  The response of an isolated granular fluid to small perturbations of the
hydrodynamic fields is considered. The corresponding linear response functions
are identified in terms of a formal solution to the Liouville equation
including the effects of the cooling reference state. These functions are
evaluated exactly in the asymptotic long wavelength limit and shown to
represent hydrodynamic modes. More generally, the linear granular Navier-Stokes
equations for the response functions and related Langevin equations are
obtained from an extension of Mori's identity. The resulting Green-Kubo
expressions for transport coefficients are compared and contrasted with those
for a molecular fluid. Next the response functions are described in terms of an
effective dynamics in the single particle phase space. A closed linear kinetic
equation is obtained formally in terms of a linear two particle functional.
This closure is evaluated for two examples: a short time Markovian
approximation, and a low density expansion on length and time scales of the
mean free time and mean free path. The former is a generalization of the
revised Enskog kinetic theory to include velocity correlations. The latter is
an extension of the Boltzmann equation to include the effects of recollisions
(rings) among the particles.
