Statistical MechanicsGeneral RelativityFRG

Scale-Dependent Classification of Einstein's Field Theory in the Statistical-Mechanical Hierarchy

Computational analysis using functional renormalization group, thermodynamic criteria, and stochastic gravity to resolve where GR sits in the microscopic-mesoscopic-macroscopic hierarchy.

🔭 Open Problem

The placement of general relativity within the three-level hierarchy of statistical mechanics -- microscopic, mesoscopic, macroscopic -- remains unresolved (Maes, 2026). GR simultaneously exhibits characteristics of all three levels: deterministic field equations (microscopic), derivability as an equation of state (macroscopic), and semiclassical fluctuation effects (mesoscopic).

This work demonstrates that GR does not admit a single classification. Its position in the hierarchy is scale-dependent, transitioning from macroscopic at infrared scales to mesoscopic and potentially microscopic at ultraviolet scales near the Planck energy.

⚙ Computational Methods

  • FRG Flow: Functional renormalization group for quantum Einstein gravity in the Einstein-Hilbert truncation. Tracks running of Newton's constant G(k) and cosmological constant Lambda(k) across momentum scales.
  • Thermodynamic Classification: Classicality parameter C(k), spectral entropy density s(k), and fluctuation-dissipation ratio R_FD(k) to determine hierarchical level at each scale.
  • Stochastic Gravity: Einstein-Langevin framework characterizing the mesoscopic regime via noise and dissipation kernels from quantum stress-energy fluctuations.
  • EFT Crossover: Quantum-corrected Newtonian potential analysis identifying where quantum corrections match classical post-Newtonian corrections at r_c ~ 0.80 L_Pl.
Macro-to-Meso Crossover
k/k0 = 2.23
Meso-to-Micro Crossover
k/k0 = 11.7
EFT Crossover Radius
0.80 L_Pl
FD Ratio (IR)
~1.0
FD Ratio (UV)
~0.5

📈 Interactive RG Flow of Gravitational Couplings

0.020 0.005

🎯 Classicality Parameter C(k)

Macroscopic (C > 10) Mesoscopic (0.3 < C < 10) Microscopic (C < 0.3)

🌡 Fluctuation-Dissipation Ratio R_FD(k)

📊 Regime Distribution Along the RG Flow

📋 Summary of Quantitative Results

QuantityValue
Initial g0 (IR)0.020
Initial lambda0 (IR)0.005
Final g (UV terminus)6.13
Final lambda (UV terminus)0.446
RG time range t[0, 2.65]
Macro-to-meso crossover k/k02.23
Meso-to-micro crossover k/k011.7
Macroscopic fraction30.3%
Mesoscopic fraction62.6%
Microscopic fraction7.1%
EFT crossover r_c (M=M_Pl)0.80 L_Pl
EFT beta_total2.40

⚡ EFT Quantum-Classical Crossover

The quantum-corrected Newtonian potential includes both classical post-Newtonian (1PN) and one-loop quantum corrections. The crossover radius where quantum corrections become comparable to classical 1PN corrections is:

Crossover Radius
r_c = 0.80 L_Pl

For astrophysical objects (M ~ M_sun ~ 10^38 M_Pl), the crossover is pushed to r_c ~ 10^-38 L_Pl, confirming GR is macroscopic at all physically accessible scales.

✨ Key Findings

Macroscopic Regime
k << M_Pl

Classical saddle-point dominates. C >> 1, R_FD ~ 1. Thermodynamic interpretation valid.

Mesoscopic Regime
k ~ M_Pl

Quantum fluctuations comparable to background. C ~ 1, FDT violated. Einstein-Langevin description required.

Microscopic Regime
k >> M_Pl

Fluctuation-dominated. C << 1, R_FD ~ 0.5. Full quantum gravity path integral needed.