| import Mathlib |
| import OAI.Analysis.HardSphere.PathBounds.CloseTailBaseTendsto |
| import OAI.Analysis.HardSphere.Conditioning.PairCloseLabels |
|
|
| 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 |
| open scoped Topology |
|
|
| lemma density_forces_two_particles {n : ℕ} (L : Length) {D : ℝ} |
| (hL : 100≤L.val) (hDeq : D=(n:ℝ)/L.val^3) (hD : 1≤D) : 2≤n := by |
| have hLp : 0<L.val^3 := pow_pos L.property _ |
| have hn : L.val^3≤(n:ℝ) := by |
| have hh : 1≤(n:ℝ)/L.val^3 := by rwa [←hDeq] |
| simpa only [one_mul] using (le_div_iff₀ hLp).mp hh |
| have hlarge : (100:ℝ)^3≤L.val^3 := by gcongr |
| have hn2 : (2:ℝ)≤n := by linarith only [hn,hlarge] |
| exact_mod_cast hn2 |
|
|
| theorem spatial_full_and_bath_close_tail : ∃ Q : ℕ, 0<Q ∧ ∀ θ : ℝ, 0<θ → |
| ∃ b Gθ : ℝ, 0<b ∧ b≤1/4 ∧ 0<Gθ ∧ |
| Tendsto (fun D : ℝ ↦ ((Gθ/b^2)*D^(-(4/3):ℝ))^(1/(Q:ℝ))) atTop (𝓝 0) ∧ |
| ∀ αmax : ℝ, ∃ D₀ : ℝ, 1≤D₀ ∧ |
| ∀ (n K : ℕ) (L : Length) (_hL : L.val=(K:ℝ)) (D α : ℝ), |
| 100≤L.val → D=(n:ℝ)/L.val^3 → D₀≤D → 1≤α → α≤αmax → |
| ∀ (g : Jet n L), IsGroundVector n L (α/D) g → ∀ A : Finset (CubeCell K), |
| (groundPositionLaw n L g {X | ∀ v∈A, θ*D≤((pairCloseLabels L (spatialCloseScale D b) (unitCenter L v) X).card:ℝ)}≤ |
| ENNReal.ofReal (((((Gθ/b^2)*D^(-(4/3):ℝ))^(1/(Q:ℝ)))^A.card))) ∧ |
| ∀ (m : ℕ) (e : Fin m ↪ Fin n), |
| ((groundPositionLaw n L g).map (fun X i ↦ X (e i))) |
| {Y | ∀ v∈A, θ*D≤((pairCloseLabels L (spatialCloseScale D b) (unitCenter L v) Y).card:ℝ)}≤ |
| ENNReal.ofReal (((((Gθ/b^2)*D^(-(4/3):ℝ))^(1/(Q:ℝ)))^A.card)) := by |
| obtain ⟨Q,hQ,htail⟩ := spatial_close_tail |
| refine ⟨Q,hQ,?_⟩ |
| intro θ hθ |
| obtain ⟨b,Gθ,hb,hb1,hGθ,hlim,hfin⟩ := htail θ hθ |
| refine ⟨b,Gθ,hb,hb1,hGθ,hlim,?_⟩ |
| intro αmax |
| obtain ⟨D₀,hD₀,hDfin⟩ := hfin αmax |
| refine ⟨D₀,hD₀,?_⟩ |
| intro n K L hL D α hL100 hDeq hD hα hαmax g hg A |
| let : Nontrivial (Fin n) := Fin.nontrivial_iff_two_le.mpr |
| (density_forces_two_particles L hL100 hDeq (hD₀.trans hD)) |
| have hfull := hDfin n K L hL D α hL100 hDeq hD hα hαmax g hg A |
| constructor |
| · simpa only [localCloseLabels_eq_pairCloseLabels] using hfull |
| · intro m e |
| exact (closeBath_marginal_le L (spatialCloseScale D b) (θ*D) e |
| (groundPositionLaw n L g) (unitCenter L) A).trans hfull |
|
|
| end HardSphere |
|
|
| end |
|
|
| end OAI |
|
|
