Active Matter systems consist of self-propelling particles (SPP) which can collectively form out-of-equilibrium stationary states which exhibit structural and/or dynamical order unseen in the equilibrium realm . Computer simulation of these systems is a fast developing field at the interface of physics, mathematics, chemistry, biology and engineering design. Examples of SPP systems found in nature are: flocks of birds, schools of fishes, bacteria colonies, etc. Recently, assemblies of robots, as models for SPP, have entered the field of study both experimentally and simulation wise . One may distinguish SPP systems of two types. In the first type the systems consist of particles interacting via the background in which they are moving. The driving forces are due to the gradients of chemical or physical factors, such as fluid flows, concentrations, temperature, light, electric and magnetic fields, etc . The systems of this type have two general classes: non-biological and biological ones. Non-biological ones include so-called active colloids , which are reaction-driven swimmers with their behaviour being influenced by colloids size, shape, and relative importance of surface activity, responsible for the chemical reaction, versus mobility (fluid media viscosity) [5, 6]. Another example is particles propelled by external fields, such as magnetic field [7, 8] or gravity . Collective behaviour of these systems (such as phase separation) is rather different compared to passive particles . The biological class ranges from active bio-molecules to organisms. Example of active behaviour in bio-molecular systems is motility of microtubules assisted by kinesin motor proteins [11, 12]. A very large part of biological systems of this class is represented by moving organisms in fluids (microswimmers ) or on substrates (cells crawling) .
The second type is formed by systems of particles, which interact via kinematic constraints imposed on their velocities. An example is a system where particles adjust the direction of their velocities to the direction of the average velocity in their neighbourhood . Realization of such constraints requires exchange of information (visual or other sensorial means) between the particles and their environment. Considerable effort in this subfield is devoted in proposing microscopic models for the particle interaction (either potential or non-potential), which could mimic collective biological behaviour of flocks of birds, schools of fishes , large groups of robots , etc. The first numerical model for coherent motion of such SPP was proposed by Vicsek et al. , which since evolved into a large subfield [16-18]. Very recently, a hybrid systems (combining the two types above) were investigated as well, where SPP with kinematic constraints (school of fish) are considered together with the effect of the surrounding flow . Apart of the particle based simulations, a large effort is being invested in deriving and proposing effective numerical methods for hydrodynamic-like equations for collective motion of SPP [20-24].