Fokker–Planck & Liouville — probability flow in statistical mechanics

The stochastic instantiation of probability-as-fluid. Fokker–Planck sets the current J = μρ − D∇ρ (drift + diffusion); its steady state is the Boltzmann distribution ρ ∝ e^(−U/kT), which makes the current vanish and whose variance equals the temperature (sealed: https://narrowhighway.com/s/8a3d0e4b4c8af33851fc97658b7af1f9631e8a770708a245234f2226f076d694). In Hamiltonian phase space, Liouville's theorem gives dρ/dt = 0 — the probability fluid is incompressible.