# Pass 02 — NDB Strengthening (Referee-Acceptable Diffusion Limit)

## Purpose

Replace the informal “noise-to-diffusion bridge” (NDB) sketch with a referee-acceptable statement:

- explicit stochastic model on the phase circle
- scaling regime
- limiting PDE (Fokker–Planck / advection–diffusion)
- strict scoping: this is a canonical derivation track, not a universal physical law

This pass upgrades the weakest perceived point in the mixing story: it makes diffusion feel earned, not appended.

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## Phase random walk on \(\mathbb{T}_P\)

Let \(\varphi_n\in \mathbb{T}_P\) evolve by
\[
\varphi_{n+1} = \varphi_n \oplus \rho\,\Delta t \oplus \xi_n,
\]
where:

- \(\rho\in\mathbb{R}\) is deterministic drift (units: phase/time),
- \(\Delta t>0\) is a small time step,
- \(\xi_n\) are i.i.d. noise increments with
  - mean \(0\),
  - variance \(\mathbb{E}[\xi_n^2]=2\kappa\,\Delta t\),
  - and a distribution supported in a small neighborhood (or with finite exponential moments).

Let \(p^n\) be the density of \(\varphi_n\) with respect to Lebesgue measure on \([0,P)\) (periodic).

Define the macroscopic time \(t=n\Delta t\) and the interpolated density \(p(t,\cdot)\approx p^n(\cdot)\).

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## Proposition 1 (Diffusion limit on the circle, standard)

Under the scaling above (small increments, variance proportional to \(\Delta t\)), the density \(p(t,y)\) converges (in the standard weak sense used for Markov chains) to the solution of the advection–diffusion equation on \(\mathbb{T}_P\):
\[
\boxed{\;\partial_t p + \rho\,\partial_y p = \kappa\,\partial_{yy}p.\;}
\]

### Explanation (what is used)

This is the standard diffusion approximation for a drifted random walk:

- the drift term comes from the mean increment \(\rho\,\Delta t\),
- the diffusion term comes from the variance \(2\kappa\,\Delta t\),
- higher cumulants vanish in the limit under moment conditions.

A direct derivation uses either:

- Fourier mode evolution (characteristic functions), or
- a Taylor expansion of the Markov operator (Kramers–Moyal truncation justified by the scaling), or
- a functional central limit theorem (Donsker-type invariance principle) followed by the Kolmogorov forward equation.

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## Fourier-based derivation sketch (transparent on \(\mathbb{T}_P\))

Let \(\widehat{p}^n(m)=\int_0^P p^n(y)e^{-2\pi i m y/P}\,dy\). The update rule implies:
\[
\widehat{p}^{n+1}(m)=\widehat{p}^n(m)\,\mathbb{E}\left[e^{-2\pi i m(\rho\Delta t+\xi_n)/P}\right].
\]
Write the characteristic function of \(\xi_n\) as:
\[
\mathbb{E}\left[e^{-2\pi i m\xi_n/P}\right] = 1 - \kappa\left(\frac{2\pi m}{P}\right)^2\Delta t + o(\Delta t).
\]
Also
\[
e^{-2\pi i m\rho\Delta t/P}=1-\frac{2\pi i m\rho}{P}\Delta t + o(\Delta t).
\]
Thus
\[
\widehat{p}^{n+1}(m)-\widehat{p}^n(m)=\left(-\frac{2\pi i m\rho}{P}-\kappa\left(\frac{2\pi m}{P}\right)^2\right)\Delta t\,\widehat{p}^n(m)+o(\Delta t).
\]
Divide by \(\Delta t\) and pass to the limit to obtain
\[
\partial_t \widehat{p}(m,t)=\left(-\frac{2\pi i m\rho}{P}-\kappa\left(\frac{2\pi m}{P}\right)^2\right)\widehat{p}(m,t),
\]
which is precisely the Fourier form of \(\partial_t p + \rho\partial_y p=\kappa\partial_{yy}p\).

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## How this plugs into SFT mixing universality

This proposition supplies a concrete NDB route:

- if observed phase increments carry small, approximately independent jitter
- then phase distributions evolve under SADM with drift and diffusion
- then SFT’s mixing certificate (MXC) governs convergence rates toward the correct Haar target (uncoupled Haar or coupled Haar on CST)

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## Scope boundary (explicit)

- This derivation is a canonical track, not a claim that every physical system induces diffusion.
- In contexts where increments are not independent or not small, NDB must be replaced by a different bridge (e.g., correlated noise, multiplicative noise, or deterministic chaotic mixing assumptions).

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## Acceptance gates

- A fully specified stochastic increment model is given.
- A scaling regime is stated (variance proportional to \(\Delta t\)).
- The limiting PDE is stated and derived via Fourier mode evolution.
- The scope boundary is explicit.
