Starting from a Langevin description of active particles that move with constant speed in infinite two-dimensional space and its corresponding Fokker-Planck equation, we develop a systematic method that allows us to obtain the coarse-grained probability density of finding a particle at a given location and at a given time in arbitrary short-time regimes. By going beyond the diffusive limit, we derive a generalization of the telegrapher equation. Such generalization preserves the hyperbolic structure of the equation and incorporates memory effects in the diffusive term. While no difference is observed for the mean-square displacement computed from the two-dimensional telegrapher equation and from our generalization, the kurtosis results in a sensible parameter that discriminates between both approximations. We carry out a comparative analysis in Fourier space that sheds light on why the standard telegrapher equation is not an appropriate model to describe the propagation of particles with constant speed in dispersive media.

We explore minimal navigation strategies for active particles in complex, dynamical, external fields, introducing a class of autonomous, self-propelled particles which we call Markovian robots (MR). These machines are equipped with a navigation control system (NCS) that triggers random changes in the direction of self-propulsion of the robots. The internal state of the NCS is described by a Boolean variable that adopts two values. The temporal dynamics of this Boolean variable is dictated by a closed Markov chain-ensuring the absence of fixed points in the dynamics-with transition rates that may depend exclusively on the instantaneous, local value of the external field. Importantly, the NCS does not store past measurements of this value in continuous, internal variables. We show that despite the strong constraints, it is possible to conceive closed Markov chain motifs that lead to nontrivial motility behaviors of the MR in one, two, and three dimensions. By analytically reducing the complexity of the NCS dynamics, we obtain an effective description of the long-time motility behavior of the MR that allows us to identify the minimum requirements in the design of NCS motifs and transition rates to perform complex navigation tasks such as adaptive gradient following, detection of minima or maxima, or selection of a desired value in a dynamical, external field. We put these ideas in practice by assembling a robot that operates by the proposed minimalistic NCS to evaluate the robustness of MR, providing a proof of concept that is possible to navigate through complex information landscapes with such a simple NCS whose internal state can be stored in one bit. These ideas may prove useful for the engineering of miniaturized robots.

The collective dynamics and structure of animal groups has attracted the attention of scientists across a broad range of fields. A variety of agent-based models have been developed to help understand the emergence of coordinated collective behavior from simple interaction rules. A common, simplifying assumption of such collective movement models, is that individual agents move with a constant speed. In this work we critically re-asses this assumption. First, we discuss experimental data showcasing the omnipresent speed variability observed in different species of live fish and artificial agents (RoboFish). Based on theoretical considerations accounting for inertia and rotational friction, we derive a functional dependence of the turning response of individuals on their instantaneous speed, which is confirmed by experimental data. We then investigate the interplay of variable speed and speed-dependent turning on self-organized collective behavior by implementing an agent-based model which accounts for both these effects. We show that, besides the average speed of individuals, the variability in individual speed can have a dramatic impact on the emergent collective dynamics: a group which differs to another only in a lower speed variability of its individuals (groups being identical in all other behavioral parameters), can be in the polarized state while the other group is disordered. We find that the local coupling between group polarization and individual speed is strongest at the order-disorder transition, and that, in contrast to fixed speed models, the group’s spatial extent does not have a maximum at the transition. Furthermore, we demonstrate a decrease in polarization with group size for groups of individuals with variable speed, and a sudden decrease in mean individual speed at a critical group size (N = 4 for Voronoi interactions) linked to a topological transition from an all-to-all to a distributed spatial interaction network. Overall, our work highlights the importance to account for fundamental kinematic constraints in general, and variable speed in particular, when modeling self-organized collective dynamics.