Director fluctuations in nematic liquid crystals; long-range correlations and the hydrodynamic equations

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Abstract
The hydrodynamic equations for a nematic liquid crystal contain the director as one of the variables. Microscopically the director corresponds to a symmetry-restoring variable, but unfortunately in a nematic there are an infinite number of these, for they can be constructed out of any even-ranked tensor of the molecular orientations. In order to find the ‘correct’ microscopic correspondence, we begin by deriving an expression for the long-ranged part of the two-particle distribution function, and hence show that all the symmetry-restoring variables have susceptibilities that diverge at long wavelength in agreement with the form predicted from Frank's expression for the free energy of distortion of a liquid crystal. This infinite number of variables, however, does not alter the total number of hydrodynamic modes from that predicted by the normal hydrodynamic equations, and we argue that the ‘true’ microscopic definition of the director is that linear combination of all the symmetry restoring variables which corresponds to these hydrodynamic modes of decay. Lastly we suggest a possible strategy for showing that the symmetry-restoring variables have a divergent susceptibility via a Bogoliubov inequality argument.

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