Appendix AA— Modal Thermodynamics from Coherence Principles

Overview

Classical thermodynamics relies on statistical ensembles of particles with fixed energy, interacting in space via potential forces.

In PBG, there are no particles or potentials—only modes, whose coherence structure defines:

Internal phase energy
Anchoring tension with the background field
Structural interference with other modes

This appendix constructs a full modal thermodynamic formalism using:

Anchoring cost $C [ψ]$ as the energy functional
Structural overlap and saturation as the entropy generator
Coherence-based ensemble distributions
A modal temperature $T$ from phase fluctuation statistics

Each mode $ψ (x) = ρ (x) e^{i ϕ (x)}$ contributes to the system energy via:

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

In an ensemble of modes ${ψ_{i}}$ , the total modal energy is:

E_{total} = \sum_{i} E [ψ_{i}]

This includes phase gradients (structure), time evolution (fluctuation), and density-based anchoring cost (entropy link).

2. Anchoring-Induced Entropy

The structural entropy density arises from the saturation constraint on modal coherence:

s (x) = \log (\frac{1}{1 - ρ_{c} (x) / ρ_{crit}})

This entropy increases with modal overlap—not randomness—and diverges as the coherence density approaches critical saturation.

Total entropy:

S = \int s (x) d^{3} x = \int \log (\frac{1}{1 - ρ_{c} (x) / ρ_{crit}}) d^{3} x

Let a modal configuration $ψ$ have anchoring energy $E [ψ]$ . Define a modal partition function:

Z = \int D ψ e^{- E [ψ] / T}

Here, $T$ is not kinetic temperature, but a measure of allowed phase fluctuation in the coherence background.

The distribution of modal configurations is:

P [ψ] = \frac{1}{Z} e^{- E [ψ] / T}

This defines modal thermodynamic equilibrium as the most probable coherence field configuration under saturation and interference constraints.

Define modal free energy as:

F = - T \log Z

And recover:

Internal energy: $U = ⟨ E [ψ] ⟩$
Entropy: $S = - \int P [ψ] \log P [ψ] D ψ$
Pressure-like terms via spatial coherence tension (see Appendix Y)

The modal free energy minimisation principle becomes:

The coherence field evolves toward configurations that minimise total anchoring cost, subject to coherence overlap and phase freedom.

Temperature $T$ emerges as a structural measure of phase fluctuation freedom:

In low-density regions: large $T$ , high phase variation
Near saturation: small $T$ , tightly constrained structure

In dynamic environments (e.g. early universe), $T$ evolves from:

T \sim {⟨ | \partial_{t} ϕ |^{2} ⟩}^{- 1}

This links modal temperature to phase velocity variance—not molecular motion.

Define modal heat capacity at constant coherence structure:

C = \frac{d U}{d T} = \frac{1}{T^{2}} (⟨ E [ψ]^{2} ⟩ - ⟨ E [ψ] ⟩^{2})

Large $C$ corresponds to systems with many competing phase configurations—e.g. near coherence turnover points.

7. Structural Equilibria and Turnover

High entropy (high $ρ_{c}$ ) triggers decoherence. As shown in Appendix R:

Entropy peak → saturation → collapse
Modal turnover → re-anchoring in low-energy modes

This implies modal thermodynamics is cyclic, not unidirectional:

Heat death corresponds to saturation
Turnover enables structural regeneration

Conclusion

PBG admits a complete thermodynamic formalism grounded in coherence structure.

Temperature reflects phase variance
Entropy arises from modal overlap
Partition functions encode ensemble coherence
Heat capacity reflects structural instability

This completes Hilbert’s 6th Problem in modal terms:

Thermodynamics derived from coherence dynamics, not statistical collision mechanics.

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

Appendix AA— Modal Thermodynamics from Coherence Principles

Overview

1. Modal Energy Functional

2. Anchoring-Induced Entropy

3. Modal Partition Function

4. Modal Free Energy

5. Modal Temperature

6. Modal Heat Capacity

7. Structural Equilibria and Turnover

Conclusion