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import OAI.Analysis.CoulombIonization.Variational.CutCoreNumber
import OAI.Analysis.CoulombIonization.Variational.CorePriceExcess
import OAI.Analysis.CoulombIonization.FormDomain.BlockEnergetics
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Filter Set Metric
open scoped BigOperators ContDiff
namespace CoulombAtom
lemma coreCoulombAt_mass_zero {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ)
(hm : formMass ψ = 0) (y : Space) : coreCoulombAt ψ y = 0 := by
have hz (s : Spins N) : (∫ x, ‖ψ.value s x‖^2) = 0 :=
(Finset.sum_eq_zero_iff_of_nonneg (fun s _ => integral_nonneg fun x => sq_nonneg _)).mp hm s
(Finset.mem_univ _)
unfold coreCoulombAt
apply Finset.sum_eq_zero
intro s _
apply Finset.sum_eq_zero
intro i _
have he := (integral_eq_zero_iff_of_nonneg (fun x => sq_nonneg _)
(hψ.1 s).norm.integrable_sq).mp (hz s)
calc
_ = ∫ _x : Configuration N, (0:ℝ) := integral_congr_ae (he.mono fun x hx => by simp only [hx,Pi.zero_apply,zero_div])
_ = 0 := integral_zero _ _
lemma normalizedCoreField_mul_mass {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ)
(Z lam : ℝ) (y : Space) :
normalizedCoreField Z lam ψ y*formMass ψ =
Z*formMass ψ/‖y‖-coreCoulombAt ψ y-lam*formMass ψ := by
by_cases hm : formMass ψ = 0
· simp only [hm,mul_zero,zero_div,coreCoulombAt_mass_zero hψ hm,sub_zero]
· unfold normalizedCoreField
rw [sub_mul,div_mul_cancel₀ _ hm]
lemma coreSlice_mass_div {N M : ℕ} (ψ : FormVector (N+M)) (t : Spins M)
(y : Configuration M) (j : Fin M) :
formMass (coreSlice ψ t y)/‖y j‖ =
∑ s : Spins N, ∫ x : Configuration N, ‖ψ.value (joinLists s t) (joinLists x y)‖^2/‖y j‖ := by
simp only [formMass,Finset.sum_div,integral_div,coreSlice]
lemma coreSlice_mass_div_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (t : Spins M) (j : Fin M) :
Integrable (fun y : Configuration M => formMass (coreSlice ψ t y)/‖y j‖) := by
simp_rw [coreSlice_mass_div]
apply integrable_finsetSum
intro s _
have hh := (integrable_join (N := N) (M := M) (nuclear_integrable
(finSumFinEquiv (Sum.inr j)) (hψ.1 (joinLists s t))
(hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _))).integral_prod_right
simpa only [joinLists_right] using hh
lemma coreSlice_cross_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (t : Spins M) (j : Fin M) :
Integrable (fun y : Configuration M => coreCoulombAt (coreSlice ψ t y) (y j)) := by
unfold coreCoulombAt
apply integrable_finsetSum
intro s _
apply integrable_finsetSum
intro i _
have hne : (finSumFinEquiv (Sum.inl i) : Fin (N+M)) ≠ finSumFinEquiv (Sum.inr j) := by
intro h
have hh := finSumFinEquiv.injective h
cases hh
have hh := (integrable_join (N := N) (M := M) (pair_integrable _ _ hne
(hψ.1 (joinLists s t)) (hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _))).integral_prod_right
simpa only [coreSlice,joinLists_left,joinLists_right] using hh
lemma integral_coreSlice_mass_div {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) :
(∑ t : Spins M, ∫ y : Configuration M, ∑ j : Fin M, formMass (coreSlice ψ t y)/‖y j‖) =
outNuclear ψ := by
have hi (s : Spins N) (t : Spins M) (j : Fin M) :
Integrable (fun y : Configuration M => ∫ x : Configuration N,
‖ψ.value (joinLists s t) (joinLists x y)‖^2/‖y j‖) := by
simpa only [joinLists_right] using (integrable_join (N := N) (M := M)
(nuclear_integrable (finSumFinEquiv (Sum.inr j)) (hψ.1 (joinLists s t))
(hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _))).integral_prod_right
simp_rw [integral_finsetSum _ (fun j _ => coreSlice_mass_div_integrable hψ _ j),
coreSlice_mass_div,integral_finsetSum _ (fun s _ => hi s _ _)]
have he (s : Spins N) (t : Spins M) (j : Fin M) :
(∫ y : Configuration M, ∫ x : Configuration N,
‖ψ.value (joinLists s t) (joinLists x y)‖^2/‖y j‖) =
∫ z, ‖ψ.value (joinLists s t) z‖^2/‖z (finSumFinEquiv (Sum.inr j))‖ := by
simpa only [joinLists_right] using integral_join (N := N) (M := M)
(nuclear_integrable (finSumFinEquiv (Sum.inr j)) (hψ.1 (joinLists s t))
(hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _))
simp_rw [he]
conv_lhs => arg 2; ext t; rw [Finset.sum_comm]
exact sum_spin_join (fun s => ∑ j : Fin M, ∫ z,
‖ψ.value s z‖^2/‖z (finSumFinEquiv (Sum.inr j))‖)
lemma integral_coreSlice_cross {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) :
(∑ t : Spins M, ∫ y : Configuration M, ∑ j : Fin M, coreCoulombAt (coreSlice ψ t y) (y j)) =
crossRepulsion ψ := by
have hi (s : Spins N) (t : Spins M) (i : Fin N) (j : Fin M) :
Integrable (fun y : Configuration M => ∫ x : Configuration N,
‖ψ.value (joinLists s t) (joinLists x y)‖^2/‖x i-y j‖) := by
have hne : (finSumFinEquiv (Sum.inl i) : Fin (N+M)) ≠ finSumFinEquiv (Sum.inr j) := by
intro h
have hh := finSumFinEquiv.injective h
cases hh
simpa only [joinLists_right,joinLists_left] using (integrable_join (N := N) (M := M)
(pair_integrable _ _ hne (hψ.1 (joinLists s t))
(hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _))).integral_prod_right
simp_rw [integral_finsetSum _ (fun j _ => coreSlice_cross_integrable hψ _ j),coreCoulombAt,coreSlice]
simp_rw [integral_finsetSum _ (fun s _ => integrable_finsetSum _ (fun i _ => hi s _ i _)),
integral_finsetSum _ (fun i _ => hi _ _ i _)]
have he (s : Spins N) (t : Spins M) (i : Fin N) (j : Fin M) :
(∫ y : Configuration M, ∫ x : Configuration N,
‖ψ.value (joinLists s t) (joinLists x y)‖^2/‖x i-y j‖) =
∫ z, ‖ψ.value (joinLists s t) z‖^2/
‖z (finSumFinEquiv (Sum.inl i))-z (finSumFinEquiv (Sum.inr j))‖ := by
have hne : (finSumFinEquiv (Sum.inl i) : Fin (N+M)) ≠ finSumFinEquiv (Sum.inr j) := by
intro h
have hh := finSumFinEquiv.injective h
cases hh
simpa only [joinLists_right,joinLists_left] using integral_join (N := N) (M := M)
(pair_integrable _ _ hne (hψ.1 (joinLists s t))
(hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _))
simp_rw [he]
conv_lhs => arg 2; ext t; rw [Finset.sum_comm]; arg 2; ext s; rw [Finset.sum_comm]
exact sum_spin_join (fun s => ∑ i : Fin N, ∑ j : Fin M, ∫ z,
‖ψ.value s z‖^2/‖z (finSumFinEquiv (Sum.inl i))-z (finSumFinEquiv (Sum.inr j))‖)
def conditionalFieldSum {N M : ℕ} (Z lam : ℝ) (ψ : FormVector (N+M))
(t : Spins M) (y : Configuration M) : ℝ :=
formMass (coreSlice ψ t y)*∑ j : Fin M,
normalizedCoreField Z lam (coreSlice ψ t y) (y j)
lemma conditionalFieldSum_ae {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (Z lam : ℝ) (t : Spins M) :
conditionalFieldSum Z lam ψ t =ᵐ[volume]
fun y => Z*(∑ j : Fin M, formMass (coreSlice ψ t y)/‖y j‖)-
(∑ j : Fin M, coreCoulombAt (coreSlice ψ t y) (y j))-
lam*M*formMass (coreSlice ψ t y) := by
apply (hψ.ae_coreSlice t).mono
intro y hy
unfold conditionalFieldSum
rw [Finset.mul_sum]
simp_rw [mul_comm (formMass _),normalizedCoreField_mul_mass hy,mul_div_assoc]
simp only [Finset.sum_sub_distrib,←Finset.mul_sum,Finset.sum_const,
Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
ring
lemma conditionalFieldSum_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (Z lam : ℝ) (t : Spins M) :
Integrable (conditionalFieldSum Z lam ψ t) := by
have hi := ((integrable_finsetSum Finset.univ (fun j _ => coreSlice_mass_div_integrable hψ t j)).const_mul Z).sub
(integrable_finsetSum Finset.univ (fun j _ => coreSlice_cross_integrable hψ t j))
exact (hi.sub ((hψ.coreSlice_mass_integrable t).const_mul (lam*M))).congr
(conditionalFieldSum_ae hψ Z lam t).symm
lemma integral_conditionalFieldSum {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (Z lam : ℝ) :
(∑ t : Spins M, ∫ y, conditionalFieldSum Z lam ψ t y) =
Z*outNuclear ψ-crossRepulsion ψ-lam*M*formMass ψ := by
have ht (t : Spins M) : (∫ y, conditionalFieldSum Z lam ψ t y) =
Z*(∫ y, ∑ j : Fin M, formMass (coreSlice ψ t y)/‖y j‖)-
(∫ y, ∑ j : Fin M, coreCoulombAt (coreSlice ψ t y) (y j))-
lam*M*(∫ y, formMass (coreSlice ψ t y)) := by
rw [integral_congr_ae (conditionalFieldSum_ae hψ Z lam t)]
have hi := (integrable_finsetSum Finset.univ (fun j _ => coreSlice_mass_div_integrable hψ t j)).const_mul Z
have hj := integrable_finsetSum Finset.univ (fun j _ => coreSlice_cross_integrable hψ t j)
rw [integral_sub (f := fun y => Z*(∑ j : Fin M, formMass (coreSlice ψ t y)/‖y j‖)-
(∑ j : Fin M, coreCoulombAt (coreSlice ψ t y) (y j)))
(g := fun y => lam*M*formMass (coreSlice ψ t y))
(hi.sub hj) ((hψ.coreSlice_mass_integrable t).const_mul (lam*M)),
integral_sub hi hj,integral_const_mul,integral_const_mul]
simp_rw [ht]
rw [Finset.sum_sub_distrib,Finset.sum_sub_distrib,←Finset.mul_sum,←Finset.mul_sum,
integral_coreSlice_mass_div hψ,integral_coreSlice_cross hψ,hψ.integral_coreSlice_mass]
theorem priced_conditional_energy_identity {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (Z lam : ℝ) :
formEnergy Z ψ+lam*(N+M)*formMass ψ =
(∑ t : Spins M, ∫ y, formEnergy Z (coreSlice ψ t y)+
lam*N*formMass (coreSlice ψ t y))+
outKinetic ψ-(∑ t : Spins M, ∫ y, conditionalFieldSum Z lam ψ t y)+
outRepulsion ψ := by
have ht (t : Spins M) : (∫ y, formEnergy Z (coreSlice ψ t y)+
lam*N*formMass (coreSlice ψ t y)) =
(∫ y, formEnergy Z (coreSlice ψ t y))+lam*N*(∫ y, formMass (coreSlice ψ t y)) := by
rw [integral_add (hψ.coreSlice_energy_integrable Z t)
((hψ.coreSlice_mass_integrable t).const_mul (lam*N)),integral_const_mul]
simp_rw [ht]
rw [Finset.sum_add_distrib,←Finset.mul_sum,hψ.integral_coreSlice_mass,
integrated_core_parts hψ Z,integral_conditionalFieldSum hψ Z lam,
formEnergy_core_out_cross]
ring
theorem fresh_cut_priced_localization_identity {L : ℕ}
(p : Fin 2 → SmoothMultiplier spaceDirections)
(hp : ∀ x, ∑ a, (p a).value x^2 = 1) {ψ : FormVector L}
(hψ : SobolevVector ψ) (Z lam : ℝ) :
formEnergy Z ψ+lam*L*formMass ψ+
(1/2:ℝ)*weightedParticleCount ψ (spatialErrorWeight p) =
(∑ b : Fin L → Fin 2, ∑ t : Spins (cutOutNumber b), ∫ y,
formEnergy Z (coreSlice (orderedCutForm p hp ψ b) t y)+
lam*cutCoreNumber b*formMass (coreSlice (orderedCutForm p hp ψ b) t y))+
(∑ b : Fin L → Fin 2, outKinetic (orderedCutForm p hp ψ b))-
(∑ b : Fin L → Fin 2, ∑ t : Spins (cutOutNumber b), ∫ y,
conditionalFieldSum Z lam (orderedCutForm p hp ψ b) t y)+
(∑ b : Fin L → Fin 2, outRepulsion (orderedCutForm p hp ψ b)) := by
have hb (b : Fin L → Fin 2) := priced_conditional_energy_identity
(orderedCutForm_sobolev p hp hψ b) Z lam
have hcard (b : Fin L → Fin 2) : (cutCoreNumber b : ℝ)+cutOutNumber b = L := by
exact_mod_cast cutNumbers_sum b
simp_rw [hcard,orderedCutForm_energy,orderedCutForm_mass] at hb
have hsum := congrArg (fun f : (Fin L → Fin 2) → ℝ => ∑ b, f b) (funext hb)
simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib] at hsum
rw [←Finset.mul_sum,spatial_cut_mass p hp hψ,spatial_cut_ims p hp hψ Z] at hsum
have hw : (∑ s : Spins L, ∑ i : Fin L, ∫ x,
spatialErrorWeight p (x i)*‖ψ.value s x‖^2) =
weightedParticleCount ψ (spatialErrorWeight p) := rfl
rw [hw] at hsum
linarith
end CoulombAtom
end
end OAI
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