  We use a fluctuating hydrodynamic approach to calculate the orientation
fluctuations correlation functions of a thermotropic nematic liquid crystal in
a nonequilibrium state induced by a stationay heat flux. Since in this
nonequilibrium stationary state the hydrodynamic fluctuations evolve on three
widely separated times scales, we use a time-scale perturbation procedure in
order to partially diagonalize the hydrodynamic matrix. The wave number and
frequency dependence of these orientation correlation functions is evaluated
and their explicit functional form on position is also calculated analytically
in and out of equilibrium. We show that for both states these correlactions are
long-ranged. This result shows that indeed, even in equilibrium there is
long-range orientational order in the nematic, consistently with the well known
properties of these systems.We also calculate the dynamic structure of the
fluid in both states for a geometry consistent with light scattering
experiments experiments. We find that as with isotropic simple fluids, the
external temperature gradient introduces an asymmetry in the spectrum shifting
its maximum by an amount proportional to the magnitude of the gradient. This
effect may be of the order of 7 per cent. Also, the width at half height may
decrease by a factor of about 10 per cent. Since to our knowledge there are no
experimental results available in the literature to compare with, the
predictions of our model calculation remains to be assessed.
