openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /LocalizedDampedBalance.lean
| import OAI.MathematicalPhysics.Boltzmann.Approximation.PositiveContributionsIntegrable | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.DampedEndpointHistories | |
| import OAI.MathematicalPhysics.Boltzmann.Collision.CollisionWaitingCone | |
| import OAI.MathematicalPhysics.Boltzmann.Collision.KernelInvariantOrder | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.WeakEntropyMarginal | |
| import OAI.MathematicalPhysics.Boltzmann.Jets.BeamNormalCharacteristicIdentity | |
| namespace OAI | |
| section | |
| /-! # Localizing the physical normal-angle source above the cold spatial cutoff -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal ProbabilityTheory | |
| open BoltzmannNonuniqueness.EuclideanFlight | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| def coldCharacteristicDomain (a η l : ℝ) : Set CharacteristicPhase := | |
| {z | a ≤ z.1 ∧ 0 < ‖transverse (z.2.1+z.1 • z.2.2)‖ ∧ | |
| Real.exp η*l < ‖transverse (z.2.1+z.1 • z.2.2)‖} | |
| theorem coldCharacteristicDomain_measurable (a η l : ℝ) : | |
| MeasurableSet (coldCharacteristicDomain a η l) := by | |
| have hx : Measurable (fun z : CharacteristicPhase => ‖transverse (z.2.1+z.1 • z.2.2)‖) := by | |
| exact ChartKernel.measurable_transverse.norm.comp (by fun_prop) | |
| exact (measurableSet_le measurable_const measurable_fst).inter | |
| ((measurableSet_lt measurable_const hx).inter (measurableSet_lt measurable_const hx)) | |
| def localizedNormalBeamKernel (b : BeamCoefficients) (ρ C a η l : ℝ) | |
| {B : Set CharacteristicPhase} (hB : MeasurableSet B) : | |
| Kernel CharacteristicPhase CharacteristicPhase := by | |
| classical | |
| exact Kernel.piecewise (coldCharacteristicDomain_measurable a η l) | |
| ((beamNormalRetainedKernel b ρ C).restrict hB) 0 | |
| instance localizedNormalBeamKernel_sFinite (b : BeamCoefficients) (ρ C a η l : ℝ) | |
| {B : Set CharacteristicPhase} (hB : MeasurableSet B) : | |
| IsSFiniteKernel (localizedNormalBeamKernel b ρ C a η l hB) := by | |
| unfold localizedNormalBeamKernel | |
| infer_instance | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| section | |
| /-! # The localized normal-angle collision law in physical flight coordinates -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal NNReal ProbabilityTheory | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| def localizedNormalCollision (b : BeamCoefficients) (ρ C T a η l : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) : | |
| Kernel PhysicalFlightState PhysicalFlightState := | |
| transportedKernel (flightCharacteristicEquiv T) | |
| (localizedNormalBeamKernel b ρ C a η l hΩ) | |
| instance localizedNormalCollision_sFinite (b : BeamCoefficients) (ρ C T a η l : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) : | |
| IsSFiniteKernel (localizedNormalCollision b ρ C T a η l hΩ) := by | |
| unfold localizedNormalCollision | |
| infer_instance | |
| theorem localizedNormalCollision_eq_restrict {C : ℝ≥0} (hC : referenceClock ≤ C) | |
| (b : BeamCoefficients) (ρ : ℝ≥0) (T a η l : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) (p : PhysicalFlightState) | |
| (hz : flightCharacteristicEquiv T p ∈ coldCharacteristicDomain a η l) | |
| (hp : 0 ≤ b.plusDensity (characteristicBeamPoint (flightCharacteristicEquiv T p))) | |
| (hm : 0 ≤ b.minusDensity (characteristicBeamPoint (flightCharacteristicEquiv T p))) | |
| (hf : characteristicBeamFrequency b (flightCharacteristicEquiv T p) ≤ (referenceClock:ℝ)) : | |
| localizedNormalCollision b ρ (ρ*C) T a η l hΩ p = | |
| (dampedDensityScaledCollision C b T ρ p).restrict ((flightCharacteristicEquiv T) ⁻¹' Ω) := by | |
| classical | |
| apply Measure.ext_of_lintegral | |
| intro f hfn | |
| let e := flightCharacteristicEquiv T | |
| rw [localizedNormalCollision,transportedKernel_lintegral _ _ p hfn, | |
| localizedNormalBeamKernel,Kernel.piecewise_apply,ite_eq_left hz,Kernel.restrict_apply, | |
| beamNormalRetainedKernel_eq_scaled hC b T ρ (e p) hp hm hf] | |
| have he := congrArg (fun K : Kernel PhysicalFlightState PhysicalFlightState => K p) | |
| (transportedKernel_restrict e (densityScaledCharacteristicKernel C b T ρ) hΩ) | |
| rw [← dampedDensityScaledCollision_transport] at he | |
| change (∫⁻ y in Ω, f (e.symm y) ∂densityScaledCharacteristicKernel C b T ρ (e p)) = | |
| ∫⁻ x, f x ∂((dampedDensityScaledCollision C b T ρ).restrict (e.measurable hΩ)) p | |
| rw [← he,transportedKernel_lintegral _ _ p hfn] | |
| rfl | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| section | |
| /-! # Local measure balances with one fixed phase cutoff -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| namespace BoltzmannNonuniqueness.WeakTime | |
| def phaseWeight (β : Phase → ℝ) (F : ℝ × Phase → ℝ) (z : ℝ × Phase) : ℝ := β z.2*F z | |
| def weightedPhaseSource (μ : Measure (ℝ × Phase)) (β : Phase → ℝ) : Measure (ℝ × Phase) := | |
| positiveDensityMeasure μ (fun z => β z.2) | |
| theorem phaseWeight_integrable {β : Phase → ℝ} (hβ : Continuous β) | |
| (hc : HasCompactSupport β) {F : ℝ × Phase → ℝ} (hi : Integrable F) : | |
| Integrable (phaseWeight β F) := by | |
| obtain ⟨C,hC⟩ := hβ.bounded_above_of_compact_support hc | |
| exact hi.bdd_mul (hβ.measurable.comp measurable_snd).aestronglyMeasurable | |
| (ae_of_all _ (fun z => hC z.2)) | |
| theorem weightedPhaseSource_finite (μ : Measure (ℝ × Phase)) [IsFiniteMeasure μ] | |
| {β : Phase → ℝ} (hβ : Continuous β) (hc : HasCompactSupport β) (hn : ∀ p, 0 ≤ β p) : | |
| IsFiniteMeasure (weightedPhaseSource μ β) := by | |
| obtain ⟨C,hC⟩ := hβ.bounded_above_of_compact_support hc | |
| apply positiveDensityMeasure_finite | |
| · exact (integrable_const C).mono' (hβ.measurable.comp measurable_snd).aestronglyMeasurable | |
| (ae_of_all _ (fun z => hC z.2)) | |
| · exact ae_of_all _ (fun z => hn z.2) | |
| theorem ae_localized_measure_interval_balance {a b : ℝ} (hab : a < b) | |
| {F : ℝ × Phase → ℝ} (hi : Integrable F (volume : Measure (ℝ × Phase))) | |
| (hn : ∀ᵐ z ∂volume, 0 ≤ F z) | |
| (μ ν : Measure (ℝ × Phase)) [IsFiniteMeasure μ] [IsFiniteMeasure ν] | |
| {β : Phase → ℝ} (hβ : ContDiff ℝ (⊤ : ℕ∞) β) | |
| (hβc : HasCompactSupport β) (hβn : ∀ p, 0 ≤ β p) | |
| (hw : ∀ φ : Phase → ℝ, ContDiff ℝ (⊤ : ℕ∞) φ → HasCompactSupport φ → | |
| (∀ p, 0 ≤ φ p) → tsupport φ ⊆ tsupport β → | |
| ∀ θ : ℝ → ℝ, ContDiff ℝ (⊤ : ℕ∞) θ → HasCompactSupport θ → tsupport θ ⊆ Ioo a b → | |
| (∫ z, (deriv θ z.1*φ z.2)*F z)+ | |
| (∫ z, θ z.1*φ z.2 ∂μ)-(∫ z, θ z.1*φ z.2 ∂ν) = 0) : | |
| ∀ᵐ s ∂volume, ∀ᵐ t ∂volume, | |
| s ∈ Ioo a b → t ∈ Ioo a b → s ≤ t → | |
| phaseSlice (phaseWeight β F) t+phaseIncrement (weightedPhaseSource ν β) s t = | |
| phaseSlice (phaseWeight β F) s+phaseIncrement (weightedPhaseSource μ β) s t := by | |
| let _ := weightedPhaseSource_finite μ hβ.continuous hβc hβn | |
| let _ := weightedPhaseSource_finite ν hβ.continuous hβc hβn | |
| apply ae_measure_interval_balance hab (phaseWeight_integrable hβ.continuous hβc hi) | |
| (hn.mono (fun z hz => mul_nonneg (hβn z.2) hz)) | |
| intro φ hφ _hφc hφn θ hθ hθc hθs | |
| have hprod : ContDiff ℝ (⊤ : ℕ∞) (fun p => φ p*β p) := hφ.mul hβ | |
| have hp := hw (fun p => φ p*β p) hprod hβc.mul_left | |
| (fun p => mul_nonneg (hφn p) (hβn p)) tsupport_mul_subset_right θ hθ hθc hθs | |
| have hβmμ : AEMeasurable (fun z : ℝ × Phase => β z.2) μ := by | |
| simpa only [Function.comp_def] using | |
| (hβ.continuous.measurable.comp measurable_snd).aemeasurable (μ := μ) | |
| have hβmν : AEMeasurable (fun z : ℝ × Phase => β z.2) ν := by | |
| simpa only [Function.comp_def] using | |
| (hβ.continuous.measurable.comp measurable_snd).aemeasurable (μ := ν) | |
| rw [show weightedPhaseSource μ β = positiveDensityMeasure μ (fun z => β z.2) from rfl, | |
| show weightedPhaseSource ν β = positiveDensityMeasure ν (fun z => β z.2) from rfl, | |
| positiveDensityMeasure_integral hβmμ (ae_of_all _ (fun z => hβn z.2)), | |
| positiveDensityMeasure_integral hβmν (ae_of_all _ (fun z => hβn z.2))] | |
| simpa only [phaseWeight,mul_assoc] using hp | |
| end BoltzmannNonuniqueness.WeakTime | |
| end | |
| end | |
| section | |
| /-! # Finite bounded physical histories are dominated by the localized actual angular histories -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal NNReal ProbabilityTheory | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| def localizedNormalPhysicalEvent (b : BeamCoefficients) (ρ C T a η l : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) : | |
| Kernel PhysicalFlightState PhysicalFlightState := | |
| localizedNormalCollision b ρ C T a η l hΩ ∘ₖ physicalWaitingKernel (Real.toNNReal C) | |
| instance localizedNormalPhysicalEvent_sFinite (b : BeamCoefficients) (ρ C T a η l : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) : | |
| IsSFiniteKernel (localizedNormalPhysicalEvent b ρ C T a η l hΩ) := by | |
| unfold localizedNormalPhysicalEvent | |
| infer_instance | |
| theorem boundedCollision_le_localized {C : ℝ≥0} (hC : referenceClock ≤ C) | |
| (b : BeamCoefficients) (ρ : ℝ≥0) (T a η l : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) | |
| {p q : PhysicalFlightState} (hq : boundedFlightCone p q) | |
| (hΩcone : ∀ r, boundedFlightCone p r → flightCharacteristicEquiv T r ∈ Ω) | |
| (hz : flightCharacteristicEquiv T q ∈ coldCharacteristicDomain a η l) | |
| (hp : 0 ≤ b.plusDensity (characteristicBeamPoint (flightCharacteristicEquiv T q))) | |
| (hm : 0 ≤ b.minusDensity (characteristicBeamPoint (flightCharacteristicEquiv T q))) | |
| (hf : characteristicBeamFrequency b (flightCharacteristicEquiv T q) ≤ (referenceClock:ℝ)) : | |
| (dampedDensityScaledCollision C b T ρ q).restrict mediumVelocityBox ≤ | |
| localizedNormalCollision b ρ (ρ*C) T a η l hΩ q := by | |
| rw [localizedNormalCollision_eq_restrict hC b ρ T a η l hΩ q hz hp hm hf] | |
| apply Measure.restrict_mono_ae | |
| filter_upwards [dampedDensityScaledCollision_ae_fixed C b T ρ q] with r hr | |
| intro hv | |
| apply hΩcone r | |
| exact ⟨hr.1 ▸ hq.1,hr.1 ▸ hq.2.1,hv,by rw [hr.1,hr.2]; exact hq.2.2.2⟩ | |
| theorem boundedHistory_le_localized {C : ℝ≥0} (hC : referenceClock ≤ C) | |
| (b : BeamCoefficients) (ρ : ℝ≥0) (T a η l : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) | |
| (p : PhysicalFlightState) (ht : 0 ≤ p.1) (hv : ‖p.2.2‖ ≤ 6) | |
| (hΩcone : ∀ q, boundedFlightCone p q → flightCharacteristicEquiv T q ∈ Ω) | |
| (hlocal : ∀ q, boundedFlightCone p q → | |
| flightCharacteristicEquiv T q ∈ coldCharacteristicDomain a η l ∧ | |
| 0 ≤ b.plusDensity (characteristicBeamPoint (flightCharacteristicEquiv T q)) ∧ | |
| 0 ≤ b.minusDensity (characteristicBeamPoint (flightCharacteristicEquiv T q)) ∧ | |
| characteristicBeamFrequency b (flightCharacteristicEquiv T q) ≤ (referenceClock:ℝ)) | |
| (n : ℕ) : | |
| ((dampedBoundedDensityScaledEvent C b T ρ)^n) p ≤ | |
| ((localizedNormalPhysicalEvent b ρ (ρ*C) T a η l hΩ)^n) p := by | |
| apply kernel_pow_le_on_invariant _ _ {q | boundedFlightCone p q} _ _ n p | |
| (boundedFlightCone_refl ht hv) | |
| · intro q hq | |
| filter_upwards [dampedBoundedDensityScaledEvent_ae_cone C b T ρ q hq.2.2.1] with r hr | |
| exact boundedFlightCone_trans hq hr | |
| · intro q hq | |
| unfold dampedBoundedDensityScaledEvent dampedDensityScaledEventKernel localizedNormalPhysicalEvent | |
| rw [← Kernel.comp_restrict,← NNReal.coe_mul,Real.toNNReal_coe] | |
| apply KernelHistoryOrder.measure_comp_mono_ae (hμ := le_rfl) | |
| filter_upwards [physicalWaitingKernel_ae_cone (ρ*C) q hq.2.2.1] with r hr | |
| have hpr := boundedFlightCone_trans hq hr | |
| obtain ⟨hz,hp,hm,hf⟩ := hlocal r hpr | |
| exact boundedCollision_le_localized hC b ρ T a η l hΩ hpr hΩcone hz hp hm hf | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| section | |
| /-! # Constant damping for weak evolution localized in characteristic phase -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| namespace BoltzmannNonuniqueness.WeakTime | |
| theorem ae_localized_damped_measure_balance {a b C : ℝ} (hab : a < b) (hC : 0 ≤ C) | |
| {F L : ℝ × Phase → ℝ} (hi : Integrable F (volume : Measure (ℝ × Phase))) | |
| (hL : Integrable L (volume : Measure (ℝ × Phase))) (hn : ∀ᵐ z ∂volume, 0 ≤ F z) | |
| (μ : Measure (ℝ × Phase)) [IsFiniteMeasure μ] | |
| {β : Phase → ℝ} (hβ : ContDiff ℝ (⊤ : ℕ∞) β) | |
| (hβc : HasCompactSupport β) (hβn : ∀ p, 0 ≤ β p) | |
| (hbound : ∀ᵐ z ∂volume, z.1 ∈ Ioo a b → β z.2*L z ≤ C*(β z.2*F z)) | |
| (hw : ∀ φ : Phase → ℝ, ContDiff ℝ (⊤ : ℕ∞) φ → HasCompactSupport φ → | |
| (∀ p, 0 ≤ φ p) → tsupport φ ⊆ tsupport β → | |
| ∀ θ : ℝ → ℝ, ContDiff ℝ (⊤ : ℕ∞) θ → HasCompactSupport θ → tsupport θ ⊆ Ioo a b → | |
| (∫ z, (deriv θ z.1*φ z.2)*F z)+ | |
| (∫ z, θ z.1*φ z.2 ∂μ)-(∫ z, (θ z.1*φ z.2)*L z) = 0) : | |
| ∀ᵐ s ∂volume, ∀ᵐ t ∂volume, | |
| s ∈ Ioo a b → t ∈ Ioo a b → s ≤ t → | |
| phaseSlice (timeWeightedDensity a b C (phaseWeight β F)) t = | |
| phaseSlice (timeWeightedDensity a b C (phaseWeight β F)) s+ | |
| phaseIncrement (timeWeightedSource a b C (weightedPhaseSource μ β)+ | |
| positiveDensityMeasure volume | |
| (dampedResidual a b C (phaseWeight β F) (phaseWeight β L))) s t := by | |
| let _ := weightedPhaseSource_finite μ hβ.continuous hβc hβn | |
| apply ae_damped_measure_interval_balance hab hC | |
| (phaseWeight_integrable hβ.continuous hβc hi) | |
| (phaseWeight_integrable hβ.continuous hβc hL) | |
| (hn.mono (fun z hz => mul_nonneg (hβn z.2) hz)) hbound | |
| intro φ hφ _hc hφn θ hθ hθc hθs | |
| have hp := hw (fun p => φ p*β p) (hφ.mul hβ) hβc.mul_left | |
| (fun p => mul_nonneg (hφn p) (hβn p)) tsupport_mul_subset_right θ hθ hθc hθs | |
| have hβm : AEMeasurable (fun z : ℝ × Phase => β z.2) μ := by | |
| simpa only [Function.comp_def] using | |
| (hβ.continuous.measurable.comp measurable_snd).aemeasurable (μ := μ) | |
| rw [show weightedPhaseSource μ β = positiveDensityMeasure μ (fun z => β z.2) from rfl, | |
| positiveDensityMeasure_integral hβm (ae_of_all _ (fun z => hβn z.2))] | |
| simpa only [phaseWeight,mul_assoc] using hp | |
| end BoltzmannNonuniqueness.WeakTime | |
| end | |
| end | |
| end OAI | |