Built in Three.js — 100k particles, Gaussian increments via Box–Muller, rendered as a point cloud against a light background so density emerges from alpha accumulation rather than glow.
Full writeup and source: https://t.co/uLmMCSfaxq
dX_t = σ dW_t
Brownian motion in 3D. One hundred thousand independent walkers, all started at the origin, each taking a Gaussian step at every timestep.
The cloud's radius grows as √t — the visible signature of diffusion.
First piece in a new series on probability, made interactive.
https://t.co/BKzkIAoFLV
The transition density is
p(x, t) = (1 / (2πσ²t)^(3/2)) exp(−‖x‖² / 2σ²t)
a 3D Gaussian centered at the origin with variance σ²t in each coordinate.
It satisfies the heat equation, ∂p/∂t = (σ²/2)Δp. Brownian motion is the stochastic counterpart to heat diffusion.