Cosmology Grand Potential Plasma Pressure Potential

The plasma pressure potential

In thermodynamics, the pressure of a system is not usually a fundamental object; it is a derived quantity. For a gas of massless particles, the pressure can be computed from statistical mechanics by summing over all possible quantum states. The standard result, derived by physicists such as Max Planck and Satyendra Nath Bose in the early 20th century, is that the pressure of radiation is proportional to the fourth power of temperature, P ∝ T⁴. This is the Stefan-Boltzmann law. The same law governs the hot plasma that filled the universe in its first moments, when particles and radiation were in equilibrium.

The Recognition Science framework's ledger, a discrete record of physical events, models this plasma through a grand-canonical ensemble, a statistical description where energy and particles can be exchanged with a reservoir. In this ensemble, the pressure is not just a number; it is a function of temperature, P(T). The framework's library, a machine-checked collection of formal theorems, proves that this pressure function has a special property: its temperature derivative is exactly the entropy density, s = dP/dT. This is the statement of the declaration plasmaPressure_potential. It is a theorem, not an assumption, because it follows from the explicit integrals that define the pressure and the entropy.

The theorem's power is that it makes two classical thermodynamic identities redundant. The Euler relation, T·s = ρ + P, and the Gibbs-Duhem relation, p' = s·T', are no longer separate postulates about the plasma. They are algebraic consequences of the potential structure. The framework proves this in its library, showing that the energy density, ρ = T·s - P, and the pressure satisfy these relations automatically. As a result, the radiation equation of state, p = ρ/3, is forced by the ensemble, not put in by hand. This is a significant simplification: the entire thermodynamic behavior of the early universe's plasma is encoded in a single function, the pressure potential.

This result is not a claim about the fundamental nature of space or time. It is a statement about the internal consistency of a statistical mechanics calculation. The framework does not claim that the grand-canonical ensemble is the only way to describe the plasma, nor does it derive the Friedmann equations that govern the expansion of the universe. Those are separate inputs. The theorem also does not claim that the plasma is in equilibrium at all times; it only applies when the potential structure holds, which is the definition of local equilibrium. The value of the theorem is that it shows how much of early-universe cosmology follows from a single, well-defined thermodynamic assumption.

/-- **THEOREM (the plasma is a potential fluid).** The temperature derivative
of the statistical-mechanical pressure is *exactly* the `radiationEntropy` of
`NeutrinoDilution` — the object previously built from the independent entropy
integrals `∫σ_B`, `∫σ_F`.  So `s = dP/dT` holds between two independently
derived statistical-mechanical quantities: the potential structure of the
plasma is a theorem, not a definition. -/
theorem plasmaPressure_potential (gB gF x : ℝ) :
    HasDerivAt (fun T => plasmaPressure gB gF T)
      (NeutrinoDilution.radiationEntropy gB gF x) x := by
  rw [NeutrinoDilution.radiationEntropy_eq]
  have hfun : (fun T => plasmaPressure gB gF T)
      = fun T => π ^ 2 / 90 * (gB + 7 / 8 * gF) * T ^ 4 :=
    funext fun T => plasmaPressure_eq gB gF T
  rw [hfun]
  have hpow : HasDerivAt (fun T : ℝ => T ^ 4) (4 * x ^ 3) x := by
    simpa using hasDerivAt_pow 4 x
  have h := hpow.const_mul (π ^ 2 / 90 * (gB + 7 / 8 * gF))
  have hval : π ^ 2 / 90 * (gB + 7 / 8 * gF) * (4 * x ^ 3)
      = 2 * π ^ 2 / 45 * (gB + 7 / 8 * gF) * x ^ 3 := by ring
  rw [hval] at h
  exact h

/-- **Euler relation (derived).** `T·s = ρ + P` is an algebraic identity of
the Legendre-transform structure — not an independent equilibrium postulate. -/
theorem potential_euler (P s : ℝ → ℝ) (x : ℝ) :
    x * s x = energyOf P s x + P x := by
  simp only [energyOf]
  ring

/-- **Gibbs–Duhem relation (derived).** Along any temperature trajectory
`T(t)`, the pressure obeys `p′ = s(T)·T′`: this is the chain rule applied to
`s = dP/dT`, not an independent postulate. -/
theorem potential_gibbs_duhem
    {P s T : ℝ → ℝ} {T' t : ℝ}
    (hP : HasDerivAt P (s (T t)) (T t))
    (hT : HasDerivAt T T' t) :
    HasDerivAt (fun u => P (T u)) (s (T t) * T') t := by
  simpa [Function.comp] using hP.comp t hT

/-- **THEOREM (radiation equation of state derived).** `p = ρ/3` is forced by
the grand-canonical integrals — it is not an input anywhere in the chain. -/
theorem plasma_eos (gB gF T : ℝ) :
    plasmaPressure gB gF T = plasmaEnergy gB gF T / 3 := by
  rw [plasmaEnergy_eq, plasmaPressure_eq]
  ring

What this page does not claim

The theorem does not claim that the grand-canonical ensemble is the only valid description of the early universe plasma. The theorem does not derive the Friedmann equations or the expansion history of the universe. The theorem does not claim the plasma is in equilibrium at all times; it applies only when the potential structure holds.

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/GrandPotential.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: