mirror of
https://github.com/blackboxprogramming/simulation-theory.git
synced 2026-03-17 06:57:15 -05:00
d228bb407a
Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com>
141 lines
3.5 KiB
Markdown
141 lines
3.5 KiB
Markdown
# 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`](../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.
|