Commit 33c73e7f authored by Felix Hilsky's avatar Felix Hilsky
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fit notation to paper

parent 0f8e6e31
......@@ -12,20 +12,22 @@
- 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”)
### Derivation of $`Q`$-Tensors
(from chapter 1 Introduction)
- 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: $`M_{ij} = ∫_{𝕊^2} p_ip_j dμ(p)`$ ($`i, j = 1, 2, 3`$)
- $`M = M^T`$, $`\operatorname{tr} M = Σ_{i=1}^3 ∫_{𝕊^2} p_i^2 dμ(p) = ∫_{𝕊^2} 1 dμ(p) = 1`$ ($`μ`$ is probability measure)
- $`M = M^T`$, $`\operatorname{Tr} M = Σ_{i=1}^3 ∫_{𝕊^2} p_i^2 dμ(p) = ∫_{𝕊^2} 1 dμ(p) = 1`$ ($`μ`$ is probability measure)
- $`e · M e = ∫_{𝕊^2} (e · p)^2 dμ(p) = ⟨\cos^2(θ)⟩`$ ($`θ`$ = 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)
- $`∫_{𝕊^2} p_i^2 dμ(p)`$ are equal ($`i=1,2,3`$) and in sum $`= \operatorname{tr} M_0 = 1`$
- $`∫_{𝕊^2} p_i^2 dμ(p)`$ are equal ($`i=1,2,3`$) and in sum $`= \operatorname{Tr} M_0 = 1`$
- 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: $`(λ_1e_1) · (λ_2^{-1} e_2) = (Q e_1)^T (Q^{-1}e_2) = e_1 · Q^T Q^{-1} e_2 = e_1 · e_2 ⇒ e_1 ⊥ e_2`$
......@@ -51,7 +53,7 @@
### Simpliest energy functional
(p. 495 = 3)
```math
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
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
```
$`a, b, c`$ constants, $`L`$ „*elastic constant*”. Physics is interested in $`L → 0`$
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