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# 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
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- *Landau-de Gennes*: $`Q = s(n \otimes n - \tfrac13\operatorname{Id})`$
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- 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
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- 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”)
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### Derivation of $`Q`$-Tensors
(from chapter 1 Introduction)
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- each point has preferred direction but can have any direction → probability measure $`μ(x, ·) : ℒ(𝕊^2) → [0,1]`$ ($`ℒ`$ being all Lebesque sets) modelling very small area around $`x`$
- symmetry modeled as $`μ(x, A) = μ(x, -A) ⇒ ⟨p⟩ = ∫_{𝕊^2} p dμ(p) = 0`$ (first moment or average)
- Tensor of second moments: $`M_{ij} = ∫_{𝕊^2} p_ip_j dμ(p)`$ ($`i, j = 1, 2, 3`$)
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  - $`M = M^T`$, $`\operatorname{Tr} M = Σ_{i=1}^3 ∫_{𝕊^2} p_i^2 dμ(p) = ∫_{𝕊^2} 1 dμ(p) = 1`$ ($`μ`$ is probability measure)
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  - $`e · M e = ∫_{𝕊^2} (e · p)^2 dμ(p) = ⟨\cos^2(θ)⟩`$ ($`θ`$ = angle between $`p`$ and $`e`$) for $`|e| =1`$)
  - if $`μ`$ is isotropic (no preferred direction): $`μ_0`$ with $`dμ_0(p) = (4π)^{-1}dA`$ with
    $`M_0 = (4π)^{-1} ∫_{𝕊^2} p \otimes p dA = \frac 13 \operatorname{Id}`$
    - $`∫_{𝕊^2} p_1 p_2 dμ(p) = 0`$ since what happens on one half-sphere is countered by the other half-sphere ($`p_2`$, $`p_3`$ are same but $`p_1`$ is of opposite sign)
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    - $`∫_{𝕊^2} p_i^2 dμ(p)`$ are equal ($`i=1,2,3`$) and in sum $`= \operatorname{Tr} M_0 = 1`$
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- Def. *de Gennes order-parameter tensor $`Q`$*: difference of second moment tensor to isotropic case $`Q = M - M_0 = ∫_{𝕊^2}(p \otimes p - \frac13 \operatorname{Id}) dμ(p)`$
  - $`Q`$ symmetric, $`\operatorname{Q} = 0`$ $`⇒`$ (spectral theorem) $`Q = λ_1 ê_1 \otimes ê_2 + λ_2 ê_2 \otimes ê_2 - (λ_1 + λ_2)ê_3 \otimes ê_3`$ ($`ê_{1,2,3}`$ orthonormal eigenvector basis, $`λ_1, λ_2`$ eigenvectors)
    - Eigenvectors for different non-zero eigenvalues are orthogonal: $`(λ_1e_1) · (λ_2^{-1} e_2) = (Q e_1)^T (Q^{-1}e_2) = e_1 · Q^T Q^{-1} e_2 = e_1 · e_2 ⇒ e_1 ⊥ e_2`$
  - two eigenvalues equal and non-zero: *uniaxial* (considered here)
  - otherwise *biaxial* (more complicated)
- Constraint on $`s`$:
  - $`s`$ defined by limiting process $`L → 0`$ and the constants $`a, b, c`$
  - but also $`Q`$ has an original definition via $`μ`$. Together:
  ```math
  Qn · n
  = n^T s (n n^T - \frac 13 \operatorname{Id}) n
  = s (n^T n n^T n - \frac13 n^T \operatorname{Id}n)
  = s(1 · 1 - \frac13 · 1) \frac23 s
  = ∫_{𝕊^2} n^T p p^T n - \frac13 n^T n dμ(p)
  = ∫_{𝕊^2} (n · p)^2 - \frac 13 dμ(p)
  = ⟨\cos^2 θ - \frac13⟩
  ⇒ s = \frac32 ⟨\cos^2 θ - \frac13⟩
  ```
  - $`⇒ -\frac12 \leq s \leq 1`$: $`s`$ says how much the molecules agree, are „in order“ (max: $`s = 1`$: perfectly ordered $`p∥n`$; $`s = -\frac12`$: $`p ⊥ n`$; $`s=0`$: $`Q = 0`$, isotropic)
  - $`⇒`$ s is *scalar order parameter associated to the tensor $`Q`$*
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  - usual assumption: $`s`$ is constant $`⇒`$ space of $`Q`$-Tensors "is" $`ℝP^2`$

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### Simpliest energy functional
(p. 495 = 3)
```math
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ℱ_{\text{LG}}[Q] = ∫_{Ω} \left(\sum_{i,j,k = 1}^3 \frac L2 Q_{ij,k} Q_{ij,k} - \frac a2 \operatorname{Tr} Q^2 - \frac b3 \operatorname{Tr} Q^3 + \frac c4 (\operatorname{Tr} Q^2)^2 \right) dx
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```
$`a, b, c`$ constants, $`L`$ „*elastic constant*”. Physics is interested in $`L → 0`$

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## Oseen-Frank
- simpler, but sometimes wrong
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- (here:) with orientation ($`n ∈ 𝕊^2`$, not $`n ∈ ℝP^2`$)
- problem: *„fake defects”*: „non-orientable line field” (places where vector field has to be rough but would be OK if we took away the orientation

### Simpliest energy functional
minimizers of $`ℱ_{\text{LG}}`$ for $`L → 0`$ are "suitably approximated" by minimizers of
```math
ℱ_{\text{OF}}[Q] = ∫_{Ω} \sum_{i, j, k = 1}^3 Q_{ij,k} Q_{ij,k} dx
```
*if* $`Q ∈ W^{1,2}`$ with $`Q`$ uniaxial almost everywhere, i.e.
```math
Q = s(n \otimes n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê_3)
```
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## 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
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  - Question if orientable calculable with integer programming problem (p. 4 = 496) (= (linear) optimization problem with only integer coefficiants)

### Notation
- $`P : 𝕊^2 → 𝒬`$ removes orientation. Orientable = in the image of $`P`$. Same for $`Q`$ only defined on $`∂Ω`$
- $`𝒬 := \{Q = s(n \otimes n - \frac13 \operatorname{ID}) | n ∈ 𝕊^2\}`$
- $`𝒬_2 := \{Q = s((n_1, n_2, 0) \otimes (n_1, n_2, 0) - \frac13 \operatorname{ID}) | n = (n_1, n_2, 0) ∈ 𝕊^2\}`$
- $`C^k`$ and Lipschitz domains are defined as having a graph as the boundary (as usual)
- $`W^{1,p}`$ defined via embedding
- $`W^{1,p}_{φ} = W^{1,p}`$ with $`φ`$ on the boundary (use $`\operatorname{tr}`$-Operator to define boundary value)
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- $`P : 𝕊^2 → 𝒬`$ removes orientation. Orientable = in the image of $`P`$. Same for $`Q`$ only defined on $`∂Ω`$
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  - for $`Q ∈ W^{1,p}(Ω, 𝒬)`$ $`ℒ^d`$-almost everywhere
  - for $`Q ∈ W^{1-\frac1p, p}(∂Ω, 𝒬)`$ orientable to $`n ∈ W^{1-\frac1p, p}(Ω, 𝕊^2)`$ $`ℋ^{d-1}`$-almost everywhere
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- in chapter 3.1 firstly for continous $`Q`$ since that's standard from topology
  - *Theorem 1*: $`𝒬_2`$ (one-dim), orientable on boundary of holes, $`Q`$ continous $`⇒`$ orientable (long proof with a lot of fiddling!)
- chapter 3.2 Theorem 2: $`Ω`$ simply connected, $`Q ∈ W^{1, p}`$, $`p \geq 2`$ (!) $`⇒`$ orientable with Sobolev-Seminorm estimate. Counterexample for $`p < 2`$
- chapter 4: different cases of $`Q`$ orientable on boundary $`⇒`$/$`⇔`$/$`⇐`$ on $`Ω`$. Sometimes $`Q: → 𝒬_2`$, sometimes $`Q`$ continous (on boundary)

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### Chapter 2 Propositions
(choose better subsection title!)

*Proposition 2*: $`Q ∈ W^{1,p}(Ω, 𝒬)`$ ($`1 \leq p \leq ∞`$) can have only two orientations.

*Proof*:
In one dimension weakly differentiable means that $`f(x) = f(a) + ∫_a^x f'(y) dy`$.
(Which is in turn equivalent to absolutely continous.)
In several dimensions this is true in each direction but only for almost all $`x_2`$, $`x_3`$.
So: $`n`$, $`m`$ have representatives such that for almost everywhere $`x_2`$, $`x_3`$: $`n`$, $`m=τn`$ are  absolutely continous, so also $`τ = τ·1 = τn·n = m·n`$.

Absolute continuity implies continuity, so $`τ`$ has a representative that is for those almost all $`x_2, x_3`$, constant.
Now use this in Fubini on a small ball in $`Ω`$ around $`y`$:
```math
∫_{B_{ε}(y)} ∂_1 τ(x) φ(x) dx = ∫_{B_{ε}(y)} τ(x) ∂_1 φ(x) dx = ∫_{B_{ε}(y)} ∂_1(τφ)(x) dx = 0
```
(first =: Fubini, second =: $`τ`$ is constant along $`x_1`$ for almost all $`x_2, x_3`$, third =: first integrated over $`x_1`$ direction, this is $`± φ(x_{1, \max}, x_2, x_3) - (± φ(x_{1, \min}, x_2, x_3)) = 0 - 0 = 0`$, then integrated over $`x_2`$, $`x_3`$, it stays 0)

Hence $`∇τ = 0`$ (since $`x_2`$, $`x_3`$ are analogous) weakly but this means that $`τ`$ is constant on $`B_{ε}(y)`$ for every $`y ∈ Ω`$ and therefore constant on $`Ω`$. Done. 

---

Trace operator is introduced by Evans but with continuity estimate for $`\operatorname{tr} : W^{1,p} → L^p`$. Here we need the stronger statement that $`\operatorname{tr}`$ maps into the [Sobolev-Slobodeckij space](https://en.wikipedia.org/wiki/Trace_operator#Characterization_using_Sobolev%E2%80%93Slobodeckij_spaces) $`W^{1-1/p, p}`$.

*Proposition 3*:

*Proof sketch*:

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## Energy functionals
- specific energy functionals are regarded (p. 11/503) because they were looked at before and Oseen-Frank was successful for them
  - conversion between energy functionals possible
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## Questions
- What is the third index $`k`$ for $`Q`$ on page 3 (= 495)?
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  - Probably it's a derivative of $`Q_{ij}`$ in direction $`k`$.
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- And why is it written as a product instead of a square$`^2`$?
- To make spectral representation of $`Q`$ fit to $`Q = s(n \otimes n - \tfrac13 \operatorname{Id})`$, we need $`s = -3 λ_1 = -3 λ_2, n = ê_3`$. How can $`s`$ then depend on $`a, b, c`$ (see formula (3) on page 3 (=495))? If that's the case, $`Q`$ is further reduced to the choice of $`n`$ but that's what we assumed as our model at the beginning. So apparently that's fine.
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- in Proposition 1 it should be $`\det Q = +\frac{2s^3}{27}`$ since the $`-`$ exists in exactly two eigenvalues: $`\det Q = λ_1 λ_2 λ_3 = (-\frac s3)(-\frac s3)(+\frac {2s}3) = +\frac{2s^3}{27}`$. Correct?

## Errors in the paper
p.5 (=497) Proposition 1 (iii): sign of $`\operatorname{det} Q`$ is wrong. Must be positive, since exactly two of the eigenvalues are negative

p. 7 (=499) Proposition 2: it says $`Q(x) ∈ W^{1,p}(Ω, 𝒬)`$ but $`(x)`$ must be erased ($`Q(x) ∈ 𝒬`$)
Also the statement of the proposition is trivial and not what the colloquial formulation says: it should be "... with $`P(n) = P(m) = Q`$ and $`n ≠ m`$ [as $`W^{1,p}`$-generalized functions, so on a non-0-Lebesgue set], we have $`m = -n`$ almost everywhere in $`Ω`$. If $`n ≠ m`$ almost everywhere, it follows straight away that $`m = -n`$ almost everywhere since pointwise $`n(x) = ± m(x)`$