Active particles which are self-propelled by converting energy into mechanical motion represent an expanding research realm in physics and chemistry. For micron-sized particles moving in a liquid (``microswimmers''), most of the basic features have been described by using the model of overdamped active Brownian motion [1]. However, for macroscopic particles or microparticles moving in a gas, inertial effects become relevant such that the dynamics is underdamped. Therefore, recently, active particles with inertia have been described by extending the active Brownian motion model to active Langevin dynamics which include inertia [2]. In this talk, recent developments of active particles with inertia (``microflyers'', ``hoppers'' or ``runners'') are summarized including: inertial delay effects between particle velocity and self-propulsion direction [3], tuning of the long-time self-diffusion by the moment of inertia [3], the influence of inertia on motility-induced phase separation and the cluster growth exponent [4], and the formation of active micelles (“rotelles”) by using inertial active surfactants. References [1] C. Bechinger, R. di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, G. Volpe, Reviews of Modern Physics 88, 045006 (2016). [2] H. Löwen, Journal of Chemical Physics 152, 040901 (2020). [3] C. Scholz, S. Jahanshahi, A. Ldov, H. Löwen, Nature Communications 9, 5156 (2018). [4] S. Mandal, B. Liebchen, H. Löwen, Physical Review Letters 123, 228001 (2019). [5] C. Scholz, A. Ldov, T. Pöschel, M. Engel, H. Löwen, Surfactants and rotelles in active chiral fluids, will be published

The Smoluchowski equation has often been used as the starting point of many continuum models of active suspensions. However, its six-dimensional nature depending on time, space and orientation requires a huge computational cost, fundamentally limiting its use for large-scale problems, such as mixing and transport of active fluids in turbulent flows. Despite the singular nature in strain-dominant flows, the generalised Taylor dispersion (GTD) theory (Frankel & Brenner 1991, J. Fluid Mech. 230:147-181) has been understood to be one of the most promising ways to reduce the Smoluchowski equation into an advection-diffusion equation, the mean drift and diffusion tensor of which rely on ‘local’ flow information only. In this talk, we will introduce an exact transformation of the Smoluchowski equation into such an advection-diffusion equation requiring only local flow information. Based on this transformation, a new advection-diffusion equation will subsequently be proposed by taking an asymptotic analysis in the limit of small particle velocity. With several examples, it will be demonstrated that the new advection-diffusion model, non-singular in strain-dominant flows, outperforms the GTD theory.