Cosmology Statistics Kernels Fermi Energy Kernel From Log Kernel

The energy kernel's derivation

In statistical mechanics, the Fermi energy kernel describes the average energy of a single fermion mode at temperature T. Written as a function of the dimensionless ratio t = E/T, it takes the form t/(eᵗ + 1). This kernel appears in the pressure and energy integrals of a fermion gas, and it encodes the Pauli exclusion principle: no two fermions can occupy the same quantum state, so the average occupation of a mode never reaches one.

The kernel is classically derived from the grand partition function. For a single mode with energy E, the partition function is the sum of Gibbs weights e^(−n·E/T) over allowed occupancies n. For fermions, the Pauli principle restricts n to the set {0, 1}, so the partition function is simply 1 + e^(−t). The average occupation is then 1/(eᵗ + 1), the Fermi–Dirac distribution, and the energy kernel is t times that occupation.

In Recognition Science, the framework's machine-checked library of formal theorems proves that this energy kernel is not an independent definition but a consequence of the log kernel. Specifically, the theorem fermiEnergyKernel_from_logKernel establishes that the Fermi energy kernel equals −t times the derivative of the Fermi log kernel, where the log kernel is the logarithm of the partition function. This is the thermodynamic identity = −d(ln Z)/dt, expressed in dimensionless variables. The derivation starts from the Gibbs weight e^(−n·t) and the occupancy set {0, 1}, and it verifies the derivative relation formally.

What the theorem does not claim is that the Gibbs weight or the occupancy set are themselves derived. The Gibbs weight e^(−βE) is a model input, the canonical-ensemble measure, and the occupancy set {0, 1} is a model input encoding Pauli exclusion. The framework's exclusion principle is formalized elsewhere as a J-cost statement, but the bridge from that certificate to the occupancy set used here remains open. The theorem also does not claim that the Fermi energy kernel is the only possible form; it shows that, given the model inputs, the kernel follows from the partition function.

The payoff is thermodynamic consistency. The pressure kernel and the energy kernel are not independent inputs to the plasma equations; they are derivative-related, exactly as the grand-canonical formalism demands. This means the framework's plasma pressure and energy, computed from the partition function in momentum space, agree with the previously defined kernels. The theorem closes a gap: what was once a definition is now a proved consequence of a single starting point.

/-- 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

/-- **THEOREM: the Fermi pressure kernel is the log partition function.**
`ln(1+e^{−t}) = ln Z_F(t)`. -/
theorem fermiLogKernel_eq_log_partition (t : ℝ) :
    fermiLogKernel t = Real.log (fermiPartition t) := by
  rw [fermiPartition_eq]
  rfl

/-- **THEOREM (Fermi–Dirac distribution).**  The ensemble-mean occupation
of a fermionic mode is `⟨n⟩_F = 1/(eᵗ+1)`, for every `t` (the two-state
sum needs no convergence condition). -/
theorem fermiOccupation_eq (t : ℝ) :
    fermiOccupation t = 1 / (Real.exp t + 1) := by
  unfold fermiOccupation boltzmannWeight
  rw [fermiPartition_eq]
  simp only [Finset.sum_range_succ, Finset.sum_range_zero,
    Nat.cast_zero, Nat.cast_one, zero_mul, one_mul, neg_zero,
    Real.exp_zero, zero_add, add_zero]
  have hy0 : Real.exp t ≠ 0 := ne_of_gt (Real.exp_pos t)
  have hpos : (0 : ℝ) < 1 + Real.exp (-t) := by positivity
  rw [Real.exp_neg]
  rw [Real.exp_neg] at hpos
  field_simp

/-- 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))

What this page does not claim

The theorem does not derive the Gibbs weight or the occupancy set from the framework's first principles. The theorem does not prove the Pauli exclusion principle itself; the bridge from the J-cost statement to the occupancy set remains open. The theorem does not claim the Fermi energy kernel is the only possible form for all fermionic systems.

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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