The interactive film is in the book's Companion

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Detailed Balance: Stationary Is Not the Same as Still

Chapter 21 · Mathematics for AI and Machine Learning · Beginner · 1:29

An interactive live film: Langevin particles keep a Gaussian stationary; a swirl in the drift leaves the density unchanged but makes the probability current J circulate, so the process is stationary without detailed balance until the swirl is removed.

Narration

The bell is a standard Gaussian density, and the dots are particles moving by Langevin dynamics: a drift up the log density, plus random kicks.

The cloud stays a standard Gaussian forever. This density is stationary: everywhere, as much probability flows in as flows out.

The rest of the narration is in the book's Companion.

More from Chapter 21

Interactive1:35Fokker-Planck: A Random Cloud, a Deterministic DensityChapter 21Preview1:28Forward Diffusion and the Heat Equation: All Distributions Become GaussianChapter 21Preview1:27Guidance Does Not Sample the Sharpened DistributionChapter 21Preview1:50Metastability and the Kramers Escape RateChapter 21Preview1:26Reverse Time: Why We Must Learn the ScoreChapter 21Preview1:34Unification: SDEs, Probability Flow ODEs, and Fokker-Planck DynamicsChapter 21Preview1:52Discretization Changes the Stationary DistributionChapter 21Preview1:29Stein Variational Gradient Descent: Pull and PushChapter 21