Detailed Balance
Chapter 21: Fokker-Planck and Distribution Dynamics — Stationary Distributions and Detailed Balance
From the book
Chapter 21: Fokker-Planck and Distribution Dynamics. In the chapter mind map this icon labels Stationary Distributions and Detailed Balance. The discussion below is excerpted and lightly edited from § Stationary Distributions and Detailed Balance in Mathematics for AI and Machine Learning.
At equilibrium (stationary state), $\partial_t p_t = 0$, yielding:
whose solution is exactly $p(\mathbf{x})$, the target distribution.
- Left side: $\nabla^2 p$
- Right side: $\nabla \cdot (p \nabla \log p) = \nabla \cdot \nabla p = \nabla^2 p$
So the equation is satisfied, confirming that $p$ is indeed a stationary solution.
What this drawing shows
What you see. Stationary weights $p^*$ fixed; paired forward and reverse flux arrows grow symmetrically, illustrating detailed balance at equilibrium.
In the mind map. Chapter 21 — Stationary Distributions and Detailed Balance. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 21: Fokker-Planck and Distribution Dynamics — see the mind-map node Stationary Distributions and Detailed Balance.
