Statistical Mechanics and Hydrodynamics of Active Fluids

In this thesis, I present work pertaining to the continuum modeling of active fluids. The main results are as follows. I derive Green-Kubo equations for the components of the viscosity in two-dimensional fluids with and without internal angular momentum. These equations illustrate the connection between the breaking of time-reversal symmetry at the microscopic level and the emergence of odd viscosity, a non-dissipative transport coefficient. They also show the potential for a new, previously unconsidered rotational viscosity in fluids in which internal spin couples to velocity. By numerically verifying the Green-Kubo equation for odd viscosity, we provide support for the use of the Onsager regression hypothesis in its derivation, where this hypothesis is applied to fluctuations about a nonequilibrium steady state. I also study the canonical active system known as Active Brownian Particles via a coarse-graining analysis. This approach indicates that in such systems, activity manifests at the continuum level in the form of a body force, rather than as an addition to the stress tensor, as has been previously assumed in the literature. It also elucidates the connection between mass currents and inter-particle alignment interactions. Taken as a whole, these results provide guidance for the project of writing down continuum descriptions of active matter, and elucidate the connections between microscopic and macroscopic behavior for nonequilibrium systems.