| import Mathlib |
| import OAI.Analysis.HardSphere.Thermodynamics.ScaledJointLowCounts |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology |
| open MeasureTheory Set |
| open scoped ENNReal |
| namespace HardSphere |
| open MeasureTheory Set Filter |
|
|
| def groundPositionLaw (N : ℕ) (L : Length) (u : Jet N L) : Measure (Configuration N L) := |
| volume.withDensity (fun X ↦ ENNReal.ofReal (‖u.1 X‖^2)) |
|
|
| namespace CellOrbital |
|
|
| lemma count_continuous {n : ℕ} (L : Length) (δ : ℝ) (c : Torus L) : |
| Continuous (count (n:=n) L δ c) := by |
| apply continuous_finsetSum |
| intro i _ |
| exact ((bump_continuous L (8*δ) c).comp (continuous_apply i)).pow 2 |
|
|
| lemma joint_one {S : Type*} [DecidableEq S] {n : ℕ} {L : Length} {h : ℝ} (hh : 0<h) |
| (c : S → Torus L) (A : Finset S) (X : Configuration n L) |
| (hX : ∀ s∈A, count L (1/100) (c s) X ≤ h) : (joint n L c h A).toFun X=1 := by |
| rw [joint_apply] |
| apply Finset.prod_eq_one |
| intro s hs |
| rw [profile_one hh (hX s hs),Complex.ofReal_one] |
|
|
| lemma groundPositionLaw_le_joint {S : Type*} [DecidableEq S] {n : ℕ} {L : Length} {h : ℝ} |
| (hh : 0<h) (c : S → Torus L) (A : Finset S) (u : Jet n L) |
| (T : Set (Configuration n L)) (hT : MeasurableSet T) |
| (hcount : ∀ X∈T, ∀ s∈A, count L (1/100) (c s) X ≤ h) : |
| groundPositionLaw n L u T ≤ ENNReal.ofReal (‖(joint n L c h A).onScalar u.1‖^2) := by |
| rw [groundPositionLaw,withDensity_apply _ hT, |
| ← ofReal_integral_eq_lintegral_ofReal |
| ((Lp.memLp u.1).integrable_norm_pow (by norm_num : (2:ℕ)≠0)).integrableOn |
| (Eventually.of_forall fun X ↦ sq_nonneg ‖u.1 X‖),IndicatorStrong.norm_sq_integral] |
| apply ENNReal.ofReal_le_ofReal |
| have hi : Integrable (fun X ↦ ‖((joint n L c h A).onScalar u.1) X‖^2) volume := |
| (Lp.memLp _).integrable_norm_pow (by norm_num) |
| calc |
| _ = ∫ X in T, ‖((joint n L c h A).onScalar u.1) X‖^2 ∂volume := by |
| apply setIntegral_congr_ae hT |
| filter_upwards [(joint n L c h A).coe_onScalar u.1] with X hX hXT |
| rw [hX,joint_one hh c A X (hcount X hXT),one_mul] |
| _ ≤ _ := setIntegral_le_integral hi (ae_of_all _ fun X ↦ sq_nonneg _) |
|
|
| theorem scaled_low_count_probability : ∃ α₀ ell D₀ Q : ℝ, |
| 0<α₀ ∧ 0<ell ∧ 0<D₀ ∧ 0<Q ∧ |
| ∀ (S : Type) [DecidableEq S] (n : ℕ) (L : Length) (D α : ℝ), |
| 2≤n → 1 ≤ L.val → D=(n:ℝ)/L.val^3 → D₀ ≤ D → |
| α₀/2 ≤ α → α ≤ 2*α₀ → |
| ∀ (g : Jet n L), IsGroundVector n L (α/D) g → |
| ∀ (c : S → Torus L), (∀ z v, z≠v → 1 ≤ dist (c z) (c v)) → |
| ∀ A : Finset S, |
| groundPositionLaw n L g {X | ∀ s∈A, count L (1/100) (c s) X < ell*D} ≤ |
| ENNReal.ofReal ((Q/D^2)^A.card) := by |
| obtain ⟨α₀,ell,D₀,Q,hα,hell,hD₀,hQ,hs⟩ := scaled_joint_low_counts |
| refine ⟨α₀,ell,D₀,Q,hα,hell,hD₀,hQ,?_⟩ |
| intro S _ n L D α hn hL hDN hDD hαl hαu g hg c hc A |
| have hh : 0<ell*D := mul_pos hell (hD₀.trans_le hDD) |
| apply (groundPositionLaw_le_joint hh c A g _ ?_ (fun X hX s hs ↦ (hX s hs).le)).trans |
| · exact ENNReal.ofReal_le_ofReal (hs S n L D α hn hL hDN hDD hαl hαu g hg c hc A) |
| · simpa only [Set.ofPred_forall] using Finset.measurableSet_biInter A (fun s _ ↦ measurableSet_lt |
| (count_continuous (n:=n) L (1/100) (c s)).measurable |
| (show Measurable (fun _ : Configuration n L ↦ ell*D) from measurable_const)) |
|
|
| end CellOrbital |
| end HardSphere |
|
|
| end |
|
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| end OAI |
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|
