In this tutorial we explore the ways to simulate self-propulsion in the
simulation software package **ESPResSo**. We consider three examples that illustrate
the properties of these systems. First, we study the concept of enhanced
diffusion of a self-propelled particle. Second, we investigate rectification in
an asymmetric geometry. Finally, we determine the flow field around a
self-propelled particle using lattice-Boltzmann simulations (LB). These three
subsections should give insight into the basics of simulating active matter
with **ESPResSo**. This tutorial assumes basic knowledge of Python and **ESPResSo**,
as well as the use of lattice-Boltzmann within **ESPResSo**. It is therefore
recommended to go through the relevant tutorials first, before attempting this one.

Active matter is a term that describes a class of systems, in which energy is constantly consumed to perform work. These systems are therefore highly out-of-equilibrium (thermodynamically) and (can) thus defy description using the standard framework of statistical mechanics. Active systems are, however, ubiquitous. On our length scale, we encounter flocks of birds, schools of fish, and, of course, humans; on the mesoscopic level examples are found in bacteria, sperm, and algae; and on the nanoscopic level, transport along the cytoskeleton is achieved by myosin motors. This exemplifies the range of length scales which the field of active matter encompasses, as well as its diversity. Recent years have seen a huge increase in studies into systems consisting of self-propelled particles, in particular artificial ones in the colloidal regime. These self-propelled colloids show promise as physical model systems for complex biological behavior (bacteria moving collectively) and could be used to answer fundamental questions concerning out-of-equilibrium statistical physics. Simulations can also play an important role in this regard, as the parameters are more easily tunable and the results ‘cleaner’ than in experiments. The above should give you some idea of the importance of the field of active matter and why you should be interested in performing simulations in it.

The `ENGINE` feature offers intuitive syntax for adding self-propulsion to
a particle. The propulsion will occur along the vector that defines the
orientation of the particle (henceforth referred to as ‘director’). In **ESPResSo**
the orientation of the particle is defined by a quaternion; this in turn
defines a rotation matrix that acts on the particle's initial orientation
(along the z-axis), which then defines the particles current orientation
through the matrix-oriented vector.
Within the `ENGINE` feature there are two ways of setting up a self-propelled
particle, with and without hydrodynamic interactions. The particle without
hydrodynamic interactions will be discussed first, as it is the simplest case.

For this type of self-propulsion the Langevin thermostat can be used. The
Langevin thermostat imposes a velocity-dependent friction on a particle.
When a constant force is applied along the director,
the friction causes the particle to attain a terminal velocity, due to the balance
of driving and friction force, see Fig. 1. The exponent with
which the particle's velocity relaxes towards this value depends on the
strength of the friction and the mass of the particle. The `ENGINE`
feature implies that rotation of the particles (the `ROTATION` feature) is
compiled into **ESPResSo**. The particle can thus reorient due to external torques or
due to thermal fluctuations, whenever the rotational degrees of freedom are
thermalized. Note that the rotation of the particles has to be enabled
explicitly via their `ROTATION` property. This ‘engine’ building block can
be connected to other particles, *e.g.*, via the virtual sites (rigid
body) to construct complex self-propelled objects.

First we import the necessary modules, define the parameters and set up the system.

In [1]:

```
%matplotlib inline
import matplotlib.pyplot as plt
```

In [2]:

```
plt.rcParams.update({'font.size': 18})
import tqdm
import numpy as np
import espressomd.observables
import espressomd.accumulators
espressomd.assert_features(
["ENGINE", "ROTATION", "MASS", "ROTATIONAL_INERTIA", "CUDA"])
```

In [3]:

```
ED_PARAMS = {'time_step': 0.01,
'box_l': 3*[10.],
'skin': 0.4,
'active_velocity': 5,
'kT': 1,
'gamma': 1,
'gamma_rotation': 1,
'mass': 0.1,
'rinertia': 3*[1.],
'corr_tmax': 100}
ED_N_SAMPLING_STEPS = 5000000
```

In [4]:

```
system = espressomd.System(box_l=ED_PARAMS['box_l'])
system.cell_system.skin = ED_PARAMS['skin']
system.time_step = ED_PARAMS['time_step']
```

- Set up a Langevin thermostat for translation and rotation of the particles.

In [5]:

```
system.thermostat.set_langevin(kT=ED_PARAMS['kT'],
gamma=ED_PARAMS['gamma'],
gamma_rotation=ED_PARAMS['gamma_rotation'],
seed=42)
```

The configuration for the Langevin-based swimming is exposed as an attribute of
the `ParticleHandle` class of **ESPResSo**, which represents a particle in the
simulation. You can either set up the self-propulsion during the creation of a
particle or at a later stage.

In [6]:

```
part_act = system.part.add(pos=[5.0, 5.0, 5.0], swimming={'v_swim': ED_PARAMS['active_velocity']},
mass=ED_PARAMS['mass'], rotation=3 * [True], rinertia=ED_PARAMS['rinertia'])
part_pass = system.part.add(pos=[5.0, 5.0, 5.0],
mass=ED_PARAMS['mass'], rotation=3 * [True], rinertia=ED_PARAMS['rinertia'])
```

Next we set up three **ESPResSo** correlators for the Mean Square Displacement (MSD), Velocity Autocorrelation Function (VACF) and the Angular Velocity Autocorrelation Function (AVACF).

In [7]:

```
pos_obs = espressomd.observables.ParticlePositions(
ids=[part_act.id, part_pass.id])
msd = espressomd.accumulators.Correlator(obs1=pos_obs,
corr_operation="square_distance_componentwise",
delta_N=1,
tau_max=ED_PARAMS['corr_tmax'],
tau_lin=16)
system.auto_update_accumulators.add(msd)
vel_obs = espressomd.observables.ParticleVelocities(
ids=[part_act.id, part_pass.id])
vacf = espressomd.accumulators.Correlator(obs1=vel_obs,
corr_operation="componentwise_product",
delta_N=1,
tau_max=ED_PARAMS['corr_tmax'],
tau_lin=16)
system.auto_update_accumulators.add(vacf)
ang_obs = espressomd.observables.ParticleAngularVelocities(
ids=[part_act.id, part_pass.id])
avacf = espressomd.accumulators.Correlator(obs1=ang_obs,
corr_operation="componentwise_product",
delta_N=1,
tau_max=ED_PARAMS['corr_tmax'],
tau_lin=16)
system.auto_update_accumulators.add(avacf)
```

No more setup needed! We can run the simulation and plot our observables.

In [8]:

```
for i in tqdm.tqdm(range(100)):
system.integrator.run(int(ED_N_SAMPLING_STEPS/100))
```

In [9]:

```
system.auto_update_accumulators.remove(msd)
msd.finalize()
system.auto_update_accumulators.remove(vacf)
vacf.finalize()
system.auto_update_accumulators.remove(avacf)
avacf.finalize()
```

In [10]:

```
taus_msd = msd.lag_times()
msd_result = msd.result()
msd_result = np.sum(msd_result, axis=2)
taus_vacf = vacf.lag_times()
vacf_result = np.sum(vacf.result(), axis=2)
taus_avacf = avacf.lag_times()
avacf_result = np.sum(avacf.result(), axis=2)
```

In [11]:

```
fig_msd = plt.figure(figsize=(10, 6))
plt.plot(taus_msd, msd_result[:, 0], label='active')
plt.plot(taus_msd, msd_result[:, 1], label='passive')
plt.xlim((taus_msd[1], None))
plt.loglog()
plt.xlabel('t')
plt.ylabel('MSD(t)')
plt.legend()
plt.show()
```

The Mean Square Displacement of an active particle is characterized by a longer ballistic regime and an increased diffusion coefficient for longer lag times. In the overdamped limit it is given by

$$ \langle r^{2}(t) \rangle = 6 D t + \frac{v^{2} \tau^{2}_{R}}{2} \left[ \frac{2 t}{\tau^{2}_{R}} + \exp\left( \frac{-2t}{\tau^{2}_{R}} \right) - 1 \right], $$where $\tau_{R} = \frac{8\pi\eta R^{3}}{k_{B} T}$ is the characteristic time scale for rotational diffusion and $ D = \frac{k_B T}{\gamma}$ is the translational diffusion coefficient.

For small times ($t \ll \tau_{R}$) the motion is ballistic

$$\langle r^{2}(t) \rangle = 6 D t + v^{2} t^{2},$$while for long times ($t \gg \tau_{R}$) the motion is diffusive

$$\langle r^{2}(t) \rangle = (6 D + v^{2}\tau_{R}) t.$$Note that no matter the strength of the activity, provided it is some finite value, the crossover between ballistic motion and enhanced diffusion is controlled by the rotational diffusion time.

The passive particle also displays a crossover from a ballistic to a diffusive motion. However, the crossover time $\tau_{C}=\frac{m}{\gamma}$ is not determined by the rotational motion but instead by the mass of the particles.

From the longterm MSD of the active particles we can define an effective diffusion coefficient $D_{\mathrm{eff}} = D + v^{2}\tau_{R}/6$. One can, of course, also connect this increased diffusion with an effective temperature. However, this apparent equivalence can lead to problems when one then attempts to apply statistical mechanics to such systems at the effective temperature. That is, there is typically more to being out-of-equilibrium than can be captured by a simple remapping of equilibrium parameters, as we will see in the second part of the tutorial.

From the autocorrelation functions of the velocity and the angular velocity we can see that the activity does not influence the rotational diffusion. Yet the directed motion for $t<\tau_{R}$ leads to an enhanced correlation of the velocity.

In [12]:

```
def acf_stable_regime(x, y):
"""
Remove the noisy tail in autocorrelation functions of finite time series.
"""
cut = np.argmax(y <= 0.) - 2
assert cut >= 1
return (x[1:cut], y[1:cut])
```

In [13]:

```
fig_vacf = plt.figure(figsize=(10, 6))
plt.plot(*acf_stable_regime(taus_vacf, vacf_result[:, 0]), label='active')
plt.plot(*acf_stable_regime(taus_vacf, vacf_result[:, 1]), label='passive')
plt.xlim((taus_vacf[1], None))
plt.loglog()
plt.xlabel('t')
plt.ylabel('VACF(t)')
plt.legend()
plt.show()
```

In [14]:

```
fig_avacf = plt.figure(figsize=(10, 6))
plt.plot(*acf_stable_regime(taus_avacf, avacf_result[:, 0]), label='active')
plt.plot(*acf_stable_regime(taus_avacf, avacf_result[:, 1]), label='passive')
plt.xlim((taus_avacf[1], None))
plt.loglog()
plt.xlabel('t')
plt.ylabel('AVACF(t)')
plt.legend()
plt.show()
```

Before we go to the second part, it is important to clear the state of the system.

In [15]:

```
def clear_system(system):
system.part.clear()
system.thermostat.turn_off()
system.constraints.clear()
system.auto_update_accumulators.clear()
system.time = 0.
```

In [16]:

```
clear_system(system)
```

In the second part of this tutorial you will consider the ‘rectifying’ properties of certain asymmetric geometries on active systems. Rectification can be best understood by considering a system of passive particles first. In an equilibrium system, for which the particles are confined to an asymmetric box with hard walls, we know that the particle density is homogeneous throughout. However, in an out-of-equilibrium setting one can have a heterogeneous distribution of particles, which limits the applicability of an ‘effective’ temperature description.

The geometry we will use is a cylindrical system with a funnel dividing two halves of the box as shown in Fig. 2.

In [17]:

```
import espressomd.shapes
import espressomd.math
```

In [18]:

```
RECT_PARAMS = {'length': 100,
'radius': 20,
'funnel_inner_radius': 3,
'funnel_angle': np.pi / 4.0,
'funnel_thickness': 0.1,
'n_particles': 500,
'active_velocity': 7,
'time_step': 0.01,
'wca_sigma': 0.5,
'wca_epsilon': 0.1,
'skin': 0.4,
'kT': 0.1,
'gamma': 1.,
'gamma_rotation': 1}
RECT_STEPS_PER_SAMPLE = 100
RECT_N_SAMPLES = 500
TYPES = {'particles': 0,
'boundaries': 1}
box_l = np.array(
[RECT_PARAMS['length'], 2*RECT_PARAMS['radius'], 2*RECT_PARAMS['radius']])
system.box_l = box_l
system.cell_system.skin = RECT_PARAMS['skin']
system.time_step = RECT_PARAMS['time_step']
system.thermostat.set_langevin(
kT=RECT_PARAMS['kT'], gamma=RECT_PARAMS['gamma'], gamma_rotation=RECT_PARAMS['gamma_rotation'], seed=42)
```

In [19]:

```
cylinder = espressomd.shapes.Cylinder(
center=0.5 * box_l,
axis=[1, 0, 0], radius=RECT_PARAMS['radius'], length=RECT_PARAMS['length'], direction=-1)
system.constraints.add(shape=cylinder, particle_type=TYPES['boundaries'])
# Setup walls
wall = espressomd.shapes.Wall(dist=0, normal=[1, 0, 0])
system.constraints.add(shape=wall, particle_type=TYPES['boundaries'])
wall = espressomd.shapes.Wall(dist=-RECT_PARAMS['length'], normal=[-1, 0, 0])
system.constraints.add(shape=wall, particle_type=TYPES['boundaries'])
```

Out[19]:

In [20]:

```
funnel_length = (RECT_PARAMS['radius']-RECT_PARAMS['funnel_inner_radius']
)/np.tan(RECT_PARAMS['funnel_angle'])
```

- Using
`funnel_length`

and the geometric parameters in`RECT_PARAMS`

, set up the funnel cone (Hint: Conical Frustum)

In [21]:

```
ctp = espressomd.math.CylindricalTransformationParameters(
axis=[1, 0, 0], center=box_l/2.)
hollow_cone = espressomd.shapes.HollowConicalFrustum(
cyl_transform_params=ctp,
r1=RECT_PARAMS['funnel_inner_radius'], r2=RECT_PARAMS['radius'],
thickness=RECT_PARAMS['funnel_thickness'],
length=funnel_length,
direction=1)
system.constraints.add(shape=hollow_cone, particle_type=TYPES['boundaries'])
```

Out[21]:

- Set up a WCA potential between the walls and the particles using the parameters in
`RECT_PARAMS`

In [22]:

```
system.non_bonded_inter[TYPES['particles'], TYPES['boundaries']].wca.set_params(
epsilon=RECT_PARAMS['wca_epsilon'], sigma=RECT_PARAMS['wca_sigma'])
```

**ESPResSo** uses quaternions to describe the rotational state of particles. Here we provide a convenience method to calculate quaternions from spherical coordinates.

- Place an equal number of swimming particles (the total number should be
`RECT_PARAMS['n_particles']`

) in the left and the right part of the box such that the center of mass is exactly in the middle. (Hint: Particles do not interact so you can put multiple in the same position) - Particles must be created with a random orientation

In [23]:

```
for i in range(RECT_PARAMS['n_particles']):
pos = box_l / 2
pos[0] += (-1)**i * 0.25 * RECT_PARAMS['length']
# https://mathworld.wolfram.com/SpherePointPicking.html
theta = np.arccos(2. * np.random.random() - 1)
phi = 2. * np.pi * np.random.random()
director = [np.sin(theta) * np.cos(phi),
np.sin(theta) * np.cos(phi),
np.cos(theta)]
system.part.add(pos=pos, swimming={'v_swim': RECT_PARAMS['active_velocity']},
director=director, rotation=3*[True])
```

In [24]:

```
com_deviations = list()
times = list()
```

- Run the simulation using
`RECT_N_SAMPLES`

and`RECT_STEPS_PER_SAMPLE`

and calculate the deviation of the center of mass from the center of the box in each sample step. (Hint: Center of mass) - Save the result and the corresponding time of the system in the lists given above.

In [25]:

```
for _ in tqdm.tqdm(range(RECT_N_SAMPLES)):
system.integrator.run(RECT_STEPS_PER_SAMPLE)
com_deviations.append(system.galilei.system_CMS()[0] - 0.5 * box_l[0])
times.append(system.time)
```

In [26]:

```
def moving_average(data, window_size):
return np.convolve(data, np.ones(window_size), 'same') / window_size
```

In [27]:

```
smoothing_window = 10
com_smoothed = moving_average(com_deviations, smoothing_window)
fig_rect = plt.figure(figsize=(10, 6))
plt.plot(times[smoothing_window:-smoothing_window],
com_smoothed[smoothing_window:-smoothing_window])
plt.xlabel('t')
plt.ylabel('center of mass deviation')
plt.show()
```

Even though the potential energy inside the geometry is 0 in every part of the accessible region, the active particles are clearly not Boltzmann distributed (homogenous density). Instead, they get funneled into the right half, showing the inapplicability of equilibrium statistical mechanics.

In [28]:

```
clear_system(system)
```

In situations where hydrodynamic interactions between swimmers or swimmers and
objects are of importance, we use the lattice-Boltzmann (LB) to propagate the
fluid's momentum diffusion. We recommend the GPU-based variant of LB in **ESPResSo**,
since it is much faster. Moreover, the current implementation of the CPU
self-propulsion is limited to one CPU. This is because the ghost-node structure
of the **ESPResSo** cell-list code does not allow for straightforward MPI parallelization
of the swimmer objects across several CPUs.

Of particular importance for self-propulsion at low Reynolds number is the fact that active systems (bacteria, sperm, algae, but also artificial chemically powered swimmers) are force free. That is, the flow field around one of these objects does not contain a monopolar (Stokeslet) contribution. In the case of a sperm cell, see Fig. 3(a), the reasoning is as follows. The whip-like tail pushes against the fluid and the fluid pushes against the tail, at the same time the head experiences drag, pushing against the fluid and being pushed back against by the fluid. This ensures that both the swimmer and the fluid experience no net force. However, due to the asymmetry of the distribution of forces around the swimmer, the fluid flow still causes net motion. When there is no net force on the fluid, the lowest-order multipole that can be present is a hydrodynamic dipole. Since a dipole has an orientation, there are two types of swimmer: pushers and pullers. The distinction is made by whether the particle pulls fluid in from the front and back, and pushes it out towards its side (puller), or vice versa (pusher), see Fig. 3(c,d).

For the setup of the swimming particles with hydrodynamics we cannot use the `v_swim`

argument anymore because it is not trivial to determine the friction acting on the particle. Instead, we have to provide the keys `f_swim`

and `dipole_length`

. Together they determine what the dipole strength and the terminal velocity of the swimmer is.
One should be careful, however, the `dipole_length`

should be at least one
grid spacing, since use is made of the LB interpolation scheme. If the length
is less than one grid spacing, you can easily run into discretization artifacts
or cause the particle not to move. This dipole length together with the
director and the keyword `pusher/puller` determines where the counter
force on the fluid is applied to make the system force free, see
Fig. 3(a) for an illustration of the setup. That is to
say, a force of magnitude `f_swim`

is applied to the particle (leading
to a Stokeslet in the fluid, due to friction) and a counter force is applied to
compensate for this in the fluid (resulting in an extended dipole flow field,
due to the second monopole). For a puller the counter force is applied in front
of the particle and for a pusher it is in the back
(Fig. 3(b)).

Finally, there are a few caveats to the swimming setup with hydrodynamic
interactions. First, the stability of this algorithm is governed by the
stability limitations of the LB method. Second, since the particle is
essentially a point particle, there is no rotation caused by the fluid
flow, *e.g.*, a swimmer in a Poiseuille flow. If the thermostat is
switched on, the rotational degrees of freedom will also be thermalized, but
there is still no contribution of rotation due to ‘external’ flow fields.
It is recommended to use an alternative means of obtaining rotations in your LB
swimming simulations. For example, by constructing a raspberry
particle.

In [29]:

```
import espressomd.lb
```

In [30]:

```
HYDRO_PARAMS = {'box_l': 3*[25],
'time_step': 0.01,
'skin': 1,
'agrid': 1,
'dens': 1,
'visc': 1,
'gamma': 1,
'mass': 5,
'dipole_length': 2,
'active_force': 0.1,
'mode': 'pusher'}
HYDRO_N_STEPS = 2000
system.box_l = HYDRO_PARAMS['box_l']
system.cell_system.skin = HYDRO_PARAMS['skin']
system.time_step = HYDRO_PARAMS['time_step']
system.min_global_cut = HYDRO_PARAMS['dipole_length']
```

- Using
`HYDRO_PARAMS`

, set up a lattice-Boltzmann fluid and activate it as a thermostat (Hint: lattice-Boltzmann)

In [31]:

```
lbf = espressomd.lb.LBFluidGPU(agrid=HYDRO_PARAMS['agrid'], dens=HYDRO_PARAMS['dens'],
visc=HYDRO_PARAMS['visc'], tau=HYDRO_PARAMS['time_step'])
system.actors.add(lbf)
system.thermostat.set_lb(LB_fluid=lbf, gamma=HYDRO_PARAMS['gamma'], seed=42)
```

In [32]:

```
box_l = np.array(HYDRO_PARAMS['box_l'])
pos = box_l/2.
pos[2] = -10.
```

- Using
`HYDRO_PARAMS`

, place particle at`pos`

that swims in`z`

-direction. The particle handle should be called`particle`

.

In [33]:

```
particle = system.part.add(
pos=pos, mass=HYDRO_PARAMS['mass'], rotation=3*[False],
swimming={'f_swim': HYDRO_PARAMS['active_force'],
'mode': HYDRO_PARAMS['mode'],
'dipole_length': HYDRO_PARAMS['dipole_length']})
```

In [34]:

```
system.integrator.run(HYDRO_N_STEPS)
```

In [35]:

```
vels = np.squeeze(lbf[:, int(system.box_l[1]/2), :].velocity)
vel_abs = np.linalg.norm(vels, axis=2)
lb_shape = lbf.shape
xs, zs = np.meshgrid(np.linspace(0.5, box_l[0] - 0.5, num=lb_shape[0]),
np.linspace(0.5, box_l[2] - 0.5, num=lb_shape[2]))
fig_vels, ax_vels = plt.subplots(figsize=(10, 6))
im = plt.pcolormesh(vel_abs.T, cmap='YlOrRd')
plt.quiver(xs, zs, vels[:, :, 0].T, vels[:, :, 2].T, angles='xy', scale=0.005)
circ = plt.Circle(particle.pos_folded[[0, 2]], 0.5, color='blue')
ax_vels.add_patch(circ)
ax_vels.set_aspect('equal')
plt.xlabel('x')
plt.ylabel('z')
cb = plt.colorbar(im, label=r'$|v_{\mathrm{fluid}}|$')
plt.show()
```

We can also export the particle and fluid data to `.vtk`

format to display the results with a visualization software like ParaView.

In [36]:

```
lbf.write_vtk_velocity('./fluid.vtk')
system.part.writevtk('./particle.vtk')
```

The result of such a visualization could look like Fig. 4.

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