Stochastic effects on solution landscapes for nematic liquid crystals

ABSTRACT

We study the effects of additive and multiplicative noise on the solution landscape of nematic liquid crystals confined to a square domain within the Landau-de Gennes framework, as well as the impact of additive noise on the symmetric radial hedgehog solution for nematic droplets. The introduction of random noise can be used to capture material uncertainties and imperfections, which are always present in physical systems. We implement random noise in our framework by introducing a $\mathit{Q}$-Wiener stochastic process to the governing differential equations. On the square, the solution landscape for the deterministic problem is well understood, enabling us to compare and contrast the deterministic predictions and the stochastic predictions, while we demonstrate that the symmetry of the radial hedgehog solution can be violated by noise. This approach of introducing noise to deterministic equations can be used to test the robustness and validity of predictions from deterministic liquid crystal models, which essentially capture idealised situations.

Graphical abstract

KEYWORDS:

• Stochastic differential equations
• nematic liquid crystals
• additive noise
• multiplicative noise

Acknowledgments

The authors would like to thank Professor Neela Nataraj for her valued suggestions and feedback on the paper, especially with regard to the numeral implementation. The authors also thank Professor Utpal Manna (IISER Trivandrum) for helpful references and discussions in the initial stages of the work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

AM is supported by the University of Strathclyde New Professors Fund, a Leverhulme Research Project Grant RPG-2021-401, an OCIAM Visiting Fellowship at the University of Oxford and a Daiwa Foundation Small Grant. The authors gratefully acknowledge funding from the Royal Society International Exchange Grant IES\R2\202068. JD’s PDRA position at the University of Strathclyde (during which time the work was completed) was funded by the EPSRC Additional Funding for Mathematical Sciences scheme.