File size: 3,804 Bytes
67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | 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
end OAI
|