6.3 Diffusion as a random walk

The diffusion equation of the previous lesson is a macroscopic law, but its origin is microscopic and statistical: it is the smooth shadow of an underlying random walk. A particle buffeted by molecular collisions takes a long sequence of uncorrelated steps, and the probability distribution of its position obeys Fick’s law exactly. This lesson makes that connection precise.

From random steps to Fick’s law

▶ The diffusion equation from a random walk Derivation

Let $p(x,t)$ be the probability density for the walker’s position. In a short time $\delta t$ it takes a random step whose distribution $G$ has zero mean and variance $\sigma^2\delta t$ . The new density is the old one convolved with the step distribution:

p(x, t+\delta t) \;=\; \int p(x', t)\,G(x - x';\,\sigma^2\delta t)\,dx'.

Expand $p(x',t)$ about $x$ to second order, writing $\Delta x = x' - x$ :

p(x',t) \;\approx\; p(x,t) + \Delta x\,p_x + \tfrac12\Delta x^2\,p_{xx}.

Insert this and use the moments of $G$ : $\int G\,d\Delta x = 1$ , $\int \Delta x\,G\,d\Delta x = 0$ (no bias), and $\int \Delta x^2\,G\,d\Delta x = \sigma^2\delta t$ . The first moment kills the $p_x$ term, leaving

p(x, t+\delta t) - p(x,t) \;\approx\; \tfrac12\sigma^2\delta t\,p_{xx}.

Divide by $\delta t$ :

\frac{\partial p}{\partial t} \;=\; \tfrac12\sigma^2\,\frac{\partial^2 p}{\partial x^2}.

This is the diffusion equation, with diffusion coefficient $D = \sigma^2/2$ . Nothing about the microscopic step distribution survives except its variance: a walk of tiny frequent steps and a walk of larger rarer ones diffuse identically if $\sigma^2/\delta t$ matches. This insensitivity to detail — a consequence of the central limit theorem (refresher: random walks and Brownian motion →) — is why diffusion is so universal.

Mean-squared displacement

The signature of normal diffusion is the way a walker’s spread grows with time. The variance of position increases linearly,

\langle x^2 \rangle = 2Dt \;\;(\text{1-D}), \qquad \langle r^2 \rangle = 4Dt \;\;(\text{2-D}), \qquad \langle r^2 \rangle = 6Dt \;\;(\text{3-D}),

with one factor of $2D$ per spatial dimension. The root-mean-squared displacement therefore grows only as $\sqrt{t}$ — diffusion is slow over long distances, a fact with sharp consequences for the timescales of the final lesson.

D1.00

t0.0

measured ⟨r²⟩—

theory 4Dt0.00

Diffusion constant D = 1.00 Number of walkers = 200

Each walker takes an independent Gaussian step every frame; the cloud spreads, and the mean-squared distance from the origin grows linearly in time with slope 4D (in 2-D; 2D in 1-D, 6D in 3-D). This linearity — not the cloud shape, not the particle paths — is the *signature* of diffusion. Below it lies the Einstein relation D = k_BT / γ that ties the diffusion constant to the microscopic friction.

Many independent walkers, each stepping randomly, collectively trace out the spreading Gaussian that solves the diffusion equation. The histogram of their positions broadens as $\sqrt{t}$ while staying centred, exactly as $\langle x^2\rangle = 2Dt$ requires.

Each 120-second run of the heavy particle's r²(t) is noisy (light blue). Averaged over many runs, the noise washes out (black). If the particle moved in a straight line at the thermal speed ⟨V²⟩ = 2kT/M, the average would grow as t² — the green parabola. It does not. Because each collision randomises direction, the average grows linearly in t, with slope 4D. Linearity is the signature of randomness.

A single such trajectory is Brownian motion: the erratic, never-settling path of a microscopic particle suspended in a fluid, kicked at random by the thermal motion of the molecules around it. The path is continuous but nowhere smooth — zoom in and it looks statistically the same at every scale. That a visible particle should jitter this way was the first direct evidence that the fluid around it is made of discrete, moving molecules, the subject of the Einstein relation two lessons on.