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Free keywords:
optimal navigation; active matter; microswimmers; optimal control theory; stochastic processes; swimming efficiency; autonomous navigation; Zermelo's problem; Finsler geometry; curved manifolds; Langevin equation
Abstract:
Motile active matter systems are composed of a collection of agents, each of which extracts energy from the surrounding environment in order to convert it into self-driven motion. At the microscopic scale, however, directed motion is hindered by the presence of stochastic fluctuations. Living microorganisms, therefore, had to develop simple yet effective propulsion and steering mechanisms in order to survive. We may turn the question of how these processes work in nature around and ask how they should work in order to perform a task in the theoretically optimal way, an issue that falls under the name of the optimal navigation problem. The first formulation of this problem dates back to the seminal work of E. Zermelo in 1931, in which he addressed the question of how to steer a ship in the presence of an external stationary wind so as to reach the destination in the shortest time. Despite the considerable progress made over the years in this context, however, there are still a number of open challenges. In this thesis, we therefore aim to generalize Zermelo’s solution by adding more and more ingredients in the description of the optimal navigation problem for microscopic active particles. First, borrowing theoretical tools from differential geometry, we here show how to extend the analytical solution of this problem to when motion occurs on curved surfaces and in the presence of arbitrary flows. Interestingly, we reveal that it can elegantly be solved by finding the geodesics of an asymmetric metric of general relativity, known as the Randers metric. Then, we study the case in which navigation happens in the presence of strong external forces. In this context, route optimization can be crucial as active particles may encounter trapping regions that would substantially slow down their progress. Comparing the exploration efficiency of Zermelo’s solution with a more trivial strategy in which the active agent always points in the same direction, here we highlight the importance of optimal path stability, which turns out to be fundamental in the design of the proper navigation strategy depending on the task at hand. We then take it a step further and include a key ingredient in the comprehensive study of optimal navigation in active matter, namely stochastic fluctuations. Although methods already exist to obtain both analytically and numerically the optimal strategies even in the presence of noise, their implementation requires the presence of an external interpreter that takes away the active agent’s autonomy. Inspired by the tactic behaviors observed in nature, we here introduce a whole new class of navigation strategies that allows an active particle to navigate semi-autonomously in a complex and noisy environment. Moreover, our study reveals that the performance of the theoretical optimal strategy can be reproduced starting from some simple principles based on symmetry and stability arguments. Finally, we lay the ground for moving toward a more realistic description of the problem. In fact, we extend the optimization problem by also considering the energetic costs involved in navigation and how these depend on the shape of the active particle itself. Remarkably, our analysis uncovers the existence of an interesting trade-off between the minimization of the arrival time at a target and the corresponding energetic cost, which in turn determines the optimal shape of an active particle.
