Active matter describes systems that convert energy from their environment into directed motion. Therefore, these systems are in intrinsic nonequilibrium, unlike their passive counterparts. From a theoretical point of view, such active systems have been modeled by agent-based models, as well as hydrodynamic approaches, which allowed for the investigation of a wide range of observed collective phenomena characterizing active matter. In this thesis we develop a microscopic field-theoretical approach to describe generic properties of active systems. This description combines the phase field crystal model with a polar order parameter and a self-propulsion term. First, we validate this approach by reproducing results obtained with corresponding agent-based models, such as binary collisions, collective migration and vortex formation. We also perform a direct comparison between our model and a microscopic phase field description of active matter. Next, we use this continuum approach to simulate some larger active systems and to analyze the coarsening process in active crystals, as well as the mechanisms leading to mobile clusters. We show the generality of our approach by extending it to binary mixtures of interacting active and passive particles. Also in this case, we first validate the model by reproducing known results, such as enhanced crystallization via active doping and the suppression of collective migration in an active bath in the presence of fixed obstacles. Interestingly, for the regime of mobile passive particles in an active bath a laning state is found, which is characterized by an alignment of the active particles that is globally nematic, but polar within each lane. This state represents a theoretical prediction feasible to be validated experimentally. Finally, we explore the field of topological active matter. We develop an agent-based model to describe self-propelled particles on curved surfaces and study the complex spatiotemporal patterns that emerge. / Aktive Materie beschreibt Systeme, die Energie aus ihrer Umgebung in gerichtete bewegung umwandeln. Im Gegensatz zur passiven Materie befinden sich diese Systeme nie im physikalischen Gleichgewicht und offenbaren dadurch interessante physikalische Phänomene. Vom theoretischen Standpunkt her wurde aktive Materie bereits simuliert, typischerweise durch agenten-basierte Modelle oder hydrodynamische Ansätze, die es ermöglichen eine Vielzahl der auftretenden kollektiven Bewegungsprinzipien zu untersuchen. In dieser Doktorarbeit entwickeln wir einen mikroskopischen Kontinuumsansatz um die generischen Eigenschaften von aktiven Systemen zu untersuchen. Unsere Beschreibung kombiniert das Phasenfeld-Kristall Modell mit einem polaren Ordnungsparameter und einem Antriebsterm. Zuerst validieren wir den Ansatz durch Reproduktion bekannter Ergebnisse agenten-basierter Modelle, wie binäre Kollisionen, kollektive Bewegung und Wirbelformationen. Des Weiteren führen wir einen direkten Vergleich zwischen unserem Modell und einer mikroskopischen Phasenfeldbeschreibung aktiver Materie durch. Danach nutzen wir den kontinuierlichen Ansatz um große aktive Systeme zu simulieren und analysieren den Vergröberungsprozess in aktiven Kristallen und Mechanismen der mobilen Aggregatbildung. Wir illustrieren die Allgemeingültigkeit unseres Simulationsansatzes durch die Erweiterung auf binäre Systeme, in denen sowohl aktive als auch passive Partikel enthalten sind. Auch in diesem Fall validieren wir das Modell durch Vergleiche mit bekannten Resultaten, wie zum Beispiel die verstärkte Kristallisation durch aktives Doping oder die Unterdrückung kollektiver Bewegung durch die Einführung von Hindernissen in einem aktiven Bad. Interessanterweise finden wir bei der Präsenz mobiler passiver Partikel in einem aktiven Bad einen Fahrspur-Zustand, in welchem die aktiven Partikel nematische Fahrspuren bilden und sich nur jeweils innerhalb einer Fahrspur nematisch polar anordnen. Dieser bisher unbekannte Zustand stellt eine theoretische Vorhersage dar, die experimentell geprüft werden kann. Schließlich begeben wir uns auf das Gebiet der topologischen aktiven Materie. Wir entwickeln ein agenten-basiertes Modell um selbst-angetriebene Partikel auf gekrümmten Oberflächen zu beschreiben und untersuchen die dabei auftretenden zeitlich und räumlich komplexen Muster.%, die dabei auftreten.

A time-delayed response of individual living organisms to information exchanged within flocks or swarms leads to the emergence of complex collective behaviors. A recent experimental setup by (Khadka et al 2018 Nat. Commun. 9 3864), employing synthetic microswimmers, allows to emulate and study such behavior in a controlled way, in the lab. Motivated by these experiments, we study a system of N Brownian particles interacting via a retarded harmonic interaction. For N 3 , we characterize its collective behavior analytically, by solving the pertinent stochastic delay-differential equations, and for N>3 by Brownian dynamics simulations. The particles form molecule-like non-equilibrium structures which become unstable with increasing number of particles, delay time, and interaction strength. We evaluate the entropy and information fluxes maintaining these structures and, to quantitatively characterize their stability, develop an approximate time-dependent transition-state theory to characterize transitions between different isomers of the molecules. For completeness, we include a comprehensive discussion of the analytical solution procedure for systems of linear stochastic delay differential equations in finite dimension, and new results for covariance and time-correlation matrices

We study the thermodynamic properties induced by non-reciprocal interactions between stochastic degrees of freedom in time- and space-continuous systems. We show that, under fairly general conditions, non-reciprocal coupling alone implies a steady energy flow through the system, i.e., non-equilibrium. Projecting out the non-reciprocally coupled degrees of freedom renders non-Markovian, one-variable Langevin descriptions with complex types of memory, for which we find a generalized second law involving information flow.We demonstrate that non-reciprocal linear interactions can be used to engineer non-monotonic memory, which is typical for, e.g., time-delayed feedback control, and is automatically accompanied with a nonzero information flow through the system. Furthermore, already a single non-reciprocally coupled degree of freedom can extract energy from a single heat bath (at isothermal conditions), and can thus be viewed as a minimal version of a time-continuous, autonomous ‘Maxwell demon’.We also show that for appropriate parameter settings, the non-reciprocal system has characteristic features of active matter, such as a positive energy input on the level of the fluctuating trajectories without global particle transport.

La motilité cellulaire est fondamentale dans de nombreux processus physiologiques, qu’ils soient normaux ou pathologiques. Cependant, bien que ces derniers impliquent la plupart du temps de nombreuses cellules se mouvant en même temps, les effets des interactions entre cellules sur leur dynamique, à la fois individuelle et collective, restent assez mal connu. Dans cette thèse, j’ai utilisé Dictyostelium discoideum à l’état végétatif pour étudier cette régulation collective de la motilité. Je me suis principalement appuyé sur une analyse minutieuse de nombreuses trajectoires cellulaires dans des situations variées pour (i) caractériser un facteur sécrété qui régule négativement la motilité cellulaire (nature chimique, voie de signalisation, dynamique de sécrétion et de réponse) et (ii) analyser et modéliser quantitativement la dynamique d’étalement de colonie cellulaires de forme, dimension et densité contrôlées. Je décris enfin un phénomène d’agrégation dynamique observé lorsque les cellules sont placées à haute densité dans un milieu nutritif / Cell motility is fundamental in many physiological, either normal or pathological, phenomena. Yet, although these most often involve several cells moving at the same time, how the interactions between cells affect both individual and collective dynamics remains a poorly understood question. In this thesis, I used vegetative Dictyostelium discoideum cells as a model to study this collective regulation of the motility. I relied mainly on the thorough analysis of numerous cell trajectories in various situations to (i) characterise a secreted factor used to down-regulate the cells’ motility (biochemical nature, response pathway, secretion and response dynamics) and (ii) quantitatively analyse and model the dynamics of spreading cell colonies of controlled initial shape, size and density. Last, I describe a dynamic aggregation phenomenon that occurs when the cells are seeded at high density in a nutrient-rich medium

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Le déplacement cohérent dirigé au sein de troupeaux, d’essaims, de nuées, prend place à toutes les échelles du vivant. En cherchant à rationaliser l’émergence de tels mouvements collectifs, les physiciens ont décrit ces assemblées comme des matériaux actifs. Ces matériaux sont formés de constituants auto-propulsés qui se déplacent spontanément dans une direction commune. Cette thèse expérimentale s’appuie sur la réalisation de troupeaux synthétiques pour explorer les propriétés de la matière active polaire dans des situations défavorables à son auto-organisation : leur dynamique en milieux désordonnés et leur réponse à des perturbations externes. Des rouleurs colloïdaux aux interactions d’alignement sont confinés au sein de dispositifs microfluidiques. Au-delà d’une densité seuil, ils forment un troupeau caractérisé par l’émergence d’un ordre en orientation de longue portée. Ces troupeaux colloïdaux font office de prototypes de la matière active polaire. Nous avons étudié la réponse d’un liquide actif polaire assemblé à partir de rouleurs colloïdaux. Nous avons montré que face à une perturbation longitudinale leur réponse est hystérétique. Nous avons expliqué théoriquement ce comportement non-linéaire et l’avons exploité pour réaliser des oscillateurs microfluidiques autonomes. Nous avons également étudié la dynamique de troupeaux colloïdaux qui se propagent dans des environnements hétérogènes. La présence d’obstacles distribués aléatoirement focalise les troupeaux le long de chemins privilégiés qui forment un réseau épars et tortueux. Augmenter le désordre conduit à la destruction du troupeau. Nous avons démontré que la suppression du mouvement collectif consiste en une transition discontinue, générique à tous les matériaux actifs polaires. / Directed collected motion within herds, swarms and flocks, is a phenomenon that takes place at all scales in living systems. Physicists have rationalized the emergence of such collective behavior. They have described these systems as active materials. These materials are assembled from self-propelled units that spontaneously move in the same direction. By experimentally studying synthetic flocks, this work uncovers some properties of polar active materials in situations that disfavor their self-organization: their dynamics in disordered environments and their response to external perturbations. Colloidal rollers with alignment interactions are confined within microfluidic devices. At high density, they spontaneously form a flock which is characterized by the emergence of orientational long-ranged order. These colloidal flocks are prototypical realizations of polar active matter. We have studied the response of a polar active liquid assembled from colloidal rollers. We have shown that they display a hysteretic response to longitudinal perturbations. We have theoretically accounted for this non-linear behavior. We have used this behavior to realize autonomous microfluidic oscillators. We have also studied the dynamics of colloidal flocks that propagate through heterogeneous environments. Randomly positioned obstacles focalize flocks along favored channels that form a sparse and tortuous network. Increasing disorder leads to the destruction of flocks. We have demonstrated that the suppression of collective motion is a discontinuous transition generic to all polar active materials.

This thesis has been concerned with a few problems in systems driven at the scale of particles. The problems dealt with here can be extended and elaborated upon in a variety of ways. In 2 we examine the dynamics of a fluid membrane in contact with a fluid containing active particles. In particular, we show that such a membrane generically enters a statistical steady state with wave-like dispersion. While the numerical results are satisfying, a one-step coarse-graining calculation, in line with [66,93], will, we expect, yield a pair of coupled stochastic differential equations (probably KPZ like at least in one dimension) with wave-like dispersion. This calculation in of interest from a theoretical point-of-view. Further, the numerical exploration of the full set of equations is also left for future work, but can be relevant to many biological systems. In 3 we show that an active fluid confined in an annular channel starts to rotate spontaneously. Further, we predict the existence of banded concentration profile. Such profiles have not yet been observed in experiments. Further, it will be interesting to study what happens to our conclusions if we include the effect of treadmilling in our calculation. In 4 we describe a solid driven by active particles. Specifically, we only concern ourselves with the polar elastomeric phase of the material. However, the questions regarding the transition into that phase are interesting and have not been explored. How exactly does a polarisation transition happen in an active polar elastomer? Is it the same as in an active nematic elastomer? What is the nature of the gelation transition in an active polar fluid? What is the dynamics of nematic defects in an elastomer? Can the presence of the elastomer prevent defect separation? We are at present trying to answer these questions. In 5 we examine the dynamics of an active fluid confined in a channel. It will be interesting to test the prediction about fluctuations in a confined active system, which we show will be normal, in experiments on highly confined actomyosin systems. In 6 we write down the coupled equations of a conformation tensor and the apolar order parameter. This is a generic framework for studying viscoelastic active fluids. A fuller study of the effect of increasing the cross-linker density in such system remains to be done, both theoretically and experimentally. In general, we have shown in the thesis that the understanding of active systems can provide a mechanistic explanation of various biological observations. However, at times the comparison between theory and biological experiments become complicated due to the inherently complicated nature of the experimental systems. Thus, for a more rigorous experimental test of the theory, it is necessary to construct cleaner reconstituted systems with possibly as few as three components. Efforts in this direction have recently borne fruit [129]. However, a complete theoretical understanding of the rich behaviour evinced in these systems is as yet lacking. We expect that the conformation tensor theory we developed in chapter 6 will provide an explanation for the anomalous rheological behaviour observed in these systems. Even in the theoretical front, lot of questions remain to be answered. The dry polar active system, described by the Toner-Tu equations have been shown to undergo a transition to a state with LRO. However, though mean-field theory predicts a second order transition [151, 152, 156], detailed numerical analysis suggests that it is actually first-order with pre-transitional solitonic bands. This has been recently examined by Chate et al. [26] who mapped it to a dynamical system, but a complete theory is still lacking. Apolar systems present another set of challenges. First, the concentration coupling with the order parameter should create similar pre-transitional effects at the order-disorder transition for this system also. This has been studied to a certain extent [133]. However, the more interesting question concerns the role of defects in apolar systems and whether they allow for the possibility of even QLRO in two dimensions. The +1/2 nematic defect has a polarity, and can thus move balistically [51, 108, 115, 149] in a dry system. However, the −1/2 defect has a three-fold symmetry [27] and its motion is thus purely diffusive. Now consider a pair of +1/2 and −1/2 defect pair that can form due to noise in the system (since it does not violate charge conservation). Depending on the configuration and the kind of activity, this defect pair can unbind at zero temperature. Unbound defects would imply that the order is short-ranged. However, it appears from detailed simulations of an agent based Vicsek-like model of active nematics, that there exists a QLRO nematic in two dimensions [111]! How does an active nematic escape being destroyed by defect unbinding? Does concentration have a major role to play? If so, does making the concentration a non-conserved, and thus fast, variable by, for example, including evaporation-deposition rules in the model studied by Chate et al. [28] destroy the QLRO? Also, does the hydrodynamic theory for Malthusian (i.e. one in which the concentration relaxes fast to a steady value) nematics show only short-ranged order, while the one in which mass is conserved show QLRO? These questions are being studied at present by simulating both the agent-based model due to Chate with evaporation-deposition and the dynamical equation for the active nematic order-parameter. These studies should clarify the role of concentration in assisting apolar order. It must be borne in mind, however, that numerical simulations of active models are more difficult than their passive counterparts due to the larger number of parameters present in the problem. In passive systems Onsager symmetry relations constrain some parameters. However, the absence of an equivalent rule for systems far away from equilibrium implies that the spatial symmetry allowed couplings will all have independent kinetic coefficients. This increases the size of the parameter space in many problems. Also, many techniques like Monte Carlo have to be carefully modified to suit such systems. A new and exciting area of research from the point of view of statistical mechanics of active systems is an examination of collective behaviour of run-and-tumble particles pioneered by Tailleur and Cates [25]. This has led to fruitful active generalisations of models of dynamic critical phenomena like model B and model H. Also, it has led to an exploration of rules for selecting a state in a region of phase coexistence – an out of equilibrium generalisation of the Maxwell construction. Another interesting avenue is building up active matter equations from microscopics. This has been done for Vicsek model by Thomas Ihle [64,65], for a simple generalisation of Vicsek-type model for both polar and apolar alignment interactions by Bertin et al. and Chate et al. [15, 16, 107], and for a model of hard rods by Marchetti et al. [10, 11]. The issues of closure still remain to be fully resolved however in deriving the macroscopic equations. A particularly exciting new system that has been recently studied extensively is a collection of chemotactic Janus particles [127]. The far-field interaction in this case does not promote polar order but state with proliferation of asters. The coarse-grained hydrodynamic equations have been derived in this case starting from a microscopic picture of colloids coated axisymetrically with a catalyst in an inhomogeneous concentration of reactants by Saha et al. [127]. Another theoretical issue that plagues the derivation of hydrodynamic equations is that of noise. So far most theories have modelled the noise as Gaussian and white, akin to equilibrium systems, but with unknown strength. However, it is likely that the noise also depends on activity, thus requiring a microscopic picture treating the active forces as stochastic quantities. It is known that multiplicative character of the noise induces interesting features at least in the case of active nematics [104]. Thus, a lot of questions need to be answered if theories of active matter have to graduate from merely offering qualitative explanations of biological experiments to becoming the prototypical theory of systems in which energy input and dissipation both occur at a scale smaller than the coarse-graining volume.