openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /LocalizedHotMassGrowth.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.EndpointHistoryMass | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.PhaseSliceLocalization | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalizedBlockComparison | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalizedDampedBalance | |
| import OAI.MathematicalPhysics.Boltzmann.Jets.GeometricLocalizedEntry | |
| namespace OAI | |
| section | |
| /-! # Converting finite angular blocks to growth of a dominated hot measure -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal ProbabilityTheory | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| theorem localizedHistoryBlockMass_eq_measure (b : BeamCoefficients) | |
| (ρ C T a η l s : ℝ) {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) | |
| {A : Set (V × V)} (hA : MeasurableSet A) (N : ℕ) (z : ℝ × WeakTime.Phase) : | |
| localizedHistoryBlockMass b ρ C T a η l s hΩ A N z = | |
| (WeakTime.phaseEndpointKernel C T ∘ₖ | |
| ∑ n ∈ Finset.range N, | |
| ((((localizedNormalBeamKernel b ρ C a η l hΩ).restrict | |
| (measurable_fst measurableSet_Ioc : MeasurableSet (Prod.fst ⁻¹' Ioc s T))) ∘ₖ | |
| WeakTime.residenceKernel C T)^n)) z (EuclideanFlight.flight T ⁻¹' A) := by | |
| let f : WeakTime.Phase → ℝ≥0∞ := (EuclideanFlight.flight T ⁻¹' A).indicator 1 | |
| have hf : Measurable f := measurable_const.indicator | |
| (hA.preimage (EuclideanFlight.flight_measurePreserving T).measurable) | |
| rw [← lintegral_indicator_one | |
| (hA.preimage (EuclideanFlight.flight_measurePreserving T).measurable)] | |
| rw [kernel_lintegral_comp_finset _ _ _ _ hf] | |
| rfl | |
| theorem localized_history_mass_growth (b : BeamCoefficients) | |
| (ρ C T a η l s : ℝ) {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) | |
| {A B : Set WeakTime.Phase} (hA : MeasurableSet A) (hB : MeasurableSet B) | |
| (N : ℕ) (g : ℝ≥0∞) (μ ν : Measure WeakTime.Phase) | |
| (hdom : WeakTime.phaseEndpointKernel C T ∘ₘ | |
| (∑ n ∈ Finset.range N, | |
| ((((localizedNormalBeamKernel b ρ C a η l hΩ).restrict | |
| (measurable_fst measurableSet_Ioc : MeasurableSet (Prod.fst ⁻¹' Ioc s T))) ∘ₖ | |
| WeakTime.residenceKernel C T)^n) : | |
| Kernel (ℝ × WeakTime.Phase) (ℝ × WeakTime.Phase)) ∘ₘ WeakTime.phaseSeed s μ ≤ ν) | |
| (hblock : ∀ p ∈ B, g ≤ localizedHistoryBlockMass b ρ C T a η l s hΩ A N (s,p)) : | |
| g*μ B ≤ ν (EuclideanFlight.flight T ⁻¹' A) := by | |
| rw [Measure.comp_assoc] at hdom | |
| exact endpoint_mass_of_pointwise _ μ ν s | |
| (hA.preimage (EuclideanFlight.flight_measurePreserving T).measurable) hB g hdom | |
| (fun p hp => (localizedHistoryBlockMass_eq_measure b ρ C T a η l s hΩ hA N (s,p)) ▸ | |
| hblock p hp) | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| section | |
| /-! # Localized normal-angle histories use one kernel for every density scale -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal ProbabilityTheory | |
| open BoltzmannNonuniqueness.ReferenceGeometry | |
| namespace BoltzmannNonuniqueness.GeometricBackground | |
| theorem localizedNormalBeamKernel_independent {θ K α γ r s : ℝ} | |
| (hθ : 0 < θ) (hK : 0 ≤ K) (hα : 0 ≤ α) (hγ : 0 ≤ γ) | |
| (hr : 0 < r) (hs : 0 < s) (η C a l : ℝ) | |
| {B : Set EuclideanFlight.CharacteristicPhase} (hB : MeasurableSet B) : | |
| localizedNormalBeamKernel (beamCoefficients hθ hα hγ η r) (K/(4*r)) C a η l hB = | |
| localizedNormalBeamKernel (beamCoefficients hθ hα hγ η s) (K/(4*s)) C a η l hB := by | |
| classical | |
| exact congrArg (fun Q : Kernel EuclideanFlight.CharacteristicPhase EuclideanFlight.CharacteristicPhase => | |
| Kernel.piecewise (coldCharacteristicDomain_measurable a η l) (Q.restrict hB) 0) | |
| (beamNormalRetainedKernel_independent hθ hK hα hγ hr hs η C) | |
| end BoltzmannNonuniqueness.GeometricBackground | |
| end | |
| end | |
| section | |
| /-! # The finite physical endpoint mass does not depend on its reference density scale -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open BoltzmannNonuniqueness.ReferenceGeometry | |
| namespace BoltzmannNonuniqueness.GeometricBackground | |
| theorem localizedHistoryBlockMass_independent {θ K α γ r r' : ℝ} | |
| (hθ : 0 < θ) (hK : 0 ≤ K) (hα : 0 ≤ α) (hγ : 0 ≤ γ) | |
| (hr : 0 < r) (hr' : 0 < r') (η C T a l s : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) | |
| (A : Set WeakTime.Phase) (N : ℕ) (z : ℝ × WeakTime.Phase) : | |
| localizedHistoryBlockMass (beamCoefficients hθ hα hγ η r) (K/(4*r)) C T a η l s hΩ A N z = | |
| localizedHistoryBlockMass (beamCoefficients hθ hα hγ η r') (K/(4*r')) C T a η l s hΩ A N z := by | |
| unfold localizedHistoryBlockMass | |
| rw [localizedNormalBeamKernel_independent hθ hK hα hγ hr hr' η C a l hΩ] | |
| end BoltzmannNonuniqueness.GeometricBackground | |
| end | |
| end | |
| section | |
| /-! # Passing a covered finite-history lower bound to the actual hot box mass -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal ProbabilityTheory | |
| open BoltzmannNonuniqueness.EuclideanFlight | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| theorem localized_hot_mass_growth (b : BeamCoefficients) (ρ C t a A R η l s κ d : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) | |
| {f : JetCopyDatum.AttainedCapSequence.HotKineticPhase → ℝ} (hf : ∀ z, 0 ≤ f z) | |
| {β : WeakTime.Phase → ℝ} (hβ₁ : ∀ p, β p ≤ 1) | |
| (hcover : ∀ p ∈ tsupport β, | |
| hotCharacteristicEquiv (s,p) ∈ JetCopyDatum.AttainedCapSequence.coldCylinderSet a A R) | |
| (hcore : ∀ p ∈ flight s ⁻¹' lateBox κ d s, β p = 1) | |
| (N : ℕ) (g : ℝ≥0∞) | |
| (hdom : WeakTime.phaseEndpointKernel C t ∘ₘ | |
| (∑ n ∈ Finset.range N, | |
| ((((localizedNormalBeamKernel b ρ C a η l hΩ).restrict | |
| (measurable_fst measurableSet_Ioc : MeasurableSet (Prod.fst ⁻¹' Ioc s t))) ∘ₖ | |
| WeakTime.residenceKernel C t)^n)) ∘ₘ | |
| WeakTime.phaseSeed s (WeakTime.phaseSlice | |
| (WeakTime.phaseWeight β (cylinderPullback a A R f)) s) ≤ | |
| WeakTime.phaseSlice (WeakTime.phaseWeight β (cylinderPullback a A R f)) t) | |
| (hpoint : ∀ p ∈ flight s ⁻¹' lateBox κ d s, | |
| g ≤ localizedHistoryBlockMass b ρ C t a η l s hΩ (lateBox κ d t) N (s,p)) : | |
| g*hotLateBoxMass f κ d s ≤ hotLateBoxMass f κ d t := by | |
| let μ := WeakTime.phaseSlice (WeakTime.phaseWeight β (cylinderPullback a A R f)) s | |
| let ν := WeakTime.phaseSlice (WeakTime.phaseWeight β (cylinderPullback a A R f)) t | |
| have hBs := (lateBox_measurable κ d s).preimage (flight_measurePreserving s).measurable | |
| have heq := cylinder_phaseSlice_eq_on (f := f) hBs hcore hcover | |
| have hm : g*μ (flight s ⁻¹' lateBox κ d s) ≤ ν (flight t ⁻¹' lateBox κ d t) := by | |
| apply localized_history_mass_growth b ρ C t a η l s hΩ | |
| (lateBox_measurable κ d t) hBs N g μ ν | |
| · simpa only [FunLike.coe_sum] using hdom | |
| · exact hpoint | |
| calc | |
| g*hotLateBoxMass f κ d s = g*μ (flight s ⁻¹' lateBox κ d s) := | |
| congrArg (fun x : ℝ≥0∞ => g*x) heq.symm | |
| _ ≤ ν (flight t ⁻¹' lateBox κ d t) := hm | |
| _ ≤ hotLateBoxMass f κ d t := cylinder_phaseSlice_le hf hβ₁ a A R t _ | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| end OAI | |