Chapter 24
Zeroth–Third Laws of Thermodynamics (Recast)

Flow Algebra

flow "Thermodynamics_0th_to_3rd_Recast"

# ---------- STATES ----------
state:
  system.U: float                    # internal energy (J)
  system.S: float                    # entropy (J/K)
  system.T: float                    # temperature (K)
  system.V: float                    # volume (m³)
  system.P: float                    # pressure (Pa)
  system.N: float                    # particle number (mol or count)
  system.μ: float                    # chemical potential (J/mol)

  # environment (boundary reservoir)
  boundary.T_env: float              # environmental temperature (K)
  boundary.P_env: float              # environmental pressure (Pa)
  boundary.S_env: float              # entropy reservoir (J/K)
  boundary.integrity.IB: float       # 0..1, coherence of system–boundary coupling

  # exchanges
  exchange.Q: float                  # heat flow (J)
  exchange.W: float                  # work (J)
  exchange.dS: float                 # entropy flow (J/K)
  exchange.mode: "reversible"|"irreversible"

  # derived quantities
  gradients.∇T: float                # temperature gradient (K/m)
  gradients.∇μ: float                # chemical potential gradient
  gradients.∇S: float                # entropy gradient (field form)
  time.t: float

# ---------- PARAMETERS ----------
params:
  kB: 1.380649e-23                  # Boltzmann constant (J/K)
  R: 8.314462618                    # gas constant (J/mol·K)
  dt: float
  relaxation.τ: float               # equilibration time constant
  irreversible.entropy_rate: float  # baseline entropy production term

# ---------- INVARIANTS ----------
invariant:
  - first_law:
      dU = δQ - δW
      δQ = T dS
      δW = P dV + μ dN
  - second_law:
      dS_total = dS_system + dS_env ≥ 0
  - zeroth_law (thermal coherence):
      if T1 = T2 = constant → thermal equilibrium
      i.e., no net ∇T ⇒ ∂μ J_coh^μ = 0 (thermal)
  - third_law (entropy limit):
      lim_{T→0} S → S₀ (finite or →0 for perfect crystal)
  - entropy_flux:
      J_S = exchange.Q / T
  - equilibrium_condition:
      ∇T = 0 and ∇μ = 0 → no net flow of energy or matter
  - coherence_relation:
      ΔS * T = ΔU - PΔV - μΔN (boundary energy balance)

# ---------- TRANSFORMS ----------
transform:

  thermal_exchange:
    do:
      # heat flow proportional to temperature difference
      exchange.Q = (boundary.T_env - system.T) * params.kB / params.dt
      system.U += exchange.Q
      system.S += exchange.Q / system.T
      exchange.dS = exchange.Q / boundary.T_env
      boundary.S_env -= exchange.dS

  mechanical_exchange:
    do:
      exchange.W = system.P * d(system.V)
      system.U -= exchange.W

  chemical_exchange:
    do:
      dN = -(∇μ * params.dt)
      system.U += system.μ * dN
      system.N += dN

  entropy_production:
    do:
      if exchange.mode == "irreversible":
        σ = params.irreversible.entropy_rate
        system.S += σ * params.dt
      boundary.integrity.IB = clamp(boundary.integrity.IB - σ*0.01, 0, 1)

  approach_equilibrium:
    do:
      # relax temperature and chemical potential toward environment
      system.T += (boundary.T_env - system.T) * exp(-params.dt / params.relaxation.τ)
      system.μ += (0 - system.μ) * exp(-params.dt / params.relaxation.τ)
      gradients.∇T = (boundary.T_env - system.T)
      gradients.∇μ = (0 - system.μ)

  tick:
    do:
      time.t += params.dt

# ---------- MONITORS ----------
monitor:
  energy_balance:
    observe: [system.U, exchange.Q, exchange.W]
    report: "Tracks internal energy balance via dU = δQ - δW."

  entropy_evolution:
    observe: [system.S, boundary.S_env, exchange.dS]
    report: "Total entropy change; checks dS_total ≥ 0 (Second Law)."

  equilibrium_state:
    observe: [system.T, boundary.T_env, gradients.∇T]
    report: "Zeroth Law: equilibrium when ∇T ≈ 0, coherence restored."

  low_temperature_limit:
    observe: [system.T, system.S]
    report: "Third Law: entropy approaches constant as T→0."

  coherence_integrity:
    observe: [boundary.integrity.IB]
    report: "Thermal coupling integrity; weakened by irreversible dissipation."

# ---------- NOTES (coherence view) ----------
note:
  - The **Zeroth Law** expresses **coherence equality**: thermal equilibrium is not a passive state but a perfectly balanced phase exchange across boundaries.
  - The **First Law** is the **energy conservation loop**: dU = δQ - δW ensures energy transforms but total curvature remains closed.
  - The **Second Law** encodes **directionality of coherence** — entropy increases as phase information disperses into finer microstates.
  - The **Third Law** defines the **limit of coherence compression**: as temperature approaches zero, fluctuations vanish, and all curvature aligns — coherence becomes absolute.
  - In HST terms:
      • Temperature = average curvature agitation (phase jitter).  
      • Entropy = total curvature spread — the field’s informational roughness.  
      • Equilibrium = uniform curvature tension; coherence fully symmetric.  
  - All four laws are **coherence conservation in disguise**: energy, entropy, and equilibrium are just different projections of one boundary condition — ∂μ J_coh^μ = 0.
  - Irreversibility marks not destruction but redistribution: a shift of coherence granularity from ordered to diffuse forms.

# ---------- WHAT THIS FLOW CAPTURES ----------
doc:
  summary: >
    Thermodynamics_0th_to_3rd_Recast unifies the classical thermal laws under
    boundary-coherence logic. It tracks how energy, entropy, and temperature
    evolve through reversible and irreversible exchanges between system and
    environment. In this framing, equilibrium is coherence equality, entropy
    is curvature dispersion, and the Third Law is the still point of perfect
    coherence — where all gradients vanish and geometry rests.