| import Mathlib |
| import OAI.Analysis.HardSphere.Orbitals.Chi |
| import OAI.Analysis.HardSphere.Geometry.DiscreteTorus |
| import OAI.Analysis.HardSphere.Geometry.BadComponent |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators Classical |
| open scoped BigOperators Classical |
| namespace HardSphere.BathSlice |
| open HardSphere.Lattice |
|
|
| lemma coordinate_dist_lift_le (L : Length) (u v : ℝ) : |
| dist (u : AddCircle L.val) (v : AddCircle L.val) ≤ |u - v| := by |
| rw [dist_eq_norm, ← AddCircle.coe_sub] |
| exact QuotientAddGroup.norm_mk_le_norm |
|
|
| lemma volume_microCube_le (L : Length) {h : ℝ} (hh : 0 ≤ h) (z : Vertex) : |
| (volume : Measure (Torus L)) (microCube L h z) ≤ ENNReal.ofReal (h^3) := by |
| let c : Point := fun j ↦ h * (z j : ℝ) + h / 2 - 1/2 |
| have hsub : microCube L h z ⊆ |
| Set.univ.pi (fun j ↦ Metric.closedBall ((c j : ℝ) : AddCircle L.val) (h/2)) := by |
| rintro x ⟨y, hy, rfl⟩ j _ |
| apply (coordinate_dist_lift_le L (y j) (c j)).trans |
| apply abs_le.mpr |
| have hl := hy.1 j |
| have hu := hy.2 j |
| constructor <;> dsimp [c] at * <;> linarith |
| calc |
| _ ≤ volume (Set.univ.pi (fun j ↦ |
| Metric.closedBall ((c j : ℝ) : AddCircle L.val) (h/2))) := measure_mono hsub |
| _ = ∏ j : Fin 3, ENNReal.ofReal (min L.val (2 * (h/2))) := by |
| rw [volume_pi_pi] |
| simp only [AddCircle.volume_closedBall] |
| _ ≤ ∏ _j : Fin 3, ENNReal.ofReal h := by |
| apply Finset.prod_le_prod |
| intro j _ |
| apply ENNReal.ofReal_le_ofReal |
| simpa only [mul_div_cancel₀ _ (by norm_num : (2 : ℝ) ≠ 0)] using |
| (min_le_right L.val (2 * (h / 2))) |
| _ = ENNReal.ofReal (h^3) := by simp [ENNReal.ofReal_pow hh] |
|
|
| def filledRegion (L : Length) (h : ℝ) (C : Finset Vertex) : Set (Torus L) := |
| ⋃ z ∈ fill C, microCube L h z |
|
|
| lemma volume_finite_microCubes_le (L : Length) {h : ℝ} (hh : 0 ≤ h) |
| (S : Finset Vertex) : |
| (volume : Measure (Torus L)) (⋃ z ∈ S, microCube L h z) ≤ |
| ENNReal.ofReal ((S.card : ℝ) * h^3) := by |
| calc |
| _ ≤ ∑ z ∈ S, volume (microCube L h z) := measure_biUnion_finset_le S _ |
| _ ≤ ∑ _z ∈ S, ENNReal.ofReal (h^3) := |
| Finset.sum_le_sum fun z _ ↦ volume_microCube_le L hh z |
| _ = _ := by simp [ENNReal.ofReal_mul, nsmul_eq_mul] |
|
|
| lemma volume_filledRegion_le_columns (L : Length) {h : ℝ} (hh : 0 ≤ h) |
| (C : Finset Vertex) (i : Fin 3) (lo hi : ℤ) |
| (hb : ∀ c ∈ C, lo ≤ c i ∧ c i ≤ hi) : |
| (volume : Measure (Torus L)) (filledRegion L h C) ≤ |
| ENNReal.ofReal ((C.card : ℝ) * ((hi - lo + 1).toNat : ℝ) * h^3) := by |
| have hsub : filledRegion L h C ⊆ |
| ⋃ z ∈ columnCover C i lo hi, microCube L h z := by |
| intro x hx |
| obtain ⟨z, hz, hx⟩ := Set.mem_iUnion₂.mp hx |
| exact Set.mem_iUnion₂.mpr ⟨z, fill_subset_columnCover C i lo hi hb hz, hx⟩ |
| calc |
| _ ≤ volume (⋃ z ∈ columnCover C i lo hi, microCube L h z) := measure_mono hsub |
| _ ≤ ENNReal.ofReal (((columnCover C i lo hi).card : ℝ) * h^3) := |
| volume_finite_microCubes_le L hh _ |
| _ ≤ _ := by |
| apply ENNReal.ofReal_le_ofReal |
| apply mul_le_mul_of_nonneg_right _ (pow_nonneg hh 3) |
| exact_mod_cast card_columnCover_le C i lo hi |
|
|
| theorem volume_small_filledRegion (L : Length) {h : ℝ} (hh : 0 ≤ h) (hh1 : h ≤ 1) |
| (C : Finset Vertex) (i : Fin 3) (lo hi : ℤ) (hlh : lo ≤ hi) |
| (hb : ∀ c ∈ C, lo ≤ c i ∧ c i ≤ hi) |
| (hspan : h * ((hi : ℝ) - lo) ≤ 1) : |
| (volume : Measure (Torus L)) (filledRegion L h C) ≤ |
| ENNReal.ofReal (3 * h^2 * C.card) := by |
| apply (volume_filledRegion_le_columns L hh C i lo hi hb).trans |
| apply ENNReal.ofReal_le_ofReal |
| have hw : ((hi - lo + 1).toNat : ℝ) = (hi : ℝ) - lo + 1 := by |
| have ht : ((hi - lo + 1).toNat : ℤ) = hi - lo + 1 := Int.toNat_of_nonneg (by omega) |
| exact_mod_cast ht |
| rw [hw] |
| have hs : h * ((hi : ℝ) - lo + 1) ≤ 3 := by nlinarith |
| have hm := mul_le_mul_of_nonneg_right hs (sq_nonneg h) |
| have hc := mul_le_mul_of_nonneg_right hm (show (0 : ℝ) ≤ C.card by positivity) |
| nlinarith |
|
|
| def witnessBox (h : ℝ) (c : Point) : Finset Vertex := |
| Finset.Icc (fun j ↦ ⌊(c j + 1/2) / h⌋ - 3) |
| (fun j ↦ ⌊(c j + 1/2) / h⌋ + 3) |
|
|
| lemma card_witnessBox (h : ℝ) (c : Point) : (witnessBox h c).card = 343 := by |
| unfold witnessBox |
| rw [Pi.card_Icc] |
| have hc (j : Fin 3) : |
| (Finset.Icc (⌊(c j + 1/2) / h⌋ - 3) (⌊(c j + 1/2) / h⌋ + 3)).card = 7 := by |
| rw [Int.card_Icc] |
| omega |
| simp only [hc, Finset.prod_const, Finset.card_univ, Fintype.card_fin] |
| norm_num |
|
|
| lemma interval_ball_witness {h r c u : ℝ} {z : ℤ} |
| (hh : 0 < h) (hr : r ≤ 2*h) |
| (hul : h * z - 1/2 ≤ u) (huu : u ≤ h * ((z : ℝ)+1) - 1/2) |
| (hud : |u - c| ≤ r) : |
| ⌊(c + 1/2) / h⌋ - 3 ≤ z ∧ z ≤ ⌊(c + 1/2) / h⌋ + 3 := by |
| have hl := (le_div_iff₀ hh).mp (Int.floor_le ((c + 1/2) / h)) |
| have hu := (div_lt_iff₀ hh).mp (Int.lt_floor_add_one ((c + 1/2) / h)) |
| obtain ⟨hd₁, hd₂⟩ := abs_le.mp hud |
| constructor |
| · have he : ((⌊(c + 1/2) / h⌋ : ℤ) : ℝ) - 3 ≤ z := by nlinarith |
| exact_mod_cast he |
| · have he : (z : ℝ) ≤ (⌊(c + 1/2) / h⌋ : ℝ) + 3 := by nlinarith |
| exact_mod_cast he |
|
|
| lemma ball_microCube_witness {L : Length} {J : ℕ} {h r : ℝ} (hh : 0 < h) |
| (hperiod : L.val = h * J) (hr : r ≤ 2*h) |
| (c : Point) {z : Vertex} {x : Torus L} |
| (hx : x ∈ microCube L h z) (hd : torusDistance L x (liftPoint L c) ≤ r) : |
| ∃ w ∈ witnessBox h c, torusProjection J w = torusProjection J z := by |
| obtain ⟨u, hu, rfl⟩ := hx |
| let k : Vertex := fun j ↦ round (L.val⁻¹ * (u j - c j)) |
| let w : Vertex := fun j ↦ z j - k j * (J : ℤ) |
| refine ⟨w, ?_, ?_⟩ |
| · apply Finset.mem_Icc.mpr |
| have hb (j : Fin 3) : |
| ⌊(c j + 1/2) / h⌋ - 3 ≤ w j ∧ w j ≤ ⌊(c j + 1/2) / h⌋ + 3 := by |
| have hdj := (CellOrbital.dist_coord_le_torusDistance L |
| (liftPoint L u) (liftPoint L c) j).trans hd |
| change dist (u j : AddCircle L.val) (c j : AddCircle L.val) ≤ r at hdj |
| rw [dist_eq_norm, ← AddCircle.coe_sub, AddCircle.norm_eq] at hdj |
| have hk : |(u j - (k j : ℝ) * L.val) - c j| ≤ r := by |
| convert hdj using 1 |
| dsimp [k] |
| ring_nf |
| apply interval_ball_witness hh hr (u := u j - (k j : ℝ) * L.val) |
| · have hl := hu.1 j |
| change h * (z j : ℝ) - 1/2 ≤ u j at hl |
| dsimp [w] |
| push_cast |
| rw [hperiod] |
| nlinarith |
| · have hl := hu.2 j |
| change u j ≤ h * ((z j : ℝ)+1) - 1/2 at hl |
| dsimp [w] |
| push_cast |
| rw [hperiod] |
| nlinarith |
| · exact hk |
| exact ⟨fun j ↦ (hb j).1, fun j ↦ (hb j).2⟩ |
| · ext j |
| simp [torusProjection, w] |
|
|
| theorem ncard_ball_bad_sites_le {L : Length} {J : ℕ} {h r : ℝ} (hh : 0 < h) |
| (hperiod : L.val = h * J) (hr : r ≤ 2*h) (y : Torus L) : |
| (torusProjection J '' {z | ∃ x ∈ microCube L h z, |
| torusDistance L x y ≤ r}).ncard ≤ 343 := by |
| let c : Point := fun j ↦ (AddCircle.equivIco L.val 0 (y j)).val |
| have hc : liftPoint L c = y := by ext j; exact AddCircle.coe_equivIco |
| have hsub : torusProjection J '' {z | ∃ x ∈ microCube L h z, |
| torusDistance L x y ≤ r} ⊆ ↑((witnessBox h c).image (torusProjection J)) := by |
| rintro v ⟨z, ⟨x, hx, hd⟩, rfl⟩ |
| obtain ⟨w, hw, he⟩ := ball_microCube_witness hh hperiod hr c hx (by simpa [hc] using hd) |
| exact Finset.mem_image.mpr ⟨w, hw, he⟩ |
| calc |
| _ ≤ (↑((witnessBox h c).image (torusProjection J)) : Set (DiscreteTorus J)).ncard := |
| Set.ncard_le_ncard hsub (Finset.finite_toSet _) |
| _ = ((witnessBox h c).image (torusProjection J)).card := Set.ncard_coe_finset _ |
| _ ≤ (witnessBox h c).card := Finset.card_image_le |
| _ = 343 := card_witnessBox h c |
|
|
| end HardSphere.BathSlice |
|
|
| end |
|
|
| end OAI |
|
|
