Using the standard Vicsek model, I show how the macroscopic transport equations can be systematically derived from microscopic collision rules. The approach starts with the exact evolution equation for the N -particle probability distribution and, after making the mean-field assumption of molecular chaos, leads to a multi-particle Enskog-type equation. This equation is treated by a non-standard Chapman–Enskog expansion to extract the macroscopic behavior. The expansion includes terms up to third order in a formal expansion parameter ϵ, and involves a fast time scale. A self-consistent closure of the moment equations is presented that leads to a continuity equation for the particle density and a Navier–Stokes-like equation for the momentum density. Expressions for all transport coefficients in these macroscopic equations are given explicitly in terms of microscopic parameters of the model. The transport coefficients depend on specific angular integrals which are evaluated asymptotically in the limit of infinitely many collision partners, using an analogy to a random walk. The consistency of the Chapman–Enskog approach is checked by an independent calculation of the shear viscosity using a Green–Kubo relation.

We observe a flocking mechanism, the emergence of a state with global polar order, in mixed systems of self-propelled particles with purely antialigning interactions, i.e., the ground state for any pair of particles is to be opposedly oriented. In binary mixtures, we find that flocking can be realized by cross-species antialigning that is dominant compared to intraspecies antialignment. While the key mechanism can be understood within a mean-field description, beyond mean-field we develop an asymptotically exact Boltzmann-scattering theory from first principles. This theory yields analytical predictions for the flocking transition and shows excellent quantitative agreement with simulations of dilute systems. For large systems, we find either microphase separation or static patterns with patches or stripes that carry different polarization orientations.

We present the results from kinetic theory for a system of self-propelled particles with alignment interactions of higher-order symmetry, particularly nematic ones. To this end, we use the Landau equation approach, a systematic approximation to the BBGKY hierarchy for small effective couplings. Our calculations are presented in a pedagogical way with the explicit goal of serving as a tutorial from a physicists’ perspective into applying kinetic theory ideas beyond mean-field to active matter systems with essentially no prerequisites and yield predictions without free parameters that are in quantitative agreement with direct agent-based simulations

The dynamical properties of two types of metal-coated dust microspheres confined in a gaseous plasma environment have been studied. It has been observed that such particles feature properties of active particles and that the particle activity changes over time and increases with the intensity of laser illumination. The dynamics of single particles as well as of small particle clusters have been analyzed quantitatively with various statistical methods to characterize their activity. Furthermore, clusters of metal-coated microspheres are observed to undergo a phase transition from an ordered to an unordered structure under increased laser illumination intensity due to the increased active motion of the particles.

The reverse perturbation method [ Phys. Rev. E 59, 4894 (1999)] for shearing simple liquids and measuring their viscosity is extended to the Vicsek model (VM) of active particles [ Phys. Rev. Lett. 75, 1226 (1995)] and its metric-free version. The sheared systems exhibit a phenomenon that is similar to the skin effect of an alternating electric current: Momentum that is fed into the boundaries of a layer decays mostly exponentially toward the center of the layer. It is shown how two transport coefficients, i.e., the shear viscosity $\nu$ and the momentum amplification coefficient $\lambda$, can be obtained by fitting this decay with an analytical solution of the hydrodynamic equations for the VM. The viscosity of the VM consists of two parts, a kinetic and a collisional contribution. While analytical predictions already exist for the former, a novel expression for the collisional part is derived by an Enskog-like kinetic theory. To verify the predictions for the transport coefficients, Green-Kubo relations were evaluated and transverse current correlations were measured in independent simulations. Not too far to the transition to collective motion, we find excellent agreement between the different measurements of the transport coefficients. However, the measured values of $\nu$ and $1-\lambda$ are always slightly higher than the mean-field predictions, even at large mean free paths and at state points quite far from the threshold to collective motion, that is, far in the disordered phase. These findings seem to indicate that the mean-field assumption of molecular chaos is much less reliable in systems with velocity-alignment rules such as the VM, compared to models obeying detailed balance such as multiparticle collision dynamics.

We consider the effect of nonreciprocity in a binary mixture of self-propelled particles with antialigning interactions, where a particle of type A reacts differently to a particle of type B than vice versa. Starting from a well-known microscopic Langevin model for the particles, setting up the corresponding exact N-particle Fokker-Planck equation, and making Boltzmann's assumptions of low density and one-sided molecular chaos, the nonlinear active Boltzmann equation with the exact collision operator is derived. In this derivation, the effect of phase-space compression and the buildup of pair correlations during binary interactions is explicitly taken into account, leading to a theoretical description beyond mean field. This extends previous results for reciprocal interactions, where it was found that orientational order can emerge in a system with purely antialigning interactions. Although the equations of motion are more complex than in the reciprocal system, the theory still leads to analytical expressions and predictions. Comparisons with agent-based simulations show excellent quantitative agreement of the dynamic and static behavior in the low-density and/or small coupling limit.