| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalObservedMass |
| import OAI.Analysis.HardSphere.Comparison.SumMoment |
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| namespace OAI |
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| section |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital HardSphere.Lattice |
| attribute [local instance] Classical.propDecidable |
| variable {K : ℕ} [NeZero K] {x y : DiscreteTorus K} |
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| def physicalGroupReference {N M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (ψ : Jet (N+1) L) (g : Fin G) (m : ℕ) (t r R C S s₀ : ℝ) (u : Time) (b : Bool) := |
| ∑flags,∑a : GridAllocation (N+1) K (G*M), |
| physicalObservedReference (x:=x) (y:=y) hL g m t r R C S u (branchParticles g a) b |
| (physicalBranchLaw hL (endpointWeightedBase ψ (2*t)) g flags a m r R C S s₀ (2*t) u) |
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| theorem physical_group_common_comparison {N M G : ℕ} {L : Length} |
| (hL : L.val=(K:ℝ)) (hK : 1 ≤ K) (h50 : 50<L.val) |
| {ψ : Jet (N+1) L} {r : ℝ} (hψ : IsGroundVector (N+1) L r ψ) |
| (g : Fin G) (m : ℕ) (t R C S s₀ h : ℝ) (u : Time) (hs : PhysicalScaleBounds m t r R C S s₀ h u) |
| (hxy : x≠y) (E : Set (PhysicalDatum (N+1) K M G L)) (hE : MeasurableSet E) |
| (ν : Kernel (PhysicalDatum (N+1) K M G L) (SimplePath x y)) [IsMarkovKernel ν] |
| (hgood : ∀d∈E,ν d {p | simplePathGood (latticeDatumBad hL g m t r R C S u d) p}=1) |
| (hM : 0<M) (δ B κ Cpath : ℝ) (hδ : 0<δ) (hB : 1 ≤ B) (hCpath : 0 ≤ Cpath) |
| (hBt : routeDensityBound t ≤ ENNReal.ofReal B) |
| (hκ : 2*(B-1)/(M:ℝ)-8*Real.log δ ≤ κ) |
| (hsmall : ENNReal.ofReal δ+(4*(49^3*(M+1)):ℕ)*(routeDensityBound t*datumCollisionBound t r h R C) ≤ 1) |
| (henc : (∫⁻d in E,∫⁻p,∫⁻p',ENNReal.ofReal (Real.exp (κ*(simpleEncounter 30 p p':ℝ))) ∂ν d ∂ν d |
| ∂physicalDatumLaw g ψ m r R C S s₀ (2*t) u) ≤ ENNReal.ofReal Cpath) : |
| ∃σ : Measure (Torus L×Torus L×Configuration N L), IsFiniteMeasure σ ∧ |
| σ univ=(physicalDatumLaw g ψ m r R C S s₀ (2*t) u) E ∧ |
| ∀b, |
| let P := physicalGroupReference (M:=M) (x:=x) (y:=y) hL ψ g m t r R C S s₀ u b |
| IsFiniteMeasure P ∧ σ≪P ∧ Integrable (fun z ↦ (σ.rnDeriv P z).toReal^2) P ∧ |
| (∫z,(σ.rnDeriv P z).toReal^2 ∂P) ≤ B*Cpath := by |
| classical |
| let I := (Fin (N+1)→Bool)×GridAllocation (N+1) K (G*M) |
| let μ : I→Measure (PhysicalDatum (N+1) K M G L) := fun i ↦ |
| physicalBranchLaw hL (endpointWeightedBase ψ (2*t)) g i.1 i.2 m r R C S s₀ (2*t) u |
| have hT : 0 ≤ 2*t := by linarith [hs.time_pos] |
| let (i : I) : IsFiniteMeasure (μ i) := |
| physicalBranchLaw_finite hL hK hψ g i.1 i.2 m R C S s₀ (2*t) hT u |
| let σ : I→Measure (Torus L×Torus L×Configuration N L) := fun i ↦ |
| physicalCommonObserved hL g m t r R C S u (branchParticles g i.2) hxy ν (μ i) E |
| let P : Bool→I→Measure (Torus L×Torus L×Configuration N L) := fun b i ↦ |
| physicalObservedReference (x:=x) (y:=y) hL g m t r R C S u (branchParticles g i.2) b (μ i) |
| have hsum : (∑i,μ i)=physicalDatumLaw g ψ m r R C S s₀ (2*t) u := by |
| rw [Fintype.sum_prod_type] |
| exact physicalBranchLaw_sum hL hK ψ g m r R C S s₀ (2*t) u |
| let F := E.indicator (fun d ↦ ∫⁻p,∫⁻p',ENNReal.ofReal (Real.exp (κ*(simpleEncounter 30 p p':ℝ))) ∂ν d ∂ν d) |
| let e : I→ℝ≥0∞ := fun i ↦ ∫⁻d,F d ∂μ i |
| have hesum : ∑i,e i ≤ ENNReal.ofReal Cpath := by |
| rw [←lintegral_finsetSum_measure,hsum] |
| simpa only [F,lintegral_indicator hE] using henc |
| have hei (i : I) : e i≠⊤ := by |
| apply ne_top_of_le_ne_top ENNReal.ofReal_ne_top |
| exact (Finset.single_le_sum (fun _ _ ↦ bot_le) (Finset.mem_univ i)).trans hesum |
| have hbranch (i : I) : |
| σ i univ=μ i E ∧ ∀b,(σ i≪P b i ∧ |
| Integrable (fun z ↦ ((σ i).rnDeriv (P b i) z).toReal^2) (P b i) ∧ |
| (∫z,((σ i).rnDeriv (P b i) z).toReal^2 ∂P b i) ≤ B*(e i).toReal) := by |
| apply physical_common_observed_estimates hL g m t r R C S hs.time_pos hs.radius_le u |
| (branchParticles g i.2) hxy ν (μ i) E hE hM (datumCollisionBound t r h R C) |
| δ B κ (e i).toReal hδ hB ENNReal.toReal_nonneg hBt hκ hsmall |
| · exact physical_branch_moment_data hL hK h50 (endpointWeightedBase ψ (2*t)) g i.1 i.2 |
| m t r R C S s₀ h u hs E hE ν hgood |
| · exact ae_restrict_of_ae (physicalBranchLaw_primary hL hK h50 (endpointWeightedBase ψ (2*t)) |
| m r R C S s₀ t u g i.1 i.2 (siteIndex y)) |
| · rw [ENNReal.ofReal_toReal (hei i)] |
| exact le_of_eq (lintegral_indicator hE _).symm |
| let (i : I) : IsFiniteMeasure (σ i) := by dsimp only [σ]; infer_instance |
| let (b : Bool) (i : I) : IsFiniteMeasure (P b i) := by dsimp only [P]; infer_instance |
| refine ⟨∑i,σ i,inferInstance,?_,?_⟩ |
| · rw [Measure.finsetSum_apply] |
| simp_rw [fun i ↦ (hbranch i).1] |
| rw [←Measure.finsetSum_apply,hsum] |
| · intro b |
| have hp : physicalGroupReference (M:=M) (x:=x) (y:=y) hL ψ g m t r R C S s₀ u b=∑i,P b i := by |
| rw [Fintype.sum_prod_type]; rfl |
| dsimp only |
| rw [hp] |
| refine ⟨inferInstance,?_,?_⟩ |
| · have H : Measure.sum σ≪∑i,P b i := Measure.absolutelyContinuous_sum_left (fun i ↦ |
| ((hbranch i).2 b).1.trans (Measure.absolutelyContinuous_of_le |
| (Finset.single_le_sum (fun _ _ ↦ bot_le) (Finset.mem_univ i)))) |
| simpa only [Measure.sum_fintype] using H |
| · have H := finite_sum_second_moment (P b) σ (fun i ↦ ((hbranch i).2 b).1) |
| (fun i ↦ ((hbranch i).2 b).2.1) |
| refine ⟨H.1,H.2.trans ?_⟩ |
| calc |
| _ ≤ ∑i,B*(e i).toReal := Finset.sum_le_sum (fun i _ ↦ ((hbranch i).2 b).2.2) |
| _ = B*(∑i,e i).toReal := by rw [←Finset.mul_sum,ENNReal.toReal_sum (fun i _ ↦ hei i)] |
| _ ≤ B*Cpath := mul_le_mul_of_nonneg_left |
| (by simpa only [ENNReal.toReal_ofReal hCpath] using ENNReal.toReal_mono ENNReal.ofReal_ne_top hesum) |
| (by linarith) |
|
|
| end HardSphere.Groups |
| end |
| end |
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| end OAI |
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