begin to read B&Z

notes/paper-notes/q4-Orientiation-Equivalence.md

+# Orientability and Energy Minimization in Liquid Crystal Models
+
+## Overview
+
+- *uniaxial nematic liquid crystals* werden modelliert
+- *Oseen-Frank*: unit vector field $`n`$ → nicht als $`ℝP^2`$ gesehen, sondern mit $`n ≠ -n`$ (ignoriert Symmetrie)
+  - wird als Standardsichtweise dargestellt
+- *Landau-de Gennes*: $`Q = s(n \otimes n - \tfrac13\operatorname{Id}`$
+- Resultat:
+  - Theorien sind gleich für einfach zusammenhängende Urbilder und $`W^{1,2}`$
+  - Unterschiede in anderen Fällen sind beschrieben:
+  - for simple energy functional, holes, various boundary conditions, difference cases are characterised (i.e. $`⇔`$)
+
+## de Gennes
+- closer to physics reality
+- $`Q`$-tensors are generally more [complex](https://arxiv.org/pdf/1409.3542.pdf), in our case just of the simple form („constrained”)
+- each point has preferred direction but can have any direction → probability measure $`μ(x, ·) : ℒ(S^2) → [0,1]`$ ($`ℒ`$ being all Lebesque sets)
+- symmetry modeled as $`μ(x, A) = μ(x, -A) ⇒ ⟨p⟩ = ∫_{S^2} p dμ(p) = 0`$ (first moment or average)
+- ... (some more steps to reach the $`Q`$ form from above)
+  - including $`-\frac12 \leq s \leq 1`$: $`s`$ says how much the molecules agree, are „in order“
+  - usual assumption: $`s`$ is constant $`⇒`$ space of $`Q`$-Tensors "is" $`ℝP^2`$
+
+## Oseen-Frank
+- simpler, but sometimes wrong
+- (here:) with orientation ($`n ∈ S^2`$, not $`n ∈ ℝP^2`$)
+- problem: *„fake defects”*: „non-orientable line field”
+
+## Compatibility
+- Oseen-Frank and de Gennes are compatible if the oriented line field (= unit vector field) can be oriented (without changing regularity)
+  - otherwise Oseen-Frank might miss a global minimizer because it is not orientable