From 7c412050570ef229dd78cbcffbf80f23728a630d Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 13 May 2023 15:42:47 +0200
Subject: Improve knowledge base

---
 source/know/concept/ficks-laws/index.md | 21 +++++++++++++--------
 1 file changed, 13 insertions(+), 8 deletions(-)

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

diff --git a/source/know/concept/ficks-laws/index.md b/source/know/concept/ficks-laws/index.md
index b205af9..8d5da7d 100644
--- a/source/know/concept/ficks-laws/index.md
+++ b/source/know/concept/ficks-laws/index.md
@@ -14,6 +14,7 @@ A diffusion process that obeys Fick's laws is called **Fickian**,
 as opposed to **non-Fickian** or **anomalous diffusion**.
 
 
+
 ## Fick's first law
 
 **Fick's first law** states that diffusing matter
@@ -21,13 +22,14 @@ 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 $$\mathrm{m}^{-2} \mathrm{s}^{-1}$$),
-whose magnitude and direction describe the "flow" of diffusing matter.
+whose magnitude and direction describes 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 $$\mathrm{m}^{-3}$$):
 
 $$\begin{aligned}
     \boxed{
-        \vec{J} = - D \: \nabla C
+        \vec{J}
+        = - D \: \nabla C
     }
 \end{aligned}$$
 
@@ -43,6 +45,7 @@ but they say more about those systems
 than about diffusion in general.
 
 
+
 ## Fick's second law
 
 To derive **Fick's second law**, we demand that matter is conserved,
@@ -65,8 +68,7 @@ $$\begin{aligned}
     = - \int_V \nabla \cdot \vec{J} \dd{V}
 \end{aligned}$$
 
-For comparison, we differentiate the definition of $$M$$,
-and exploit that the integral ignores $$t$$:
+For comparison, we can also just differentiate the definition of $$M$$ directly:
 
 $$\begin{aligned}
     \dv{M}{t}
@@ -74,7 +76,8 @@ $$\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**:
+Above, we have two valid expressions for $$\idv{M}{t}$$,
+which must be equal, so stripping the integrals leads to this **continuity equation**:
 
 $$\begin{aligned}
     \pdv{C}{t}
@@ -101,6 +104,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 
+
 ## Fundamental solution
 
 Fick's second law has exact solutions for many situations,
@@ -110,11 +114,12 @@ where the initial concentration $$C(x, 0)$$ is
 a [Dirac delta function](/know/concept/dirac-delta-function/):
 
 $$\begin{aligned}
-    C(x, 0) = \delta(x - x_0)
+    C(x, 0)
+    = \delta(x - x_0)
 \end{aligned}$$
 
-According to Fick's second law,
-the concentration's time evolution of $$C$$ turns out to be:
+By solving Fick's second law with this initial condition,
+$$C$$'s time evolution turns out to be:
 
 $$\begin{aligned}
     H(x - x_0, t)
-- 
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