Cosmology Phase Space Reduction Plasma Pressure From Phase Space

The pressure from momentum space

In thermal physics, the pressure of a gas of light or other massless particles is usually written down as a formula: P = (π²/90)(g_B + 7/8 g_F)T⁴, where T is temperature and g_B, g_F count the particle types. This is the Stefan-Boltzmann law, and its T⁴ dependence has been measured in laboratories for over a century. The question is whether that formula is a postulate or a derived result.

The Recognition Science library's phase-space module answers that question. It starts with the grand-canonical integral over all momentum states in three dimensions: P = (g/(2π)³) ∫ d³k T·K(‖k‖/T), where K is a kernel that encodes whether the particles are bosons or fermions. The theorem plasmaPressure_from_phaseSpace proves that this three-dimensional integral is exactly equal to the previously assumed one-dimensional form, g/(2π²) · T⁴ · ∫ t² K(t) dt. The angular factor 4π/(2π)³ = 1/(2π²) comes from integrating over the sphere of directions, and the T⁴ comes from rescaling momenta by temperature. Neither is put in by hand.

The proof is unconditional: it uses only the radial reduction of a Haar integral, the volume of the unit ball in ℝ³ (which is 4π/3), and a change of variables. No integrability hypotheses are needed. The same reduction works for energy density, giving the companion theorem plasmaEnergy_from_phaseSpace. Chaining these with standard integral values yields the closed forms: pressure equals (π²/90)(g_B + 7/8 g_F)T⁴ and energy density equals (π²/30)(g_B + 7/8 g_F)T⁴, the factor 7/8 being the fermionic correction.

The T⁴ exponent itself is structural. A separate theorem proves that in any number of spatial dimensions d, the phase-space density scales as T^(d+1). Since the framework has already proved that the spatial dimension is 3, the Stefan-Boltzmann exponent 4 is D+1 with D=3. The framework's library states this as: the T⁴ of radiation thermodynamics is the dimension theorem wearing thermodynamic clothes.

What the declaration does not claim is just as important. The mode density 1/(2π)³ per unit volume, the massless dispersion E = ‖k‖, and the choice of Bose or Fermi statistics kernels remain model inputs, not derived results. The theorem proves the reduction of the integral, not the physical content of those inputs. It also does not prove the value of the spatial dimension; that is a separate upstream theorem. The reduction is a mathematical identity about integrals, not a statement about what particles exist or how they interact.

/-- **THEOREM (pressure from phase space).**  The 3D grand-canonical
pressure integral with the Bose/Fermi log kernels *is* the
`GrandPotential.plasmaPressure` that was previously definitional: its
`g/(2π²)·T⁴` prefactor is now derived. -/
theorem plasmaPressure_from_phaseSpace (gB gF : ℝ) {T : ℝ} (hT : 0 < T) :
    phaseSpaceDensity 3 gB T boseLogKernel
      + phaseSpaceDensity 3 gF T fermiLogKernel
      = GrandPotential.plasmaPressure gB gF T := by
  rw [phaseSpaceDensity_reduction gB hT, phaseSpaceDensity_reduction gF hT]
  unfold GrandPotential.plasmaPressure boseLogKernel fermiLogKernel
  rfl

/-- **THEOREM (T-scaling in general dimension).**  In `d ≠ 0` spatial
dimensions the phase-space density scales as `T^(d+1)`:

  `P_d(T) = T^(d+1) · P_d(1)`.

The proof never evaluates the unit-ball volume — the scaling is pure
dimensional analysis of the measure `d^d k` against the substitution
`k = T·t`.  The exponent is structural: one power of `T` per momentum
dimension plus one from the kernel prefactor. -/
theorem phaseSpaceDensity_T_scaling (d : ℕ) (hd : d ≠ 0) (g : ℝ) {T : ℝ}
    (hT : 0 < T) (K : ℝ → ℝ) :
    phaseSpaceDensity d g T K = T ^ (d + 1) * phaseSpaceDensity d g 1 K := by
  haveI : Nonempty (Fin d) := ⟨⟨0, Nat.pos_of_ne_zero hd⟩⟩
  unfold phaseSpaceDensity
  have h1 := MeasureTheory.integral_fun_norm_addHaar
    (μ := (volume : Measure (EuclideanSpace ℝ (Fin d))))
    (fun y => T * K (y / T))
  have h2 := MeasureTheory.integral_fun_norm_addHaar
    (μ := (volume : Measure (EuclideanSpace ℝ (Fin d))))
    (fun y => (1 : ℝ) * K (y / 1))
  simp only [finrank_euclideanSpace, Fintype.card_fin, smul_eq_mul,
    nsmul_eq_mul, div_one, one_mul] at h1 h2
  simp only [div_one, one_mul]
  rw [h1, h2]
  have hswap : (fun y : ℝ => y ^ (d - 1) * (T * K (y / T)))
      = fun y : ℝ => T * (y ^ (d - 1) * K (y / T)) := by
    funext y; ring
  rw [hswap, integral_const_mul, radial_scale_pow (d - 1) K hT,
    Nat.sub_add_cancel (Nat.one_le_iff_ne_zero.mpr hd)]
  ring

/-- **Closed form from the 3D integral.**  Chaining the reduction with the
Mellin-transform integral values: the phase-space pressure is
`(π²/90)(g_B + (7/8)g_F)·T⁴` — Stefan–Boltzmann with the fermionic `7/8`,
now derived end-to-end from the momentum-space integral. -/
theorem phaseSpacePressure_closed_form (gB gF : ℝ) {T : ℝ} (hT : 0 < T) :
    phaseSpaceDensity 3 gB T boseLogKernel
      + phaseSpaceDensity 3 gF T fermiLogKernel
      = π ^ 2 / 90 * (gB + 7 / 8 * gF) * T ^ 4 := by
  rw [plasmaPressure_from_phaseSpace gB gF hT,
    GrandPotential.plasmaPressure_eq]

/-- Grand-canonical phase-space integral of one massless sector in `d`
spatial dimensions: degeneracy `g`, temperature `T`, dimensionless kernel
`K` evaluated at `E/T` with `E = ‖k‖`, and mode density `1/(2π)^d`.

* Pressure: `K(t) = −ln(1−e^{−t})` (Bose) or `ln(1+e^{−t})` (Fermi),
  from `±T·ln Z` per mode.
* Energy density: `K(t) = t/(eᵗ∓1)`, i.e. `E·n(E/T)` per mode rescaled
  by `T`. -/
noncomputable def phaseSpaceDensity (d : ℕ) (g T : ℝ) (K : ℝ → ℝ) : ℝ :=
  g / (2 * π) ^ d * ∫ k : EuclideanSpace ℝ (Fin d), T * K (‖k‖ / T)

What this page does not claim

The theorem does not derive the mode density 1/(2π)³ or the dispersion E = ‖k‖; these remain model choices. The theorem does not prove the value of the spatial dimension; that is a separate upstream result. The theorem does not establish the physical existence of any particular particle species or their interactions. The theorem does not derive the Bose or Fermi statistics kernels from first principles.

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

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