| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.ParticleGradSq |
|
|
| 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.SphericalCoordinates |
| open scoped BigOperators |
|
|
| noncomputable def activeLabels {α ι : Type*} [Fintype α] (σ : α → ι) : Finset α := by |
| classical |
| exact Finset.univ.filter (fun i ↦ ∃ j, j≠i ∧ σ j=σ i) |
|
|
| lemma mem_activeLabels {α ι : Type*} [Fintype α] (σ : α → ι) (i : α) : |
| i∈activeLabels σ ↔ ∃ j, j≠i ∧ σ j=σ i := by |
| classical |
| simp only [activeLabels,Finset.mem_filter,Finset.mem_univ,true_and] |
|
|
| lemma inactive_card_le {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq α] (σ : α → ι) : |
| (activeLabels σ)ᶜ.card≤Fintype.card ι := by |
| classical |
| apply Finset.card_le_card_of_injOn σ (by simp) |
| intro a ha b _ hab |
| by_contra hne |
| have ha' : a∉activeLabels σ := Finset.mem_compl.mp ha |
| apply ha' |
| simp only [activeLabels,Finset.mem_filter,Finset.mem_univ,true_and] |
| exact ⟨b,(fun h ↦ hne h.symm),hab.symm⟩ |
|
|
| lemma active_card_lower {α ι : Type*} [Fintype α] [Fintype ι] (σ : α → ι) : |
| Fintype.card α-Fintype.card ι≤(activeLabels σ).card := by |
| classical |
| have h := inactive_card_le σ |
| rw [Finset.card_compl] at h |
| have hc := (activeLabels σ).card_le_univ |
| omega |
|
|
| lemma sector_capacity {n : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {F : (Fin (n+1) → Point) → E} (hF : ContDiff ℝ 1 F) |
| {r l : ℝ} (hr : 0<r) (hl : 0≤l) {ι : Type*} [Fintype ι] |
| (b : ι → Point) (σ : Fin (n+1) → ι) |
| (hz : ∀X a c, a≠c → sqRadius (X a-X c)≤r^2 → F X=0) : |
| ((n+1-Fintype.card ι : ℕ) : ℝ≥0∞)* |
| (∫⁻ X, ENNReal.ofReal (‖F X‖^2) ∂Measure.pi (fun a ↦ volume.restrict (cube (b (σ a)) l))) ≤ |
| ENNReal.ofReal (Real.sqrt 3*l^3/r)* |
| ∑ i, (∫⁻ X, ENNReal.ofReal (particleGradSq F i X) |
| ∂Measure.pi (fun a ↦ volume.restrict (cube (b (σ a)) l))) := by |
| classical |
| let μ := Measure.pi (fun a ↦ volume.restrict (cube (b (σ a)) l)) |
| let M := ∫⁻ X, ENNReal.ofReal (‖F X‖^2) ∂μ |
| let G := fun i ↦ ∫⁻ X, ENNReal.ofReal (particleGradSq F i X) ∂μ |
| let C := ENNReal.ofReal (Real.sqrt 3*l^3/r) |
| have hi (i : Fin (n+1)) (hi : i∈activeLabels σ) : M≤C*G i := by |
| obtain ⟨j,hji,hs⟩ := (mem_activeLabels σ i).mp hi |
| obtain ⟨k,rfl⟩ := Fin.exists_succAbove_eq hji |
| exact sector_coordinate_capacity hF hr hl (fun a ↦ b (σ a)) i k (congrArg b hs) hz |
| have hs := Finset.sum_le_sum hi |
| have hcard := active_card_lower σ |
| rw [Fintype.card_fin] at hcard |
| calc |
| _ ≤ ((activeLabels σ).card : ℝ≥0∞)*M := mul_le_mul_of_nonneg_right (by exact_mod_cast hcard) (by positivity) |
| _ = ∑ _ ∈ activeLabels σ, M := by simp |
| _ ≤ ∑ i ∈ activeLabels σ, C*G i := hs |
| _ = C*∑ i ∈ activeLabels σ, G i := (Finset.mul_sum _ _ _).symm |
| _ ≤ C*∑ i, G i := by gcongr; exact Finset.subset_univ _ |
| _ = _ := rfl |
|
|
| end HardSphere.SphericalCoordinates |
|
|
| end |
|
|
| end OAI |
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|
