  We construct Langevin equations describing the fluctuations of the tensor
order parameter $Q_{\alpha\beta}$ in nematic liquid crystals by adding noise
terms to time-dependent variational equations that follow from the
Ginzburg-Landau-de Gennes free energy. The noise is required to preserve the
symmetry and tracelessness of the tensor order parameter and must satisfy a
fluctuation-dissipation relation at thermal equilibrium. We construct a noise
with these properties in a basis of symmetric traceless matrices and show that
the Langevin equations can be solved numerically in this basis using a
stochastic version of the method of lines. The numerical method is validated by
comparing equilibrium probability distributions, structure factors and dynamic
correlations obtained from these numerical solutions with analytic predictions.
We demonstrate excellent agreement between numerics and theory. This
methodology can be applied to the study of phenomena where fluctuations in both
the magnitude and direction of nematic order are important, as for instance in
the nematic swarms which produce enhanced opalescence near the
isotropic-nematic transition or the problem of nucleation of the nematic from
the isotropic phase.
