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159357f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 | import Mathlib
import OAI.Analysis.CoulombIonization.FieldAnalysis.ShellMeanStatistic
import OAI.Analysis.CoulombIonization.Variational.FullSliceStatistics
import OAI.Analysis.CoulombIonization.FormDomain.InnerLinearCap
namespace OAI
noncomputable section
namespace CoulombAtom
section
open MeasureTheory Set Filter
open scoped BigOperators
attribute [local irreducible] oneBodySquareTotal graphFormVector fermionGraph
lemma shell_closed_inner_term_integrable {N : ℕ} {psi : FormVector N}
(hpsi : SobolevVector psi) (p : SmoothMultiplier spaceDirections) (Z : ℝ)
{h a : ℝ} (ha : 0 < a) (hh : h ≤ a) (hp : ∀ y, p.value y ≠ 0 → 2*a ≤ ‖y‖)
(y : Space) (S : ℝ) (s : Spins N) :
Integrable (fun x => (p.value y^2/S)*(closedRawInnerField Z h x y*‖psi.value s x‖^2)) := by
by_cases hz : p.value y = 0
· simp only [hz,zero_pow (by decide : 2 ≠ 0),zero_div,zero_mul]
exact integrable_zero _ _ _
have hf : Measurable (fun x : Configuration N => closedRawInnerField Z h x y) :=
by simpa only [Function.comp_def,id_eq] using
(closedRawInnerField_measurable N Z h).comp (measurable_id.prodMk measurable_const)
exact (rawFormPair_integrable hpsi hf
(closedRawInnerField_bound Z ha (by linarith) (hp y hz)) s).const_mul _
lemma fullSlicePair_shellMean {N M : ℕ} {psi : FormVector (N+M)}
(p : SmoothMultiplier spaceDirections) (Z : ℝ) {h a : ℝ}
(ha : 0 < a) (hh : h ≤ a) (hp : ∀ y, p.value y ≠ 0 → 2*a ≤ ‖y‖)
(hs : SplitPositionSupport psi (fun z => p.value z = 0) (fun z => h < ‖z‖))
(t : Spins M) (v : Configuration M) (hpsi : SobolevVector (coreSlice psi t v))
(hm : formMass (coreSlice psi t v) ≠ 0) :
fullSlicePair psi (shellMeanStatistic p Z ha hh hp) t v =
∑ i : Fin M, (p.value (v i)^2/(oneBodySquareTotal p).value v)*
rawFormPair (coreSlice psi t v) (fun x => closedRawInnerField Z h x (v i)) := by
have hv := hs.outer_of_mass_ne_zero t v hm
have he (s : Spins N) :
(∫ x : Configuration N, shellMeanField p Z h (joinLists x v)*‖(coreSlice psi t v).value s x‖^2) =
∑ i : Fin M, (p.value (v i)^2/(oneBodySquareTotal p).value v)*
(∫ x : Configuration N, closedRawInnerField Z h x (v i)*‖(coreSlice psi t v).value s x‖^2) := by
calc
_ = ∫ x : Configuration N, ∑ i : Fin M,
(p.value (v i)^2/(oneBodySquareTotal p).value v)*
(closedRawInnerField Z h x (v i)*‖(coreSlice psi t v).value s x‖^2) := by
apply integral_congr_ae
apply ae_of_all
intro x
dsimp only
by_cases hz : (coreSlice psi t v).value s x = 0
· simp only [hz,norm_zero,zero_pow (by decide : 2 ≠ 0),mul_zero,Finset.sum_const_zero]
· rw [shellMeanField_join p Z h x v (hs s t x v hz).1 hv]
simp only [Finset.sum_div,Finset.sum_mul]
apply Finset.sum_congr rfl
intro i _
ring
_ = ∑ i : Fin M, ∫ x : Configuration N,
(p.value (v i)^2/(oneBodySquareTotal p).value v)*
(closedRawInnerField Z h x (v i)*‖(coreSlice psi t v).value s x‖^2) :=
integral_finsetSum _ (fun i _ => shell_closed_inner_term_integrable hpsi p Z ha hh hp _ _ s)
_ = _ := by simp only [integral_const_mul]
change (∑ s : Spins N, ∫ x : Configuration N,
shellMeanField p Z h (joinLists x v)*‖(coreSlice psi t v).value s x‖^2) = _
simp_rw [he]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro i _
rw [←Finset.mul_sum]
rfl
lemma fullSlicePair_shellMean_le {N M : ℕ} {psi : FormVector (N+M)}
(p : SmoothMultiplier spaceDirections) {Z lam h a : ℝ}
(hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) (ha : h ≤ a)
(hp : ∀ y, p.value y ≠ 0 → 2*a ≤ ‖y‖)
(hs : SplitPositionSupport psi (fun z => p.value z = 0) (fun z => h < ‖z‖))
(t : Spins M) (v : Configuration M) (hpsi : SobolevVector (coreSlice psi t v)) :
fullSlicePair psi (shellMeanStatistic p Z (hh.trans_le ha) ha hp) t v ≤
(h/a)*Real.sqrt (innerCapStatistic (coreSlice psi t v) Z lam h)*
Real.sqrt (formMass (coreSlice psi t v)) := by
let C := (h/a)*Real.sqrt (innerCapStatistic (coreSlice psi t v) Z lam h)*
Real.sqrt (formMass (coreSlice psi t v))
have hap : 0 < a := hh.trans_le ha
have hC : 0 ≤ C := by dsimp [C]; positivity
by_cases hm : formMass (coreSlice psi t v) = 0
· rw [fullSlicePair_eq_zero_of_mass _ t v hpsi hm]
exact hC
rw [fullSlicePair_shellMean p Z (hh.trans_le ha) ha hp hs t v hpsi hm]
have hS := oneBodySquareTotal_nonneg p v
calc
_ ≤ ∑ i : Fin M, (p.value (v i)^2/(oneBodySquareTotal p).value v)*C := by
apply Finset.sum_le_sum
intro i _
by_cases hi : p.value (v i) = 0
· simp only [hi,zero_pow (by decide : 2 ≠ 0),zero_div,zero_mul,le_refl]
· have hy := hp _ hi
have hr := rawFormPair_closedRawInnerField (Z := Z) hpsi (hh.trans_le ha)
(show h+a ≤ ‖v i‖ by linarith) hm
rw [hr]
exact mul_le_mul_of_nonneg_left (actual_inner_linear_statistic hpsi hZ hlam hh ha hy)
(div_nonneg (sq_nonneg _) hS)
_ = ((oneBodySquareTotal p).value v/(oneBodySquareTotal p).value v)*C := by
rw [←Finset.sum_mul,←Finset.sum_div,oneBodySquareTotal_value]
_ ≤ C := by
by_cases hz : (oneBodySquareTotal p).value v = 0
· rw [hz,div_zero,zero_mul]; exact hC
· rw [div_self hz,one_mul]
end
open MeasureTheory Filter
open scoped BigOperators
namespace CoreObservationEnsemble
lemma fullAverage_shellMean_le (E : CoreObservationEnsemble)
(p : SmoothMultiplier spaceDirections) {Z lam h a : ℝ}
(hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) (ha : h ≤ a)
(hp : ∀ y, p.value y ≠ 0 → 2*a ≤ ‖y‖)
(hs : E.supported (fun z => p.value z = 0) (fun z => h < ‖z‖)) :
E.fullAverage (shellMeanStatistic p Z (hh.trans_le ha) ha hp) ≤
(h/a)*Real.sqrt (E.innerCapAverage Z lam h)*Real.sqrt E.mass := by
let I := Σ i : E.Branch, Spins (E.graph i).outSize
let X : I → Type := fun i => Configuration (E.graph i.1).outSize
let H : ∀ i : I, X i → ℝ := fun i v => innerCapStatistic (coreSlice (E.graph i.1).vector i.2 v) Z lam h
let W : ∀ i : I, X i → ℝ := fun i v => formMass (coreSlice (E.graph i.1).vector i.2 v)
have hH (i : I) : Integrable (H i) := innerCapStatistic_coreSlice_integrable (E.graph i.1).sobolev i.2 hZ hlam hh
have hW (i : I) : Integrable (W i) := (E.graph i.1).sobolev.coreSlice_mass_integrable i.2
have hnH (i : I) (v : X i) : 0 ≤ H i v := innerCapStatistic_nonneg ..
have hnW (i : I) (v : X i) : 0 ≤ W i v := formMass_nonneg _
have hc := sum_integral_sqrt_mul_sqrt_le (fun i : I => (volume : Measure (X i))) H W hH hW hnH hnW
let F := shellMeanStatistic p Z (hh.trans_le ha) ha hp
have hi (i : I) :
(∫ v : X i, fullSlicePair (E.graph i.1).vector F i.2 v) ≤
(h/a)*(∫ v : X i, Real.sqrt (H i v)*Real.sqrt (W i v)) := by
rw [←integral_const_mul]
apply integral_mono_ae (fullSlicePair_integrable (E.graph i.1).sobolev F i.2)
((integrable_sqrt_mul_sqrt (hH i) (hW i) (hnH i) (hnW i)).const_mul _)
filter_upwards [(E.graph i.1).sobolev.ae_coreSlice i.2] with v hv
simpa only [mul_assoc] using fullSlicePair_shellMean_le p hZ hlam hh ha hp (hs i.1) i.2 v hv
have hsum := Finset.sum_le_sum (s := Finset.univ) (fun i _ => hi i)
rw [←Finset.mul_sum] at hsum
have heF : (∑ i : I, ∫ v : X i, fullSlicePair (E.graph i.1).vector F i.2 v) = E.fullAverage F := by
rw [Fintype.sum_sigma]
exact Finset.sum_congr rfl (fun i _ => fullSlicePair_fubini (E.graph i).sobolev F)
have heH : (∑ i : I, ∫ v : X i, H i v) = E.innerCapAverage Z lam h := by
rw [Fintype.sum_sigma]
rfl
have heW : (∑ i : I, ∫ v : X i, W i v) = E.mass := by
rw [Fintype.sum_sigma]
exact Finset.sum_congr rfl (fun i _ => (E.graph i).sobolev.integral_coreSlice_mass)
rw [heF] at hsum
rw [heH,heW] at hc
exact hsum.trans (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hc (div_nonneg hh.le (hh.trans_le ha).le))
end CoreObservationEnsemble
theorem exists_actual_shellMean_constant : ∃ C : ℝ, 1 ≤ C ∧
∀ (Z lam : ℝ) (N : ℕ) (psi : FormVector N), SobolevFermion psi →
formMass psi = 1 → 0 ≤ Z → 0 < lam → ∀ D h u : ℝ, 0 ≤ D → 0 < h → h < u/4 →
max (corePriceExcess Z lam psi) 0 ≤ D →
∀ p : SmoothMultiplier spaceDirections,
(∀ y, p.value y ≠ 0 → u/2 ≤ ‖y‖ ∧ ‖y‖ ≤ 4*u) →
rawFormPair psi (shellMeanField p Z h) ≤
(4*h/u)*Real.sqrt (C*(innerScale D h+lam^2)) := by
obtain ⟨L,hL⟩ := exists_scaled_shell_stencil
obtain ⟨C,hC,hbound⟩ := exists_actual_observed_innerCap_constant (by norm_num : (0:ℝ)<1/2) (beta := 4) L
refine ⟨C,hC,?_⟩
intro Z lam N psi hpsi hm hZ hlam D h u hD hh hu he p hp
have hup : 0 < u := by linarith
obtain ⟨ys,hlen,hys,hcover⟩ := hL u hup
let E := (CoreObservationEnsemble.initial psi hpsi).observeCells ys
have hs := CoreObservationEnsemble.annular_stencil_support psi hpsi hup ys (fun y hy => (hys y hy).1) hcover
have hsupport : E.supported (fun z => p.value z = 0) (fun z => h < ‖z‖) := by
intro i
exact (hs i).mono (fun z hz => by by_contra hn; exact hz (hp z hn)) (fun z hz => hu.trans_le hz)
have hap : 0 < u/4 := by positivity
have hha : h ≤ u/4 := hu.le
have hpa : ∀ y, p.value y ≠ 0 → 2*(u/4) ≤ ‖y‖ := fun y hy => by linarith [(hp y hy).1]
have hmean := E.fullAverage_shellMean_le p hZ hlam.le hh hha hpa hsupport
have hmass : E.mass = 1 := (CoreObservationEnsemble.observeCells_mass ..).trans ((CoreObservationEnsemble.initial_mass ..).trans hm)
have hav : E.innerCapAverage Z lam h ≤ C*(innerScale D h+lam^2) :=
hbound Z lam N psi hpsi hm hZ hlam D h u hD hh (by linarith) he ys hlen (by
intro y hy; constructor <;> nlinarith [(hys y hy).1,(hys y hy).2])
have heq : E.fullAverage (shellMeanStatistic p Z hap hha hpa) = rawFormPair psi (shellMeanField p Z h) := by
rw [CoreObservationEnsemble.observeCells_fullAverage,CoreObservationEnsemble.initial_fullAverage]
rfl
rw [heq,hmass,Real.sqrt_one,mul_one] at hmean
calc
_ ≤ (h/(u/4))*Real.sqrt (E.innerCapAverage Z lam h) := hmean
_ ≤ (h/(u/4))*Real.sqrt (C*(innerScale D h+lam^2)) :=
mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hav) (div_nonneg hh.le hap.le)
_ = _ := by congr 1; field_simp
end CoulombAtom
end
end OAI
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