# An Introduction to Modeling Approaches of Active Matter

L. Hecht  Institut für Physik kondensierter Materie, Technische Universität
Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany  J. C. Ureña  Institut
für Physik kondensierter Materie, Technische Universität Darmstadt,
Hochschulstr. 8, 64289 Darmstadt, Germany  B. Liebchen<EMAIL_ADDRESS>darmstadt.de Institut für Physik kondensierter Materie, Technische Universität
Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany

(August 27, 2024)

###### Contents

  1. I Introduction
  2. II Wet and dry models: The role of the solvent in active matter
  3. III Dry active particles: The active Brownian particle model and its alternatives
  4. IV Continuum theories for dry active matter
  5. V Hydrodynamics of microswimming: Low Reynolds number and Stokes flow
  6. VI Modeling hydrodynamics at the many-particle level
  7. VII Continuum theories of microswimmers with hydrodynamic interactions

##  I Introduction

This article is based on lecture notes for the Marie Curie Training school
“Initial Training on Numerical Methods for Active Matter”. It provides an
introductory overview of modeling approaches for active matter and is
primarily targeted at PhD students (or other readers) who encounter some of
these approaches for the first time. The aim of the article is to help put the
described modeling approaches into perspective.

We begin with a brief discussion of the role of the solvent in (soft) active
matter, which is followed by an introduction to “dry particle-only models”,
such as the active Brownian particle model, before coming to models for wet
active matter, which (explicitly) include a solvent, and continuum
descriptions for the collective behavior of many active particles.

##  II Wet and dry models: The role of the solvent in active matter

Models of active matter can be classified into “dry” and “wet” models. The
former class of models involves only equations of motion for “particles”
whereas the latter involves an explicit description of a solvent in addition
to the embedded active particles, which ensures momentum conservation, as will
be discussed below.

Dry models are naturally used to describe active systems which do not involve
a liquid solvent, such as granular particles on vibrating plates [1, 2, 3, 4,
5, 6], self-vibrating granular particles [7, 8, 9], bacteria gliding on a
rigid surface [10], flocks of birds, animal herds, swarms of locusts [11, 12,
13, 14, 15], human crowds [16, 17], or flying drones [18]. However, they are
also frequently used as simplified descriptions of active matter systems
involving a solvent which is only effectively represented and commonly acts as
a thermal bath leading to fluctuations in the equations of motion of the
individual particles. In contrast, wet models are used to describe
microswimmers such as synthetic active colloids [19, 20, 21], droplet swimmers
[22, 23, 24, 25], and biological microorganisms like bacteria [26, 10, 27],
algae [28], or sperm cells [26], including their interaction with the
surrounding solvent and the corresponding cross interactions among different
microswimmers. A particular example of a wet system at larger scales, i.e.,
beyond the soft matter realm, which microswimmers belong to, can be found in
schools of fish [13, 29]. To understand the applicability regime of the
various active matter models, it is instructive to first discuss the impact of
the solvent on active systems:

  1. (i)

Fluctuations and dissipation: Active particles are typically orders of
magnitude larger than the molecules of the surrounding solvent and are subject
to collisions with the latter. This leads to fluctuations in their motion,
analogously to Brownian motion of “passive” colloids in equilibrium [cf.
trajectory in Fig. 1 (a)]. The trajectory of an isolated active particle is
then typically given by the combination of ballistic motion due to self
propulsion and fluctuations due to collisions with the solvent molecules. As
can be seen in Fig. 1 (b), the motion of an active particle is not straight
because the collisions of the solvent molecules with the active particles
feature both a radial and a tangential component. The latter induces a
stochastic turn of the particle orientation and, hence, reorientation of the
self-propulsion direction, which is called rotational Brownian motion or
rotational diffusion. Following the fluctuation-dissipation theorem from
statistical mechanics, these fluctuations are necessarily linked to
dissipation occurring, e.g., in the form of Stokes drag for spherical
particles. For microswimmers, i.e., for (active) particles at the microscale,
dissipation normally dominates over inertia. Hence, the motion is overdamped.
These effects of the solvent, namely translational diffusion, rotational
diffusion, and dissipation, are the only effects of the solvent which are
typically taken into account in dry models, such as the active Brownian
particle (ABP) model, which we will discuss further below.

Figure 1: (Top panel) Experimental trajectories of a Janus colloid showing (a)
passive and (b) active Brownian motion (kindly provided by J. R. Howse; see
Ref. [19] for experimental details). (Bottom panel) Exemplary trajectories
obtained in the overdamped regime from simulations of (c) the passive Brownian
particle (PBP) model, (d) the active Brownian particle (ABP) model, (e) the
run-and-tumble particle (RTP) model, (f) the active Ornstein-Uhlenbeck
particle (AOUP) model, and (g) the chiral active particle (CAP) model. Note
that an isolated AOUP is equivalent to an underdamped PBP with
$m/\gamma=\tau_{\text{p}}$ (see Sec. III).

  2. (ii)

Momentum conservation: Physically, in the absence of external fields or
boundaries, the overall momentum of an active system has to be conserved. For
example, when a microorganism or an active Janus colloid moves forward, there
is necessarily a counter-propagating solvent flow such that the overall
momentum of the active particle and the surrounding solvent is conserved
(swimming in vacuum is impossible). Thus, the solvent not only acts as a bath
providing fluctuations and drag but also ensures momentum conservation.

  3. (iii)

Hydrodynamic interactions: The solvent mediates hydrodynamic interactions
among different active particles. These arise because the flow pattern induced
by each active particle as a consequence of its swimming acts onto all other
particles in the system. These solvent-mediated interactions are often long-
ranged. In particular, in the absence of external forces, they often decay as
$1/r^{2}$ for force-dipole swimmers, such as various bacteria or algae
(explicit measurements of the flow field exist, e.g., for E. coli bacteria
[30]), and as $1/r^{3}$ for source-dipole swimmers, such as Paramecium [31] or
(idealized) Janus colloids with a uniform surface mobility [32, 33]. However,
they can be weakened or decay faster in the presence of a substrate or other
boundaries [34].

  4. (iv)

Hydrodynamic boundary effects: If the active particles are in contact with
boundaries, such as a glass substrate, which is frequently used in experiments
with active colloids, or another liquid-solid or liquid-air interface, the
solvent can lead to additional interesting effects. An example of these is
constituted by osmotic flows at fluid-solid interfaces, such as those induced
by auto-phoretic particles [35] or by some modular swimmers involving ion-
resins [36, 37]. At fluid-air interfaces, active particles can cause Marangoni
flows [38, 39, 40, 41], which act on all particles in the system and can
elicit interesting collective behaviors [42, 43, 44, 45].

##  III Dry active particles: The active Brownian particle model and its
alternatives

Active Brownian particle model:
One of the simplest and most popular models to describe active particles is
the active Brownian particle (ABP) model [46, 47, 48, 49, 50, 51, 52],
originally introduced to describe the motion of colloidal particles which
smoothly change their self-propulsion direction due to rotational diffusion.
It treats the solvent as a bath providing only fluctuations and drag without
ensuring momentum conservation and, at least in its most commonly used form,
without accounting for hydrodynamic interactions among particles. The ABP
model does not explicitly describe the mechanism leading to self propulsion
either, which arises through the interactions of the active particles with the
surrounding solvent (or with a substrate), but simply replaces it with an
effective force that drives the particle forward. Microscopically, this is not
correct because, as discussed above, microswimmers are force free, but it
leads to a simple generic model for the dynamics of active particles, which
stays agnostic on many details of the specific underlying realization. In its
simplest form, the ABP model in two-dimensional space is defined by the
overdamped Langevin equations

$\displaystyle\frac{\text{d}\vec{r}_{i}(t)}{\text{d}t}=$
$\displaystyle\leavevmode\nobreak\
v_{0}\vec{p}_{i}(t)-\frac{1}{\gamma}\nabla_{\vec{r}_{i}}U+\sqrt{2D}\vec{\xi}_{i}(t),$
(1) $\displaystyle\frac{\text{d}\phi_{i}(t)}{\text{d}t}=$
$\displaystyle\sqrt{2D_{\text{R}}}\eta_{i}(t),$ (2)

where $\vec{r}_{i}=(x_{i},y_{i})$ and $\phi_{i}$ are the position and the
orientation angle of the $i$-th spherical ABP, respectively, $v_{0}$ is the
self-propulsion speed, $\gamma$ is the Stokes drag coefficient,
$U=\sum_{i<j}u\left(\left|\vec{r}_{i}-\vec{r}_{j}\right|\right)$ is the
interaction energy with interaction potential $u(r)$, $D$ and $D_{\text{R}}$
are the translational and rotational diffusion coefficients, respectively, and
$\vec{\xi}_{i}(t)=(\xi_{x,i}(t),\xi_{y,i}(t))$ and $\eta_{i}(t)$ represent
Gaussian white noise with unit variance and zero mean. The self-propulsion
direction is given by $\vec{p}_{i}=(\cos{\phi_{i}},\sin{\phi_{i}})$.

It is instructive to first consider a single ABP, i.e., $U=0$. Different from
a passive Brownian particle (PBP) [cf. Fig. 1 (c)], the trajectory of an ABP
is characterized by an initial period of directed motion followed by a
randomization of the self-propulsion direction due to rotational diffusion
[cf. Fig. 1 (d)]. The initial directed motion persists for a time
$\tau_{\text{p}}=1/D_{\text{R}}$ (persistence time) over a distance
$l_{\text{p}}=v_{0}\tau_{\text{p}}$ (persistence length). Accordingly, the
average displacement of an ABP reads [48, 49, 50]

$\left\langle\vec{r}_{i}(t)-\vec{r}_{i}(0)\right\rangle=l_{\text{p}}\left(1-e^{-t/\tau_{\text{p}}}\right)\vec{p}_{i}(0).$
(3)

Therefore, an ABP moves, on average, over a distance $l_{\text{p}}$ along its
initial orientation $\vec{p}_{i}(0)$ before its orientation is randomized,
which rationalizes the term “persistence lenth”. Additionally, the mean square
displacement (MSD) of an ABP (in two-dimensional space) reads [19, 48, 49, 50]

$\displaystyle\left\langle(\vec{r}_{i}(t)-\vec{r}_{i}(0))^{2}\right\rangle=$
$\displaystyle
2l^{2}_{\text{p}}\left(\frac{t}{\tau_{\text{p}}}-1+e^{-t/\tau_{\text{p}}}\right)$
$\displaystyle+4Dt,$ (4)

which provides valuable insight into the different dynamical regimes of the
ABP model. Three regimes are observed when expanding Eq. (III) for short,
intermediate, and long times. The motion of an ABP is initially diffusive with
diffusion coefficient $D$ for $t\ll D/v_{0}^{2}$. For $D/v_{0}^{2}\ll
t\ll\tau_{\text{p}}$, a ballistic regime which represents directed motion due
to the activity of the particle comes about. Finally, for
$t\gg\tau_{\text{p}}$, the motion is again diffusive with “active diffusion
coefficient” $D_{\text{A}}=D+l^{2}_{\text{p}}/(2\tau_{\text{p}})$. These three
regimes are shown in Fig. 2.

The relative importance of activity in comparison with diffusion can be
characterized by the Péclet number $\text{Pe}=v_{0}/\sqrt{2DD_{\text{R}}}$
[49]. Assuming spherical active particles of radius $R$ and diameter
$\sigma=2R$, the Stokes-Einstein relation yields
$D=\frac{4}{3}R^{2}D_{\text{R}}$, and thus the Péclet number can be rewritten
in terms of the particle diameter (as customarily done in the literature [53,
54, 55]) as $\text{Pe}=\sqrt{3/2}v_{0}/(\sigma D_{\text{R}})$ or
$\text{Pe}=v_{0}\sigma/(\sqrt{6}D)$.

Figure 2: Mean square displacement (MSD) over time in the ABP, RTP, AOUP, and
CAP models calculated based on 1000 realizations of the non-dimensionalized
equations in Tab. 1. Simulation parameters: Time step $\Delta
t/\tau_{\text{p}}=10^{-3}$, $\text{Pe}=10$ for the ABP and CAP models and
$\tilde{\omega}=\pi 10^{2}$ for the CAP model. In the RTP model, the
probability that a tumbling event occurs during a certain time step is given
by a Poisson distribution with rate $\lambda=\lambda_{\text{t}}\Delta
t=10^{-3}$. The dashed vertical lines are placed at
$t/\tau_{\text{p}}=D/(v_{0}^{2}\tau_{\text{p}})$ (left) and
$t/\tau_{\text{p}}=1$ (right) separating the different regimes exhibited by
the particles in each model. The reader is referred to Refs. [56, 57] for more
details on the MSD of the CAP model.

For sufficiently dense ensembles of active particles with a sufficiently large
Péclet number and purely repulsive interactions arising because, e.g., the
individual particles cannot overlap, the ABP model predicts a spectacular
phenomenon known as motility-induced phase separation (MIPS) [58, 21, 59, 60,
50, 61]. A sequence of snapshots of the state of an ensemble of ABPs which
interact via the purely repulsive Weeks-Chandler-Anderson (WCA) potential [62]
and for which MIPS occurs is shown in Fig. 3 (a)–(d). Initially, the ensemble
is uniformly distributed. For suitable parameters (large Péclet number and
high density), the uniform state loses stability and the particles aggregate
in small clusters. These clusters grow following the coarsening law shown in
Fig. 3 (e) until a single macrocluster, which coexists with a low-density
active gas, is eventually formed. Overall, while phase separation in
equilibrium generally requires inter-particle attractions, active systems can
phase separate even in their complete absence [63, 60]. The mechanism
underlying MIPS is shown in Fig. 3 (f): When particles collide, they block
each other until their orientations are randomized and they can separate from
each other. Broadly, MIPS occurs if the active particles are fast and numerous
enough for collisions with existing clusters to occur more often than
particles in these clusters leave them due to rotational diffusion.

Figure 3: (a)–(d) Evolution of motility-induced phase separation (MIPS) for an
ensemble of $N=39200$ ABPs of diameter $\sigma$ with number density
$\rho\sigma^{3}=0.6$, Péclet number $\text{Pe}=200/\sqrt{2}$, and time step
$\Delta t/\tau_{\text{p}}=5\times 10^{-6}$. The ABPs interact via the purely
repulsive Weeks-Chandler-Anderson (WCA) potential [62] with strength
$\epsilon=10k_{\text{B}}T$. (e) Time evolution of the mean cluster size $L(t)$
for the ABP model (adapted from Ref. [53] \- Published by The Royal Society of
Chemistry). (f) Schematic of the mechanism which leads to MIPS (with
permission from Ref. [21] \- © 2013 by the American Physical Society).

The ABP model can also be generalized to account for inertial effects [50, 64,
65, 66, 67], which is used, e.g., to model active granular particles on
vibrating plates [1, 2].

Alternatives to the ABP model:
Several alternative models have been designed that have a similar scope to
that of the ABP model in the sense that they also treat the solvent as a bath
which only provides fluctuations and drag rather than accounting for momentum
conservation and hydrodynamic interactions.

Run-and-tumble model: The run-and-tumble particle (RTP) model [68, 69, 70, 71]
was originally introduced to describe the characteristic motion patterns of
certain bacteria such as E. coli [72, 73, 74], but it has now advanced to a
standard model for the description of active particles. (In fact, the first
theory for MIPS was formulated for RTPs [68] and MIPS has been observed in
simulations of this model as well [69].) In contrast to ABPs, RTPs alternate
running periods, during which the self-propulsion direction remains unchanged,
with idealized tumbling events, in which the orientation of the particles is
randomized [cf. Fig. 1 (e)]. The equations of motion for the $i$-th RTP read

$\displaystyle\frac{\text{d}\vec{r}_{i}(t)}{\text{d}t}=$
$\displaystyle\leavevmode\nobreak\
v_{0}\vec{p}_{i}(t)-\frac{1}{\gamma}\nabla_{\vec{r}_{i}}U,$ (5)
$\displaystyle\frac{\text{d}\phi_{i}(t)}{\text{d}t}=$
$\displaystyle\sum_{n}\Delta\phi_{n}\delta(t-T_{n}),$ (6)

where the parameters of Eq. (5) are defined as in Eq. (1). The values of
$\Delta\phi_{n}$ are uniformly distributed between $0$ and $2\pi$, with
tumbling events taking place at discrete times $T_{n}$ [71]. In practice, the
times $T_{n}$ are chosen either randomly with $\langle
T_{\text{n+1}}-T_{\text{n}}\rangle=\lambda^{-1}_{\text{t}}$ (and, e.g.,
tumbling events following a Poisson distribution, which leads to exponentially
distributed times between tumbling events, as originally found for E. coli
[75]) or equally spaced. In any case, the (mean) tumbling rate
$\lambda_{\text{t}}$ is fixed, yielding a persistence time
$\tau_{\text{p}}=1/\lambda_{\text{t}}$, which plays the role of the (mean)
time between tumbling events.

Remarkably, the many-particle dynamics following from the RTP and the ABP
models turn out to be equivalent at coarse-grained scales if
$(d-1)D_{\text{R}}=\lambda_{\text{t}}$, where $d>1$ is the spatial dimension
[69, 70]. At the single-particle level, this is also reflected by the MSD in
Fig. 2 showing almost identical time evolution for RTPs and ABPs, with the
exception that RTPs always show a ballistic behavior for
$t\ll\tau_{\text{p}}$, since tumbling events are statistically unlikely on
this timescale and translational diffusion is not considered. The latter can
also be taken into account, resulting in the emergence of a diffusive regime
for $t\ll D/v^{2}_{0}$, as in the ABP model, where $D$ is the translational
diffusion coefficient.

Active Ornstein-Uhlenbeck model: Another alternative to the ABP model is the
active Ornstein-Uhlenbeck particle (AOUP) model [76, 77, 78, 79], which has
certain advantages compared with the ABP model regarding the theoretical
description of the many-body dynamics of dry active particles. This is due to
the fact that the AOUP model avoids the strongly nonlinear dependence of the
center-of-mass motion on the particle orientation, which is present in the ABP
model [cf. Eqs. (1) and (2)], by using colored noise to generate self
propulsion. The equation of motion for particle $i$ in the AOUP model (in the
overdamped regime) reads

$\frac{\text{d}\vec{r}_{i}(t)}{\text{d}t}=\vec{v}_{0,i}(t)-\frac{1}{\gamma}\nabla_{\vec{r}_{i}}U,$
(7)

where $\gamma$ is the Stokes drag coefficient and $U$ is the total interaction
potential. Whereas the self-propulsion speed $v_{0}$ remains constant for a
single particle in the ABP and RTP model, it evolves with time in the AOUP
model according to

$\tau_{\text{p}}\frac{\text{d}\vec{v}_{0,i}(t)}{\text{d}t}=-\vec{v}_{0,i}(t)+\sqrt{2D}\vec{\xi}_{i}(t),$
(8)

where $\tau_{\text{p}}$ is the persistence time and $\vec{\xi}_{i}(t)$ is
Gaussian white noise with unit variance and zero mean. As a result, the
velocity components of an isolated AOUP are represented by colored Gaussian
noise111Whereas white noise is delta-correlated in time, the correlation
function of colored noise takes finite values for finite time differences.
with correlation function $\left\langle
v_{0,i}^{(\alpha)}(0)v_{0,j}^{(\beta)}(t)\right\rangle=\delta_{ij}\delta_{\alpha\beta}(D/\tau_{\text{p}}){\rm
e}^{-t/\tau_{\text{p}}}$ between components $\alpha$ and $\beta$ of particles
$i$ and $j$. Note that a single particle in the AOUP model, i.e., $U=0$, is
formally identical to an underdamped passive Brownian particle with
$m/\gamma=\tau_{\text{p}}$ [cf. Fig. 1 (f)]. Hence, a single AOUP shows a
ballistic regime for $t\ll\tau_{\text{p}}$ followed by a diffusive regime for
$t\gg\tau_{\text{p}}$, as shown in Fig. 2.

Since it involves colored noise, the AOUP model does not permit formulation of
an exact Fokker-Planck equation for the corresponding probability
distribution. However, it is still possible to derive an approximate Fokker-
Planck equation for the many-body dynamics, which does not depend on the
particle orientation but only on the particle positions [78, 79, 76, 80]. MIPS
has also been reported for the AOUP model [77, 79] suggesting that it provides
a useful alternative for the description the many-body dynamics of active
particles although the single-particle properties significantly differ from
those of the ABP and RTP model.

Chiral particle model: A further class of models describes chiral active
particles (CAPs) [50, 81, 56, 82, 83, 57, 84, 85, 86, 87], which experience an
additional effective torque arising from an anisotropy in their shape or
propulsion mechanism. For an isolated CAP, this leads to circular trajectories
in the limit of zero noise, whereas the orientation angle of the $i$-th CAP in
the presence of noise evolves according to

$\displaystyle\frac{\text{d}\phi_{i}(t)}{\text{d}t}$
$\displaystyle=\omega+\sqrt{2D_{\text{R}}}\eta_{i}(t),$ (9)

where $\omega$ is a constant angular velocity. As in the ABP model, the
position $\vec{r}_{i}$ of the $i$-th CAP generally evolves with time according
to Eq. (1). An exemplary trajectory of a CAP and the time evolution of its MSD
are shown in Figs. 1 (g) and 2, respectively. Examples of circle swimmers
include E. coli bacteria near surfaces and interfaces [88, 89], sperm cells
[90, 91] and artificial microswimmers such as L-shaped particles [92],
“spherical-cap particles” near a substrate [93], and asymmetric Quincke
rollers [94].

An overview of the models introduced thus far is provided in Tab. 1. This
table is intended as a guide to numerically implementing the previously
described models on a single-particle level. To this end, the equations of
motion are presented in dimensionless form.

Table 1: Dimensionless equations of motion and parameters of a single active
particle in the ABP, RTP, AOUP and CAP models. The variables
$\vec{\mathfrak{r}}$, $\vec{\mathfrak{v}_{0}}$, and $\mathfrak{t}$ shown in
the equations of the second column have been non-dimensionalized by rescaling
the original dimensional variables with respect to the natural time and length
scales shown in the fourth column: $\vec{\mathfrak{r}}=\vec{r}/l$,
$\vec{\mathfrak{v}_{0}}=\vec{v}_{0}\tau_{\text{p}}/l$,
$\mathfrak{t}=t/\tau_{\text{p}}$. The dot over these variables denotes the
derivative with respect to the dimensionless time $\mathfrak{t}$. Pe is the
Péclet number [49]. DRY ACTIVE PARTICLE MODELS FOR A SINGLE PARTICLE
---
Model | Equations of motion | Parameters | Natural units
ABP |  $\dot{\vec{\mathfrak{r}}}(\mathfrak{t})=\vec{p}(\mathfrak{t})+\text{Pe}^{-1}\vec{\xi}(\mathfrak{t})$ |  $\text{Pe}=\frac{v_{0}}{\sqrt{2DD_{\text{R}}}}$ |  Time scale: $\tau_{\text{p}}=D^{-1}_{\text{R}}$
$\dot{\phi}(\mathfrak{t})=\sqrt{2}\eta(\mathfrak{t})$ |  Length scale: $l=l_{\text{p}}=v_{0}D^{-1}_{\text{R}}$
RTP |  $\dot{\vec{\mathfrak{r}}}(\mathfrak{t})=\vec{p}(\mathfrak{t})$ |  None222for equally spaced $T_{n}$ or Poisson-distributed tumbling events as found in E. coli [75] and without translational diffusion; with the latter, the equation of motion for the position reads $\dot{\vec{\mathfrak{r}}}(\mathfrak{t})=\vec{p}(\mathfrak{t})+\text{Pe}^{-1}\vec{\xi}(\mathfrak{t})$ with $\text{Pe}=v_{0}/\sqrt{2D\lambda_{\text{t}}}$. |  Time scale: $\tau_{\text{p}}=\lambda^{-1}_{\text{t}}$
$\dot{\phi}(\mathfrak{t})=\sum_{n}\Delta\phi_{n}\delta(\mathfrak{t}-\tilde{T}_{n})$ |  Length scale: $l=l_{\text{p}}=v_{0}\lambda^{-1}_{\text{t}}$
AOUP |  $\dot{\vec{\mathfrak{r}}}(\mathfrak{t})=\vec{\mathfrak{v}}_{0}(\mathfrak{t})$ |  None |  Time scale: $\tau_{\text{p}}$
$\dot{\vec{\mathfrak{v}}}_{0}(\mathfrak{t})=-\vec{\mathfrak{v}}_{0}(\mathfrak{t})+\sqrt{2}\vec{\xi}(\mathfrak{t})$ |  Length scale: $l=\sqrt{D\tau_{\text{p}}}$
CAP |  $\dot{\vec{\mathfrak{r}}}(\mathfrak{t})=\vec{p}(\mathfrak{t})+\text{Pe}^{-1}\vec{\xi}(\mathfrak{t})$ |  $\text{Pe}=\frac{v_{0}}{\sqrt{2DD_{\text{R}}}}$ |  Time scale: $\tau_{\text{p}}=D^{-1}_{\text{R}}$
$\dot{\phi}(\mathfrak{t})=\tilde{\omega}+\sqrt{2}\eta(\mathfrak{t})$ |  $\tilde{\omega}=\omega\tau_{\text{p}}$ |  Length scale: $l=l_{\text{p}}=v_{0}D^{-1}_{\text{R}}$

Monte Carlo simulations: A final example to describe isotropic dry active
particles is based on kinetic Monte Carlo simulations [95, 96, 97], where the
displacements of the particles are correlated in time. Namely, the
displacement during a certain time step is drawn from a Gaussian distribution
whose mean equals the displacement in the previous time step.

Figure 4: Simulation snapshots of active particles with alignment interactions
[cf. Eqs. (10) and (11)] for $D=0$ and (a) $\omega=0$ (smooth variant of the
Vicsek model) and (b), (c) $\omega>0$ (chiral active particles). Colors
represent particle orientation angles such that equally colored particles are
aligned or phase-synchronized (with permission from Ref. [81] \- © 2017 by the
American Physical Society).

Models with explicit alignment interactions: Thus far, we have focused our
discussion on isotropic active particles, i.e., on particles without explicit
alignment interactions. The most popular model for describing (dry) active
particles with (polar) alignment interactions is the Vicsek model [98, 99],
which accounts for self-propelled particles (“birds”) that align their
orientation with that of their neighbors. A generalized continuous-time
variant of the Vicsek model comprising CAPs with alignment interactions can be
defined by

$\displaystyle\frac{\text{d}\vec{r}_{i}(t)}{\text{d}t}=$
$\displaystyle\leavevmode\nobreak\
v_{0}\vec{p}_{i}(t)+\sqrt{2D}\vec{\xi}_{i}(t),$ (10)
$\displaystyle\frac{\text{d}\phi_{i}(t)}{\text{d}t}=$
$\displaystyle\leavevmode\nobreak\ \omega+\frac{K}{\pi R^{2}_{0}}\sum_{j\in
S_{R_{0}}^{(i)}}\sin(\phi_{j}-\phi_{i})$
$\displaystyle+\sqrt{2D_{\text{R}}}\eta_{i}(t),$ (11)

where $\omega$ is an angular velocity, $K$ is the strength of the alignment
interactions (for K=0 this model reduces to the CAP model) and the sum is
calculated over all particles within a circle $S_{R_{0}}^{(i)}$ of radius
$R_{0}$ centered at the position of particle $i$ [100, 101, 102, 81]. The
hallmark of this model is that particles tend to follow the orientation of
their neighbors, which can induce polar order, e.g., in the form of the
traveling bands shown in Fig. 4 (a) for $\omega=0$. When considering CAPs with
polar interactions (Eq. (11) with $\omega>0$), one finds two remarkable
phenomena: The formation of rotating macrodroplets with late-time sizes
comparable to the system size, which is indicative of phase separation, at low
angular velocity [cf. Fig. 4 (b)]; and a pattern of rotating microflocks
exhibiting phase synchronization and a self-selected length scale at high
angular velocity [cf. Fig. 4 (c)].

Another class of models with alignment interactions accounts for nematic
alignment interactions [103, 104, 105, 106], which arise in systems of head-
tail symmetric particles, such as (self-propelled) rods [107] featuring apolar
interactions.

Applicability regime of dry active particle models:
The ABP model and its alternatives are commonly used to perform particle-based
simulations of active particles and also as a starting point for the
formulation of continuum theories, as we shall discuss hereunder. These models
have proven useful when applied to, e.g., the following problems concerning
active particles:

  1. (i)

When we are concerned with single active particle flow fields, the ABP model
has been very successful, e.g., to predict correlation functions in close
agreement with experiments of Janus colloids [108].

  2. (ii)

When hydrodynamic interactions play a minor role such as for certain active
colloids, their many-body behavior is reasonably well described by ABPs [21].
Similarly, when hydrodynamic interactions are dominated by other interactions
such as, e.g., phoretic interactions of autophoretic colloids with a near-
uniform surface mobility, the ABP model serves as a useful starting point for
the derivation of simple models with effective phoretic pair interactions
[33].

  3. (iii)

When a solvent is absent but fluctuations are still relevant as, e.g., for
granular particles on vibrating plates, where quasi-deterministic chaos arises
and leads to effective randomness, which can be described as Brownian noise,
the ABP model can be used as a numerical model [4, 6].

  4. (iv)

The ABP model is also useful for fundamental theoretical explorations, e.g.,
when we are more interested in the fundamental consequences of activity on the
collective behavior of active particles rather than in the specific link to
experimental realizations.

Advantages and limitations:
Compared with most “wet” models, a key advantage of the ABP model and its
alternatives is their simplicity from both a conceptual and a computational
viewpoint. In particular, these models allow one to simulate very large
ensembles of active particles (state-of-the-art simulations often use
$10^{5}-10^{7}$ particles [109, 64, 110, 53, 111, 63]). One key limitation of
these models regarding the description of soft active matter systems is that
they do not account for momentum conservation and often not for hydrodynamic
interactions either. This can be particularly relevant for the description of
the collective behavior or for describing single microswimmers near walls. The
ABP model is popular when simulating the collective behavior of autophoretic
active colloids as well. Here, beside hydrodynamic interactions, also phoretic
interactions can play a crucial role and are also neglected by the standard
ABP model [33], which can however be extended to take them into account [112,
113, 114, 115, 116, 117, 118].

##  IV Continuum theories for dry active matter

To understand the collective behavior of (dry) active particles, one often
uses continuum models, which can be used for a purely theoretical analysis or
a numerical analysis based on continuum simulations. In general, one can
distinguish between (i) phenomenological and (ii) microscopic theories.

  * (i)

Phenomenological theories: This class is often based on an identification of
the relevant “slow variables” (e.g., the density field $\rho(\vec{r},t)$ in
the case of isotropic active systems with particle number conservation or the
density field and polarization density for polar active systems with polar
alignment interactions) and on writing down all terms which are allowed by
symmetry and conservation laws up to a certain order. Accordingly, these
theories are sometimes called Landau theories. A key advantage of
phenomenological theories is that they predict the structure of the field
equations essentially based on symmetry, conservation laws, and dimensionality
of the system without requiring any reference to the details of the underlying
particle system (such as the precise form of the interactions). Thus, these
field theories are sometimes called “generic” in this sense and can even be
formulated (and numerically solved) if no underlying particle-based model is
known. However, phenomenological field theories do not provide information
about the values of the coefficients. Thus, one often treats all occurring
coefficients as independent parameters and studies the phenomenology of the
field equations as a function of all these parameters. A related important
drawback of this approach is that it then remains unclear if there is an
underlying particle-based model or realization which leads to the
corresponding parameter values. A specific example of a phenomenological
theory is discussed further below.

  * (ii)

Microscopic theories: In contrast to phenomenological theories, microscopic
theories involve a systematic derivation of the field equations typically from
the underlying equations of motion for the individual active particles. This
approach yields equations of motion for the relevant fields, which directly
follow from the underlying particle-based model. Thus, in contrast to the
former class of theories, one advantage of this second approach is that one
obtains, in addition to the structure of equations, an explicit link between
the coefficients of the particle-based model and the continuum theory. This
typically leads to a (much) smaller number of independent parameters than one
would obtain from phenomenological approaches. Another advantage is that,
following the microscopic approach, terms which are allowed by symmetry cannot
be missed, which has happened for various standard models of active matter in
the past when following the phenomenological approach.

In the following, we will illustrate both approaches based on specific
examples for isotropic and polar active systems.

Example: Phenomenological theory for isotropic active matter
Collective phenomena of isotropic active matter, such as phase separation, can
be described, e.g., by the phenomenological active model B+, which is based on
the common model B that describes phase separation in equilibrium systems
[119]. Here, the density field $\rho(\vec{r},t)$ is assumed to be the only
slow variable of the system and the order parameter $\phi$ is related to it by
the linear transformation
$\phi=(2\rho-\rho_{\text{H}}-\rho_{\text{L}})/(\rho_{\text{H}}-\rho_{\text{L}})$,
where $\rho_{\text{H}}$ and $\rho_{\text{L}}$ denote the density at the low-
density and the high-density critical point, respectively [120, 121]. The
active model B+ is given by the equations

$\displaystyle\frac{\partial\phi}{\partial t}=$
$\displaystyle-\nabla\cdot\left[-M\nabla\left(\frac{\delta\mathcal{F}}{\delta\phi}+\lambda|\nabla\phi|^{2}\right)\right.$
$\displaystyle\left.+\zeta
M\left(\nabla^{2}\phi\right)\nabla\phi+\sqrt{2D}\vec{\Lambda}\right],$ (12)
$\displaystyle\mathcal{F}[\phi]=$
$\displaystyle\int\text{d}^{3}r\,\left[\frac{a}{2}\phi^{2}+\frac{b}{4}\phi^{4}+\frac{K}{2}|\nabla\phi|^{2}\right].$
(13)

Here, the free-energy functional $\mathcal{F}$ is approximated up to the order
$\phi^{4}$ and up to square-gradient terms [122]. Equation (12) has the form
of a continuity equation, and hence it ensures particle number conservation,
whereas reaction terms are not allowed in Eq. (12). The order parameter $\phi$
is subject to a Gaussian white noise field $\vec{\Lambda}(\vec{r},t)$ with
zero mean and unit variance. The diffusion coefficient is denoted by $D$ and
$M$ is the mobility of the active particles. For active particles, the time-
reversal symmetry (TRS) is broken locally. This fact is included in the active
model B+ by the additional terms proportional to $\lambda$ and $\zeta$. The
active model B+ describes the phase separation behavior of isotropic active
matter and predicts two types of patterns: The first one is characterized by
phase separation into a dense and a dilute phase and the additional occurrence
of vapor bubbles inside the dense phase, which are continuously created and
move to the surface of the dense phase [cf. Fig. 5 (a)]. The second pattern is
characterized by the emergence of dense clusters that do not grow beyond a
certain characteristic size [cf. Fig. 5 (b)]. The coefficients
$a,b,K,\lambda,\zeta$ are not known in this phenomenological approach and are
treated as parameters of the model. Thus, there is no obvious connection to
particle-based models such as the ABP model, whereas in microscopic theories
all parameters are directly related to the underlying particle-based model, as
will be discussed next.

Figure 5: Numerical results of the active model B+: (a) Coexistence of a
liquid phase (yellow to red colors) comprising continuously created vapor
bubbles and a gas phase (black and purple). (b) Phase separation into a dense
(yellow to red colors) and a dilute phase (black and purple). Dense clusters
stabilize at a certain cluster size in the steady state (taken from Ref.
[122]).

Example: Microscopic theories for isotropic active matter
There are several approaches to developing microscopic theories. To exemplify
one of them, we consider a system of $N$ active particles without alignment
interactions. Then, we write down the Smoluchowski equation for the
$N$-particle probability density and integrate out variables to obtain the
one-particle density field. This approach has been used, for instance, to
formulate a microscopic theory of MIPS in overdamped ABPs with positions
$\vec{r}_{i}$ and orientations $\varphi_{i}$ [123]. Let
$\mathbf{X}=\\{\vec{r}_{1},...,\vec{r}_{N},\varphi_{1},...,\varphi_{N}\\}$
denote the state of the $N$-particle system. The corresponding Smoluchowski
equation [124] for the joint probability distribution $\psi_{N}(\mathbf{X},t)$
reads

$\displaystyle\frac{\partial\psi_{N}}{\partial t}=$
$\displaystyle{\sum\limits_{k=1}^{N}}\nabla_{\vec{r}_{k}}\cdot\left[\frac{\left(\nabla_{\vec{r}_{k}}U\right)}{\gamma}-v_{0}\vec{p}_{k}+D\nabla_{\vec{r}_{k}}\right]\psi_{N}$
$\displaystyle+D_{\text{R}}{\sum\limits_{k=1}^{N}}\frac{\partial^{2}\psi_{N}}{\partial\varphi_{k}^{2}},$
(14)

with
$U=\sum_{k<k^{\prime}}u\left(\left|\vec{r}_{k}-\vec{r}_{k^{\prime}}\right|\right)$,
interaction potential $u(r)$, and self propulsion along $\vec{p}_{k}$ with
speed $v_{0}$. The Smoluchowski equation ensures probability conservation and
its physical interpretation is illustrated in Fig. 6. Starting from Eq. (14),
one usually derives an equation of motion for the one-particle probability
distribution $\psi_{1}(\vec{r}_{1},\varphi_{1},t)$ by integration. Due to the
pair interactions, the resulting equation still contains terms which include
the two-particle probability distribution. Similarly, one can derive an
equation for the two-particle probability distribution, which then includes
the three-particle distributions, and so on, leading to a hierarchy of coupled
differential equations that have to be closed by a suitable closure scheme
[125, 126]. Afterwards, an equation of motion for the particle density
$\rho(\vec{r},t)$ can be derived by integrating over the orientation
$\varphi$, which typically couples again to higher moments and leads to a
second hierarchy of equations, which has again to be closed using a suitable
closure scheme. To study phase separation, one possible approximation to avoid
the first type of hierarchy is to assume that the density varies slowly in
space such that the local density is constant within the range of the
interaction potential resulting in an effective self-propulsion speed
$v(\rho)=v_{0}-\zeta\rho$ with constant $\zeta$. This density-dependent self-
propulsion speed effectively accounts, to some extend, for the net effect of
the repulsive interactions, namely the slowdown of active particles in regions
of high density. The result of this microscopic approach fits well to computer
simulations of ABPs and predicts MIPS in overdamped ABPs [123].

Figure 6: Interpretation of the Smoluchowski equation [cf. Eq. (14)] for an
ABP as probability conservation law. The contribution of each particle to the
probability current can be decomposed into a translational and a rotational
current. The former includes translational diffusion and a drift term due to
the interaction potential as well as the self-propulsion velocity, whereas the
latter considers rotational diffusion. The Smoluchowski equation can then be
interpreted as a continuity equation ensuring probability conservation.

An alternative approach, sometimes called the “Dean approach” [127], is based
on an explicit coarse-graining of the Langevin equations for the individual
particles. This approach has been applied in several works, e.g., to describe
MIPS in systems of RTPs [68], pattern formation in self-propelled particles
with alignment interactions [101], collective phenomena in systems of CAPs
[81], pattern formation in systems of phoretically interacting active colloids
[112], or active systems showing nematic order [128]. Here, one uses Itô
calculus [127, 129, 130] to deduce a stochastic differential equation, which
involves multiplicative noise, for the (fluctuating) combined probability
$f(\vec{r},\varphi,t)=\sum_{i=1}^{N}\delta(\vec{r}_{i}(t)-\vec{r})\delta(\varphi_{i}-\varphi)$
to find one particle with orientation $\varphi$ at position $\vec{r}$ at time
$t$. To derive the one-particle density field $\rho(\vec{r},t)$, one can then,
for example, choose to neglect the multiplicative noise term (mean field) and
derive a hierarchy of equations in a similar way to the Smoluchowski approach.

Example: Microscopic theories for polar active matter
The aforementioned continuum theories for dry active matter were focused on
isotropic active matter that can be described by only considering the density
field. However, if the particles feature alignment interactions such as in the
Vicsek model, polar order can arise. Thus, describing these systems
additionally requires the consideration of the mean local orientation of the
particles by means of a polarization density $\vec{p}(\vec{r},t)$.
Corresponding theories for the density field and the polarization density can
be derived based on the Smoluchowski approach or the Dean approach. Another
approach, which is aimed at describing the collective behavior of the Vicsek
model (which is discrete in time in its original formulation) and is given by
Ref. [131], is based on the Liouville equation for the $N$-particle
probability density $\psi_{N}(\mathbf{X},t)$ and applied to the well-known
Vicsek model [98]. Within this model, the particles only interact during a
collision event by aligning their orientation to that of their next neighbors
and the orientation is subject to Gaussian white noise. Under the assumption
that the particles are uncorrelated prior to a collision, the $N$-particle
density is written as a product of one-particle densities, which is a good
approximation if the noise strength is large and if the mean-free path between
two collisions is larger than the interaction radius. Then, the one-particle
probability distribution is obtained by integration. However, the solution
contains complicated collision integrals that are approximated using the
Chapman-Enskog expansion [132], which takes the stationary state as a
reference and expands around it in powers of the gradients. Finally, this
leads to a set of two coupled differential equations for $\rho$ and $\vec{p}$.
This set of equations is similar to that of the phenomenological Toner-Tu
model [11] except for additional gradient terms, which occur only in the
microscopic approach.

Independently of whether a theory is phenomenological or microscopic, the
relevant field equations can then be studied based on various analytical and
numerical techniques ranging from perturbation theories, linear stability
analyses, or dynamical renormalization group calculations in the presence of
additional noise terms to explicit numerical solutions based on, e.g., finite
difference, finite volume, or finite element methods.

##  V Hydrodynamics of microswimming: Low Reynolds number and Stokes flow

The ABP model and its alternatives do not resolve the self-propulsion
mechanism, but instead involve an effective force to phenomenologically model
the resulting directed motion. To understand and describe the self-propulsion
mechanism of a microswimmer, one has to explicitly model the flow field
produced by the microswimmer and its interaction with the body of the swimmer.

Microhydrodynamics: Let us now briefly discuss the basic equations which are
involved in the modeling of a single microswimmer. While swimming at the
macroscale involves inertia and leads to flow fields which are described by
the Navier-Stokes equation, microswimmers have to employ swimming mechanisms
which work even in the absence of inertia since, at the microscale, viscous
effects dominate over inertial effects. This is quantified by the Reynolds
number, which measures the relative importance of inertial and viscous forces
and is given by $\mathrm{Re}=(\rho Lv)/\eta$, where the numerator represents
the product of the fluid density, a characteristic length scale, and a typical
flow speed, whereas the denominator contains the solvent viscosity. For
microswimmers, $\mathrm{Re}\ll 1$: For E. coli bacteria in water, for example,
we have $L\sim 3\,\mu\text{m},\leavevmode\nobreak\ v\sim
30\,\mu\text{m/s},\leavevmode\nobreak\ \eta=0.001\,\text{Pa s}$, and
$\rho=1\,\text{g/cm}^{3}$ [49]. Thus, $\text{Re}\sim 10^{-5}-10^{-4}\ll 1$ and
inertial effects can safely be neglected. For comparison, phenomena occurring
at high Reynolds numbers, such as turbulence, often occur at $\text{Re}\sim
10^{3}-10^{4}$ [133].

At low Reynolds number, the Navier-Stokes equation reduces to the Stokes
equation, which describes “creeping flow” and reads

$\eta\nabla^{2}\vec{u}-\nabla p+\vec{f}=0,$ (15)

where $\vec{u}(\vec{r},t)$ and $p(\vec{r},t)$ are the solvent velocity field
and the pressure field, respectively, and $\vec{f}(\vec{r},t)$ is the force
density representing the forces exerted by the microswimmers on the solvent.
The Stokes equation is typically complemented by the incompressibility
condition $\nabla\cdot\vec{u}=0$ leading to a complete set of equations to
determine $\vec{u}(\vec{r},t)$ and $p(\vec{r},t)$ for a given
$\vec{f}(\vec{r},t)$ and given boundary conditions. Notably, the Stokes
equation does not contain any time derivatives, and therefore, the solvent
responds instantaneously to the applied forces (no motion would take place
once the forcing term is switched off), which reflects the absence of inertia.
Accordingly, the swimming mechanism of scallops, which move by periodically
opening and closing their shells, would not work at low Reynolds number [cf.
Fig. 7 (a)]. Likewise, any other mechanism based on reciprocal motions would
not lead to directed motion. This is Purcell’s scallop theorem [134].

Figure 7: (a) Motion of a scallop. By quickly closing and slowly opening the
two shells, the scallop produces a net flow and starts moving. At low Reynolds
number, the net displacement is zero for this reciprocal motion (taken from
Ref. [135]). (b) Schematic of the non-reciprocal motion of an E. coli
bacterium (taken from Ref. [136]). (c) Electron microscope image of
Chlamydomonas reinhardtii algae showing the flagella producing self propulsion
by non-reciprocal motion (taken from Ref. [137]).

The general procedure to model microswimmers which move by body-shape
deformations (or squirmers) at low Reynolds numbers consists in solving the
Stokes equation with appropriate boundary conditions for the solvent velocity
field $\vec{u}$ on the surface of the microswimmers. This yields the solvent
velocity field $\vec{u}$, from which the stress tensor
$\mathbf{\sigma}=\eta\left(\nabla\otimes\vec{u}+(\nabla\otimes\vec{u})^{\text{T}}\right)$
can be obtained. The latter then allows one to calculate the total force
$\vec{F}=\int_{S}\text{d}S\,\mathbf{\sigma}(\vec{r},t)\hat{n}$ and the torque
$\vec{T}=\int_{S}\text{d}S\,\vec{r}\times(\mathbf{\sigma}(\vec{r},t)\hat{n})$
which act on the microswimmer, where $S$ and $\text{d}S$ denote the surface of
the microswimmer and a differential element of it, respectively. Then, for a
solid particle, the rigidity condition

$\vec{u}(\vec{r})=\vec{v}+\vec{\omega}\times\vec{r},\quad\vec{r}\in S$ (16)

is typically assumed to apply at the surface $S$ of the particle and links the
particle velocity $\vec{v}$ and angular velocity $\vec{\omega}$ to $\vec{F}$
and $\vec{T}$. Finally, the torque-free ($\vec{T}=0$) and force-free
($\vec{F}=0$) conditions allow one to solve for $\vec{v}$ and $\vec{\omega}$
[49, 138]. Since microswimmers often deform in a cyclic way, the net
displacement during one cycle of period $T$ is given by
$\int_{0}^{T}\text{d}t\,\vec{v}(t)$, which is zero for reciprocal movement in
the regime of low Reynolds numbers [34]. Thus, non-reciprocal body-shape
deformations are required to produce directed motion. Two examples of
biological microswimmers that self propel by non-reciprocal motion are
demonstrated in Fig. 7 (b) and (c). A minimal microswimmer model can be
constructed, e.g., based on three spheres connected by two arms, which
periodically change their length (three-sphere swimmer) [139, 140, 141, 142,
143] or based on two spheres which can contract or expand radially and are
connected by an elastic arm [140, 144, 145].

##  VI Modeling hydrodynamics at the many-particle level

In ensembles of microswimmers, each of them generates a specific flow pattern
which typically decays slowly in space and leads to long-ranged hydrodynamic
cross interactions among different microswimmers as well as to hydrodynamic
(self) interactions with walls and interfaces. These hydrodynamic interactions
are typically not included in models of dry active matter such as the ABP
model and its alternatives. One way of simulating several interacting
microswimmers is to explicitly model the detailed self-propulsion mechanism of
each microswimmer, i.e., to alternately solve the Stokes equation with the
microswimmer-solvent boundary conditions for all swimmers simultaneously and
to propagate the swimmers based on the force- and torque-free conditions.
While such an approach is conceptually relatively simple and accurate in
principle, it creates a huge numerical effort and typically becomes unfeasible
even for moderately large microswimmer ensembles. In the following, we briefly
discuss some alternative approaches, which allow for more efficient numerical
descriptions of microswimmer ensembles.

Minimal models and hydrodynamic far-field interactions:
To model the dynamics of large microswimmer ensembles, an explicit modeling of
the solvent flow including the detailed particle-solvent boundary conditions
occurring in real microswimmers is often numerically so demanding that very
large system sizes remain unreachable. Therefore, one often looks for a
compromise between the ABP model, which neglects hydrodynamic interactions and
momentum conservation altogether, and an explicit modeling of the self-
propulsion mechanism of all interacting microswimmers in a given ensemble. One
common approach involves formulating hydrodynamically consistent minimal
models for the collective behavior of microswimmers, where one does not
explicitly describe the self-propulsion mechanism of each microswimmer but
replaces each microswimmer with a simpler representative that creates a
similar (far-field) flow pattern. To this end, one uses a multipole expansion
of the flow field (similar to that used, e.g., in electrodynamics) [146, 147,
148, 149] and only considers the leading-order terms. In the simplest case,
these are the so-called “singularity solutions” of the Stokes equation (e.g.,
the flow field of a force dipole), which are then used to replace the flow
field created by each microswimmer and are equivalent to the far-field flow
pattern generated by the actual microswimmer to be modeled. For example, it is
well known that E. coli bacteria produce essentially the same far-field flow
pattern as a force dipole (pusher) [30] and Chlamydomonas algae produce a far-
field flow pattern which can be represented by the flow field produced by an
oscillatory force dipole [150]. Let us briefly discuss three common
singularity solutions of the Stokes equation:

  * (i)

Point force (“Stokeslet”): The flow generated by a point force
$\vec{f}_{\text{p}}=f\hat{e}\delta(\vec{r}-\vec{r}_{0})$ placed at position
$\vec{r}_{0}$ and pointing along the direction $\hat{e}$ is similar to the
far-field flow of a particle that is driven by an external force [138, 147].
By setting $\vec{f}=\vec{f}_{\text{p}}$ in the Stokes equation [cf. Eq. (15)],
the resulting velocity field reads

$\vec{u}_{\text{PF}}(\vec{r})=\frac{f}{8\pi\eta
r}\left[\hat{e}+\left(\hat{r}\cdot\hat{e}\right)\hat{r}\right],$ (17)

where $r=|\vec{r}|$, $\hat{r}=\vec{r}/r$, and $\eta$ denotes the viscosity of
the solvent. The velocity field is shown in Fig. 8 (a). Since microswimmers
are force free (momentum conservation), the Stokeslet solution alone is
unsuitable to represent them.

Figure 8: Illustration of the velocity field $\vec{u}(\vec{r})$ of (a) a point
force, (b) a force dipole, and (c) a source dipole.

  * (ii)

Force dipole: The far-field solution of the Stokes equation in the presence of
two point forces
$\vec{f}_{+}=f\hat{e}\delta(\vec{r}-\vec{r}_{0}-(l/2)\hat{e})$ and
$\vec{f}_{-}=-f\hat{e}\delta(\vec{r}-\vec{r}_{0}+(l/2)\hat{e})$, which are
separated by a distance $l$, reads

$\vec{u}_{\text{FD}}(\vec{r})=\frac{fl}{8\pi\eta
r^{2}}\left[3\left(\hat{e}\cdot\hat{r}\right)^{2}-1\right]\hat{r}$ (18)

in the limit $l\rightarrow 0$ or at distances $r\gg l$ and it is represented
in Fig. 8 (b) for $f>0$ [138, 147]. These force dipoles push fluid molecules
in the forward and backward directions. Hence, microswimmers that show this
kind of far-field flow pattern are called “pushers”, whereas the case of
$f<0$, where all flow field lines are reverted, corresponds to a “puller”.

  * (iii)

Source dipole: The point-force and force-dipole solutions are obtained by
solving the Stokes equation together with the incompressibility condition
$\nabla\cdot\vec{u}=0$. In the presence of sources of solvent molecules, the
Stokes equation is unchanged, but the incompressibility condition changes to
$\nabla\cdot\vec{u}(\vec{r})=s(\vec{r})$, where $s(\vec{r})$ denotes the
source density [147]. While a point source is of limited relevance (it would
lead to a net flow of solvent molecules entering or leaving the domain), the
source dipole is an important singular solution to the Stokes equation. Its
source density consists of two point sources
$s_{+}(\vec{r})=Q\delta(\vec{r}-\vec{r}_{0}-(l/2)\hat{e})$ (source of solvent
molecules) and $s_{-}(\vec{r})=-Q\delta(\vec{r}-\vec{r}_{0}+(l/2)\hat{e})$
(sink of solvent molecules) that are separated by a distance $l$, where $Q>0$
denotes the magnitude of the source densities. The corresponding solution to
the Stokes equation in the limit $l\rightarrow 0$ reads [147]

$\vec{u}_{\text{SD}}(\vec{r})=\frac{Ql}{4\pi
r^{3}}\left[3\left(\hat{e}\cdot\hat{r}\right)\hat{r}-\hat{e}\right]$ (19)

and its velocity field is demonstrated in Fig. 8 (c).

Since self-propelled particles are force free, the simplest representation of
active particles by singularity solutions of the Stokes equation is given by
source and force dipoles. Examples of simulations of microswimmer models based
on these singularity solutions comprise, e.g., studies of motile suspensions
of active rod-like particles [151], of the dynamics of a single molecule
composed of microswimmers [152], of RTPs with hydrodynamic interactions [153],
or of microswimmers near boundaries [154].

To simulate microswimmers based on singularity solutions of the Stokes
equation, one often models the external fluid velocity field $\vec{u}$ as a
sum of all microswimmer singularity solutions and applies certain boundary
conditions on the surface of each microswimmer. The velocity $\vec{v}$ of each
microswimmer is then calculated using the force-free and torque-free
conditions based on the stress tensor, as previously discussed for a single
microswimmer, via numerical integration. To obtain self propulsion, one shifts
the force or source dipole away from the center of the particles [151, 152].
Moreover, one can also combine the singularity solutions with numerical
solvers such as the Lattice-Boltzmann method [155] discussed below. Beside
simulations, the force and source dipole models are used to develop continuum
theories for active matter with hydrodynamic interactions, which we will
discuss in the last section of this article.

Squirmer models:
An alternative (not necessarily unrelated) approach to formulate
hydrodynamically consistent models of microswimmers is to consider squirmers,
i.e., spherical particles with a prescribed solvent flow along the surface
(without explicitly modeling the origin of the latter) [156, 157, 158, 159,
160, 146, 161, 162, 163, 164, 165]. On the surface of the squirmer particle,
the vertical fluid velocity is set to zero and the tangential surface velocity
is prescribed by a series of first derivatives of Legendre polynomials, which
can be used, e.g., to model the net effect of autophoresis, which leads to a
slip velocity across the surface of Janus particles [166]. The squirmer model
has been used in several works, e.g., in combination with the lattice-
Boltzmann method [163, 167, 168] or multi-particle collision dynamics
simulations [156, 157, 159, 160, 161, 162, 165, 164].

In contrast to the ABP model and its alternatives, microswimmer models based
on combinations of singularity solutions of the Stokes equation or on
squirmers are momentum conserving and can correctly describe hydrodynamic
interactions at large inter-particle distances for a given active system.
However, they do not necessarily account for the correct hydrodynamic near-
field interactions and are therefore mainly useful to model active systems at
low density (squirmer models, when used to represent Janus particles, may
serve as an exception, which is expected to correctly describe hydrodynamic
interactions down to distances on the order of the slip length [166]). These
effective models are often used also as a starting point for continuum
theories as briefly discussed further below.

Explicit simulations of the solvent:
In the following, we briefly introduce several numerical methods which are
frequently used in active matter physics to explicitly determine the flow
field and to simulate hydrodynamic interactions, often beyond the far-field
approximation.

_Lattice-Boltzmann method:_ One popular method to solve fluid dynamics
problems is the lattice-Boltzmann method (LBM), where one solves the Boltzmann
equation instead of the (Navier-)Stokes equation and exploits the fact that
the latter equation can be derived from the former [169, 170, 171, 172, 173,
174, 163, 175]. Interestingly, the Boltzmann equation is numerically often
more convenient when combined with suitable approximations. It describes the
particle distribution function $f(\vec{r},\vec{v},t)$, which is the density of
particles with velocity $\vec{v}$ at position $\vec{r}$ and time $t$. With the
so-called collision operator $\Omega(f)$, the Boltzmann equation reads [176,
170]

$\frac{\partial f}{\partial
t}+\vec{v}\cdot\nabla_{\vec{r}}f+\frac{\vec{F}}{m}\cdot\nabla_{\vec{v}}f=\Omega(f),$
(20)

where $m$ denotes the mass of the particles and $\vec{F}$ is the external
force field acting on them. The second term on the left-hand side describes
advection of the particles with velocity $\vec{v}$, whereas the third term
describes external forces acting on the solvent particles and affecting their
velocity. The source term on the right-hand side of Eq. (20) describes the
local redistribution of the solvent particles due to collisions. This
collision operator is often approximated by
$\Omega(f)=-(f-f_{\text{eq}})/\tau$, which describes the relaxation of the
distribution $f$ towards the equilibrium distribution $f_{\text{eq}}$ on the
time scale $\tau$ and is known as the Bhatnagar–Gross–Krook (BGK) collision
operator [177]. In the LBM, the continuous Boltzmann equation [i.e., Eq. (20)]
is discretized in position, velocity, and time and numerically solved on a
lattice with spacing $\Delta x$ at discrete times with time step $\Delta t$.
The velocity $\vec{v}$ can only take discrete values $\vec{c}_{i}$, which are
given by a discrete set $\\{\vec{c}_{i},w_{i}\\}$ with weights $w_{i}$. The
discretized Boltzmann equation is then solved numerically as discussed, e.g.,
in Ref. [170]. To simulate microswimmers that, e.g., create directed motion
through body-shape deformations, one often describes the microswimmer surface
as a set of boundary links that define a closed surface and solves the
discretized Boltzmann equation together with suitable boundary conditions
[170].

_Multi-particle collision dynamics:_
Another popular approach to simulate the dynamics of microswimmers is based on
multi-particle collision dynamics (MPCD), where, in contrast to the LBM, the
solvent is represented by point-like particles which have continuous positions
and velocities [178, 179, 180, 181, 182, 183, 184]. To model active particles,
one usually combines the MPCD method for the solvent molecules with molecular
dynamics (MD) simulations of the active particles, which are coupled to the
solvent and are represented either as a single particle or by a quasi-
continuous distribution of particles which are connected with (time-dependent)
springs and represent the surface of a (deformable) microswimmer [185]. The
MPCD method has been used in several works to investigate, e.g., chemotactic
Janus colloids [186], active particles with phoretic interactions [187],
dynamics of active particles in chemically active media [188], the motion of
squirmers [162, 164, 157], the influence of hydrodynamic interactions on phase
separation in systems of microswimmers [160], collective behavior of sperm
cells [189], and active particles in filament networks [190].

_Dissipative-particle dynamics:_
Another coarse-grained approach to modeling the solvent is given by
dissipative-particle dynamics (DPD) simulations. Here, each DPD particle
represents a small solvent region and, similar to the MPCD simulations, the
positions and velocities of the DPD particles take continuous values. The DPD
particles interact via three types of effective forces: A weak conservative
force models the soft repulsion of the solvent molecules, a dissipative force
models the friction, and a random force accounts for thermal fluctuations.
Knowing these forces, Newton’s equation of motion is solved for the DPD
particles to obtain the hydrodynamics of the solvent [191, 192, 193]. This
model has been adapted, e.g., to active suspensions [194] and to model the
self-propulsion of Janus colloids [195].

_Microscopic solvent simulations:_
Finally, beside the previously discussed mesoscale-simulation methods,
particle-based simulations of the solvent molecules based on direct MD
simulations, which allow one to resolve very small spatial and temporal
scales, are possible. Nevertheless, these simulations are computationally very
intense, which makes it impossible to study systems of the microscale over
time scales of seconds, which are relevant to most active matter systems.
Still, this explicit modeling of the solvent has been successfully used to
model a self-propelled particle in a Lennard-Jones solvent [196].

Overall, the LBM, MPCD, and the DPD methods are mesoscale simulation methods,
which can be applied to many hydrodynamic problems in soft and active matter
physics and beyond. Since the DPD method is based on particles moving in
continous space, it avoids lattice artifacts and allows simulations capturing
much larger length and time scales than typically possible in MD simulations.
However, DPD simulations include a large number of parameters (in order to
model the different forces), which have to be chosen carefully. The MPCD
method, on its part, which models the net effect of individual collisions
rather than accounting for every collision event, is computationally very
efficient and can be efficiently parallelized. This applies also to the LBM,
which numerically solves the Boltzmann equation and is well suited, e.g., for
implementing complex (moving) boundaries [170].

##  VII Continuum theories of microswimmers with hydrodynamic interactions

Based on the previously discussed effective microswimmer models, continuum
theories for large ensembles of particles can be formulated which explicitly
account for hydrodynamic interactions at least at low density. These theories
describe wet active matter and can be formulated, e.g., based on the puller
and pusher solutions of the Stokes equation. One popular approach to account
for hydrodynamic far-field interactions is to write down the (overdamped)
equations of motion for the position and orientation of each microswimmer,
which couple with the overall fluid velocity field. The contribution of each
microswimmer to the overall velocity field is modeled by singularity solutions
of the Stokes equation such as force or source dipoles (which can be
superimposed due to the linearity of the Stokes equation). One then derives a
continuity equation for the $N$-particle probability density, which typically
takes the form of a Fokker-Planck equation [197, 198, 199, 200]. From here,
one can proceed in a similar way to that of microscopic theories for dry
active matter in order to derive an equation of motion for the one-particle
density. Since the described approach to formulate continuum theories for wet
active matter is based on the singularity solutions of the Stokes equation,
which only describe the far-field flow pattern of active particles, near-field
hydrodynamic effects are not included in this approach. However, although
complicated in practice, one can go beyond the far-field regime in principle,
e.g., by using superimposed singularity solutions to represent the flow field
contribution due to each swimmer or by starting with squirmer models.

Let us finally mention that one can alternatively formulate phenomenological
minimal models of wet active matter. Following a similar spirit to the case of
dry active matter, these models are generic in the sense that they are largely
based on considerations of symmetry, conservation laws, and dimensionality and
do not refer to details such as the specific self-propulsion mechanism, which
is employed by the microswimmers. One example of such a minimal model for wet
active matter is given by the phenomenological active model H [201], which
accounts for momentum conservation. It is based on the active model B [121]
and is closely related to the model H for equilibrium systems [119]. The
active model H addresses the phase separation behavior of wet active matter
and couples the generalized density field $\phi(\vec{r},t)$ to the velocity
field $\vec{v}(\vec{r},t)$ of the solvent. The general idea is that diffusive
dynamics of the active particles take place in the moving frame of the solvent
and the velocity field of the solvent is given by the corresponding Navier-
Stokes equation. There are also phenomenological models for specific phenomena
such as bacterial turbulence, which are based on phenomenological equations to
describe the fluid velocity field [202].

More generally, there is a large number of alternative approaches to
formulating continuum theories for microswimmers. Readers interested in
further details are referred to Refs. [103, 146, 34].

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