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Chapter 27
Carnot Cycle and Coherence Efficiency

Thermal and Entropic Flows
Author: Mark Anthony Higgins
© Mark Anthony Higgins 11 October 2025
Revised Date: 23 February 2026
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Flow Algebra 
flow "Carnot Cycle and Coherence Efficiency" v1.0


# ---------- PURPOSE ----------
# Models the ideal reversible Carnot cycle, extending it through HST coherence logic.
# Efficiency is interpreted as coherence retention — the fraction of boundary energy that
# remains phase-aligned between hot and cold reservoirs.

# ---------- PARAMETERS ----------
params:
  units: "SI"
  T_hot:   positive        # hot reservoir temperature [K]
  T_cold:  positive        # cold reservoir temperature [K]
  gamma:   positive = 1.4  # heat capacity ratio (ideal-gas guide)
  n:       positive = 1.0  # moles of working substance
  R:       positive = 8.314462618
  IB:      0.0..1.0        # boundary coherence index (1 = fully reversible)
  include_adiabatic_checks: true

# ---------- STATE ----------
state WorkingFluid:
  T: positive
  p: positive
  V: positive
  S: real
  U: real

state Reservoirs:
  T_hot:   params.T_hot
  T_cold:  params.T_cold

state Ports:
  Qdot_hot:  real     # heat absorbed from hot reservoir
  Qdot_cold: real     # heat rejected to cold reservoir
  Wdot_out:  real     # work output rate
  sigma:     nonneg   # entropy production rate

# ---------- IDEAL CARNOT RELATIONS ----------
invariants:
  # Reversible heat ratios
  Q_hot / T_hot = Q_cold / T_cold
  eta_carnot = 1 - (T_cold / T_hot)

  # First law (power form)
  ddt(WorkingFluid.U) = Ports.Qdot_hot + Ports.Qdot_cold - Ports.Wdot_out

  # Entropy balance (perfect coherence)
  if params.IB == 1:
    Ports.Qdot_hot / T_hot + Ports.Qdot_cold / T_cold = 0
    Ports.sigma = 0

  # Coherence-adjusted entropy production (IB < 1)
  if params.IB < 1:
    Ports.sigma = (1 - params.IB) * abs(Ports.Qdot_hot / T_hot)
    Ports.Qdot_cold / T_cold = - Ports.Qdot_hot / T_hot + Ports.sigma

# ---------- COHERENCE EFFICIENCY ----------
transforms:
  # Coherence-based efficiency factor
  eta_carnot = 1 - (params.T_cold / params.T_hot)
  eta_coh    = params.IB * eta_carnot

  # Interpret eta_coh as the fraction of energy maintaining boundary coherence
  Wdot_out = eta_coh * Ports.Qdot_hot

  # Optional field guide (adiabatic compression/expansion)
  if params.include_adiabatic_checks:
    invariant: T * V^(params.gamma - 1) = const

# ---------- INVARIANT SET (SUMMARY) ----------
invariants summary:
  eta_carnot = 1 - T_cold / T_hot
  eta_coh    = IB * eta_carnot
  sigma      = (1 - IB) * |Qdot_hot| / T_hot

# ---------- MONITORS ----------
monitors:
  efficiency_bounds:
    assert eta_coh <= eta_carnot + 1e-9
    assert Ports.sigma >= 0
  coherence_check:
    log.debug = "IB = ${params.IB} → η_coh = ${round(eta_coh,4)}"

# ---------- SCENES ----------
scenes:
  reversible_reference:
    given params:
      T_hot:  700.0
      T_cold: 300.0
      IB:     1.0
    expect:
      eta_coh = 1 - T_cold / T_hot
      Ports.sigma = 0
      comment: "Perfect coherence — reversible Carnot limit reached."

  partial_coherence_example:
    given params:
      T_hot:  700.0
      T_cold: 300.0
      IB:     0.85
    expect:
      eta_coh = 0.85 * (1 - T_cold / T_hot)
      Ports.sigma > 0
      comment: "Entropy production reflects coherence loss."

# ---------- NOTES ----------
notes:
  - In the classical Carnot cycle, reversibility means σ = 0 and efficiency depends only on T_hot/T_cold.
  - In the coherence formulation, η is reduced by the boundary coherence factor IB.
  - IB quantifies how faithfully energy phase is transferred between thermal boundaries.
  - η_coh thus represents both thermodynamic and structural fidelity.
Overview

The Carnot cycle represents the theoretical limit of heat-to-work conversion — a perfectly reversible exchange between two temperature boundaries.  In Flow Algebra form, it becomes the baseline coherence cycle, where every joule of energy remains phase-aligned between hot and cold reservoirs.

Efficiency is not just thermal here — it’s coherence efficiency, expressing how much of the boundary’s ordered tension survives without scattering

Core Relations

\(\eta_{\mathrm{Carnot}} = 1 - \tfrac{T_{\mathrm{cold}}}{T_{\mathrm{hot}}}, \quad \frac{T_{\mathrm{hot}}}{Q_{\mathrm{hot}}} = \frac{T_{\mathrm{cold}}}{Q_{\mathrm{cold}}}, \quad \sigma = 0.\)

These define the reversible ideal: no entropy generation (σ=0\sigma=0) and perfect phase continuity between reservoirs.
Any deviation from this balance marks a loss of coherence — a physical signature of entropy in HST terms.

Coherence Efficiency

Holographic coherence reframes Carnot’s limit through the boundary coherence index IB∈[0,1]I_B \in [0,1].
It measures the fraction of energy that remains phase-matched across the boundary:

$$ \eta_{\mathrm{coh}} = I_B \left( 1 - \frac{T_{\mathrm{cold}}}{T_{\mathrm{hot}}} \right). $$

  • \( I_B = 1:\) perfect boundary symmetry → fully reversible cycle.
  • \( I_B < 1:\) partial coherence → energy leakage into phase noise → entropy generation.
  • \( I_B = 0:\) total decoherence → all input energy thermalised as noise.

This simple multiplier extends Carnot efficiency into the real world, where geometry, material, and temporal mismatch introduce measurable coherence loss.

Physical Interpretation

$$ \sigma = (1 - I_B)\, \frac{|\dot{Q}_{\mathrm{hot}}|}{T_{\mathrm{hot}}}. $$

linking entropy production directly to the degree of boundary misalignment.
Thus, entropy is not disorder, but a measurable curvature mismatch between coupled boundaries.

Why Coherence Efficiency Matters
  • It embeds reversibility and material realism in one symbol.
  • It quantifies how microstructure and timing alter performance, even when \( T_{\mathrm{hot}} \) and \( T_{\mathrm{cold}} \) are fixed.
  • It allows the Carnot boundary to exist as a spectrum — from coherent quantum contact to diffusive thermal barrier.
  • It bridges thermodynamics and information: perfect coherence = zero information loss.
Example Calculations
CaseT hot (K)T cold ​ (K)I B ​ηcoh​
Ideal reversible
700
300
1.00
0.571
Moderate coherence
700
300
0.85
0.486
Low coherence
700
300
0.60
0.343

As \( I_B \) drops, the thermal gradient remains the same — yet less of that potential is realised as structured work.

The rest disperses as curvature noise (σ\sigma), measurable in the energy spectrum as broadband thermal radiation or surface scattering.

Coherence View of the Carnot Limit

n HST terms, the Carnot boundary corresponds to zero differential curvature across the coherence membrane.
Work and heat become mirror flows across that surface: one structured, one distributed.
Reversibility is thus not a time symmetry, but a coherence symmetry — the moment when energy transfer preserves total phase integrity.

$$ \eta_{\mathrm{coh}} = I_B \left( 1 - \frac{T_{\mathrm{c}}}{T_{\mathrm{h}}} \right), \qquad \sigma = (1 - I_B)\, \frac{|\dot{Q}|}{T_{\mathrm{h}}}. $$

Summary

The classical Carnot cycle defines the thermal boundary of possibility.
The Coherence Carnot extends that line inward — into structure, timing, and phase fidelity.
It tells us that reversibility is not an absolute, but a function of how well a boundary remembers itself.
When IB→1I_B \to 1, the system becomes its own memory — the true reversible state where energy, geometry, and information are one.

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