Appendix AE — Derivation 30: Continuum Mechanics from Modal Anchoring

Appendix AE — Derivation 30

Continuum Mechanics of Coherence Media

Quick guide
Coherence density $ρ_{c}$ plays the rôle of mass density,
phase gradient $\nabla ϕ$ becomes velocity/strain,
anchoring cost generates stress and pressure,
decoherence supplies dissipation.
Only the three substrate constants ${α, β, γ}$ appear.

1 · Coherence flow variables

Symbol	Meaning	Unit
$ρ_{c} (x, t)$	total modal coherence density	(dimension-less)
$ϕ (x, t)$	phase field	rad
$v_{ϕ} \equiv \nabla ϕ$	phase velocity	m $^{- 1}$
$J_{c} \equiv ρ_{c} v_{ϕ}$	coherence current	m $^{- 1}$

2 · Continuity equation

Structural coherence is conserved except where modes decohere:

\begin{matrix} (2.1) & \partial_{t} ρ_{c} + \nabla \cdot J_{c} = - Γ_{dec} (ρ_{c}) \end{matrix}

$Γ_{dec}$ (saturation-triggered rate) carries units s $^{- 1}$ .

3 · Anchoring stress tensor

The coarse-grained Lagrangian density (from Appendix A)

L = \frac{1}{2} γ ρ_{c} (\partial_{t} ϕ)^{2} - \frac{1}{2} α ρ_{c} | \nabla ϕ |^{2} - β (ρ_{c}) ρ_{c}

leads to the modal stress tensor via the standard Euler–Piola derivative

\begin{matrix} (3.1) & T_{i j} = \frac{\partial L}{\partial (\partial_{i} ϕ)} \partial_{j} ϕ - δ_{i j} L = α ρ_{c} \partial_{i} ϕ \partial_{j} ϕ + P_{anchor} δ_{i j}, \end{matrix}

with anchoring pressure

P_{anchor} (ρ_{c}) = β (ρ_{c}) ρ_{c} .

Units check: $α ρ_{c} | \nabla ϕ |^{2}$ → J m⁻³ ✔.

4 · Momentum-like evolution

Define phase momentum density

p = ρ_{c} \nabla ϕ .

Its dynamics follow from $\partial_{t} (\partial L / \partial (\partial_{t} ϕ))$ :

\begin{matrix} (4.1) & \partial_{t} p + \nabla \cdot T = - \nabla (ρ_{c} B^{2}) - f_{dec} \end{matrix}

The bias term $- \nabla (ρ_{c} B^{2})$ is a “force” from the external coherence field $B$ (Helmholtz kernel).
$f_{dec} = η v_{ϕ}$ (see § 5) represents decoherence drag.

5 · Viscous-like dissipation

Near saturation, small shear produces structural drag.
We model a modal viscosity proportional to the local decoherence rate:

\begin{matrix} (5.1) & η (ρ_{c}) = η_{0} \frac{ρ_{c}}{ρ_{crit}} \frac{Γ_{dec}}{Γ_{0}}, f_{dec} = \nabla \cdot [η (\partial_{i} v_{j} + \partial_{j} v_{i} - \frac{2}{3} δ_{i j} \nabla \cdot v_{ϕ})] . \end{matrix}

$η_{0}, Γ_{0}$ are derived prefactors tabulated in Appendix Y.

6 · Elastic analogy (small-strain limit)

Set the phase displacement $u_{i} \equiv ϕ$ (in radians).
Small gradients give strain

ε_{i j} = \frac{1}{2} (\partial_{i} u_{j} + \partial_{j} u_{i}) .

Taylor-expanding $β (ρ_{c})$ about equilibrium ρ̄ and matching (3.1) yields Lamé-type coefficients

\begin{matrix} (6.1) & λ = ρ_{c} β^{'} ({\bar{ρ}}_{c}), μ = α {\bar{ρ}}_{c} . \end{matrix}

(Derivation snippet moved to Footnote 1.)

7 · Governing set (collecting results)

Continuity

\partial_{t} ρ_{c} + \nabla \cdot (ρ_{c} \nabla ϕ) = - Γ_{dec} .

Momentum

\partial_{t} (ρ_{c} \nabla ϕ) + \nabla \cdot T = - \nabla (ρ_{c} B^{2}) - f_{dec} .

Anchoring field

α \nabla^{2} B - β B + ρ_{c} = 0 .

This is a fluid–elastic hybrid wholly derived from coherence dynamics.

Footnote 1 — Small-strain derivation of $(λ, μ)$

Write $ρ_{c} = {\bar{ρ}}_{c} + δ ρ$ , expand $β (ρ_{c})$ to 1st order:
$β (ρ_{c}) = β ({\bar{ρ}}_{c}) + β^{'} ({\bar{ρ}}_{c}) δ ρ$ .
Using $δ ρ ≃ - {\bar{ρ}}_{c} \nabla \cdot u$ and retaining quadratic terms in $\partial_{i} u_{j}$ , (3.1) reduces to the isotropic Hooke form with
$λ, μ$ given in (6.1).

Take-aways

Mass density $\to$ coherence density
Velocity/strain $\to$ phase gradient
Stress $\to$ anchoring tension
Viscosity $\to$ decoherence drag

No extra constants beyond $α, β, γ$ .
Hence classic continuum behaviour is an emergent facet of modal anchoring.

Appendix AD - Chiral Anchoring | [Index](./Appendix Master) | Appendix AF - Anomalous Magnetic Moment