
on the full one-body level is rather superficial, it becomes most prominent
when integrating out the angular dependence of the full one-body PDF
$p(\mathbf{x},\theta,t)$. We find that chirality can change the sign of all
activity-induced coefficients of the AMB+, $\lambda$, and $\xi$, as well as
the Cahn-Hilliard coefficient $k(\varrho)$. A recent work, which directly
focused on the field-theoretical description of ACPs, could not report this
phenomenon [89]. The presumably far-reaching consequences of sign-changing
coefficients in a field-theoretical description of active matter may not
necessarily be pointed out explicitly.

The systematic analysis presented in this work allows us to further include
the effect of nematic order on the time-evolution of the mean-particle PDF,
but it would amount to a field-theoretical description at an even higher order
than the AMB+ (which is already at fourth order in density gradients). This
could be, nevertheless, an interesting future work from the perspective of
ACPs, as in the nematic order chirality contributes differently than at the
order of polarization. It would be interesting to investigate the robustness
of sign changes caused by these higher-order contributions.

Inspired by a rich behaviour of complex macroscopic phenomena in active
matter, the next step would be to couple the chirality to the spatial
interaction of the particles [102, 103]. Motivated from an orientation-
dependent potential of the form
$\hat{\mathbf{e}}(\theta_{i})\cdot\hat{\mathbf{e}}(\theta_{j})$ for particles
$i$ and $j$ [104], we could consider additional aligning interactions together
with active chirality from an analytic perspective. A possible attempt might
be to consider an additional term in the Langevin description for the angle,
of the form of $\alpha\sum_{i\neq
j}\sin(\theta_{i}-\theta_{j})\Theta(R-|\mathbf{x}_{i}-\mathbf{x}_{j})$ in the
restricted spatial domain. Here $\alpha$ sets the strength of the alignment of
particle $i$ with all neighbouring particles within the distance $R$, as
specified by the Heaviside function $\Theta(\cdot)$. A recent work considered
the numerical effects of such alignment for phenomena like motility-induced
phase separation and the flocking of active chiral particles [81]. It would be
interesting to see the analytical implications of such additional alignment
interactions coupled with steric spatial interactions and their effect on
field-theoretical descriptions.

E. K. thanks Pierluigi Muzzeddu and Hidde Vuijk for valuable discussions on
the topic. The authors further acknowledge support by the Deutsche
Forschungsgemeinschaft (E. K. and R. M. trough DFG grant No. SPP 2332 -
492009952 and A. S. through DFG grant No. SH 1275/5-1)

##  Appendix A Evaluation of the Integral (2.2.1)

In the main text, in Eq. (2.2.1), we are left with evaluating a boundary
integral, where integration and differentiation are with respect to different
particle labels. Hence a use of the Gaussian divergence theorem is not
possible straight-forwardly. Instead, we use an extended version of the
Reynolds transport theorem, which usually allows for the time-differentiation
of an integral quantity, where the integration volume $V$ itself is time-
dependent $V=V(t)$. For an arbitrary space- and time-dependent function
$f=f(\mathbf{x},t)$, the theorem in this context reads:

$\frac{\partial}{\partial t}\int_{V(t)}\mathrm{d}\mathbf{x}\
f=\int_{V(t)}\mathrm{d}\mathbf{x}\ \frac{\partial f}{\partial
t}+\int_{\partial V(t)}\mathrm{d}\mathrm{S}_{\mathbf{x}}\
\left(\mathbf{n}\cdot\mathbf{v}_{\partial V(t)}\right)\ f,$ (50)

where $\mathbf{v}_{\partial{V(t)}}=\mathbf{v}_{\partial{V(t)}}(\mathbf{x},t)$
is the velocity of an element of the moving boundary $\partial V(t)$ and
$\mathrm{d}\mathrm{S}_{\mathbf{x}}\ \mathbf{n}$ is the outward area element of
the boundary at time $t$.

This theorem can be extended to cases, where the integration volume $V$ is
space-dependent $V=V(\mathbf{x})$ (but of a constant shape) and we are
interested in taking the divergence of the integral with respect to that
coordinate $\mathbf{x}$. For a vector-valued function
$\mathbf{f}=\mathbf{f}(\mathbf{x},\mathbf{y},t)$, the extend transport theorem
reads:

$\nabla_{\mathbf{x}}\cdot\int_{V(\mathbf{x})}\mathrm{d}\mathbf{y}\
\mathbf{f}=\int_{V(\mathbf{x})}\mathrm{d}\mathbf{y}\
\nabla_{\mathbf{x}}\cdot\mathbf{f}+\int_{\partial
V(\mathbf{x})}\mathrm{d}\mathrm{S}_{\mathbf{y}}\
\mathbf{n}_{\mathbf{y}}\cdot\mathbf{f}.$ (51)

The proof of this relation can be found in Ref. [65]. We will apply this
theorem to the reduced configuration space
$\Lambda(\chi_{1})=\Omega\setminus\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})\times[0,2\pi)$,
where the space-dependence is ion
$\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})$, the disk of radius
$\varepsilon_{d}$ centered at $\mathbf{x}_{1}$. Therefore, only
$\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})$ contributes a moving
boundary, and hence a boundary integral in 51. We now apply the extended
Reynolds transport theorem to evaluate Eq. (2.2.1). Note here that
$P_{2}(t)=P_{2}(\chi_{1},\chi_{2},t)$ for a shortness of notation.

$\displaystyle\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\nabla_{\chi_{1}}\cdot\bigg{[}\mathbf{D}\nabla_{\chi_{1}}-\mathbf{f}(\theta_{1})\bigg{]}P_{2}(t)$
$\displaystyle=\nabla_{\chi_{1}}\cdot\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\bigg{[}\mathbf{D}\nabla_{\chi_{1}}-\mathbf{f}(\theta_{1})\bigg{]}P_{2}(t)$
$\displaystyle\quad-\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\cdot\bigg{[}D_{T}\nabla_{1}-v\hat{\mathbf{e}}(\theta_{1})\bigg{]}P_{2}(t).$
(52)

The moving boundary only arises in the spatial part of
$\chi_{1}=(\mathbf{x}_{1},\theta_{1})$, and thus we are only left with the
spatial part of the diffusion matrix
$\mathbf{D}=\mathrm{diag}(D_{T},D_{T},D_{R})$ and drift term
$\mathbf{f}(\theta_{1})=(v\hat{\mathbf{e}}(\theta_{1}),\omega)^{\mathrm{T}}$.
We again apply the extended theorem for the first term in the evaluated
integral

$\displaystyle\nabla_{\chi_{1}}\cdot\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\bigg{[}\mathbf{D}\nabla_{\chi_{1}}P_{2}(t)\bigg{]}=\nabla_{\chi_{1}}\cdot\mathbf{D}\nabla_{\chi_{1}}\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
P_{2}(t)$
$\displaystyle\quad-\nabla_{1}\cdot\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\ \bigg{[}D_{T}\ P_{2}(t)\bigg{]}.$ (53)

We are left with evaluating the originating boundary integral to combine it
with the boundary integral of (52). Therefore we jointly use the divergence
theorem and the transport theorem, paying respect to the specific integration
volumes in the following steps. First we we applied the divergence theorem on
the volume $\Omega\setminus\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})$
“backwards”

$\displaystyle-\nabla_{1}\cdot\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\bigg{[}D_{T}\
P_{2}(t)\bigg{]}=-D_{T}\nabla_{1}\cdot\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\nabla_{2}P_{2}(t)$
$\displaystyle\quad+D_{T}\nabla_{1}\cdot\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\Omega}\
\mathrm{d}\mathrm{S}_{2}\ \mathbf{n}_{2}P_{2}(t).$ (54)

Using the generalized transport theorem, the first integral on the right-hand
side of Eq. (A) can be rewritten as

$\displaystyle-D_{T}\nabla_{1}\cdot\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\nabla_{2}P_{2}(t)=-D_{T}\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\nabla_{1}\cdot\nabla_{2}P_{2}(t)$ $\displaystyle\quad-
D_{T}\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\cdot\nabla_{2}P_{2}(t).$ (55)

Note again here that only $\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})$ is
space-dependent and thus contributes a boundary term in the generalized
transport theorem. Since integration and differentiation are with respect to
the same particle label in the first integral on the right-hand side of Eq.
(A), we can apply the divergence theorem “forwards”

$\displaystyle-D_{T}\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\nabla_{1}\cdot\nabla_{2}P_{2}(t)=-D_{T}\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\Omega}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\cdot\nabla_{1}P_{2}(t)$ $\displaystyle\quad-
D_{T}\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\cdot\nabla_{1}P_{2}(t).$ (56)

Note here that when applying the divergence theorem the box-boundary
$\partial\Omega$ contributes a surface integral. As the box-boundary does not
explicitly depend on $\mathbf{x}_{1}$, the partial differential operator
$\nabla_{1}$ can be moved outside the integral (without creating another
boundary term) and we observe that it cancels with the second integral on the
right-hand side of Eq. (A). We are thus left with the two integrals on the
inner boundary $\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})$ and have
found that

$\displaystyle-\nabla_{1}\cdot\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\bigg{[}D_{T}\ P_{2}(t)\bigg{]}$
$\displaystyle=-D_{T}\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\cdot\left(\nabla_{1}+\nabla_{2}\right)P_{2}(t).$ (57)

The rewriting of the boundary term of Eq. (A) finally enables us to combine
the result with the term in Eq. (52). Using the definition of the full one
body PDF $p(\chi_{1},t)=\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
P_{2}(\chi_{1},\chi_{2},t)$, Eq. (52) thus becomes

$\displaystyle\int_{\Lambda(\chi_{1})}\mathrm{d}\chi_{2}\
\nabla_{\chi_{1}}\cdot\bigg{[}\mathbf{D}\nabla_{\chi_{1}}-\mathbf{f}(\theta_{1})\bigg{]}P_{2}(t)=\nabla_{\chi_{1}}\cdot\bigg{[}\mathbf{D}\nabla_{\chi_{1}}-\mathbf{f}(\theta_{1})\bigg{]}p(\chi_{1},t)$
$\displaystyle\quad-\int_{\partial\mathrm{B}_{\varepsilon_{d}}(\mathbf{x}_{1})}\mathrm{d}\mathrm{S}_{2}\
\mathbf{n}_{2}\
\bigg{[}D_{T}\left(2\nabla_{1}-\nabla_{2}\right)+v\hat{\mathbf{e}}(\theta_{1})\bigg{]}P_{2}(t),$
(58)

which constitutes the result of Eq. (2.2.1) in the main text.

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