  Statistical mechanics provides the foundation for describing complex
materials using only a few thermodynamic variables. No such framework currently
exists far from equilibrium. In this Letter, we demonstrate how thermodynamics
emerges far from equilibrium, using fluids composed of active spinners as a
case study. Activity gives rise to a single effective temperature that
parameterizes both the equation of state and the emergent Boltzmann statistics.
The same effective temperature, renormalized by velocity correlations, controls
the linear response through canonical Green-Kubo relations for both the
familiar shear viscosity and the odd (or Hall) viscosity observed in chiral
fluids. The full frequency dependence of these viscosities can be derived
analytically by modeling the active-spinner fluid as a random walker undergoing
cyclotron motion in shear-stress space. More generally, we provide a
first-principles derivation of the Green-Kubo relations valid for a broader
class of fluids far from equilibrium. Besides advancing non-equilibrium
thermodynamics, our work demonstrates in silico a non-invasive microrheology of
active fluids.
