Coarse Brownian dynamics for nematic liquid crystals: Bifurcation, projective integration, and control via stochastic simulation

We demonstrate how time integration of stochastic differential equations (i.e., Brownian dynamics simulations) can be combined with continuum numerical analysis techniques to analyze the dynamics of liquid crystalline polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the approach analyzes the (unavailable in closed form) “coarse” macroscopic equations, estimating the necessary quantities through appropriately initialized, short “bursts” of Brownian dynamics simulation. Through this approach, both stable and unstable branches of the equilibrium bifurcation diagram are obtained for the Doi model of LCPs and their “coarse stability” is estimated. Additional macroscopic computational tasks enabled through this approach, such as coarse projective integration and coarse stabilizing controller design, are also demonstrated.