11.3 Random walks and Brownian motion

A random walk is the simplest stochastic process: a particle takes a step in a random direction at each tick of a clock. The construction is trivial; the consequences are deep. The continuum limit of a random walk is Brownian motion, the canonical model of diffusion. The same equation appears in heat conduction, particle physics, finance, and the Brownian-motion lesson of Sound 1.3. All of it descends from “many tiny independent steps.”

The 1-D random walk

Consider a particle on the integer line, starting at $X_0 = 0$ . At each time step $n = 1, 2, 3, \ldots$ , the particle takes a step of size 1 in a random direction:

X_n \;=\; X_{n-1} + \xi_n, \qquad \xi_n = \pm 1 \text{ with equal probability}.

The steps $\xi_n$ are independent and identically distributed (i.i.d.) Bernoulli-like random variables with $\mathbb{E}[\xi_n] = 0$ and $\mathrm{Var}[\xi_n] = 1$ .

The position after $N$ steps is

X_N \;=\; \sum_{n=1}^N \xi_n.

By linearity of expectation:

\mathbb{E}[X_N] \;=\; 0.

The walker has no preferred direction. But the spread about zero is non-trivial:

▶ Mean-square displacement: ⟨X_N²⟩ = N

We want $\mathbb{E}[X_N^2]$ . Expand the square:

X_N^2 \;=\; \left( \sum_{n=1}^N \xi_n \right)^2 \;=\; \sum_{n=1}^N \xi_n^2 \;+\; 2 \sum_{m < n} \xi_m \xi_n.

Take expectations:

\mathbb{E}[X_N^2] \;=\; \sum_{n=1}^N \mathbb{E}[\xi_n^2] \;+\; 2 \sum_{m < n} \mathbb{E}[\xi_m \xi_n].

Each $\xi_n^2 = (\pm 1)^2 = 1$ always, so $\mathbb{E}[\xi_n^2] = 1$ . For the cross terms, independence of $\xi_m$ and $\xi_n$ (for $m \neq n$ ) implies $\mathbb{E}[\xi_m \xi_n] = \mathbb{E}[\xi_m]\, \mathbb{E}[\xi_n] = 0 \cdot 0 = 0$ . So

\mathbb{E}[X_N^2] \;=\; \sum_{n=1}^N 1 \;=\; N.

The root-mean-square (RMS) displacement is therefore

\sqrt{\mathbb{E}[X_N^2]} \;=\; \sqrt{N}.

After $N$ steps, the typical walker has wandered a distance $\sim \sqrt{N}$ from the start — not $N$ . This is the universal signature of diffusion: distance grows as the square root of time (when each step happens in one tick).

The variance grows linearly in $N$ , but the distance grows as $\sqrt{N}$ . A million-step walker has wandered $\sim 1000$ steps from the start, not a million. Random motion is dramatically slower than directed motion — that’s the whole point of diffusion.

The same calculation in higher dimensions: each component is an independent random walk; the squared distance is the sum of squared components; $\mathbb{E}[|\mathbf{X}_N|^2] = N$ for any dimension. The walker spreads as $\sqrt{N}$ regardless of how many directions it can move in.

mode:

step count N = 100 walks shown = 5

Each step is independent and equally likely to go left/right (1-D) or in one of the four cardinal directions (2-D). The expected position is zero — the walks have no preferred direction — but the expected *squared* displacement grows linearly with the number of steps: ⟨X_N²⟩ = N. So the typical walker after N steps is a distance ≈ √N from the origin, marked by the red dashed envelope or circle. The histogram on the right shows the final-position distribution over 5000 independent walks; it is approximately Gaussian by the CLT.

The interactive shows several walkers simultaneously, plus a histogram of where 5000 ensemble walkers end up after $N$ steps. Three things to absorb:

Individual walks look nothing alike. Each is a wild squiggle; their final positions span a wide range.
The ensemble is statistical. The distribution of final positions (right panel) is symmetric, centred at zero, with width $\sqrt{N}$ — exactly the CLT prediction.
The dashed envelope (1-D) or circle (2-D) shows $\pm\sqrt{N}$ . Most walkers end up inside it; very few stray much beyond it.

The continuum limit: Brownian motion

Take the random walk and let the step size shrink and step rate grow together: $\Delta x \to 0$ , $\Delta t \to 0$ , with $\Delta x^2 / \Delta t = 2 D$ held fixed (the diffusion coefficient $D$ ). The discrete random walk becomes Brownian motion $B(t)$ , a continuous-time stochastic process with three defining properties:

$B(0) = 0$ .
For any $0 \leq s < t$ , the increment $B(t) - B(s)$ is Gaussian with mean 0 and variance $2 D (t - s)$ .
Increments over non-overlapping intervals are independent.

The variance grows linearly in time: $\mathbb{E}[B(t)^2] = 2 D t$ . The RMS displacement after time $t$ is $\sqrt{2 D t}$ . The probability density at time $t$ is the heat-equation Green’s function:

p(x, t) \;=\; \frac{1}{\sqrt{4 \pi D t}}\, \exp\!\left( -\frac{x^2}{4 D t} \right),

a Gaussian spreading at rate $\sqrt{2 D t}$ . This is one and the same as the heat equation solution we built earlier — Brownian motion is the stochastic dual of the heat equation, and the heat equation governs the probability density of a Brownian particle.

This duality is one of the most beautiful results in mathematical physics. The same equation $\partial_t p = D \partial_x^2 p$ describes:

The temperature distribution in a metal bar (Fourier, 1822).
The probability density of a Brownian particle (Einstein, 1905).
The diffusion of dye molecules in water.
The price of a financial option (Black–Scholes, 1973).
The smoothing of an image by Gaussian blur.

All have the same generator — the second-derivative Laplacian — and all are continuum limits of i.i.d. random walks.

The Einstein relation

Einstein’s 1905 derivation of the diffusion equation from kinetic theory established a remarkable connection. If a Brownian particle of mass $m$ is suspended in a fluid at temperature $T$ , and the fluid exerts a viscous drag force $-\zeta v$ on the particle (with $\zeta$ a friction coefficient), then the diffusion coefficient is

\boxed{\;D \;=\; \frac{k_B T}{\zeta}.\;}

This is the Einstein relation (also called the Stokes–Einstein relation when $\zeta = 6 \pi \eta a$ for a spherical particle of radius $a$ in a fluid of viscosity $\eta$ ). It says: the random motion of the particle (parametrised by $D$ ) and the deterministic dissipation of its velocity (parametrised by $\zeta$ ) are linked by temperature. Fluctuation and dissipation are two faces of the same underlying microscopic motion.

The Einstein relation was the experimental key to Avogadro’s number. Jean Perrin’s 1908 measurements of Brownian motion in pollen grains let him compute $k_B$ , and hence $N_A = R/k_B$ , settling the long-standing question of whether atoms actually existed. (Modern value: $N_A \approx 6.022 \times 10^{23}$ .) Einstein’s 1905 paper on Brownian motion was one of his “annus mirabilis” four, alongside special relativity, the photoelectric effect, and mass-energy equivalence.

Brownian motion has unusual properties

Brownian motion is continuous (you can plot a path without lifting the pen) but nowhere differentiable — the velocity is undefined at every instant. The classical derivative of a Brownian path diverges; only stochastic notions of “derivative” (Itô calculus, Stratonovich calculus) make sense.

The total path length of Brownian motion over any time interval is infinite: a walker covers infinite distance in finite time. This sounds paradoxical until you remember that “distance” includes the back-and-forth wiggles — the displacement is finite (and grows as $\sqrt{t}$ ), but the arc length of the wiggle is unbounded.

These pathologies are why classical calculus does not apply to Brownian motion, and why stochastic calculus is its own subject. Most physical applications get away with averaged statistics — mean, variance, autocorrelation — without ever invoking pathwise calculus, and the bookshelf follows the same shortcut.

What we use this for

Random walks and Brownian motion underwrite a lot of the bookshelf:

Brownian motion of air molecules (Sound 1.3) — the source of thermal pressure fluctuations and the molecular-scale origin of sound. The mean-square velocity of an air molecule at room temperature is $\sqrt{3 k_B T / m} \approx 500$ m/s.
Thermal noise in any system at temperature $T$ — Johnson–Nyquist noise across a resistor, the source of “white noise” in audio, the noise floor of every electronic measurement.
Diffusion processes in biological systems — neurotransmitter release at a synapse, ion movement across a cell membrane, oxygen transport in the blood.
Molecular relaxation in air (Sound 10.2) — the absorption of sound by O₂ and N₂ involves diffusive transfer of energy between translational and vibrational modes.
The Black–Scholes equation for option pricing — a heat equation in disguise, with stock-price log-returns following Brownian motion.

The next lesson, 11.4 — Poisson processes, turns to counting processes — random events arriving at a fixed rate — and the Poisson distribution that governs their statistics.