Active Matter¶

Table of Contents¶

  1. Introduction
  2. Active particles
  3. Enhanced Diffusion
  4. Rectification
  5. Hydrodynamics of self-propelled particles
  6. Further reading

Introduction¶

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. The 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 particles¶

Active matter is a term that describes a class of systems, in which agents (particles) constantly consume energy to perform work, for example in the form of directed motion. These systems are therefore highly out-of-equilibrium and 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.

Active Particles in ESPResSo¶

The ENGINE feature offers intuitive syntax for adding self-propulsion to a particle. Propulsion will occur along the vector that defines the orientation of the particle (henceforth referred to as ‘director’). Within the ENGINE feature there are two ways of setting up a self-propelled particle, with and without coupling to a fluid.

Self-Propulsion without Hydrodynamics¶

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 timescale on 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. The ‘engine’ building block can be connected to other particles, e.g., via the virtual sites (rigid body) to construct complex self-propelled objects.

Fig. 1: A balance of the driving force in the direction defined by the ‘director’ unit vector and the friction due to the Langevin thermostat results in a constant terminal velocity.

Enhanced Diffusion¶

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

In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
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", "WALBERLA", "VIRTUAL_SITES_RELATIVE"])
In [2]:
ED_PARAMS = {'time_step': 0.01,
             'box_l': 3*[10.],
             'skin': 0.4,
             'f_swim': 5,
             'kT': 1,
             'gamma': 1,
             'gamma_rotation': 1,
             'mass': 0.1,
             'rinertia': 3*[1.],
             'corr_tmax': 100}
ED_N_SAMPLING_STEPS = 5000000
In [3]:
system = espressomd.System(box_l=ED_PARAMS['box_l'])
system.cell_system.skin = ED_PARAMS['skin']
system.time_step = ED_PARAMS['time_step']

Exercise¶

  • Use ED_PARAMS to set up a Langevin thermostat for translation and rotation of the particles.
In [4]:
# SOLUTION CELL
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.

Exercise¶

  • Set up one active and one passive particle, call them part_act and part_pass (Hint: see the docs)
  • Use ED_PARAMS for the necessary parameters
In [5]:
# SOLUTION CELL
part_act = system.part.add(pos=[5.0, 5.0, 5.0], swimming={'f_swim': ED_PARAMS['f_swim']},
                           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 [6]:
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 [7]:
for i in tqdm.tqdm(range(100)):
    system.integrator.run(int(ED_N_SAMPLING_STEPS/100))

100%|██████████| 100/100 [00:47<00:00,  2.12it/s]
In [8]:
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 [9]:
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 [10]:
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 could be tempted to also connect this increased diffusion with an effective temperature. However, this apparent equivalence leads 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 [11]:
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 [12]:
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 [13]:
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 [14]:
def clear_system(system):
    system.part.clear()
    system.thermostat.turn_off()
    system.constraints.clear()
    system.auto_update_accumulators.clear()
    system.time = 0.
In [15]:
clear_system(system)

Rectification¶

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 where noninteracting particles are confined to any volume with hard walls as the only potential energy contribution, we know that the particle density is homogeneous throughout the whole available space according to the Boltzmann factor $e^{U(x)/k_BT}$. In an out-of-equilibrium setting however, asymmetric boundaries can lead to a heterogeneous distribution of particles in the steady state.

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

Fig. 2: Sketch of the rectifying geometry which we simulate for this tutorial.
In [16]:
import espressomd.shapes
import espressomd.math
In [17]:
RECT_PARAMS = {'length': 100,
               'radius': 20,
               'funnel_inner_radius': 3,
               'funnel_angle': np.pi / 4.0,
               'funnel_thickness': 0.1,
               'n_particles': 500,
               'f_swim': 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 [18]:
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[18]:
<espressomd.constraints.ShapeBasedConstraint at 0x76b498426350>
In [19]:
funnel_length = (RECT_PARAMS['radius']-RECT_PARAMS['funnel_inner_radius']
                 )/np.tan(RECT_PARAMS['funnel_angle'])

Exercise¶

  • Using funnel_length and the geometric parameters in RECT_PARAMS, set up the funnel cone (Hint: Use a Conical Frustum)
In [20]:
# SOLUTION CELL
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[20]:
<espressomd.constraints.ShapeBasedConstraint at 0x76b498427020>

Exercise¶

  • Set up a WCA potential between the walls and the particles using the parameters in RECT_PARAMS
In [21]:
# SOLUTION CELL
system.non_bonded_inter[TYPES['particles'], TYPES['boundaries']].wca.set_params(
    epsilon=RECT_PARAMS['wca_epsilon'], sigma=RECT_PARAMS['wca_sigma'])

Exercise¶

  • 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 should be created with a random orientation
In [22]:
# SOLUTION CELL
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={'f_swim': RECT_PARAMS['f_swim']},
                    director=director, rotation=3*[True])
In [23]:
com_deviations = list()
times = list()

Exercise¶

  • 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 [24]:
# SOLUTION CELL
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)

100%|██████████| 500/500 [00:46<00:00, 10.67it/s]
In [25]:
def moving_average(data, window_size):
    return np.convolve(data, np.ones(window_size), 'same') / window_size
In [26]:
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.

Hydrodynamics of self-propelled particles¶

In this final part of the tutorial we simulate and visualize the flow field around a self-propelled swimmer.

Of particular importance for self-propulsion at low Reynolds number is the fact that active systems are net force free. This is because the particles have to generate their propulsion on their own as opposed to being dragged by an external force. As a consequence, the far field flow around a swimmer does not contain a monopolar (Stokeslet) contribution. In the case of an E. Coli cell, see Fig. 3(a), the reasoning is as follows: The corkscrew tail pushes against the fluid and the fluid pushes against the tail, which generates the forward motion. At the same time the head experiences drag, pushing against the fluid and being pushed back against by the fluid. In total, both the swimmer and the fluid experience no net force even though the swimmer moves. 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 basic types of swimmer: pushers and pullers. The distinction is made by whether the particle pulls fluid in from the front (puller, Fig. 3(b,d)) or pushes fluid away from the back (pusher, Fig. 3(a,c)).

Fig. 3: (a) Illustration of a E. Coli cell modeled using our two-point swimmer code. The head is represented by a solid particle, on which a force is acting (black arrow). In the fluid a counter force is applied (white arrow). This generates a pusher-type particle. (b) Illustration of the puller-type Chlamydomonas algae, also represented by our two-point swimmer. (c,d) Sketch of the flow-lines around the swimmers: (c) pusher and (d) puller.

In situations where hydrodynamic interactions between swimmers or swimmers and objects are of importance, we use lattice-Boltzmann (LB) to simulate the fluid. We recommend the GPU-based variant of LB in ESPResSo, since it is much faster. Moreover, the current implementation of the CPU self-propulsion with hydrodynamics is limited to one core.

ESPResSo implements swimming coupled with hydrodynamics by applying a force to the particle along its director as in the 'dry' case above, just with a different thermostat that provides the friction. Additionally, it also gives you the tools to apply a counterforce to the fluid that implicitly models the propulsion mechanism (flagella, cilia, etc). This is done by creating a virtual particle that follows the swimmer particle in a fixed relative configuration, i.e., it moves at constant distance with the swimmer and also adapts to the swimmer rotation. The orientation of the virtual particle must be pointing in the opposite direction as the swimmer. The virtual particle then marked as "is_engine_force_on_fluid", which results in it not experiencing any friction but instead applying the propulsion force f_swim to the fluid. espressomd.swimmer_helper.add_dipole_particle is a helper function that performs these steps for you and creates a pusher or puller type swimmer for you.

The keyword mode (possible values: "pusher", "puller") determines where the counterforce on the fluid is applied, see again Fig. 3(c, d).

Caveats of the algorithm are:

  • One should make sure that the dipole length is at least one lattice Boltzmann grid spacing, since velocities are interpolated from that grid. If the length is less than one grid spacing, you can easily run into discretization artifacts or cause the particle not to move.
  • The stability of the swimming algorithm is governed by the stability limitations of the LB method.
  • Since the particle is essentially a point particle, there is no rotation caused by shear flow, e.g., a swimmer in a Poiseuille flow. If the thermostat is switched on, the rotational degrees of freedom will be thermalized, but there is 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 [27]:
import espressomd.lb
import espressomd.swimmer_helpers
In [28]:
clear_system(system)
In [29]:
HYDRO_PARAMS = {'box_l': 3*[35],
                'time_step': 0.01,
                'skin': 1,
                'agrid': 1,
                'dens': 0.1,
                'visc': 5,
                'gamma': 2,
                'mass': 1,
                'dipole_length': 4,
                'f_swim': 0.01,
                'dipole_particle_type': 1
                }

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']

Exercise¶

  • Using HYDRO_PARAMS, set up a lattice-Boltzmann fluid and activate it as a thermostat (Hint: lattice-Boltzmann)
In [30]:
# SOLUTION CELL
lbf = espressomd.lb.LBFluidWalberla(agrid=HYDRO_PARAMS['agrid'],
                                    density=HYDRO_PARAMS['dens'],
                                    kinematic_viscosity=HYDRO_PARAMS['visc'],
                                    tau=HYDRO_PARAMS['time_step'])
system.lb = lbf
system.thermostat.set_lb(LB_fluid=lbf, gamma=HYDRO_PARAMS['gamma'], seed=42)
In [31]:
box_l = np.array(HYDRO_PARAMS['box_l'])
pos = box_l/2.
pos[2] = -15.

Exercise¶

  • Using HYDRO_PARAMS, place particle at pos that swims in z-direction. The particle handle should be called particle.
  • Using espressomd.swimmer_helpers.add_dipole_particle and HYDRO_PARAMS, create a dipole particle that applies the propulsion force to the fluid.
In [32]:
# SOLUTION CELL
director = np.array([0,0,1])
particle = system.part.add(
    pos=pos, 
    director=director,
    mass=HYDRO_PARAMS['mass'], rotation=3*[False],
    swimming={'f_swim': HYDRO_PARAMS['f_swim']})
espressomd.swimmer_helpers.add_dipole_particle(system, particle, HYDRO_PARAMS['dipole_length'], HYDRO_PARAMS['dipole_particle_type'])
Out[32]:
<espressomd.particle_data.ParticleHandle at 0x76b49847d6d0>
In [33]:
system.integrator.run(HYDRO_N_STEPS)

WARNING: Recalculating forces, so the LB coupling forces are not included in the particle force the first time step. This only matters if it happens frequently during sampling. in function void Thermostat::Thermostat::lb_coupling_deactivate() (/builds/espressomd/espresso/src/core/thermostat.cpp:91) on node 0
In [34]:
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')
vels = np.copy(vels)
ax_vels.streamplot(xs, zs, vels[:, :, 0].T, vels[:, :, 2].T, density=1.5)
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 [35]:
import pathlib
vtk_base_dir = pathlib.Path('./vtk_out').resolve()
vtk_identifier = 'RESULTS_FLOW_FIELD'
vtk_outdir = vtk_base_dir / vtk_identifier
vtk_outdir.mkdir(parents=True, exist_ok=True)
lb_vtk = espressomd.lb.VTKOutput(lb_fluid=lbf, 
                                 identifier=vtk_identifier,
                                 observables=["velocity_vector"],
                                 base_folder=vtk_base_dir.as_posix(),
                                 prefix='lb_velocity')
lbf.add_vtk_writer(vtk=lb_vtk)
for i in range(HYDRO_N_STEPS // 100):
    system.integrator.run(100)
    system.part.writevtk(vtk_outdir / f'position_{i}.vtk', types=[0])
    lb_vtk.write()

Further reading¶

[1] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic. Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proc. Natl. Acad. Sci., 105:1232, 2008.
[2] Y. Katz, K. Tunstrøm, C.C. Ioannou, C. Huepe, and I.D. Couzin. Inferring the structure and dynamics of interactions in schooling fish. Proc. Nat. Acad. Sci., 108(46):18720, 2011.
[3] D. Helbing, I. Farkas, and T. Vicsek. Simulating dynamical features of escape panic. Nature, 407:487, 2000.
[4] J. Zhang, W. Klingsch, A. Schadschneider, and A. Seyfried. Experimental study of pedestrian flow through a T-junction. In V. V. Kozlov, A. P. Buslaev, A. S. Bugaev, M. V. Yashina, A. Schadschneider, and M. Schreckenberg, editors, Traffic and Granular Flow ’11, page 241. Springer (Berlin/Heidelberg), 2013.
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