Cosmology Statistics Kernels Fermi Energy Kernel Eq Occupation

The Fermi energy kernel

In statistical mechanics, the Fermi–Dirac distribution gives the average number of fermions occupying a quantum state with energy E at temperature T: 1/(e^(E/T)+1). The energy kernel t/(eᵗ+1), where t = E/T is a dimensionless ratio, is the product of this occupation number with t. This quantity appears throughout plasma physics and cosmology as the integrand for energy density calculations.

The framework's machine-checked library of formal theorems proves that this energy kernel is not an independent input but a derived consequence. The theorem fermiEnergyKernel_eq_occupation establishes that the previously defined energy kernel equals t times the Fermi occupation number. Starting from the grand partition function of a single mode at zero chemical potential, Z_F = 1 + e^(−t), the proof evaluates the ensemble mean occupancy and multiplies by t. The only inputs are the Gibbs weight e^(−n·t) for each n-quanta microstate and the occupancy set {0,1} for fermions, which encodes Pauli exclusion.

The derivation also shows thermodynamic consistency. The Fermi occupation number equals −d(ln Z_F)/dt, and the energy kernel equals −t times this derivative. In physical variables, the mean energy per mode satisfies ⟨E⟩ = −∂_β ln Z, exactly as the grand-canonical formalism requires. This means the pressure and energy kernels are derivative-related, not independent assumptions.

In Recognition Science, this theorem is a capstone that connects the framework's statistical foundations to plasma physics. The plasma energy now starts from the single-mode partition function summed over momentum space, rather than from a postulated kernel. What the theorem does not claim is that the Gibbs weight itself is derived; it remains a model input. The bridge from the framework's J-cost statement of the exclusion principle to the occupancy set {0,1} used here also remains open.

/-- **THEOREM: the Fermi energy kernel is `t·⟨n⟩_F`.** -/
theorem fermiEnergyKernel_eq_occupation (t : ℝ) :
    fermiEnergyKernel t = t * fermiOccupation t := by
  rw [fermiOccupation_eq]
  unfold PhaseSpaceReduction.fermiEnergyKernel
  ring

/-- **THEOREM (grand-canonical consistency, Fermi).**
`d/dt[ln(1+e^{−t})] = −⟨n⟩_F(t)`, for every `t`. -/
theorem fermiLogKernel_hasDerivAt (t : ℝ) :
    HasDerivAt fermiLogKernel (-(fermiOccupation t)) t := by
  have h1 : HasDerivAt (fun s : ℝ => -s) (-1) t := (hasDerivAt_id t).neg
  have h2 : HasDerivAt (fun s : ℝ => Real.exp (-s))
      (Real.exp (-t) * (-1)) t := (Real.hasDerivAt_exp (-t)).comp t h1
  have h3 : HasDerivAt (fun s : ℝ => 1 + Real.exp (-s))
      (0 + Real.exp (-t) * (-1)) t := (hasDerivAt_const t 1).add h2
  have hpos : (0 : ℝ) < 1 + Real.exp (-t) := by positivity
  have h4 := h3.log (ne_of_gt hpos)
  have heq : (0 + Real.exp (-t) * (-1)) / (1 + Real.exp (-t))
      = -(fermiOccupation t) := by
    rw [fermiOccupation_eq]
    have hy0 : Real.exp t ≠ 0 := ne_of_gt (Real.exp_pos t)
    rw [Real.exp_neg]
    rw [Real.exp_neg] at hpos
    field_simp
    ring
  rw [← heq]
  exact h4

/-- Fermi version: `t/(eᵗ+1) = −t·(d/dt) ln Z_F`. -/
theorem fermiEnergyKernel_from_logKernel (t : ℝ) :
    fermiEnergyKernel t = -t * deriv fermiLogKernel t := by
  rw [(fermiLogKernel_hasDerivAt t).deriv, fermiEnergyKernel_eq_occupation]
  ring

/-- Gibbs weight of the `n`-quanta microstate of a mode at dimensionless
energy `t = E/T` (equivalently `β·E`), chemical potential zero:
`w_n = e^{−n·t}`. -/
noncomputable def boltzmannWeight (t : ℝ) (n : ℕ) : ℝ := Real.exp (-(n * t))

/-- Fermionic single-mode grand partition function: Pauli-restricted
occupancy `n ∈ {0,1}`, `Z_F(t) = Σ_{n<2} e^{−n·t}`. -/
noncomputable def fermiPartition (t : ℝ) : ℝ :=
  ∑ n ∈ Finset.range 2, boltzmannWeight t n

What this page does not claim

The Gibbs weight e^(−βE) is derived within this module; it is a model input. The bridge from the framework's J-cost exclusion principle to the occupancy set {0,1} is proved; it remains open. The theorem applies to bosons without the positivity condition t > 0.

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

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