Abstract
For an active system on a solid substrate, only the number of components in the system is conserved. The momentum is not conserved due to friction on the solid substrate. When the components have a mean orientation, the rotational invariance of the system is broken. The hydrodynamic variables to consider are the density and direction of the orientational order. The order is nematic if there is a well-defined orientation of the components but no preferred direction along this orientation. If one of the two directions is privileged along the orientation, we speak of polar order.
The hydrodynamic theory of active matter was initiated by John Toner and Yuhai Tu, and was further developed by Sriram Ramaswamy and colleagues. It uses conservation of the number of components and symmetry invariances to determine the evolution of density and orientation. Among the predictions of this theory, the most spectacular are the existence of giant density fluctuations and the existence of propagative acoustic waves, even though the medium is dissipative. These waves are possible in a dissipative medium because of the injection of energy due to the system's activity. Another important result due to Toner and Tu is that for a two-dimensional system, long-range ordered phases exist, which is not possible for a system at thermodynamic equilibrium.
Finally, the lecture presented a study of topological defects in orientational order. Active systems are characterized by spontaneous movement of defects, either translational or rotational, even if the system is not polar.
