Appendix F — Derivation 6: Thermodynamic Structure from Coherence

Overview

Classical thermodynamics interprets entropy as a measure of disorder and defines temperature through kinetic energy distributions. Statistical mechanics models systems via ensembles of microstates.

In modal dynamics, there are no particles or microstates. Instead, thermodynamic behaviour arises from the distribution, overlap, and stability of coherence structures.

This appendix derives entropy and equilibrium from anchoring tension, coherence saturation, and modal turnover—without invoking probability or randomness.

1. Modal Ensemble: Structured, Not Stochastic

A modal system consists of a set of coherence functions $\{{\psi }_i(x,t)\}$ coexisting within a shared coherence field $B(x)$.

Each mode:

• Contributes anchoring cost $\mathcal{C}[{\psi }_i]$
• Interacts with others through overlap and saturation
• May persist, decohere, or reorganise depending on structure

The system evolves to minimise total anchoring cost:

Where $\mathrm{\Gamma }$ represents cross-mode interference and saturation effects.

2. Entropy as Modal Overlap

In classical systems, entropy measures the number of compatible configurations. In modal dynamics, entropy arises from the structural indistinguishability of coherence profiles within a given anchoring domain.

Let ${\rho }_c(x)$ represent the coherence density at point $x$—the number of overlapping modes per unit volume.

We define the local modal entropy density:

Where:

• ${\rho }_{\text{crit}}$ is the saturation threshold beyond which anchoring fails
• $s(x)\to \mathrm{\infty }$ as ${\rho }_c\to {\rho }_{\text{crit}}$

This entropy is not a measure of disorder, but of anchoring pressure: the structural cost of supporting multiple overlapping coherence modes.

3. Modal Temperature

Temperature is defined not by energy, but by turnover pressure: the rate at which modes must reorganise to maintain coherence under increasing saturation.

We define modal temperature $T$ as the local rate of anchoring instability:

This diverges at saturation and vanishes in low-density regions. Systems equilibrate by flattening this derivative—i.e., equalising turnover tension.

4. Thermalisation Without Statistics

Thermal equilibrium is reached when:

• $\mathrm{\nabla }T(x)=0$ (uniform modal turnover pressure)
• Modes reorganise to distribute coherence density evenly
• Anchoring cost is minimised under structural constraints

No stochastic averaging is required.
No microstate counting is invoked.
Thermalisation emerges from bias-driven reorganisation of coherent structure.

5. Modal Heat and Transfer

The equivalent of heat flow is coherence redistribution. When one region becomes overpopulated:

• Decoherence releases tension
• Phase structure diffuses
• Coherence re-anchors elsewhere

The analogue of heat capacity is the anchoring flexibility—how many modes can be added before coherence pressure destabilises the region.

6. Second Law

The second law becomes structural:

A modal system will evolve toward a configuration that maximises coherence support and minimises anchoring stress.

This matches traditional entropy increase—but replaces randomness with deterministic phase evolution.

Conclusion

Thermodynamic behaviour is not statistical.
It is the emergent geometry of modes rearranging themselves to preserve coherence under collective strain.

Appendix E | [Index](./Appendix Master) | Appendix G