| import Mathlib |
| import OAI.Analysis.CoulombIonization.Localization.PacketInsert |
| import OAI.Analysis.CoulombIonization.Variational.SpatialRotation |
| import OAI.Analysis.CoulombIonization.Variational.TruncatedCoulomb |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CoulombAtom |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open Set Filter Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter |
| open scoped Topology |
| open MeasureTheory Filter |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators InnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators Pointwise |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open scoped Convolution |
| open scoped RealInnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open Filter |
| open scoped Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff BigOperators |
| open MeasureTheory Filter Set |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open scoped BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter Set Metric TopologicalSpace |
| open scoped Topology BigOperators |
| open MeasureTheory Filter Set Metric TopologicalSpace |
| open scoped Topology BigOperators |
| open MeasureTheory Filter Set Metric TopologicalSpace |
| open scoped Topology BigOperators |
| open MeasureTheory Filter Set Metric TopologicalSpace |
| open scoped Topology BigOperators |
| open MeasureTheory Filter Set Metric |
| open scoped Topology BigOperators |
| open MeasureTheory Filter Set Metric |
| open scoped Topology BigOperators |
|
|
| def packetCloud (p : ℕ → Space) (r : ℝ) (k : ℕ) (x : Space) : ℝ := |
| ∑ i ∈ Finset.range k, packetDensity (p i) r x |
|
|
| lemma packetCloud_nonneg (p : ℕ → Space) (r : ℝ) (k : ℕ) (x : Space) : |
| 0 ≤ packetCloud p r k x := |
| Finset.sum_nonneg (fun i _ => packetDensity_nonneg (p i) r x) |
|
|
| lemma packetCloud_integrable (p : ℕ → Space) {r : ℝ} (hr : 0 < r) (k : ℕ) : |
| Integrable (packetCloud p r k) := by |
| exact integrable_finsetSum _ (fun i _ => packetDensity_integrable (p i) hr) |
|
|
| lemma packetCloud_memLp (p : ℕ → Space) {r : ℝ} (hr : 0 < r) (k : ℕ) : |
| MemLp (packetCloud p r k) (5/3) := by |
| exact memLp_finsetSum _ (fun i _ => packetDensity_memLp (p i) hr) |
|
|
| lemma packetCloud_exists {p : ℕ → Space} {r : ℝ} {k : ℕ} {x : Space} |
| (hx : packetCloud p r k x ≠ 0) : ∃ i < k, packetDensity (p i) r x ≠ 0 := by |
| by_contra hh |
| push Not at hh |
| exact hx (Finset.sum_eq_zero (fun i hi => hh i (Finset.mem_range.mp hi))) |
|
|
| lemma packetCloud_bound (p : ℕ → Space) {r : ℝ} (hr : 0 < r) (k : ℕ) |
| (hsep : ∀ i j, i < j → j < k → 3*r ≤ ‖p i-p j‖) (x : Space) : |
| packetCloud p r k x ≤ packetDensityConstant/r^3 := by |
| by_cases hx : packetCloud p r k x = 0 |
| · rw [hx]; positivity [packetDensityConstant_pos] |
| obtain ⟨i,hi,hix⟩ := packetCloud_exists hx |
| have unique (j : ℕ) (hj : j ∈ Finset.range k) (hji : j ≠ i) : packetDensity (p j) r x = 0 := by |
| by_contra hjx |
| have h1 := packetDensity_support (p i) hr hix |
| have h2 := packetDensity_support (p j) hr hjx |
| have h3 : 3*r ≤ ‖p i-p j‖ := by |
| rcases lt_or_gt_of_ne hji with hji | hij |
| · rw [norm_sub_rev]; exact hsep j i hji hi |
| · exact hsep i j hij (Finset.mem_range.mp hj) |
| have ht := norm_sub_le_norm_sub_add_norm_sub (p i) x (p j) |
| rw [norm_sub_rev (p i) x] at ht |
| linarith |
| unfold packetCloud |
| rw [Finset.sum_eq_single_of_mem i (Finset.mem_range.mpr hi) unique] |
| exact packetDensity_bound (p i) x hr |
|
|
| lemma packetCloud_support (p : ℕ → Space) {r : ℝ} (hr : 0 < r) (k : ℕ) |
| {a : ℝ} (hp : ∀ i < k, ‖p i‖ ≤ a) {x : Space} (hx : packetCloud p r k x ≠ 0) : |
| ‖x‖ ≤ a+r := by |
| obtain ⟨i,hi,hix⟩ := packetCloud_exists hx |
| have h1 := packetDensity_support (p i) hr hix |
| have ht := norm_add_le (x-p i) (p i) |
| rw [sub_add_cancel] at ht |
| have h2 := hp i hi |
| linarith |
|
|
| lemma packetCloud_potential_bound (p : ℕ → Space) {r a : ℝ} (hr : 0 < r) (ha : 0 ≤ a) (k : ℕ) |
| (hsep : ∀ i j, i < j → j < k → 3*r ≤ ‖p i-p j‖) |
| (hp : ∀ i < k, ‖p i‖ ≤ a) (z : Space) (hz : ‖z‖ ≤ a) : |
| (∫ x : Space, packetCloud p r k x/‖z-x‖) ≤ |
| packetDensityConstant/r^3 * (2*a+2*r)^2 * unitCoulombIntegral := by |
| have hL : 0 < 2*a+2*r := by linarith |
| have hi := CoulombAnalysis.tfPotential_integrable (packetCloud_integrable p hr k) |
| (packetCloud_memLp p hr k) z |
| have hk : Integrable (fun x : Space => truncatedCoulomb (2*a+2*r) (z-x)) := |
| (integrable_comp_sub_left _ z).mpr (truncatedCoulomb_integrable _) |
| calc |
| _ ≤ ∫ x : Space, (packetDensityConstant/r^3)*truncatedCoulomb (2*a+2*r) (z-x) := by |
| apply integral_mono hi (hk.const_mul _) |
| intro x |
| change packetCloud p r k x / ‖z-x‖ ≤ |
| (packetDensityConstant/r^3)*truncatedCoulomb (2*a+2*r) (z-x) |
| by_cases hx : packetCloud p r k x = 0 |
| · rw [hx,zero_div] |
| exact mul_nonneg (div_nonneg packetDensityConstant_pos.le (pow_nonneg hr.le _)) |
| (truncatedCoulomb_nonneg _ _) |
| · have hxs := packetCloud_support p hr k hp hx |
| have ht := norm_sub_le z x |
| have hmem : z-x ∈ ball (0 : Space) (2*a+2*r) := by |
| rw [mem_ball_zero_iff]; linarith |
| rw [truncatedCoulomb,indicator_of_mem hmem,mul_one_div] |
| exact div_le_div_of_nonneg_right (packetCloud_bound p hr k hsep x) (norm_nonneg _) |
| _ = _ := by |
| rw [integral_const_mul,integral_sub_left_eq_self,truncatedCoulomb_integral hL] |
| ring |
|
|
| theorem packetPairEnergy_bound (p : ℕ → Space) {r a : ℝ} (hr : 0 < r) (ha : 0 ≤ a) (k : ℕ) |
| (hsep : ∀ i j, i < j → j < k → 3*r ≤ ‖p i-p j‖) |
| (hp : ∀ i < k, ‖p i‖ ≤ a) : |
| packetPairEnergy p k ≤ (k:ℝ)*(packetDensityConstant/r^3*(2*a+2*r)^2*unitCoulombIntegral) := by |
| have hterm (j : ℕ) (hj : j ∈ Finset.range k) : |
| (∑ i ∈ Finset.range j, 1/‖p i-p j‖) ≤ |
| packetDensityConstant/r^3*(2*a+2*r)^2*unitCoulombIntegral := by |
| have hjk := Finset.mem_range.mp hj |
| have he : (∑ i ∈ Finset.range j, 1/‖p i-p j‖) = |
| ∫ x : Space, packetCloud p r j x/‖p j-x‖ := by |
| simp only [packetCloud,Finset.sum_div] |
| rw [integral_finsetSum] |
| · apply Finset.sum_congr rfl |
| intro i hi |
| have his := hsep i j (Finset.mem_range.mp hi) hjk |
| rw [packetDensity_potential (p i) hr (by rw [norm_sub_rev]; linarith),norm_sub_rev] |
| · intro i _ |
| exact CoulombAnalysis.tfPotential_integrable (packetDensity_integrable (p i) hr) |
| (packetDensity_memLp (p i) hr) (p j) |
| rw [he] |
| exact packetCloud_potential_bound p hr ha j |
| (fun i l hil hl => hsep i l hil (by omega)) (fun i hi => hp i (by omega)) (p j) (hp j hjk) |
| calc |
| _ ≤ ∑ _j ∈ Finset.range k, packetDensityConstant/r^3*(2*a+2*r)^2*unitCoulombIntegral := |
| Finset.sum_le_sum hterm |
| _ = _ := by simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul] |
|
|
|
|
| open MeasureTheory Filter Set Metric |
| open scoped Topology BigOperators |
|
|
|
|
| def packetGridPoint {n : ℕ} (r : ℝ) (v : Fin 3 → Fin n) : Space := |
| WithLp.toLp 2 (fun t => (3*r)*(v t : ℝ)) |
|
|
| lemma packetGridPoint_apply {n : ℕ} (r : ℝ) (v : Fin 3 → Fin n) (t : Fin 3) : |
| packetGridPoint r v t = (3*r)*(v t : ℝ) := rfl |
|
|
| lemma packetGridPoint_norm {n : ℕ} {r : ℝ} (hr : 0 < r) (v : Fin 3 → Fin n) : |
| ‖packetGridPoint r v‖ ≤ 9*r*n := by |
| have hb (t : Fin 3) : |packetGridPoint r v t| ≤ 3*r*n := by |
| rw [packetGridPoint_apply,abs_of_nonneg (by positivity)] |
| exact mul_le_mul_of_nonneg_left (Nat.cast_le.mpr (v t).isLt.le) (by positivity) |
| have hs := EuclideanSpace.norm_sq_eq (packetGridPoint r v) |
| have hsq : ‖packetGridPoint r v‖^2 ≤ 3*(3*r*n)^2 := by |
| rw [hs] |
| calc |
| _ ≤ ∑ _t : Fin 3, (3*r*n)^2 := by |
| apply Finset.sum_le_sum |
| intro t _ |
| simpa only [Real.norm_eq_abs] using (sq_le_sq₀ (abs_nonneg _) (by positivity)).mpr (hb t) |
| _ = _ := by simp |
| have hp : 0 ≤ 9*r*n := by positivity |
| nlinarith [sq_nonneg (3*r*n),norm_nonneg (packetGridPoint r v)] |
|
|
| lemma packetGridPoint_separated {n : ℕ} {r : ℝ} (hr : 0 < r) |
| (v w : Fin 3 → Fin n) (hvw : v ≠ w) : 3*r ≤ ‖packetGridPoint r v-packetGridPoint r w‖ := by |
| obtain ⟨t,ht⟩ : ∃ t, v t ≠ w t := by |
| by_contra hh |
| push Not at hh |
| exact hvw (funext hh) |
| have hcast : 1 ≤ |(v t : ℝ)-(w t : ℝ)| := by |
| rcases lt_or_gt_of_ne (show (v t).val ≠ (w t).val from fun hh => ht (Fin.ext hh)) with h | h |
| · have hh : (v t : ℝ)+1 ≤ (w t : ℝ) := by exact_mod_cast h |
| rw [abs_of_nonpos (by linarith : (v t : ℝ)-(w t : ℝ) ≤ 0)] |
| have := hh |
| linarith |
| · have hh : (w t : ℝ)+1 ≤ (v t : ℝ) := by exact_mod_cast h |
| rw [abs_of_nonneg (by linarith : 0 ≤ (v t : ℝ)-(w t : ℝ))] |
| have := hh |
| linarith |
| have hb := PiLp.norm_apply_le (packetGridPoint r v-packetGridPoint r w) t |
| change |(3*r)*(v t : ℝ)-(3*r)*(w t : ℝ)| ≤ _ at hb |
| rw [← mul_sub,abs_mul,abs_of_pos (by positivity : 0 < 3*r)] at hb |
| nlinarith |
|
|
| lemma packetGrid_card (n : ℕ) : Fintype.card (Fin 3 → Fin n) = n^3 := by simp |
|
|
| def packetGrid (n : ℕ) (r : ℝ) (i : ℕ) : Space := |
| if h : i < Fintype.card (Fin 3 → Fin n) then |
| packetGridPoint r ((Fintype.equivFin (Fin 3 → Fin n)).symm ⟨i,h⟩) else 0 |
|
|
| lemma packetGrid_norm (n : ℕ) {r : ℝ} (hr : 0 < r) (i : ℕ) : |
| ‖packetGrid n r i‖ ≤ 9*r*n := by |
| unfold packetGrid |
| split_ifs |
| · exact packetGridPoint_norm hr _ |
| · simp only [norm_zero]; positivity |
|
|
| lemma packetGrid_separated (n : ℕ) {r : ℝ} (hr : 0 < r) |
| (i j : ℕ) (hij : i < j) (hj : j < n^3) : |
| 3*r ≤ ‖packetGrid n r i-packetGrid n r j‖ := by |
| have hj' : j < Fintype.card (Fin 3 → Fin n) := by rwa [packetGrid_card] |
| have hi' : i < Fintype.card (Fin 3 → Fin n) := lt_trans hij hj' |
| simp only [packetGrid,dite_eq_left hi',dite_eq_left hj'] |
| apply packetGridPoint_separated hr |
| intro hh |
| have he := (Fintype.equivFin (Fin 3 → Fin n)).symm.injective hh |
| have : i = j := congrArg Fin.val he |
| omega |
| |
| |
| open MeasureTheory Filter Set Metric |
| open scoped Topology BigOperators |
| |
| end CoulombAtom |
| |
| end |
| |
| end OAI |
| |
