Modellierung angetriebener Partikelsystemen in zwei Dimensionen

Birds fly, cells crawl and bacteria swim. Each of these individuals
has their own propulsion mechanism leading to a persistent motion,
hence they all belong to the class referred to as propelled
particle systems. Propelled particles in large number typically
exhibit impressive self-organization processes such as the flocking
motion of birds, the coherent motion of cell colonies and the
swarming of bacteria. The emergence of collective motion in these
non-equilibrium systems constitutes a ubiquitous phenomenon in
nature—and perhaps one of the most fascinating. One reason for this
might be the emergence of highly dynamic, coherently moving spatial
patterns such as clusters, swirls or waves, and the fact that the
patterns commonly extend over length scales much larger than the
size of the individuals. To elucidate the physical principles
underlying the collective motion of these particles, numerous
theoretical studies have been devoted to model propelled particle
systems by approaching the problem on all levels of description.
These range from particle-based simulations to kinetic theory and
hydrodynamic models. However, these models were typically inspired
by the idea of universality and tended to analyze the generic
non-equilibrium phenomena in propelled particle systems, thereby
obscuring a one-to-one relation to experimental studies. This
thesis focusses on theoretical modeling approaches for propelled
particle systems that are either in qualitative or quantitative
agreement with recent observations and mea- surements in
experimental propelled particle systems. Two experimental systems
are specifically considered: The actin gliding assay, where
molecular motors, immobilized on a substrate, propel actin
filaments, and the polar vibrated disk assay, where disk-like gran-
ular particles with a built-in polar asymmetry are driven by
vertical vibrations. By means of various theoretical tools
including rule-based automaton models, numerical solution of
Newtonian equations of motion, and kinetic theory, the following
central new findings and insights within the field of propelled
particle systems were discovered: Anomalously strong curvature
fluctuations of single actin filaments moving in the glid- ing
assay arise from two different interactions with the molecular
motors. The motors either “Push” or “Hold” locally, giving rise to
persistent movement or localized jams, which, in turn, lead to
pronounced curvature kinks. Interestingly, both excitation and
relaxation of curvature originates from these interactions, and it
is shown that the impact of thermal fluctuations on the curvature
distribution is negligible compared to these active fluctua- tions.
At high densities, filaments in the gliding assay form beautiful
patterns of coherent motion such as clusters, swirls and waves.
These patterns were shown to be triggered by local
“ferromagnetic”-like alignment interactions between the filaments.
In the later stages of pattern formation, hydrodynamic interactions
become relevant. Coherently moving clus- ters induce a back-flow in
the overlying fluid, mediating a ‘repulsive’ cluster–cluster or
cluster–boundary interaction. With the addition of crosslinking
molecules to the gliding assay at high filament densities, an
absorbing state comprised of open and closed rings can form. The
assembly dynamics is fully understood in terms of a competition
between merging events of filaments and filament growth that
freezes the curvature. Specifically, by means of an appropriate
particle-based model, the statistical properties of the system,
such as the characteristics of the ring radii distribution and the
ratio of open to closed rings as a function of the system’s noise
level, was qualitatively reproduced. The vibrated polar disk assay
has a size of only about 20 particle diameters —a fact which
precludes definitive conclusions on the nature of the underlying
phase transition to a polarized state of coherent motion. By means
of a microscopic model, the experimental single particle motion,
the details of binary collisions and the collective dynamics in the
confined geometry were quantitatively reproduced. Specifically, we
matched all properties of the persistent random walk such as
average speed, amplitude and spectrum of orien- tational and
velocity fluctuations. Agreement between the characteristics of the
collisions described by the model and those measured in the
experiment, were verified by comparing the probability
distributions for collision extension and time. Finally, collective
properties were studied and likened by considering the average
polarization within some restricted area, again confirming a very
good agreement between model and experiment. The quan- titive match
of all details of the experimental dynamics allowed us to use our
models to scale up the vibrated polar disk assay in-silico, proving
that a long-range polar ordered state would develop in the vibrated
disk assay in the absence of boundaries. Moreover, a generic
automaton model for propelled particles was employed to analyze the
onset of collective motion in time. The central finding is that
collective motion close to the transition is induced by nucleation
of a cluster of sufficiently large mass, and not by a wide-spread
coarsening process of polarized domains. In particle-conserving,
propelled particle systems wave-like patterns generically emerge
close to the phase boundary. Considering kinetic theory, there are
clear indications that wave-like patterns cannot exist in the
absence of particle conservation. Moreover, kinetic theory for
propelled particle systems is found to be restricted to weak
aligning systems, whereby post-collision angles are only slightly
reduced with respect to pre-collision orientations. This conclusion
was obtained by extending kinetic theory for propelled particle
systems with respect to significant qualitative features of
collisions ob- served in experimental propelled particle systems.
Since kinetic theory predicts disorder for regimes in which real
(experimental) systems exhibit order, the inherent restrictions of
kinetic theory to weak aligning systems could be elucidated.
Furthermore, using a set of Newtonian equations of motion for
propelled, dissipative col- loids, we found that near the phase
boundary, the microscopic states from which collective motion
develops are not free of orientational correlations, i.e. the
assumption of molecular chaos that is commonly applied in kinetic
theory is not valid for propelled particle sys- tems at the onset
of collective motion. Most importantly, the ensuing correlations at
the onset are —for the aforementioned system— a qualitative
prerequisite for kinetic theory to predict a phase transition to
collective motion at all. This conclusion was made pos- sible by
quantitatively connecting the details of the microscopic collision
process with the mesoscopic kinetic description, in turn allowing
for a quantitative test of the predictions of kinetic theory. If
the aforementioned correlations are implemented into the kinetic
ap- proach, the prediction of kinetic theory for the phase boundary
quantitatively coincides with the one obtained from the underlying
microscopic Newtonian particle dynamics in the regime of low
packing fractions. Finally, an appropriate particle-based model for
propelled particle systems at large packing fractions was analyzed
in detail. Upon characterizing the degree of bond orienta- tional
and translational order, the following generic states were
identified: An unpolarized active crystal state with long-ranged
orientational and translational order, and a polycrys- talline
state, which coherently flows and is composed of hexagonal domains.
It was shown that the underlying ordering transitions are
defect–mediated. Most of the results of this thesis exemplify the
importance of a one-to-one comparison between theoretical models
and experimental studies in order to advance our understanding of
propelled particle systems. Rather than adopting a generic
approach, the central goal of this thesis is to develop both a
bottom-up modeling framework as well as an experiment- specific
one. This formulation advances our understanding of the physics of
two specific experimental propelled particle systems and hopefully
will serve as a starting point for investigations of the dynamics
in other active systems such as moving cells and bacteria.