From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/ficks-laws/index.md | 44 ++++++++++++++++-----------------
 1 file changed, 22 insertions(+), 22 deletions(-)

(limited to 'source/know/concept/ficks-laws')

diff --git a/source/know/concept/ficks-laws/index.md b/source/know/concept/ficks-laws/index.md
index f978ef5..b205af9 100644
--- a/source/know/concept/ficks-laws/index.md
+++ b/source/know/concept/ficks-laws/index.md
@@ -20,10 +20,10 @@ as opposed to **non-Fickian** or **anomalous diffusion**.
 moves from regions of high concentration to regions of lower concentration,
 at a rate proportional to the difference in concentration.
 
-Let $\vec{J}$ be the **diffusion flux** (with unit $\mathrm{m}^{-2} \mathrm{s}^{-1}$),
+Let $$\vec{J}$$ be the **diffusion flux** (with unit $$\mathrm{m}^{-2} \mathrm{s}^{-1}$$),
 whose magnitude and direction describe the "flow" of diffusing matter.
 Formally, Fick's first law predicts that the flux
-is proportional to the gradient of the concentration $C$ (with unit $\mathrm{m}^{-3}$):
+is proportional to the gradient of the concentration $$C$$ (with unit $$\mathrm{m}^{-3}$$):
 
 $$\begin{aligned}
     \boxed{
@@ -31,7 +31,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $D$ (with unit $\mathrm{m}^{2}/\mathrm{s}$)
+Where $$D$$ (with unit $$\mathrm{m}^{2}/\mathrm{s}$$)
 is known as the **diffusion coefficient** or **diffusivity**,
 and depends on both the medium and the diffusing substance.
 
@@ -47,17 +47,17 @@ than about diffusion in general.
 
 To derive **Fick's second law**, we demand that matter is conserved,
 i.e. the diffusing species is not created or destroyed anywhere.
-Suppose that an arbitrary volume $V$ contains an amount $M$ of diffusing matter,
-distributed in space according to $C(\vec{r})$, such that:
+Suppose that an arbitrary volume $$V$$ contains an amount $$M$$ of diffusing matter,
+distributed in space according to $$C(\vec{r})$$, such that:
 
 $$\begin{aligned}
     M
     \equiv \int_V C \dd{V}
 \end{aligned}$$
 
-Over time $t$, matter enters/leaves $V$.
-Let $S$ be the surface of $V$, and $\vec{J}$ the diffusion flux,
-then $M$ changes as follows, to which we apply the divergence theorem:
+Over time $$t$$, matter enters/leaves $$V$$.
+Let $$S$$ be the surface of $$V$$, and $$\vec{J}$$ the diffusion flux,
+then $$M$$ changes as follows, to which we apply the divergence theorem:
 
 $$\begin{aligned}
     \dv{M}{t}
@@ -65,8 +65,8 @@ $$\begin{aligned}
     = - \int_V \nabla \cdot \vec{J} \dd{V}
 \end{aligned}$$
 
-For comparison, we differentiate the definition of $M$,
-and exploit that the integral ignores $t$:
+For comparison, we differentiate the definition of $$M$$,
+and exploit that the integral ignores $$t$$:
 
 $$\begin{aligned}
     \dv{M}{t}
@@ -74,14 +74,14 @@ $$\begin{aligned}
     = \int_V \pdv{C}{t} \dd{V}
 \end{aligned}$$
 
-Both $\idv{M}{t}$ are equal, so stripping the integrals leads to this **continuity equation**:
+Both $$\idv{M}{t}$$ are equal, so stripping the integrals leads to this **continuity equation**:
 
 $$\begin{aligned}
     \pdv{C}{t}
     = - \nabla \cdot \vec{J}
 \end{aligned}$$
 
-From Fick's first law, we already have an expression for $\vec{J}$.
+From Fick's first law, we already have an expression for $$\vec{J}$$.
 Substituting this into the continuity equation yields
 the general form of Fick's second law:
 
@@ -92,8 +92,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Usually, it is assumed that $D$ is constant
-with respect to space $\vec{r}$ and concentration $C$,
+Usually, it is assumed that $$D$$ is constant
+with respect to space $$\vec{r}$$ and concentration $$C$$,
 in which case Fick's second law reduces to:
 
 $$\begin{aligned}
@@ -105,8 +105,8 @@ $$\begin{aligned}
 
 Fick's second law has exact solutions for many situations,
 but the most important one is arguably the **fundamental solution**.
-Consider a 1D system (for simplicity) with constant diffusivity $D$,
-where the initial concentration $C(x, 0)$ is
+Consider a 1D system (for simplicity) with constant diffusivity $$D$$,
+where the initial concentration $$C(x, 0)$$ is
 a [Dirac delta function](/know/concept/dirac-delta-function/):
 
 $$\begin{aligned}
@@ -114,7 +114,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 According to Fick's second law,
-the concentration's time evolution of $C$ turns out to be:
+the concentration's time evolution of $$C$$ turns out to be:
 
 $$\begin{aligned}
     H(x - x_0, t)
@@ -127,11 +127,11 @@ as a consequence of
 the [central limit theorem](/know/concept/central-limit-theorem/):
 the diffusion behaviour is a sum of many independent steps
 (i.e. molecular collisions).
-The standard deviation is $\sqrt{2 D t}$,
-meaning that the distance of a diffusion is proportional to $\sqrt{t}$.
+The standard deviation is $$\sqrt{2 D t}$$,
+meaning that the distance of a diffusion is proportional to $$\sqrt{t}$$.
 
-This solution $H$ is extremely useful,
-because any initial concentration $C(x, 0)$ can be written as
+This solution $$H$$ is extremely useful,
+because any initial concentration $$C(x, 0)$$ can be written as
 a convolution of itself with a delta function:
 
 $$\begin{aligned}
@@ -142,7 +142,7 @@ $$\begin{aligned}
 
 In other words, any function is a linear combination of delta functions.
 Fick's second law is linear,
-so the overall solution $C(x, t)$ is the same combination of fundamental solutions $H$:
+so the overall solution $$C(x, t)$$ is the same combination of fundamental solutions $$H$$:
 
 $$\begin{aligned}
     C(x, t)
-- 
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