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import OAI.Analysis.CoulombIonization.Variational.RotationCoreKernelIntegrable
import OAI.Analysis.CoulombIonization.Localization.PacketCloud
namespace OAI
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
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)
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
end OAI
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