Underdamped Langevin and HMC: Momentum-Driven Exploration

Standard Langevin dynamics is overdamped. Friction is so strong that any velocity vanishes at once, so the chain explores the way a random walk does.

Look at the red path, starting at the left end of this long, narrow valley. Each step is a small kick, and the displacement grows only like the square root of the number of steps. It takes hundreds of steps to reach the dashed line on the far side.

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