| import Mathlib |
| import OAI.Analysis.CoulombIonization.FormDomain.FermionDensityMass |
|
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter Complex TopologicalSpace |
| open scoped Topology InnerProductSpace ENNReal |
| open MeasureTheory Filter Complex |
| open scoped Topology BigOperators ComplexConjugate FourierTransform SchwartzMap ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff SchwartzMap FourierTransform ENNReal |
| open MeasureTheory Filter |
| open scoped ContDiff InnerProductSpace Topology |
| open MeasureTheory Filter |
| open scoped ENNReal |
| namespace CoulombAtom.Pauli |
| open CoulombPauli CoulombPackets CoulombLT |
|
|
| def density {N : ℕ} (ψ : FormVector (N+1)) (hψ : ∀ s, MemLp (ψ.value s) 2) (x : Space) : ℝ := |
| (oneParticleDensity ψ hψ x).toReal |
|
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| lemma density_nonneg {N : ℕ} (ψ : FormVector (N+1)) (hψ : ∀ s, MemLp (ψ.value s) 2) (x : Space) : |
| 0 ≤ density ψ hψ x := ENNReal.toReal_nonneg |
|
|
| lemma density_integrable {N : ℕ} (ψ : FormVector (N+1)) (hψ : ∀ s, MemLp (ψ.value s) 2) : |
| Integrable (density ψ hψ) := fermionDensity_real_integrable (toMany ψ hψ) |
|
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| lemma density_mass {N : ℕ} (ψ : FormVector (N+1)) (hψ : FormAdmissible ψ) : |
| (∫ x, density ψ hψ.1 x) = (N+1:ℝ) := by |
| change (∫ x, (fermionDensity (toMany ψ hψ.1) x).toReal) = _ |
| rw [fermionDensity_real_mass,toMany_norm_sq,hψ.2.2.2.2.1,mul_one] |
|
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| lemma density_power_integrable {N : ℕ} (ψ : FormVector (N+1)) (hψ : FormAdmissible ψ) : |
| Integrable (fun x => density ψ hψ.1 x^(5/3:ℝ)) := by |
| have hm := (fermionDensity_aemeasurable (toMany ψ hψ.1)).pow_const (5/3:ℝ) |
| have hb := form_lieb_thirring ψ hψ |
| have hn : (∫⁻ x, oneParticleDensity ψ hψ.1 x^(5/3:ℝ)) ≠ ⊤ := |
| ne_top_of_le_ne_top ENNReal.ofReal_ne_top hb |
| have hi := integrable_toReal_of_lintegral_ne_top hm hn |
| change Integrable (fun x => (fermionDensity (toMany ψ hψ.1) x).toReal ^ (5/3:ℝ)) |
| simpa only [← ENNReal.toReal_rpow] using hi |
|
|
| lemma density_power_bound {N : ℕ} (ψ : FormVector (N+1)) (hψ : FormAdmissible ψ) : |
| (∫ x, density ψ hψ.1 x^(5/3:ℝ)) ≤ |
| 32*(N+1:ℝ) + (2048/3:ℝ) * |
| (∑ s : Spins (N+1), ∑ i : Fin (N+1), ∑ a : Fin 3, |
| ∫ x : Configuration (N+1), ‖ψ.gradient s i a x‖^2) := by |
| have hm := (fermionDensity_aemeasurable (toMany ψ hψ.1)).pow_const (5/3:ℝ) |
| have hb := form_lieb_thirring ψ hψ |
| have hn : (∫⁻ x, oneParticleDensity ψ hψ.1 x^(5/3:ℝ)) ≠ ⊤ := |
| ne_top_of_le_ne_top ENNReal.ofReal_ne_top hb |
| have he := integral_toReal hm (ae_lt_top' hm hn) |
| simp only [← ENNReal.toReal_rpow] at he |
| change (∫ x, (fermionDensity (toMany ψ hψ.1) x).toReal ^ (5/3:ℝ)) ≤ _ |
| rw [he] |
| exact (ENNReal.toReal_le_toReal hn ENNReal.ofReal_ne_top).2 hb |>.trans_eq |
| (ENNReal.toReal_ofReal (by positivity)) |
| |
| lemma toMany_coordinate_lintegral {N : ℕ} (ψ : FormVector N) |
| (hψ : ∀ s, MemLp (ψ.value s) 2) (i : Fin N) (w : Space → ℝ) |
| (hw : Measurable w) : |
| (∫⁻ q, ENNReal.ofReal (w (q i).1) * ENNReal.ofReal (‖toMany ψ hψ q‖^2) |
| ∂configMeasure N oneMeasure) = |
| ∑ s : Spins N, ∫⁻ x : Configuration N, |
| ENNReal.ofReal (w (x i)) * ENNReal.ofReal (‖ψ.value s x‖^2) := by |
| have hm : AEMeasurable (fun p : Configuration N × Spins N => |
| ENNReal.ofReal (w (p.1 i)) * ENNReal.ofReal (‖ψ.value p.2 p.1‖^2)) |
| ((volume : Measure (Configuration N)).prod Measure.count) := |
| ((hw.comp ((measurable_pi_apply i).comp measurable_fst)).ennreal_ofReal.aemeasurable).mul |
| ((memLp_spin ψ.value hψ).aestronglyMeasurable.norm.pow 2).aemeasurable.ennreal_ofReal |
| calc |
| _ = ∫⁻ q, ENNReal.ofReal (w (q i).1) * |
| ENNReal.ofReal (‖ψ.value (fun j => (q j).2) (fun j => (q j).1)‖^2) |
| ∂configMeasure N oneMeasure := by |
| apply lintegral_congr_ae |
| filter_upwards [toMany_ae ψ hψ] with q hq |
| rw [hq] |
| _ = ∫⁻ p : Configuration N × Spins N, |
| ENNReal.ofReal (w (p.1 i)) * ENNReal.ofReal (‖ψ.value p.2 p.1‖^2) |
| ∂volume.prod Measure.count := |
| (splitSpin_preserving N).lintegral_comp_emb (splitSpin N).measurableEmbedding |
| (fun p : Configuration N × Spins N => ENNReal.ofReal (w (p.1 i)) * ENNReal.ofReal (‖ψ.value p.2 p.1‖^2)) |
| _ = _ := by |
| rw [lintegral_prod_symm _ hm] |
| simp only [lintegral_fintype,Measure.count_singleton,mul_one] |
| |
| lemma density_coulomb_lintegral {N : ℕ} (ψ : FormVector (N+1)) (hψ : FormAdmissible ψ) : |
| (∫⁻ x, ENNReal.ofReal (‖x‖⁻¹) * oneParticleDensity ψ hψ.1 x) = |
| ENNReal.ofReal (∑ s : Spins (N+1), ∑ i : Fin (N+1), |
| ∫ x : Configuration (N+1), ‖ψ.value s x‖^2 / ‖x i‖) := by |
| change (∫⁻ x : Space, ENNReal.ofReal (‖x‖⁻¹) * fermionDensity (toMany ψ hψ.1) x) = _ |
| rw [density_weighted_lintegral (toMany ψ hψ.1) (fun x : Space => ‖x‖⁻¹) measurable_norm.inv] |
| trans ∑ i : Fin (N+1), ∑ s : Spins (N+1), ∫⁻ x : Configuration (N+1), |
| ENNReal.ofReal (‖x i‖⁻¹) * ENNReal.ofReal (‖ψ.value s x‖^2) |
| · exact Finset.sum_congr rfl (fun i _ => |
| toMany_coordinate_lintegral ψ hψ.1 i (fun x : Space => ‖x‖⁻¹) measurable_norm.inv) |
| have hi (i : Fin (N+1)) (s : Spins (N+1)) : |
| (∫⁻ x : Configuration (N+1), ENNReal.ofReal (‖x i‖⁻¹) * |
| ENNReal.ofReal (‖ψ.value s x‖^2)) = |
| ENNReal.ofReal (∫ x : Configuration (N+1), ‖ψ.value s x‖^2 / ‖x i‖) := by |
| rw [ofReal_integral_eq_lintegral_ofReal (hψ.2.2.2.2.2.1 s i) |
| (Filter.Eventually.of_forall (fun x => by positivity))] |
| apply lintegral_congr |
| intro x |
| rw [← ENNReal.ofReal_mul (by positivity : 0 ≤ ‖x i‖⁻¹)] |
| congr 1 |
| simp only [div_eq_mul_inv,mul_comm] |
| simp_rw [hi] |
| rw [Finset.sum_comm] |
| have hnuc (s : Spins (N+1)) (i : Fin (N+1)) : |
| 0 ≤ ∫ x : Configuration (N+1), ‖ψ.value s x‖^2 / ‖x i‖ := |
| integral_nonneg (fun _ => by positivity) |
| have heach (s : Spins (N+1)) : |
| (∑ i : Fin (N+1), ENNReal.ofReal (∫ x : Configuration (N+1), ‖ψ.value s x‖^2 / ‖x i‖)) = |
| ENNReal.ofReal (∑ i : Fin (N+1), ∫ x : Configuration (N+1), ‖ψ.value s x‖^2 / ‖x i‖) := |
| (ENNReal.ofReal_sum_of_nonneg (fun i _ => hnuc s i)).symm |
| simp_rw [heach] |
| exact (ENNReal.ofReal_sum_of_nonneg (fun s _ => Finset.sum_nonneg |
| (fun i _ => hnuc s i))).symm |
| |
| lemma density_coulomb_integral {N : ℕ} (ψ : FormVector (N+1)) (hψ : FormAdmissible ψ) : |
| Integrable (fun x => density ψ hψ.1 x / ‖x‖) ∧ |
| (∫ x, density ψ hψ.1 x / ‖x‖) = |
| ∑ s : Spins (N+1), ∑ i : Fin (N+1), |
| ∫ x : Configuration (N+1), ‖ψ.value s x‖^2 / ‖x i‖ := by |
| have hm : AEMeasurable (fun x => ENNReal.ofReal (‖x‖⁻¹) * oneParticleDensity ψ hψ.1 x) := |
| measurable_norm.inv.ennreal_ofReal.aemeasurable.mul |
| (fermionDensity_aemeasurable (toMany ψ hψ.1)) |
| have he := density_coulomb_lintegral ψ hψ |
| have hn : (∫⁻ x, ENNReal.ofReal (‖x‖⁻¹) * oneParticleDensity ψ hψ.1 x) ≠ ⊤ := by |
| rw [he]; exact ENNReal.ofReal_ne_top |
| have hv (x : Space) : |
| (ENNReal.ofReal (‖x‖⁻¹) * oneParticleDensity ψ hψ.1 x).toReal = density ψ hψ.1 x/‖x‖ := by |
| rw [ENNReal.toReal_mul,ENNReal.toReal_ofReal (by positivity)] |
| exact mul_comm _ _ |
| have hit := integrable_toReal_of_lintegral_ne_top hm hn |
| have hei := integral_toReal hm (ae_lt_top' hm hn) |
| simp only [hv] at hit hei |
| refine ⟨hit, ?_⟩ |
| rw [hei,he,ENNReal.toReal_ofReal (by positivity)] |
| end CoulombAtom.Pauli |
|
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| open MeasureTheory Set Metric |
| open scoped ENNReal |
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| end |
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| end OAI |
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