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Date: Tue, 7 Sep 2021 09:47:25 +0200
Subject: Expand knowledge base

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+---
+title: "Fick's laws"
+firstLetter: "F"
+publishDate: 2021-09-05
+categories:
+- Physics
+- Mathematics
+
+date: 2021-03-06T16:12:35+01:00
+draft: false
+markup: pandoc
+---
+
+# Fick's laws
+
+**Fick's laws of diffusion** govern the majority of diffusion processes,
+where a certain "impurity" substance redistributes itself through a medium over time.
+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
+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.
+Formally, Fick's first law predicts that the flux
+is proportional to the gradient of the concentration $C$ (with unit $\mathrm{m}^{-3}$):
+
+$$\begin{aligned}
+    \boxed{
+        \vec{J} = - D \: \nabla C
+    }
+\end{aligned}$$
+
+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 second law
+
+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:
+
+$$\begin{aligned}
+    M
+    \equiv \int_V C \dd{V}
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \dv{M}{t}
+    = - \int_S \vec{J} \cdot \dd{\vec{S}}
+    = - \int_V \nabla \cdot \vec{J} \dd{V}
+\end{aligned}$$
+
+For comparison, we differentiate the definition of $M$,
+and exploit that the integral ignores $t$:
+
+$$\begin{aligned}
+    \dv{M}{t}
+    = \dv{t} \int_V C \dd{V}
+    = \int_V \pdv{C}{t} \dd{V}
+\end{aligned}$$
+
+Both $\dv*{M}{t}$ 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 $\vec{J}$.
+Substituting this into the continuity equation yields
+the general form of Fick's second law:
+
+$$\begin{aligned}
+    \boxed{
+        \pdv{C}{t}
+        = \nabla \cdot \Big( D \: \nabla C \Big)
+    }
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \pdv{C}{t} = D \: \nabla^2 C
+\end{aligned}$$
+
+
+## Fundamental solution
+
+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
+a [Dirac delta function](/know/concept/dirac-delta-function/):
+
+$$\begin{aligned}
+    C(x, 0) = \delta(x - x_0)
+\end{aligned}$$
+
+According to Fick's second law,
+the concentration's time evolution of $C$ turns out to be:
+
+$$\begin{aligned}
+    H(x - x_0, t)
+    \equiv C(x, t)
+    = \frac{1}{\sqrt{4 \pi D t}} \exp\!\Big( \!-\!\frac{(x - x_0)^2}{4 D t} \Big)
+\end{aligned}$$
+
+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).
+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
+a convolution of itself with a delta function:
+
+$$\begin{aligned}
+    C(x, 0)
+    = (C * \delta)(x)
+    = \int_{-\infty}^\infty C(x_0, 0) \: \delta(x - x_0) \dd{x_0}
+\end{aligned}$$
+
+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 $H$:
+
+$$\begin{aligned}
+    C(x, t)
+    = (C * H)(x)
+    &= \int_{-\infty}^\infty C(x_0, 0) \: H(x - x_0, t) \dd{x_0}
+    \\
+    &= \int_{-\infty}^\infty \frac{1}{\sqrt{4 \pi D t}} \exp\!\Big( \!-\!\frac{(x - x_0)^2}{4 D t} \Big) \: C(x_0, 0) \dd{x_0}
+\end{aligned}$$
+
+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.
+
+
+
+## References
+1. U.F. Thygesen,
+   *Lecture notes on diffusions and stochastic differential equations*,
+   2021, Polyteknisk Kompendie.
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