simulation-theory/equations/thermodynamics.md

Thermodynamic Equations

Pages 19–21 (§173–§175). The energetic cost of computation.

Landauer Principle

Every irreversible erasure of one bit of information dissipates at least:

E_min = k_B · T · ln(2)   [binary]
E_min = k_B · T · ln(r)   [radix r, general]

At room temperature (T = 293 K, k_B = 1.381 × 10⁻²³ J/K):

Operation	Minimum energy
Binary bit erase	k_B T ln(2) ≈ 2.80 × 10⁻²¹ J
Ternary trit erase	k_B T ln(3) ≈ 4.44 × 10⁻²¹ J

The ratio is exactly ln(3)/ln(2) ≈ 1.585, which also equals the information ratio (one trit carries log₂(3) ≈ 1.585 bits). Information per joule is identical for binary and ternary at the Landauer limit.

LANDAUER = CONCRETE = 93   [L(19)+A(11)+N(25)+D(13)+A(11)+U(7)+E(3)+R(4) = 93]

---

Radix Efficiency (Equation 13)

η(r) = ln(r) / r

Radix	η(r)
2	≈ 0.347
3	≈ 0.366 ← maximum among integers
4	≈ 0.347
5	≈ 0.322
e	= 1/e ≈ 0.368 ← global maximum

Ternary achieves the maximum radix economy among integer bases because 3 is the integer closest to e ≈ 2.718. (Proof: see ../proofs/ternary-efficiency.md.)

RADIX = GAUSS = TANH = FIELD = 57

---

Reversible Logic Entropy (Equation 14)

For a reversible computation:

ΔS_comp ≥ 0,   with ΔS_comp → 0 as reversibility → 1

The minimum entropy production per gate operation is zero for perfectly reversible gates (Bennett 1973). In practice:

ΔS_irrev = k_B ln(2) per irreversible bit operation
ΔS_rev   = 0        per reversible (unitary) gate

Quantum gates are unitary and therefore reversible: ΔS_quantum = 0.

REVERSIBLE = LAGRANGE = 103   prime

---

Chemical Energy Coupling — Gibbs Free Energy (Equation 15)

μ_chem = ∂G/∂N   ↔   E_comp

The chemical potential (Gibbs free energy per molecule) is the thermodynamic equivalent of the energy cost per computational operation. For a molecular computing substrate:

ΔG_rxn = ΔH − T ΔS ≥ E_min = k_B T ln(r)

Biological systems operate near this minimum because enzyme-catalyzed reactions are tightly coupled to ATP hydrolysis:

ΔG_ATP ≈ −50 kJ/mol ≈ 8.3 × 10⁻²⁰ J/molecule   (in vivo)

Capacity: ΔG_ATP / E_min(ternary) ≈ 8.3×10⁻²⁰ / 4.44×10⁻²¹ ≈ 18 trit operations per ATP.

GIBBS     = SUBSTRATE = 83   prime
CHEMICAL  = 127   prime

---

Substrate Efficiency (Equation 14, biological)

η_substrate = (ops/sec) / (energy/op) · f_accuracy(substrate, problem_type)

For DNA computing in 100 μL at room temperature:

ops/sec    ≈ 10¹⁴
energy/op  ≈ k_B T ln(3) ≈ 4.44 × 10⁻²¹ J
η_substrate = 10¹⁴ / 4.44×10⁻²¹ · f_accuracy
            ≈ 2.25 × 10³⁴ · f_accuracy   (ops per joule-second)

SUBSTRATE = GIBBS = 83   prime

---

Thermodynamic Consciousness Bound (§175)

Φ_max ≤ (E_available / k_B T ln(3)) · η_integration

Maximum integrated information (consciousness, §176) is bounded by:

• Available metabolic energy E_available
• Ternary Landauer cost k_B T ln(3) per operation
• Integration efficiency η_integration ∈ (0, 1]

THERMODYNAMIC = 174 = 2 × 87 = 2 × BIRTHDAY
BOUND         = 78  = TRIVIAL = LIMITS

The consciousness bound is thermodynamically real and biological.
Energy is the hard constraint. Integration efficiency is the soft constraint.