openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /EntropyCollisionIntegrability.lean
| import OAI.MathematicalPhysics.Boltzmann.Compactness.PeriodicJointBudgetLimit | |
| namespace OAI | |
| /-! # Separate collision integrability from entropy and dissipation budgets -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped ENNReal | |
| namespace BoltzmannNonuniqueness.PeriodicApproximation | |
| open CollisionCompactness | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [SFinite μ] | |
| theorem normalized_collisions_of_entropy {g : X × Velocity → ℝ} | |
| (hgm : Measurable g) (hgn : ∀ p, 0 ≤ g p) | |
| (hE : Integrable (entropyGauge g) (μ.prod volume)) | |
| (hD : rateDissipation μ g < ⊤) : | |
| Integrable (fun p => gainDensity g p/(1+g p/maxwellian p.2)) (μ.prod volume) ∧ | |
| Integrable (fun p => lossDensity g p/(1+g p/maxwellian p.2)) (μ.prod volume) ∧ | |
| ∀ᵐ p ∂μ.prod (volume : Measure Velocity), | |
| Integrable (fun q => postRate g (collisionGrouping.symm (p,q))) | |
| ((volume : Measure Velocity).prod sphereArea) ∧ | |
| Integrable (fun q => preRate g (collisionGrouping.symm (p,q))) | |
| ((volume : Measure Velocity).prod sphereArea) := by | |
| have hI := integrable_of_entropyGauge hgm hgn hE | |
| apply normalized_gainLoss_and_fibers hgm hgn hI _ hD | |
| have hL := linearMoment_integrable hgm hgn hE | |
| convert hL.sub hI using 1 | |
| funext p | |
| simp only [linearMoment,Pi.sub_apply] | |
| ring | |
| end BoltzmannNonuniqueness.PeriodicApproximation | |
| end | |
| end OAI | |