---
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.