  A formal derivation of linear hydrodynamics for a granular fluid is given.
The linear response to small spatial perturbations of the homogeneous reference
state is studied in detail using methods of non-equilibrium statistical
mechanics. A transport matrix for macroscopic excitations in the fluid is
defined in terms of the response functions. An expansion in the wavevector to
second order allows identification of all phenomenological susceptibilities and
transport coefficients through Navier-Stokes order, in terms of appropriate
time correlation functions. The transport coefficients in this representation
are the generalization to granular fluids of the familiar Helfand and
Green-Kubo relations for normal fluids. The analysis applies to a wide range of
collision rules. Important differences in both the analysis and results from
those for normal fluids are identified and discussed. A scaling limit is
described corresponding to the conditions under which idealized inelastic hard
sphere models can apply. Further details and interpretation are provided in the
paper following this one, by specialization to the case of smooth, inelastic
hard spheres with constant coefficient of restitution.
