Making Physics
Making Physics / Thermal & Statistical / Random Walk & Diffusion
Chapter 15

Random Walk & Diffusion

Can deterministic macroscopic behavior — diffusion — emerge from pure randomness, and does the macroscopic law remember the microscopic step?

Interactive · live simulation

Thousands of walkers from the origin. Switch the step rule between lattice, Gaussian, and Pearson — the cloud spreads the same way and the measured ⟨r²⟩/4t lands on the same D. Turn on drift to see advection-diffusion.

Random Walk & Diffusion — simulation render
fig 1. The displacement histogram fills a Gaussian by the central limit theorem — even the discrete lattice walk, confined to integer sites, smooths into a bell curve once the step count is large.

Three rules, one law

Lattice steps, Gaussian increments, and fixed-length Pearson jumps are very different microscopically. Yet with matched parameters they all reach the same diffusion coefficient and the same \langle r^2\rangle \propto t growth. The microscopic rule sets only the number D; the form of the law is universal — the random-walk face of the central limit theorem.

Mean-square displacement growing linearly in time: the variance sits on the 4Dt
fig 2. Mean-square displacement growing linearly in time: the variance sits on the 4Dt line for the whole run — the defining signature of diffusion.

Linear, not ballistic

The signature of diffusion is that the spread grows linearly in time: \langle |r-\langle r\rangle|^2\rangle = 4Dt in 2D, with D = a^2/4\tau for fixed-length steps. Not the t^2 of a moving particle, not saturating — linear. The lab measures D = \text{variance}/4t live and lands it on the formula.

Advection-diffusion: a constant drift slides the mean downstream at vt while the
fig 3. Advection-diffusion: a constant drift slides the mean downstream at vt while the cloud spreads around it exactly as it would with no drift.

Brownian motion's skeleton

This is the bridge from the gas lab to real Brownian motion: a heavy particle jostled by a gas takes effectively random steps, with D = k_B T/\gamma by the Einstein relation. Strip away the forces, keep the randomness, and diffusion is what remains.

This chapter is drawn from the physics-lab study notes and renders. The longer write-ups and project essays live on the blog.

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