Random walks and diffusion: Fick's two laws

One wanderer is unpredictable, a crowd of wanderers is not

Intuition

Imagine someone leaving a pub who takes each step in a random direction. You cannot say where they will be after a hundred steps. Now imagine a thousand people doing the same. Some go left, some go right, and on average they stay about where they started. But they spread out. The more time passes, the wider the spread. That predictable spreading is what the lecture builds on.

The random-walk model describes Brownian motion. The actual position of a particular particle on a particular walk is unpredictable. Its probability distribution can still be worked out accurately. When many particles do random walks, the ensemble behaves predictably. The average of the position $x$ stays around 0. But if you square each value of $x$ before averaging, the result rises roughly linearly with time. Slide 9 shows this with about 30 grey traces: the average of $x$ stays within about −30 to +40, and the average of $x^2$ climbs roughly in a straight line to about 13,000 at $t = 25{,}000$ (values read from the axes).

Random steps add up to net flow from high to low concentration

Intuition

Put a drop of ink in still water. Every ink particle is just jiggling at random. But the ink starts in one small crowded spot and there is no ink elsewhere. Random jiggling from a crowded spot sends more particles out than in. So the ink spreads until the colour is even.

Diffusion is transport of matter that arises because of concentration differences. It goes in a particular direction, yet the motion of each particle is random. The lecture’s answer is to look at the macroscopic behaviour of the collection instead of the microscopic motion of one particle. The net effect of the random motion is that the concentration throughout the vessel evens out. So macroscopically there is net movement of matter from high concentration to low concentration, while at the microscopic level every particle moves completely at random.

Fick’s laws turn the picture into two equations

To describe diffusion, the lecture defines the diffusion flux $J_x$ (mol s⁻¹): the net amount of substance moving across a surface perpendicular to the $x$ direction per time interval. Fick (1855) found that the flux is proportional to the surface area $A$ and to the concentration gradient, $-(dc/dx)$.

$J_x = -DA\frac{dc}{dx}$

This is Fick’s first law. $D$ is the diffusion coefficient. The slide does not define it, but it is the number that says how fast the substance spreads. The minus sign means the flow goes downhill, from high $c$ to low $c$. On slide 13, concentration is flat and high on the left, falls linearly in the middle (that is the gradient), then is flat and low on the right.

The flux into a region depends on the gradients at its two ends. To get the rate of change of concentration at one place, Fick uses differentials.

$\frac{dc}{dt} = D\frac{d^2c}{dx^2}$

This is Fick’s second law. On slide 14, a local minimum on a wavy concentration profile fills up (“concentration rises with time”) and a local maximum drains (“concentration falls with time”). In plain words, bumps flatten and dips fill.

Exam detail

Learn both laws with every symbol: $J_x$ in mol s⁻¹, $A$ the area, $dc/dx$ the gradient, $D$ the diffusion coefficient, $c$, $t$ and $x$. Know which law gives a flux (first) and which gives a change in concentration with time (second).

Whether a walker comes home depends on the number of dimensions

The slide 15 cartoon says: a drunk man will find his way home, since 2-d random walks are recurrent. A drunk bird may get lost forever, since 3-d random walks are transient. The slide attributes this to Shizuo Kakutani. The learning outcomes also ask for a consequence of 3-d transience in chemical systems, but the slides do not state one.

Pitfall

Do not mix up the two averages. The average of $x$ stays near zero. The average of $x^2$ grows with time. If someone says “the particles move away on average”, they mean the spread, not the average position.

Optional aside

The learning outcomes also mention relating $D$ to temperature and particle properties, and relating the root-mean-squared displacement to $D$ and time. Lecture 29’s slides give no equation for those. Lecture 30 uses the time to diffuse a distance $x$ as $x^2/2D$ (see the kinetics topic).

Recall

How can every particle move randomly and yet the concentration still ends up even?

Where it is crowded, more particles wander out than wander in. Where it is empty, more wander in than out. The random steps balance only once the crowd is even.