Improve knowledge base

1 files changed, 25 insertions, 29 deletions

diff --git a/source/know/concept/ficks-laws/index.md b/source/know/concept/ficks-laws/index.md
index 8d5da7d..20bc50b 100644
--- a/source/know/concept/ficks-laws/index.md
+++ b/source/know/concept/ficks-laws/index.md
@@ -21,10 +21,12 @@ 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 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 $$\mathrm{m}^{-3}$$):
+is proportional to the gradient of the concentration $$C(\vec{r})$$
+(with unit $$\mathrm{m}^{-3}$$):
 
 $$\begin{aligned}
     \boxed{
@@ -37,12 +39,9 @@ Where $$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.
 
-Fick's first law is a general physical principle,
-which was discovered experimentally,
-and thus does not have a general derivation.
-Proofs for specific systems do exist,
-but they say more about those systems
-than about diffusion in general.
+Fick's first law is an empirical physical principle,
+and therefore does not have a general derivation,
+although proofs for specific systems do exist.
 
 
 
@@ -59,12 +58,12 @@ $$\begin{aligned}
 \end{aligned}$$
 
 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:
+Let $$\partial V$$ be the surface of $$V$$, and $$\vec{J}$$ the diffusion flux,
+then $$M$$ changes as follows, applying the divergence theorem:
 
 $$\begin{aligned}
     \dv{M}{t}
-    = - \int_S \vec{J} \cdot \dd{\vec{S}}
+    = - \int_{\partial V} \vec{J} \cdot \dd{\vec{S}}
     = - \int_V \nabla \cdot \vec{J} \dd{V}
 \end{aligned}$$
 
@@ -91,7 +90,7 @@ the general form of Fick's second law:
 $$\begin{aligned}
     \boxed{
         \pdv{C}{t}
-        = \nabla \cdot \Big( D \: \nabla C \Big)
+        = \nabla \cdot \Big( D \, \nabla C \Big)
     }
 \end{aligned}$$
 
@@ -100,7 +99,8 @@ with respect to space $$\vec{r}$$ and concentration $$C$$,
 in which case Fick's second law reduces to:
 
 $$\begin{aligned}
-    \pdv{C}{t} = D \: \nabla^2 C
+    \pdv{C}{t}
+    = D \, \nabla^2 C
 \end{aligned}$$
 
 
@@ -108,7 +108,7 @@ $$\begin{aligned}
 ## Fundamental solution
 
 Fick's second law has exact solutions for many situations,
-but the most important one is arguably the **fundamental solution**.
+but the most important one is arguably the **fundamental solution** $$H$$.
 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/):
@@ -118,8 +118,8 @@ $$\begin{aligned}
     = \delta(x - x_0)
 \end{aligned}$$
 
-By solving Fick's second law with this initial condition,
-$$C$$'s time evolution turns out to be:
+By solving Fick's second law with this initial condition (details omitted),
+we find that $$C$$ obeys:
 
 $$\begin{aligned}
     H(x - x_0, t)
@@ -127,13 +127,12 @@ $$\begin{aligned}
     = \frac{1}{\sqrt{4 \pi D t}} \exp\!\Big( \!-\!\frac{(x - x_0)^2}{4 D t} \Big)
 \end{aligned}$$
 
-This result is a normalized Gaussian,
-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).
+This result is a normalized Gaussian:
+diffusion is a sum of many independent molecular collisions,
+so the [central limit theorem](/know/concept/central-limit-theorem/) applies,
+hence this result.
 The standard deviation is $$\sqrt{2 D t}$$,
-meaning that the distance of a diffusion is proportional to $$\sqrt{t}$$.
+meaning that the expected 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
@@ -146,22 +145,19 @@ $$\begin{aligned}
 \end{aligned}$$
 
 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 $$H$$:
+Fick's second law is linear, so the overall solution $$C(x, t)$$
+is the same combination of fundamental solutions $$H$$:
 
 $$\begin{aligned}
     C(x, t)
     = (C * H)(x)
     &= \int_{-\infty}^\infty C(x_0, 0) \: H(x - x_0, t) \dd{x_0}
-    \\
-    &= \int_{-\infty}^\infty \frac{1}{\sqrt{4 \pi D t}} \exp\!\Big( \!-\!\frac{(x - x_0)^2}{4 D t} \Big) \: C(x_0, 0) \dd{x_0}
 \end{aligned}$$
 
 This technique is analogous to using
 the [impulse response](/know/concept/impulse-response/)
-of a linear operator to extrapolate all its inhomogeneous solutions.
-The difference is that here, we used the initial condition
-instead of the forcing function.
+of a linear operator to extrapolate all its inhomogeneous solutions,
+but here we used the initial condition instead of the forcing function.