I'm trying to work through Karl Friston's mathematical derivation of the Free Energy Principle from Langevin Dynamics (see this paper). I'm confused about the part at the end of page 8 where he uses the Ao/Helmholtz decomposition to rewrite the flow, $f(x)$ from the Langevin equation $\frac{dx}{dt} = f(x) + \omega$, into $f(x) = (\Gamma - Q) \nabla_x \ln(p^{\ *}(x))$ at non-equilibrium steady state (NESS). The probability density over states at NESS (i.e. NESS Density) is denoted $p^{\ *}(x)$.
What I understand so far:
- The Helmholtz decomposition splits a vector field into the sum of an irrotational (curl-free) vector field $\Gamma$ and a solenoidal (divergence-free) vector field $Q$.
- NESS is different from ESS because there's an arrow of time.
- I get why $f(x) = (\Gamma - Q) \nabla_x \ln(p^{\ *}(x)) \implies \frac{dp^{\ *}(x)}{dt} = 0$. The explanation in Appendix 11.1 was pretty clear.
What I don't understand:
- How exactly does the solenoidal flow prevent the system from "relaxing into an ESS"? The Helmholtz decomposition was supposed to answer this question, but I don't see how it does.
- How do "the separable ‘dissipative’ (noise) and ‘solenoidal’ components [...] perform a gradient descent" here? I only know about gradient descent from machine learning and I assume that the term means something different in this context.
I think I'm probably missing some prerequisite concepts and/or background, so any suggestions as to where I could learn what I need to learn to understand this would be greatly appreciated.
(EDIT: Removed some sub-questions.)
