Long-Range Order and Dynamic Structure Factor of a Nematic under a   Thermal Gradient
  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. 
R. F. Rodr\'iguez and H. H\'ijar