| import Mathlib |
| import OAI.Analysis.CoulombIonization.Variational.RotationCoreKernelIntegrable |
| import OAI.Analysis.CoulombIonization.Localization.PacketCloud |
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| namespace OAI |
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| noncomputable section |
|
|
| 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 |
| open MeasureTheory Filter Set Metric |
| open scoped Topology BigOperators |
| open MeasureTheory Filter Set Metric |
| open scoped Topology BigOperators |
| namespace CoulombAtom |
|
|
| def gridPacketRadius (B : ℝ) (n : ℕ) : ℝ := B/(20*(n:ℝ)) |
| def packetKineticConstant : ℝ := 400*radialPacketKinetic |
| def packetRepulsionConstant : ℝ := 128000*packetDensityConstant*unitCoulombIntegral |
|
|
| lemma packetKineticConstant_nonneg : 0 ≤ packetKineticConstant := by |
| unfold packetKineticConstant |
| exact mul_nonneg (by norm_num) radialPacketKinetic_nonneg |
|
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| lemma packetRepulsionConstant_nonneg : 0 ≤ packetRepulsionConstant := by |
| unfold packetRepulsionConstant |
| positivity [packetDensityConstant_pos,unitCoulombIntegral_nonneg] |
|
|
| lemma gridPacketRadius_pos {B : ℝ} (hB : 0 < B) {n : ℕ} (hn : 0 < n) : |
| 0 < gridPacketRadius B n := div_pos hB (by positivity) |
|
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| lemma gridPacketRadius_mul {B : ℝ} {n : ℕ} (hn : 0 < n) : |
| gridPacketRadius B n * n = B/20 := by |
| have hn' : (n:ℝ) ≠ 0 := (Nat.cast_pos.mpr hn).ne' |
| unfold gridPacketRadius |
| field_simp |
|
|
| lemma gridPacketRadius_le {B : ℝ} (hB : 0 < B) {n : ℕ} (hn : 0 < n) : |
| gridPacketRadius B n ≤ B/20 := by |
| have hn' : (1:ℝ) ≤ n := Nat.one_le_cast.mpr hn |
| have hr := gridPacketRadius_pos hB hn |
| have he := gridPacketRadius_mul (B := B) hn |
| nlinarith |
| |
| lemma packetGrid_ball {B : ℝ} (hB : 0 < B) {n : ℕ} (hn : 0 < n) (i : ℕ) : |
| ‖packetGrid n (gridPacketRadius B n) i‖ ≤ 9*B/20 := by |
| calc |
| _ ≤ 9*gridPacketRadius B n*n := packetGrid_norm n (gridPacketRadius_pos hB hn) i |
| _ = _ := by rw [mul_assoc,gridPacketRadius_mul hn]; ring |
| |
| lemma packetGrid_repulsion {B : ℝ} (hB : 0 < B) {n : ℕ} (hn : 0 < n) : |
| packetPairEnergy (packetGrid n (gridPacketRadius B n)) (n^3) ≤ |
| packetRepulsionConstant*(n:ℝ)^6/B := by |
| let r := gridPacketRadius B n |
| have hr : 0 < r := gridPacketRadius_pos hB hn |
| have hrB : r ≤ B := by have := gridPacketRadius_le hB hn; dsimp [r]; linarith |
| have hp (i : ℕ) (_hi : i < n^3) : ‖packetGrid n r i‖ ≤ B := by |
| have := packetGrid_ball hB hn i |
| dsimp only [r] |
| linarith |
| have hb := packetPairEnergy_bound (packetGrid n r) hr hB.le (n^3) |
| (packetGrid_separated n hr) hp |
| have hs : (2*B+2*r)^2 ≤ 16*B^2 := by nlinarith |
| have hn' : (n:ℝ) ≠ 0 := (Nat.cast_pos.mpr hn).ne' |
| calc |
| _ ≤ (n^3:ℕ)*(packetDensityConstant/r^3*(2*B+2*r)^2*unitCoulombIntegral) := hb |
| _ ≤ (n^3:ℕ)*(packetDensityConstant/r^3*(16*B^2)*unitCoulombIntegral) := by |
| gcongr |
| exact unitCoulombIntegral_nonneg |
| exact div_nonneg packetDensityConstant_pos.le (pow_nonneg hr.le _) |
| _ = _ := by |
| dsimp [r,gridPacketRadius,packetRepulsionConstant] |
| push_cast |
| field_simp |
| ring |
| |
| theorem grid_packet_screening {N : ℕ} {ψ : FormVector N} (hψ : SobolevFermion ψ) |
| (y : Space) {B : ℝ} (hB : 0 < B) (hnuc : B ≤ ‖y‖) |
| (hc : FormAvoids ψ (ball y B)) {n : ℕ} (hn : 0 < n) |
| {Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam) : |
| (n:ℝ)^3*(Z*formMass ψ/‖y‖-coreCoulombAt ψ y-lam*formMass ψ) ≤ |
| (formEnergy Z ψ + lam*N*formMass ψ-priceEnergy (energy Z) lam*formMass ψ) + |
| formMass ψ*(packetKineticConstant*(n:ℝ)^5/B^2+packetRepulsionConstant*(n:ℝ)^6/B) := by |
| let r := gridPacketRadius B n |
| have hr : 0 < r := gridPacketRadius_pos hB hn |
| have hrB : r ≤ B/20 := gridPacketRadius_le hB hn |
| have hp (i : ℕ) (_hi : i < n^3) : ‖packetGrid n r i‖+2*r ≤ B := by |
| have := packetGrid_ball hB hn i |
| dsimp only [r] at * |
| linarith |
| have hnp (i : ℕ) (_hi : i < n^3) : ‖packetGrid n r i‖+r < ‖y‖ := by |
| have := packetGrid_ball hB hn i |
| dsimp only [r] at * |
| linarith |
| have hh := averaged_many_packet_screening hψ y (packetGrid n r) hr (n^3) hc hp |
| (packetGrid_separated n hr) hnp hZ hlam |
| have hpair := packetGrid_repulsion hB hn |
| have hm := formMass_nonneg ψ |
| have hh' := mul_le_mul_of_nonneg_left hpair hm |
| have he : (n:ℝ)^3*(radialPacketKinetic/r^2) = packetKineticConstant*(n:ℝ)^5/B^2 := by |
| dsimp [r,gridPacketRadius,packetKineticConstant] |
| field_simp |
| ring |
| push_cast at hh hh' |
| simp only [r] at hh hh' he |
| nlinarith |
|
|
| end CoulombAtom |
|
|
| open MeasureTheory Filter Set Metric |
| open scoped Topology BigOperators |
|
|
| end |
|
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| end OAI |
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