Bacterial suspensions has become a prototype of non-equilibrium matter, where energy is supplied and dissipated at the particle scale. Indeed, each swimmer moves in the fluid balancing the thrust provided by its flagella with the viscous drag. In the past years, a systematic effort has resulted in the construction of kinetic equations to describe the motion of bacterial suspensions. Here, the object of study is the distribution function f(r,n,t), giving the number of bacteria at position r, swimming along the director n, at time t. These kinetic equations have streaming terms describing the free swim, Boltzmann-like terms for the mutual alignment due to collisions, Fokker-Planck term describe the rotational diffusion of the director, and a Lorentz-like term for the tumbling process, in which the new director is chosen at random. Finally, the flows generated by their swim lead to hydrodynamic interactions which, being long-ranged, are included with Vlasov-like mean field terms. Applying standard and ad-hoc methods on these kinetic equations, the properties of bacterial suspensions at low and moderate densities have been studied, begin able to describe the diffusion, active rheology, flocking, chemotactic response, among other relevant processes.
A. Puglisi