Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation Diagrams via Stochastic Simulation

TL;DR

This paper introduces a novel method combining stochastic differential equation integration with bifurcation analysis to study liquid crystalline polymers, avoiding explicit closure models and enabling detailed stability and bifurcation insights.

Contribution

It presents a new approach that integrates Brownian dynamics simulations with continuum bifurcation analysis to analyze complex liquid crystal models without explicit closure.

Findings

Successfully obtained stable and unstable bifurcation branches.

Estimated coarse stability of the bifurcation diagram.

Enabled additional macroscopic computational tasks like projective integration.

Abstract

We demonstrate how time-integration of stochastic differential equations (i.e. Brownian dynamics simulations) can be combined with continuum numerical bifurcation 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.