Helmholtz decomposition of flow at non-equilibrium steady state

Question

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.)

Answer 1

I wrote up the full derivation in detail here: https://shianghuacheng.notion.site/Explaining-the-Free-Energy-Principle-to-my-past-self-118fcfd4925d8035ba35c1609bd81243.

Answer 2

Note that the key fact (apprently not mentioned in the Friston paper) is that $Q$ is skew symmetric matrix. If you plug Ping Ao's decomposition into the difffusion equation you get $\partial_\mu Q_{\mu\nu} \partial_\nu p^*$ which is zero because of the antisymmetry.