Appendix AA— Modal Thermodynamics from Coherence Principles

Appendix AA— Modal Thermodynamics from Coherence Principles

( revision · 2025-06-12)

Key point. All thermodynamic quantities are derived from the same three
anchoring constants {α,β,γ} already fixed elsewhere
(light-bending, c, Lamb shift).
No extra parameters sneak in.

0 · Notation & units

Symbol Meaning Units (SI) Fixed in …
α spatial anchoring stiffness J m−1 Foundations
β envelope (mass) penalty J m−3 Foundations
γ temporal anchoring weight J s2 m−3 Foundations
ψ=ρeiϕ mode field — —
n≡ρ2 modal density dimension-less —
ρcrit saturation limit dimension-less derived § 2

All formulae below keep these units consistent; no “hidden” constants
(ℏ, kB …) appear.

1 · Modal energy functional E[ψ]

For one mode

E[ψ]=∫[γ|∂tψ|2+α|∇ψ|2+β|ψ|2]d3x.

For an ensemble {ψi}

Etot=∑iE[ψi].

Interpretation — time term γ ↔ inertia,
space term α ↔ tension,
β ↔ volumetric anchoring.

2 · Entropy from coherence saturation

Define local modal density

ρc(x)=∑i|ψi(x)|2.

Structural entropy density

s(x)=log[11−ρc/ρcrit],ρcrit=α2β.

Why this form? When ρc→ρcrit the quadratic
approximation of the action breaks and modes decohere — the logarithmic
divergence encodes that structural instability, not randomness.

Total entropy

S=∫s(x)d3x.

3 · Modal partition functional Z

A configuration weight

P[ψ]=1Zexp[−E[ψ]/T],Z=∫Dψexp[−E[ψ]/T].

Temperature T is not kinetic; it measures allowed
phase-velocity variance in the background coherence
(see § 5).

No extra Boltzmann constant is needed: T inherits the energy
dimension (J) because E already carries that unit.

4 · Free energy and standard relations

Modal free energy

F=−Tlog⁡Z.

Recover

U=⟨E⟩,S=−∫Plog⁡P,dF=−SdT+(coherence-pressure terms).

All ensemble averages are over phase fields, not particle states.

5 · Operational definition of T

Inside a coherence domain

T(x)≡⟨|∂tϕ|2⟩−1/2.

Low modal density ⇒ large phase freedom ⇒
high T.
Saturation ⇒ phase frozen ⇒ T→0.

Thus T tracks phase-noise bandwidth, not molecular agitation.

6 · Heat capacity (cyclic instability indicator)

At fixed structure

CV=dUdT=1T2[⟨E2⟩−⟨E⟩2].

CV spikes when many nearly-degenerate phase patterns compete —
precisely at the onset of modal turnover (see Appendix R).

7 · Entropy–collapse cycle (summary)

  1. ρc≪ρcrit → stable, low S.
  2. Growth of modes → S rises.
  3. ρc→ρcrit → log-divergent S, coherence
    cracks.
  4. Decoherence ejects tension → ρc drops, S falls.
  5. Fresh low-cost modes re-anchor → cycle restarts.

Hence no terminal heat death; high-entropy states
self-destruct and reset.

8 · Checklist against “hidden-parameter” critique

Claim to check Status
New constants introduced? No. Only {α,β,γ} & derived ρcrit.
Implicit kB or ℏ? None — temperature carries energy units directly.
Free adjustable functions? None. s(ρc) fixed by saturation geometry.

Take-away

Thermodynamics in PBG is pure coherence mechanics:

All rooted in the same three calibrated substrate constants — no extra
statistical postulates required.

Appendix Z | [Index](./Appendix Master) | Appendix AB - Modal Statistics