Cosmology Statistics Kernels Bose Log Kernel Eq Log Partition

The log kernel as a partition function

In statistical mechanics, the grand partition function for a single mode at zero chemical potential is a sum over all possible occupation numbers n of the Gibbs weight e^(−nE/T), with t = E/T as a dimensionless variable. For bosons, n can be any nonnegative integer, so this sum is a geometric series. The theorem boseLogKernel_eq_log_partition states that the previously defined Bose log kernel, written as −ln(1−e^(−t)), equals the natural logarithm of this partition function. This is not a new physical assumption; it is a proved identity that connects two quantities that were originally introduced separately.

The proof is direct. The partition function for bosons is Z_B = (1−e^(−t))⁻¹, and taking the logarithm gives exactly −ln(1−e^(−t)). The theorem is tagged THEOREM because it is derived from the definition of the partition function and the Gibbs weight, both of which are MODEL choices. The same derivation works for fermions, where the occupancy set is {0, 1} and the partition function is Z_F = 1 + e^(−t), giving ln(1+e^(−t)) as the corresponding log kernel.

This identity matters because it shows that the pressure kernels used in the framework's cosmological calculations are not arbitrary inputs. They are the natural logarithms of partition functions, exactly as the grand-canonical formalism demands. The framework's library proves this for both bosons and fermions, and also shows that the energy kernels are related to the log kernels by derivatives, so that pressure and energy are not independent but derivative-related. The plasma pressure and energy in the framework's η_B chain now start from sums of e^(−nE/T) in momentum space, grounding the earlier phase-space definitions in the partition function.

What this theorem does not claim is equally important. It does not derive the Gibbs weight e^(−βE) itself, which remains a MODEL choice, the canonical-ensemble measure. It does not derive the occupancy sets: the bosonic set ℕ and the fermionic set {0,1} are also MODEL choices. The framework's exclusion principle is formalized as a J-cost statement elsewhere, but the bridge from that certificate to the occupancy set {0,1} used here remains OPEN. The theorem only establishes the identity between the log kernel and the logarithm of the partition function, given those choices.

/-- **THEOREM: the Bose pressure kernel is the log partition function.**
`−ln(1−e^{−t}) = ln Z_B(t)`.  The kernel that was a definition in
`PhaseSpaceReduction` is the grand-canonical `ln Z` of one mode. -/
theorem boseLogKernel_eq_log_partition {t : ℝ} (ht : 0 < t) :
    boseLogKernel t = Real.log (bosePartition t) := by
  rw [bosePartition_eq ht, Real.log_inv]
  rfl

/-- **Bose partition function** (geometric series): for `t > 0`,
`Z_B(t) = (1 − e^{−t})⁻¹`. -/
theorem bosePartition_eq {t : ℝ} (ht : 0 < t) :
    bosePartition t = (1 - Real.exp (-t))⁻¹ := by
  unfold bosePartition
  simp only [boltzmannWeight_pow]
  exact tsum_geometric_of_lt_one (le_of_lt (Real.exp_pos _))
    (exp_neg_lt_one ht)

/-- **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

/-- **CAPSTONE (pressure).**  The plasma pressure of the η_B chain equals
the phase-space integral of `T·ln Z_mode(E/T)` — the grand-canonical
pressure `P = (T/V)·ln Z` — with `Z_B = Σ_{n∈ℕ} e^{−nE/T}` and
`Z_F = Σ_{n∈{0,1}} e^{−nE/T}`.  The log kernels are gone as inputs; only
the Gibbs weight and the occupancy sets remain. -/
theorem plasmaPressure_from_partitionFunction (gB gF : ℝ) {T : ℝ}
    (hT : 0 < T) :
    phaseSpaceDensity 3 gB T (fun t => Real.log (bosePartition t))
      + phaseSpaceDensity 3 gF T (fun t => Real.log (fermiPartition t))
      = GrandPotential.plasmaPressure gB gF T := by
  have hB := phaseSpaceDensity_congr_pos gB T hT
    (fun t => Real.log (bosePartition t)) boseLogKernel
    (fun t ht => (boseLogKernel_eq_log_partition ht).symm)
  have hF := phaseSpaceDensity_congr_pos gF T hT
    (fun t => Real.log (fermiPartition t)) fermiLogKernel
    (fun t _ => (fermiLogKernel_eq_log_partition t).symm)
  rw [hB, hF]
  exact PhaseSpaceReduction.plasmaPressure_from_phaseSpace gB gF hT

What this page does not claim

This theorem does not derive the Gibbs weight e^(−βE) itself, which remains a MODEL choice. This theorem does not derive the occupancy sets for bosons or fermions, which are also MODEL choices. The bridge from the framework's J-cost exclusion principle to the fermionic occupancy set {0,1} remains OPEN.

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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