Commit 33c73e7f by Felix Hilsky

### 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 $L → 0$ ... ...
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!