Particles and fields in minimal hydrodynamic models for active turbulence

We provide an overview of theoretical developments in the rapidly developing field of active-fluid turbulence. After a summary of the partial differential equations (PDEs) for active fluids with scalar, polar, and nematic order parameters, we concentrate on a minimal description that employs the Toner-Tu-Swift-Hohenberg (TTSH) model. We present illustrative results —numerical, theoretical, and rigorous— for TTSH turbulence. In particular, we present statistical properties of such turbulence in both Eulerian and Lagrangian frameworks. We end with a perspective of new trends in the field of active-fluid turbulence.