![]() Instead of equations of motion, there exist restrictions on the types of braids that can form. The topological information associated with this braid discretely encodes the state of the system and the dynamics. Here, defect trajectories wind about one-another to form a geometric braid. We propose a minimal model for certain regimes of ANMT dynamics based on topological braids. Models of ANMT dynamics range in complexity from PDEs for the evolution of the director and velocity fields, to ODEs for the motion of topological defects in the director field. Indeed, we will show that the behavior of ANMT systems is particularly well modeled by certain tools from dynamical systems (foliations, braids, and the Nielsen-Thurston classification theorem). ![]() Active nematic microtubule (ANMT) systems can have turbulent (i.e. The canonical example of such an active nematic is a system of microtubule bundles confined to 2D 5. A nematic phase is characterized by local orientational alignment of the agents (represented by a director field, which encodes how the orientation angle of agents varies spatially). rod-like or disk-like), the active system can exhibit a time-dependent nematic phase. ![]() When these agents have asymmetries in their shapes (e.g. Janus particles), convert energy consumed by individual agents into global flows with interesting emergent behaviour. bacterial suspensions, bird flocks) or engineered (e.g. We will look at the dynamics of four +1/2 defects on a sphere as a case study, using both simulations and a reinterpretation of experimental data from the literature.Īctive matter systems, whether biological (e.g. Indeed, we conjecture that the emergent defect dynamics are often optimal in that they give braids which maximize the, suitably normalized, topological entropy. Since microtubule bundles, an extensile system, stretch out exponentially in time, the resultant defect movement must correspond to braids with positive topological entropy. ![]() In particular, we consider the topological entropy of braids, which quantifies how chaotic the associated flow must be. We introduce a minimal model of ANMT systems based on the topological properties of these braids. As these defects wind about each other, their trajectories trace out braids. For 2D active nematic microtubule (ANMT) systems, these flows are largely characterized by the dynamics of mobile defects in the nematic director field. In active matter systems, energy consumed at the small scale by individual agents gives rise to emergent flows at large scales. Mount Holyoke College, South Hadley, MA, United States. ![]()
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