Cosmology Partition Kernels Bose Log Kernel From Partition

The Bose log kernel

In statistical mechanics, a partition function is a sum over all possible states of a system, weighting each state by a Boltzmann factor. For a single bosonic mode, where any number of particles can occupy the same state, this sum is a geometric series. The theorem establishes that the natural logarithm of this partition function, written as log of the sum over n of exp(-t)^n, equals the Bose log kernel used in the framework's phase space reduction.

The proof is direct. The partition function for a bosonic mode is the geometric series Z_B = Σ x^n = (1 - x)⁻¹, where x = exp(-t) and t > 0 so the series converges. Taking the logarithm gives log Z_B = -log(1 - x), which is precisely the definition of the Bose log kernel. The theorem states this equality formally, with the condition that t is positive, ensuring the series converges.

This result matters because it removes a definitional choice. The kernel is not an arbitrary function selected for convenience; it is forced by the partition function derived from the occupancy range. The physical input is that bosons can occupy a state any number of times, represented by summing over all natural numbers. The framework's library proves this in a machine-checked way, so the equality holds as a matter of formal mathematics.

What the theorem does not claim is equally important. It does not derive the partition function from first principles; it takes the occupancy range as given. It does not prove that the Bose-Einstein distribution is correct physics; it shows that the mean occupancy computed from the partition function equals 1/(e^t - 1), the standard result. The theorem is about the identity between the log of the partition function and the kernel, not about the physical origin of the occupancy range.

/-- `log Z_B` is exactly the Bose log kernel used in
`PhaseSpaceReduction.boseLogKernel` (hence in the pressure/entropy integrals
upstream): the kernel is no longer a definitional choice but the log of the
derived partition function. -/
theorem boseLogKernel_from_partition (t : ℝ) (ht : 0 < t) :
    Real.log (∑' n : ℕ, Real.exp (-t) ^ n)
      = PhaseSpaceReduction.boseLogKernel t := by
  rw [bose_partition_tsum t ht, Real.log_inv,
      PhaseSpaceReduction.boseLogKernel]

/-- The Bose single-mode grand partition function: occupancies range over all
of `ℕ`, and the Boltzmann-weighted sum is the geometric series with ratio
`x = exp (-t) < 1`. -/
theorem bose_partition_hasSum (t : ℝ) (ht : 0 < t) :
    HasSum (fun n : ℕ => Real.exp (-t) ^ n) (1 - Real.exp (-t))⁻¹ := by
  have hξpos : (0 : ℝ) < Real.exp (-t) := Real.exp_pos _
  have hξlt : Real.exp (-t) < 1 := Real.exp_lt_one_iff.mpr (by linarith)
  apply hasSum_geometric_of_norm_lt_one
  rw [Real.norm_eq_abs, abs_of_pos hξpos]
  exact hξlt

/-- The Bose single-mode grand partition function: occupancies range over all
of `ℕ`, and the Boltzmann-weighted sum is the geometric series with ratio
`x = exp (-t) < 1`. -/
theorem bose_partition_hasSum (t : ℝ) (ht : 0 < t) :
    HasSum (fun n : ℕ => Real.exp (-t) ^ n) (1 - Real.exp (-t))⁻¹ := by
  have hξpos : (0 : ℝ) < Real.exp (-t) := Real.exp_pos _
  have hξlt : Real.exp (-t) < 1 := Real.exp_lt_one_iff.mpr (by linarith)
  apply hasSum_geometric_of_norm_lt_one
  rw [Real.norm_eq_abs, abs_of_pos hξpos]
  exact hξlt

What this page does not claim

The theorem does not derive the partition function from first principles; it takes the occupancy range as given. The theorem does not prove that the Bose-Einstein distribution is correct physics; it shows the mean occupancy equals 1/(e^t - 1). The theorem does not claim that the framework's cosmology chain is complete or that the partition function layer is the only input.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: