Cosmology Occupation Energy Fermi Energy Kernel Eq

The Fermi energy kernel

In statistical physics, the energy density of a gas of fermions at temperature T is computed by integrating, over all frequencies, an integrand that combines the energy of a mode with the average number of particles occupying that mode. For fermions, the occupation number is the Fermi-Dirac distribution, 1/(e^t + 1), where t is a dimensionless energy. The integral of t^3 times this occupation number, over t from zero to infinity, gives the total energy density of a fermion gas.

The declaration fermi_energy_kernel_eq in the framework's machine-checked library of formal theorems proves that this integrand can be rewritten as t^3 times a ratio of two finite sums. The ratio is the occupation number derived from a two-state partition function, which is the partition function for a system where each mode can hold at most one particle, the Pauli restriction. The theorem states, for any positive t, that t^3 / (e^t + 1) equals t^3 times the ratio of the sum over n from 0 to 1 of n times e^(-t)^n to the sum over n from 0 to 1 of e^(-t)^n. This is a formal identity, proved in the framework's library, not a physical measurement.

The theorem is one step in a longer chain. The framework derives the occupation number from a partition function, then forms the energy integrand as t^3 times that occupation number, and then proves the integral of the fermion integrand is exactly 7/8 of the integral of the boson integrand. That 7/8 ratio is a separate theorem, energy_ratio_seven_eighths, which follows from the integral values π^4/15 and 7π^4/120. The declaration fermi_energy_kernel_eq itself only establishes the identity of the integrand; it does not compute the integral or the ratio.

In the framework, this identity is part of a derivation that connects the partition function to thermal energy densities. The framework's library proves these statements with a machine-checked proof, meaning the steps are verified by a computer. The declaration does not claim that the framework derives the value of the fine-structure constant, does not claim that the Fermi energy density matches any measured cosmological value, and does not claim that the 7/8 ratio is a new physical prediction. It is a formal identity about the structure of the energy integrand.

/-- The Fermi energy integrand equals `t³` × (the Pauli-restricted occupation
number derived from the two-state partition function). -/
theorem fermi_energy_kernel_eq (t : ℝ) (_ht : 0 < t) :
    t ^ 3 / (Real.exp t + 1)
      = t ^ 3 * ((∑ n : Fin 2, ((n : ℕ) : ℝ) * Real.exp (-t) ^ (n : ℕ))
          / (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))) := by
  rw [PartitionKernels.fermi_occupation t]
  ring

/-- **The 7/8 ratio, stated at the partition-function level**: the ratio of
the thermal energy integrals, with each integrand written as
`t³ × ⟨n⟩` (occupation numbers from the derived partition functions), is
exactly `7/8`. -/
theorem energy_ratio_seven_eighths :
    (∫ t in Ioi (0 : ℝ),
        t ^ 3 * ((∑ n : Fin 2, ((n : ℕ) : ℝ) * Real.exp (-t) ^ (n : ℕ))
          / (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))))
      / (∫ t in Ioi (0 : ℝ),
        t ^ 3 * ((∑' n : ℕ, (n : ℝ) * Real.exp (-t) ^ n)
          / (∑' n : ℕ, Real.exp (-t) ^ n)))
      = 7 / 8 := by
  have hf : (∫ t in Ioi (0 : ℝ),
      t ^ 3 * ((∑ n : Fin 2, ((n : ℕ) : ℝ) * Real.exp (-t) ^ (n : ℕ))
        / (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))))
      = ∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1) := by
    refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
    exact (fermi_energy_kernel_eq t ht).symm
  have hb : (∫ t in Ioi (0 : ℝ),
      t ^ 3 * ((∑' n : ℕ, (n : ℝ) * Real.exp (-t) ^ n)
        / (∑' n : ℕ, Real.exp (-t) ^ n)))
      = ∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1) := by
    refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
    exact (bose_energy_kernel_eq t ht).symm
  rw [hf, hb]
  exact FermionWeightIntegral.fermi_div_bose_integral

/-- The Fermi energy integrand equals `t³` × (the Pauli-restricted occupation
number derived from the two-state partition function). -/
theorem fermi_energy_kernel_eq (t : ℝ) (_ht : 0 < t) :
    t ^ 3 / (Real.exp t + 1)
      = t ^ 3 * ((∑ n : Fin 2, ((n : ℕ) : ℝ) * Real.exp (-t) ^ (n : ℕ))
          / (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))) := by
  rw [PartitionKernels.fermi_occupation t]
  ring

What this page does not claim

The declaration does not claim to derive the fine-structure constant. The declaration does not claim the Fermi energy density matches any measured cosmological value. The declaration does not claim the 7/8 ratio is a new physical prediction.

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

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