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Weber, Christoph (2013): Modellierung angetriebener Partikelsystemen in zwei Dimensionen, Modelling propelled particle system in: von Partikeln zu Feldern, from particles to fields. Dissertation, LMU München: Faculty of Physics



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.