Brownian skier on the Sochi DEM
The skier follows an overdamped Langevin dynamics on a digital elevation model of the Roza Khutor to Sirius corridor:
dx = -\nabla h(x)\,dt + \sigma\,dW
The drift term moves the point downhill; the Brownian term adds controllable noise, so larger \sigma makes the trajectory explore the terrain instead of sliding along one deterministic descent line.
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1 Fokker–Planck particle fields
The pre-rendered videos below start 4464 particles from a uniform grid over the same map and evolve their independent Langevin trajectories for 10 seconds at 60 fps. The batched trajectory step is computed with jax.vmap. The noise values are uniformly spaced from \sigma = 0 to \sigma = 15.
\sigma = 0: particles left, KDE right
\sigma = 2.5: particles left, KDE right
\sigma = 5: particles left, KDE right
\sigma = 7.5: particles left, KDE right
\sigma = 10: particles left, KDE right
\sigma = 12.5: particles left, KDE right
\sigma = 15: particles left, KDE right