Appendix Y — Derivation 25: Continuum Mechanics of Coherence Media

(Built on Foundations v1.0 • 2025-06-10)

Large-scale PBG systems—galactic discs, dense plasmas, saturation fronts—
are best described by a continuum limit.
Here we derive the fluid-like balance laws that follow directly from the
locked anchoring Lagrangian

1 Field variables

Define coherence current

2 Continuity with decoherence sink

Conservation of total modal content:

A phenomenological sink ${\mathrm{\Gamma }}_{\text{dec}}$
models turnover as ${\rho }_c\to {\rho }_{\text{crit}}$.

3 Anchoring stress tensor

Spatial part of $\mathcal{L}$ yields energy density

Define

Units: α [J m⁻¹]·m⁻²·m⁻³ → J m⁻³ (stress) ✔︎.

4 Momentum-like balance

Introduce modal “momentum density”
$begin:math:text$\mathbf p=\rho_c\,\nabla\phi$end:math:text$.
Variation of the action with respect to $\varphi$ gives

Right-hand side is the anchoring bias gradient plus a decoherence
drag ${\mathbf{f}}_{\text{dec}}$ (parallel to $\mathbf{p}$).

5 Decoherence and saturation model

A minimal choice,

with one phenomenological constant ${\gamma }_0$,
ensures divergence as ${\rho }_c\to {\rho }_{\text{crit}}$.

6 Closure: continuum PBG equations

1. Coherence continuity (2.1)
2. Momentum balance  (4.1)
3. Helmholtz field   $(\alpha {\mathrm{\nabla }}^2-\beta )B=-{\rho }_c$
   (static limit of Foundations §1)

Together they form a structurally reactive fluid-elastic system:
phase gradients play the role of velocity, β-pressure supplies bulk
compressibility, and anchoring bias drives “gravitational” motion.

Notes & next steps

• Set ${\gamma }_0$ by matching laboratory decoherence times; all other
  coefficients are fixed constants.
• Add viscosity-like term if phase-shear dissipation is needed for galaxy
  disc simulations.
• Compare with MHD equations to map PBG coherence flow onto plasma
  behaviour.

Last edited 2025-06-10 • dimensional checks vs Foundations v1.0 complete

Appendix X - Coherence Kernel | [Index](./Appendix Master) | Appendix Z