We use a fluctuating hydrodynamics (FH) approach to study the fluctuations of the hydrodynamic variables of a thermotropic nematic liquid crystal ( NLC ) in a nonequilibrium steady state ( NESS ). This NESS is produced by an externally imposed temperature gradient and a uniform gravity field. We calculate analytically the equilibrium and nonequilibrium seven modes of the NLC in this NESS . These modes consist of a pair of sound modes, one orientation mode of the director and two visco-heat modes formed by the coupling of the shear and thermal modes. We find that the nonequilibrium effects produced by the external gradients only affect the longitudinal modes. The analytic expressions for the visco-heat modes show explicitly how the heat and shear modes of the NLC are coupled. We show that they may become propagative, a feature that also occurs in the simple fluid and suggests the realization of new experiments. We show that in equilibrium and in the isotropic limit of the NLC , our modes reduce to well-known results in the literature. For the NESS considered, we point out the differences between our modes and those reported by other authors. We close the chapter by proposing the calculation of other physical quantities that lend themselves to a more direct comparison with possible experiments for this system.
Part of the book: Non-Equilibrium Particle Dynamics