Pressure and phase separation in active matter

Self-propelled particles are ubiquitous in nature, where they are found at all scales, from molecular motors to bacteria to birds flocks and fish schools. These systems are prototypical of "active matter" and have attracted lots of interest recently, due to their rich phenomenology. In this talk I will investigate two aspects of active matter: 1) The mechanical pressure of a non-equilibrium fluid is unambiguously defined as the force per unit area it exerts on its containing vessel. Surprisingly, and contrary to what happens for equilibrium systems, we will show that this pressure is not a function of the fluid itself, but depends on the nature of its container: there is no equation of state for the pressure of active fluids, which leads to counter-intuitive phenomena. 2) Hard-core repulsion or other interactions which tend to slow down self-propelled particles give rise to effective cohesive forces in active systems, even in abscence of any explicit attractive interaction. This can lead to a motility-induced phase separation and we will see how one can understand its phase equilibria.