Bifurcation Analysis of Liquid Crystal Phase Transitions

TL;DR

This paper discusses bifurcation analysis as a method to study phase transitions in liquid crystals, linking phenomenological theories with statistical mechanics through examples and applications.

Contribution

It introduces bifurcation analysis as a practical tool for understanding liquid crystal phase transitions, bridging phenomenological and statistical approaches.

Findings

Bifurcation analysis reveals the relation between particle interactions and phase transition properties.

Tutorial example demonstrates the technique's application to liquid crystal models.

Discussion of realistic models shows the method's versatility and complementarity to simulations.

Abstract

These lectures focus on bifurcation analysis as a tool for studying phase transitions that occur in models of liquid-crystalline systems. We show how this approach bridges the gap between the phenomenological Landau theory and the --- often intractable --- full statistical mechanical treatment. Employing a `toy model' as a tutorial example the various ingredients of the technique are presented. Special attention is paid to the way in which one obtains information on the relation between the characteristics of the assumed interparticle interactions (shape, symmetry ...) and global properties of the phase transitions (order, symmetry of resultant phases ...). Finally, a few more involved examples are discussed indicating how the approach can be applied to more realistic models and how it can serve as a complement to simulations.