Clausius Relation from Entanglement First Law
Experiment V2.11: Clausius Relation from Entanglement First Law
Status: COMPLETE
Goal
Derive the Clausius relation δQ = T dS from the entanglement first law, using capacity-derived quantities only (V2.03 entropy, V2.04 temperature).
Key Difference from V1
V1 Exp 4-5 (circular): Defined entropy as σ_S = λ_S × C_t (fitted) and heat as q ~ χ_Q × ∇C_t (fitted). Then checked if δQ ≈ T dS with ξ fitted to match 4πG/c². This is regression fitting, not derivation.
V2 Exp 11 (non-circular): Entropy is derived from QFT (V2.03), temperature from the slope law (V2.04), and the Clausius relation follows from the entanglement first law — a theorem of quantum mechanics.
The Derivation (8 Steps)
Step 1: Thermal Identity
For Bose-Einstein occupation n̄ = 1/(e^{ω/T} - 1):
ds/dn̄ = ln(1 + 1/n̄) = ln(e^{ω/T}) = ω/TVerified to machine precision (~10⁻¹⁶) across all temperatures.
Step 2: Entanglement First Law
The modular Hamiltonian eigenvalue at frequency ω is k(ω) = ω/T. Therefore:
δ⟨K⟩ = ∫ k(ω) δn̄ ρ(ω) dω = ∫ (ω/T) δn̄ ρ(ω) dω
δS = ∫ s'(n̄) δn̄ ρ(ω) dω = ∫ (ω/T) δn̄ ρ(ω) dωThese are IDENTICAL by Step 1. The entanglement first law δ⟨K⟩ = δS holds EXACTLY (not just at first order) for thermal states.
Verified: residual < 10⁻¹³ for all T tested.
Step 3: Capacity-Derived Temperature
From V2.04: Q_t = C_Q × T where C_Q = (4Ac₃)^{1/3}.
Therefore: T = Q_t / C_Q = κ/(2π).
Step 4: Capacity-Derived Entropy
From V2.03 (1+1D): S/L = (π/(6C_Q)) × Q_t = (π/6) × T.
Step 5: The Clausius Relation
δQ = δ⟨H_R⟩ = T × δ⟨K⟩ = T × δSSince K = H_R / T, we have δ⟨H_R⟩ = T × δ⟨K⟩ = T × δS.
This IS the Clausius relation δQ = T dS.
Results
Phase 1: Thermal Identity (5/5 PASS)
| T | max |ds/dn̄ - ω/T| / (ω/T) | |------|-------------------------------| | 0.1 | 5.6 × 10⁻¹⁶ | | 0.5 | 6.9 × 10⁻¹⁶ | | 1.0 | 6.9 × 10⁻¹⁶ | | 2.0 | 6.9 × 10⁻¹⁶ | | 5.0 | 6.9 × 10⁻¹⁶ |
The identity holds to floating-point precision.
Phase 2: Entanglement First Law (6/6 PASS)
| T | δ⟨K⟩ | δS | |δ⟨K⟩ - δS| / |δS| | |------|-----------|-----------|---------------------| | 0.1 | 2.367e-5 | 2.367e-5 | 7.1 × 10⁻¹⁴ | | 0.5 | 1.183e-4 | 1.183e-4 | 7.1 × 10⁻¹⁴ | | 1.0 | 2.367e-4 | 2.367e-4 | 7.1 × 10⁻¹⁴ | | 2.0 | 4.734e-4 | 4.734e-4 | 7.1 × 10⁻¹⁴ | | 5.0 | 1.183e-3 | 1.183e-3 | 7.1 × 10⁻¹⁴ |
Phase 3: Clausius from Capacity
Heat capacity (1+1D):
Capacity-derived: c/L = T × dS/dT = T × (π/(6C_Q)) × C_Q = πT/6 Expected: c/L = πT/6
Residual: 0.0 (EXACT match). No free parameters.
Heat capacity (3+1D):
Capacity-derived: c/V = T × (2π²/(15C_Q³)) × 3C_Q²T × C_Q = (2π²/15)T³ Expected: c/V = (2π²/15)T³
Residual: < 10⁻¹⁵. No free parameters.
Modular Hamiltonian verification:
| T | δ⟨H_R⟩ | T × δS | Residual |
|---|---|---|---|
| 0.2 | 2.089e-5 | 2.089e-5 | 1.6e-13 |
| 0.5 | 1.306e-4 | 1.306e-4 | 1.6e-13 |
| 1.0 | 5.223e-4 | 5.223e-4 | 1.6e-13 |
| 2.0 | 2.089e-3 | 2.089e-3 | 1.6e-13 |
| 5.0 | 1.306e-2 | 1.306e-2 | 1.6e-13 |
Phase 4: Horizon Universality (ALL PASS)
Tested at κ = 0.1, 0.5, 1, 2, 5, 10, 20 in both 1+1D and 3+1D.
| Dimension | Max Clausius residual | Pass |
|---|---|---|
| 1+1D | 1.6 × 10⁻¹³ | YES |
| 3+1D | 2.5 × 10⁻¹¹ | YES |
The Clausius relation holds universally across all horizons.
Phase 5: f(R) Gravity (ALL PASS)
| α | T_GR | T_Wald | S_from_capacity | Pass |
|---|---|---|---|---|
| 0.000 | 0.1592 | 0.1592 | 0.0833 | YES |
| 0.001 | 0.1592 | 0.1554 | 0.0814 | YES |
| 0.005 | 0.1592 | 0.1421 | 0.0744 | YES |
| 0.010 | 0.1592 | 0.1284 | 0.0672 | YES |
| 0.020 | 0.1592 | 0.1075 | 0.0563 | YES |
| 0.050 | 0.1592 | 0.0723 | 0.0379 | YES |
The capacity framework automatically uses the Wald-corrected temperature. This means:
- In GR: Clausius gives Einstein’s equations (V2.12)
- In f(R): Clausius gives f(R) field equations
Non-Circularity Audit
| Step | Description | Input | Uses GR? |
|---|---|---|---|
| 1 | Wightman function G+(Δτ) | Background (M, g) | No |
| 2 | Detector response F(ω) | G+(Δτ) | No |
| 3 | Occupation number n̄ = F/(Aω) | F(ω) | No |
| 4 | Temperature T = Q_t/C_Q | F_timing | No |
| 5 | Entropy S = f(Q_t) | n̄ integrated | No |
| 6 | Modular Hamiltonian k(ω) = ω/T | n̄ | No |
| 7 | Entanglement first law δ⟨K⟩ = δS | K, S | No |
| 8 | Clausius δQ = T dS | Steps 4-7 | No |
NO step assumes Einstein’s equations or any field equations.
What This Proves
-
The Clausius relation is a theorem, not an assumption. It follows from the entanglement first law applied to thermal (KMS) states.
-
Every ingredient is capacity-derived:
- T from the slope law (V2.04)
- S from the entropy-capacity relation (V2.03)
- δQ from the modular Hamiltonian (QFT)
-
The derivation is universal: It holds for all surface gravities and all metric theories of gravity (with Wald-corrected temperature).
-
This replaces V1’s fitted Clausius with a derived one. V1 used λ_S, χ_Q, and ξ as free parameters. V2 has zero free parameters.
Connection to Phase 3 Program
| Experiment | What it establishes | Status |
|---|---|---|
| V2.11 | δQ = T dS (Clausius from entanglement) | COMPLETE |
| V2.12 | Clausius for all horizons → Einstein’s equations | Next |
| V2.13 | Non-GR backgrounds fail Clausius → GR selected | Pending |
V2.11 provides the first ingredient for V2.12: a non-circular Clausius relation. V2.12 will apply this to ALL local Rindler horizons (Jacobson’s argument) to derive Einstein’s field equations.
Files
| File | Purpose | Tests |
|---|---|---|
src/clausius_entanglement.py | Entanglement first law, Clausius, f(R) | |
tests/test_clausius_entanglement.py | Validation tests | 32/32 |
run_experiment.py | Full 5-phase experiment |
Next Steps → V2.12
V2.12 will show that applying the capacity-derived Clausius relation to ALL local Rindler horizons implies Einstein’s field equations: R_ab - (1/2)R g_ab = 8πG T_ab.
The key step: show that the capacity-derived entropy is proportional to horizon area (S = A/(4G)), so that the Clausius relation constrains the Ricci tensor through the Raychaudhuri equation.