Cosmology Radiation Entropy Relation Bose Entropy Eq Four Thirds Energy

The 4/3 entropy relation

In the thermodynamics of radiation, the entropy density s of a gas of massless particles is related to its energy density ρ and temperature T by s = (4/3)ρ/T. This is a standard result in statistical mechanics, usually obtained by combining thermodynamic identities with the Stefan-Boltzmann law. The Recognition Science framework's machine-checked library of formal theorems contains a proof that derives this relation from the microscopic entropy functional of quantum statistics, rather than assuming it as an input.

The central theorem, named bose_entropy_eq_four_thirds_energy, states that for a gas obeying Bose-Einstein statistics, the integral of the entropy integrand over all energies equals exactly 4/3 times the integral of the energy density integrand. In symbols, ∫₀^∞ σ_B(x) dx = (4/3) ∫₀^∞ x³/(eˣ−1) dx, where x = E/T is the dimensionless energy. The proof splits the entropy integrand into the energy kernel plus a logarithmic kernel, expands the logarithm using the Mercator series, and evaluates the resulting Mellin transform at s = 3 to obtain Γ(3)ζ(4) = π⁴/45. A parallel theorem, fermi_entropy_eq_four_thirds_energy, establishes the same 4/3 relation for Fermi-Dirac statistics, with the energy integral replaced by x³/(eˣ+1).

The framework's proof also derives the fermionic entropy weight: the ratio of the Fermi entropy integral to the Bose entropy integral is exactly 7/8, matching the familiar ratio of energy densities. The entropy coefficient 2π²/45, which appears in the standard formula s_γ = (2π²/45)gT³, is obtained from the functional as well. These results are proved in the library with no unproved axioms beyond the standard logical ones.

What the theorem does not do is identify the physical entropy density with the ideal-gas entropy functional, nor does it set the phase-space normalization g/(2π²) or the units ħ = c = k_B = 1. Those are definitional bridges, chosen as models, not derived. The theorem establishes the functional and numerical content of the 4/3 relation and the 7/8 weight, but the physical interpretation of the entropy density as a thermodynamic quantity remains a modeling choice.

/-- **THEOREM (s = (4/3)ρ/T, Bose).** The Bose entropy integral is exactly
`4/3` of the Bose energy integral.  This is the dimensionless content of the
thermodynamic relation `s = (4/3)·ρ/T` for a massless boson gas, derived from
the microscopic entropy functional (never assumed). -/
theorem bose_entropy_eq_four_thirds_energy :
    (∫ t in Ioi (0 : ℝ), boseEntropyIntegrand t)
      = 4 / 3 * ∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1) := by
  rw [bose_entropy_integral_value, FermionWeightIntegral.bose_integral_value]
  ring

/-- **THEOREM (s = (4/3)ρ/T, Bose).** The Bose entropy integral is exactly
`4/3` of the Bose energy integral.  This is the dimensionless content of the
thermodynamic relation `s = (4/3)·ρ/T` for a massless boson gas, derived from
the microscopic entropy functional (never assumed). -/
theorem bose_entropy_eq_four_thirds_energy :
    (∫ t in Ioi (0 : ℝ), boseEntropyIntegrand t)
      = 4 / 3 * ∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1) := by
  rw [bose_entropy_integral_value, FermionWeightIntegral.bose_integral_value]
  ring

/-- **THEOREM (s = (4/3)ρ/T, Fermi).** The Fermi entropy integral is exactly
`4/3` of the Fermi energy integral: the same 4/3 law holds for a massless
fermion gas. -/
theorem fermi_entropy_eq_four_thirds_energy :
    (∫ t in Ioi (0 : ℝ), fermiEntropyIntegrand t)
      = 4 / 3 * ∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1) := by
  rw [fermi_entropy_integral_value, FermionWeightIntegral.fermi_integral_value]
  ring

/-- **THEOREM (7/8 at the entropy layer).** The Fermi entropy integral is
exactly `7/8` of the Bose one: the fermionic statistics weight of
`EntropyPerPhoton.fermionWeight` holds directly for entropy, not only for
energy. -/
theorem fermi_div_bose_entropy :
    (∫ t in Ioi (0 : ℝ), fermiEntropyIntegrand t)
      / (∫ t in Ioi (0 : ℝ), boseEntropyIntegrand t) = 7 / 8 := by
  rw [bose_entropy_integral_value, fermi_entropy_integral_value]
  rw [div_eq_iff (by positivity)]
  ring

What this page does not claim

The theorem does not prove the physical identification of entropy density with the ideal-gas functional. The theorem does not derive the phase-space normalization constant g/(2π²). The theorem does not set the units ħ = c = k_B = 1.

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