Fluctuating Nematodynamics using the Stochastic Method of Lines

TL;DR

This paper develops a stochastic numerical method to simulate the fluctuations of the nematic order parameter tensor in liquid crystals, accurately capturing thermal noise effects and validated against theoretical predictions.

Contribution

It introduces a Langevin equation framework with symmetry-preserving noise and a stochastic method of lines for numerical solutions, enabling detailed fluctuation studies in nematic systems.

Findings

Numerical solutions match analytic equilibrium distributions.

The method accurately reproduces structure factors and dynamic correlations.

Validated approach for studying fluctuation-driven phenomena in nematics.

Abstract

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.