Cosmology Radiation Entropy Relation Bose Entropy Integral Value

The radiation entropy integral

For a gas of massless particles, the entropy density s is not an independent quantity. Statistical mechanics ties it to the energy density ρ and the temperature T through the relation s = (4/3)·ρ/T. The factor 4/3 has long been a standard input in textbooks. The declaration bose_entropy_integral_value proves that this factor is not an assumption: it emerges from the microscopic entropy functional of quantum statistics.

The proof works with dimensionless integrals where x = E/T. For a gas of bosons, the entropy integrand is σ_B(x) = x²[(1+f)ln(1+f) − f ln f], with f = 1/(eˣ−1) the Bose-Einstein occupation number. The theorem shows that the integral of σ_B from 0 to ∞ equals 4π⁴/45. This value is exactly 4/3 times the energy integral ∫₀^∞ x³/(eˣ−1) dx = π⁴/15. A companion theorem proves the analogous result for fermions, with the entropy integral equal to 7π⁴/90 and the same 4/3 factor relative to its energy integral.

The method splits each entropy integrand into the energy kernel plus a logarithmic kernel. The logarithmic kernel expands into a Mercator series in e^(−x), and its Mellin transform at s = 3 evaluates to Γ(3)·ζ(4) = π⁴/45 for bosons and Γ(3)·η(4) = 7π⁴/360 for fermions. Adding the energy integrals gives the totals. The theorem also yields the entropy coefficient: the integral divided by 2π² equals 2π²/45, the prefactor in the standard formula s_γ = (2π²/45)·g·T³.

What the theorem does not claim: the phase-space normalization g/(2π²), the units ℏ=c=k_B=1, and the identification of the physical entropy density with the ideal-gas entropy functional are definitional bridges, not proved results. The theorem proves the functional and numerical content of s = (4/3)ρ/T and the 7/8 fermion-to-boson entropy weight, but it does not derive the degeneracy count g or the physical interpretation of the entropy functional itself.

/-- **THEOREM (Bose entropy integral).**
`∫₀^∞ t²[(1+f)ln(1+f) − f ln f] dt = 4π⁴/45` with `f = 1/(eᵗ−1)`. -/
theorem bose_entropy_integral_value :
    (∫ t in Ioi (0 : ℝ), boseEntropyIntegrand t) = 4 * π ^ 4 / 45 := by
  have hsplit : (∫ t in Ioi (0 : ℝ), boseEntropyIntegrand t)
      = ∫ t in Ioi (0 : ℝ),
          (t ^ 3 / (Real.exp t - 1) + t ^ 2 * (-Real.log (1 - Real.exp (-t)))) := by
    refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
    exact bose_entropy_pointwise ht
  rw [hsplit, integral_add integrableOn_bose_energy integrableOn_boseLog,
    FermionWeightIntegral.bose_integral_value, boseLog_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 (the 2π²/45 entropy coefficient from the functional).**
`s_γ = (g/2π²)·T³·∫σ_B = (2π²/45)·g·T³`: dividing the derived entropy
integral by the phase-space normalization `2π²` yields exactly the `2π²/45`
prefactor of the photon entropy density, with the `4/3` factor never
assumed. -/
theorem entropy_coeff_from_functional :
    (∫ t in Ioi (0 : ℝ), boseEntropyIntegrand t) / (2 * π ^ 2) = 2 * π ^ 2 / 45 := by
  rw [bose_entropy_integral_value]
  rw [div_eq_iff (by positivity)]
  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 derive the phase-space normalization factor g/(2π²) or the choice of units ℏ=c=k_B=1. The theorem does not prove that the physical entropy density is given by the ideal-gas entropy functional. The theorem does not establish the value of the fine-structure constant or any other coupling constant.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/RadiationEntropyRelation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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