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Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com>
141 lines
3.5 KiB
Markdown
141 lines
3.5 KiB
Markdown
# Thermodynamic Equations
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> Pages 19–21 (§173–§175). The energetic cost of computation.
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## Landauer Principle
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Every irreversible erasure of one bit of information dissipates at least:
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```
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E_min = k_B · T · ln(2) [binary]
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E_min = k_B · T · ln(r) [radix r, general]
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```
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At room temperature (T = 293 K, k_B = 1.381 × 10⁻²³ J/K):
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| Operation | Minimum energy |
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|-----------|----------------|
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| Binary bit erase | k_B T ln(2) ≈ 2.80 × 10⁻²¹ J |
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| Ternary trit erase | k_B T ln(3) ≈ 4.44 × 10⁻²¹ J |
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The ratio is exactly ln(3)/ln(2) ≈ 1.585, which also equals the information ratio
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(one trit carries log₂(3) ≈ 1.585 bits). Information per joule is identical for
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binary and ternary at the Landauer limit.
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```
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LANDAUER = CONCRETE = 93 [L(19)+A(11)+N(25)+D(13)+A(11)+U(7)+E(3)+R(4) = 93]
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```
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---
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## Radix Efficiency (Equation 13)
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```
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η(r) = ln(r) / r
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```
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| Radix | η(r) |
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|-------|--------|
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| 2 | ≈ 0.347 |
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| 3 | ≈ 0.366 ← maximum among integers |
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| 4 | ≈ 0.347 |
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| 5 | ≈ 0.322 |
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| e | = 1/e ≈ 0.368 ← global maximum |
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Ternary achieves the maximum radix economy among integer bases because 3 is the
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integer closest to e ≈ 2.718. (Proof: see [`../proofs/ternary-efficiency.md`](../proofs/ternary-efficiency.md).)
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```
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RADIX = GAUSS = TANH = FIELD = 57
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```
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---
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## Reversible Logic Entropy (Equation 14)
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For a reversible computation:
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```
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ΔS_comp ≥ 0, with ΔS_comp → 0 as reversibility → 1
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```
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The minimum entropy production per gate operation is zero for perfectly reversible gates
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(Bennett 1973). In practice:
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```
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ΔS_irrev = k_B ln(2) per irreversible bit operation
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ΔS_rev = 0 per reversible (unitary) gate
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```
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Quantum gates are unitary and therefore reversible: `ΔS_quantum = 0`.
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```
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REVERSIBLE = LAGRANGE = 103 prime
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```
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---
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## Chemical Energy Coupling — Gibbs Free Energy (Equation 15)
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```
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μ_chem = ∂G/∂N ↔ E_comp
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```
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The chemical potential (Gibbs free energy per molecule) is the thermodynamic equivalent
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of the energy cost per computational operation. For a molecular computing substrate:
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```
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ΔG_rxn = ΔH − T ΔS ≥ E_min = k_B T ln(r)
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```
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Biological systems operate near this minimum because enzyme-catalyzed reactions are
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tightly coupled to ATP hydrolysis:
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```
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ΔG_ATP ≈ −50 kJ/mol ≈ 8.3 × 10⁻²⁰ J/molecule (in vivo)
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```
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Capacity: ΔG_ATP / E_min(ternary) ≈ 8.3×10⁻²⁰ / 4.44×10⁻²¹ ≈ 18 trit operations per ATP.
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```
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GIBBS = SUBSTRATE = 83 prime
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CHEMICAL = 127 prime
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```
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---
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## Substrate Efficiency (Equation 14, biological)
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```
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η_substrate = (ops/sec) / (energy/op) · f_accuracy(substrate, problem_type)
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```
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For DNA computing in 100 μL at room temperature:
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```
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ops/sec ≈ 10¹⁴
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energy/op ≈ k_B T ln(3) ≈ 4.44 × 10⁻²¹ J
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η_substrate = 10¹⁴ / 4.44×10⁻²¹ · f_accuracy
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≈ 2.25 × 10³⁴ · f_accuracy (ops per joule-second)
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```
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```
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SUBSTRATE = GIBBS = 83 prime
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```
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---
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## Thermodynamic Consciousness Bound (§175)
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```
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Φ_max ≤ (E_available / k_B T ln(3)) · η_integration
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```
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Maximum integrated information (consciousness, §176) is bounded by:
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- Available metabolic energy E_available
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- Ternary Landauer cost k_B T ln(3) per operation
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- Integration efficiency η_integration ∈ (0, 1]
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```
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THERMODYNAMIC = 174 = 2 × 87 = 2 × BIRTHDAY
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BOUND = 78 = TRIVIAL = LIMITS
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```
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The consciousness bound is thermodynamically real and biological.
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Energy is the hard constraint. Integration efficiency is the soft constraint.
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