  We calculate the light scattering spectrum of the anisotropic diffusion of
impurities (dye) in a compressible nematic solvent. We use a fluctuating
hydrodynamic description when the system is in fully thermodynamic equilibrium
and in a non-equilibrium steady state ($NESS$) induced by a dye-concentration
gradient. We find that the equilibrium spectrum is symmetric (Lorentzian) with
respect to the frequency shifts, but anisotropic through its explicit
dependence on the ratio of the parallel and normal diffusion coefficients of
the dye. The values of these coefficients were taken from experimental
measurements of diffusion of methylred and nitrozo di-methyl aniline in a
$MBBA$ solvent. We find that the compressibility of the solvent increases the
maximum and the width at half height of the Rayleigh peak, with respect to the
incompressible case \cite{suspension1}. This increase varies between 12% and
25%, respectively, when the impurities concentrations is the range of 1% - 5%.
The $NESS$ induces a coupling between the concentration fluctuations of the dye
and the hydrodynamic fluctuations of the solvent. In this case the
compressibility effects may increase the maximum and the width of the central
peak up to $% 25%$, for values of the concentration gradient four orders of
magnitude smaller than those considered in the incompressible case. For the
nonequilibrium Brillouin part of the spectrum we find that the intensities of
the sound propagation modes are unequal and one of the peaks shrinks in the
same amount as the other increases. This asymmetry increases linearly with the
magnitude of the solute concentration gradient. The maximum difference between
the nonequilibrium and equilibrium contributions to the Brillouin spectrum for
various values of the external gradient is also estimated.
