Appendix R — Derivation 18: Modal Entropy and Heat Death Reversal

(Locked v1.0 – saturation constants inserted 2025-06-10)

Update summary
– Saturation density and turnover rate fixed by
Saturation Constants for Modal Thermodynamics.
– All symbolic $ρ_{crit}$ , $γ_{0}$ now carry numbers.
– Units match locked constants
$α = 0.090 034 J m^{- 1}$ ,
$β = 5.30 \times 10^{- 54} J m^{- 3}$ ,
$γ = 1.000 228 \times 10^{- 18} J s^{2} m^{- 3}$ .

1 Ensemble energy

For one mode

E [ψ] = \int [γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}] d^{3} x .

For an ensemble ${ψ_{i}}$

C_{tot} = \sum_{i} E [ψ_{i}] + \int Γ ({ψ_{i}}) d^{3} x,

with $Γ$ the interference / saturation term.

2 Entropy density

Locked saturation density

ρ_{crit} = 6.0 \times 10^{44} m^{- 3} .

Local coherence density

ρ_{c} (x) = \sum_{i} | ψ_{i} (x) |^{2} .

Define

s (x) = \ln [\frac{1}{1 - ρ_{c} / ρ_{crit}}] .

$s \to \infty$ as $ρ_{c} \to ρ_{crit}$ .

Calibrated turnover rate

γ_{0} = 8.2 \times 10^{- 8} s^{- 1} .

Γ_{dec} = γ_{0} \frac{ρ_{c}^{2}}{ρ_{crit} - ρ_{c}} .

Choose $k_{B} = 1$ (unit convention) and set

T (x) = \frac{ρ_{c}}{ρ_{crit} - ρ_{c}} .

4 Balance laws

4.1 Coherence continuity

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

4.2 Momentum balance

\partial_{t} (ρ_{c} \nabla ϕ) + \nabla \cdot [α ρ_{c} \nabla ϕ \nabla ϕ + \frac{1}{2} β ρ_{c} I] = - \nabla (ρ_{c} B^{2}) - Γ_{dec} \frac{\nabla ϕ}{| \nabla ϕ |} .

All coefficients are fixed; no free parameters.

5 Heat capacity (uniform patch)

Take $λ = ρ_{c} / ρ_{crit}$ constant:

C = \frac{d ⟨ E ⟩}{d T} = \frac{λ}{(1 - λ)^{2}},

diverging at saturation ( $λ \to 1$ ) and vanishing when
$λ \to 0$ .

6 Structural second law

A modal ensemble evolves toward uniform turnover pressure
$\nabla T = 0$ while minimising total anchoring cost.

Entropy growth is deterministic redistribution, not stochastic mixing.

7 Cosmic end-state cycle (quantitative sketch)

With the now-locked

ρ_{crit} = 6.0 \times 10^{44} m^{- 3}, γ_{0} = 8.2 \times 10^{- 8} s^{- 1},

the large-scale coherence medium evolves through a repeatable cycle:

Stage	Density range	Key equations	Time-scale example
Dilute void	$ρ_{c} ≪ ρ_{crit}$	$T ≃ ρ_{c} / ρ_{crit}$ , $Γ_{dec} \approx 0$	Cosmological: $≳$ Gyr
Growth / crowding	$ρ_{c} \sim 0.1 ρ_{crit}$	continuity + external bias drives inflow	$\sim$ 10⁷ yr (cluster accretion)
Near saturation	$ρ_{c} \to 0.9 ρ_{crit}$	$$\Gamma_{\text{dec}} = \gamma_0\frac{ \rho_c^{2} }{ \rho_{\text{crit}}-\rho_c }$$ diverges	example: $$\tau = \Gamma_{\text{dec}}^{-1} \approx 18;\text{days}$$
Turn-over burst	$ρ_{c} > ρ_{crit}$ locally	anchoring fails ⇒ modes decohere; energy ejected	burst duration $≲$ months
Re-coherence	$ρ_{c} \to 0.01 ρ_{crit}$	$T$ drops $\to$ new low-tension modes nucleate	recovery $\sim$ 10⁶ yr

Because $T$ and $s$ reset after each turnover, entropy is cyclic:

s (ρ_{c} ≪ ρ_{crit}) \to s_{max} \to collapse \to s (ρ_{c} ≪ ρ_{crit})

Hence the universe never settles into a permanent “heat death.”
Regions saturate, collapse, and seed fresh coherent structure on
timescales set by $γ_{0}$ and the local density field.

Simulation roadmap:
notebooks/turnover-3d.ipynb (in preparation) will integrate equations
(continuity, momentum, sink) on a $256^{3}$ grid to verify the cycle
durations quoted above.

Appendix Q - Galaxy-Scale Lensing | [Index](./Appendix Master) | Appendix S - Chiral Anchoring

Appendix R — Modal Thermodynamics from Coherence Principles