| import Mathlib |
| import OAI.Analysis.CoulombIonization.FormDomain.WeakDerivativeSmulL2 |
| import OAI.Analysis.CoulombIonization.FormDomain.GradientVector |
|
|
| 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 oneParticleDensity {N : ℕ} (ψ : FormVector (N+1)) |
| (hψ : ∀ s, MemLp (ψ.value s) 2) : Space → ℝ≥0∞ := |
| fermionDensity (toMany ψ hψ) |
|
|
| theorem form_lieb_thirring {N : ℕ} (ψ : FormVector (N+1)) |
| (hψ : FormAdmissible ψ) : |
| (∫⁻ x, oneParticleDensity ψ hψ.1 x ^ (5/3:ℝ)) ≤ |
| ENNReal.ofReal (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 hn : ‖toMany ψ hψ.1‖^2 = 1 := |
| (toMany_norm_sq ψ hψ.1).trans hψ.2.2.2.2.1 |
| have ha (i j : Fin (N+1)) (hij : i ≠ j) : |
| permute (Equiv.swap i j) (toMany ψ hψ.1) = -toMany ψ hψ.1 := by |
| rw [toMany_permute ψ hψ.1 (Equiv.swap i j) (hψ.2.2.2.1 _)] |
| simp [Equiv.Perm.sign_swap hij] |
| have hw (i : Fin (N+1)) (a : Fin 3) := form_hasFermionGradient ψ hψ i a |
| simp only [EuclideanSpace.basisFun_apply] at hw |
| have h := fermion_lieb_thirring (toMany ψ hψ.1) (gradientMany ψ hψ) hw hn ha |
| simp only [gradientMany_norm_sq] at h |
| have hs : (∑ s : Spins (N+1), ∑ i : Fin (N+1), ∑ a : Fin 3, |
| ∫ x : Configuration (N+1), ‖ψ.gradient s i a x‖^2) = |
| ∑ i : Fin (N+1), ∑ a : Fin 3, ∑ s : Spins (N+1), |
| ∫ x : Configuration (N+1), ‖ψ.gradient s i a x‖^2 := by |
| rw [Finset.sum_comm] |
| exact Finset.sum_congr rfl (fun i _ => Finset.sum_comm) |
| simpa only [oneParticleDensity,hs] using h |
| end CoulombAtom.Pauli |
|
|
| namespace CoulombLT |
| open CoulombPauli CoulombPackets |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
|
|
| lemma coordinateDensity_aemeasurable {N : ℕ} |
| (ψ : fermionL2 (V := V) (N+1)) (i : Fin (N+1)) : |
| AEMeasurable (fun x => ∑ s : Fin 2, ∫⁻ y, |
| ENNReal.ofReal (‖splitFermion i ψ ((x,s),y)‖^2) |
| ∂(configMeasure N (spinSpaceMeasure (V := V)))) volume := by |
| have hm := ((Lp.memLp (splitFermion i ψ)).aestronglyMeasurable.norm.pow 2).aemeasurable.ennreal_ofReal |
| have h1 := hm.lintegral_prod_right' |
| have h2 := h1.lintegral_prod_right' |
| simpa only [lintegral_fintype,Measure.count_singleton,mul_one,Pi.pow_apply] using h2 |
|
|
| lemma fermionDensity_aemeasurable {N : ℕ} (ψ : fermionL2 (V := V) (N+1)) : |
| AEMeasurable (fermionDensity ψ) volume := by |
| exact Finset.aemeasurable_fun_sum Finset.univ (fun i _ => coordinateDensity_aemeasurable ψ i) |
|
|
| lemma coordinate_density_weighted {N : ℕ} (ψ : fermionL2 (V := V) (N+1)) |
| (i : Fin (N+1)) (w : V → ℝ) (hw : Measurable w) : |
| (∫⁻ x, ENNReal.ofReal (w x) * (∑ s : Fin 2, ∫⁻ y, |
| ENNReal.ofReal (‖splitFermion i ψ ((x,s),y)‖^2) |
| ∂(configMeasure N (spinSpaceMeasure (V := V))))) = |
| ∫⁻ q, ENNReal.ofReal (w (q i).1) * ENNReal.ofReal (‖ψ q‖^2) |
| ∂(configMeasure (N+1) (spinSpaceMeasure (V := V))) := by |
| let ν := configMeasure N (spinSpaceMeasure (V := V)) |
| have hm : AEMeasurable (fun p : (V × Fin 2) × (Fin N → V × Fin 2) => |
| ENNReal.ofReal (w p.1.1) * ENNReal.ofReal (‖splitFermion i ψ p‖^2)) |
| ((spinSpaceMeasure (V := V)).prod ν) := |
| ((hw.comp (measurable_fst.comp measurable_fst)).ennreal_ofReal.aemeasurable).mul |
| ((Lp.memLp (splitFermion i ψ)).aestronglyMeasurable.norm.pow 2).aemeasurable.ennreal_ofReal |
| have hp := splitAt_preserving (μ := spinSpaceMeasure (V := V)) i |
| have ha := pull_ae (splitAt i).symm (hp.symm (splitAt i)) ψ |
| calc |
| _ = ∫⁻ x, ∫⁻ s : Fin 2, ∫⁻ y, |
| ENNReal.ofReal (w x) * ENNReal.ofReal (‖splitFermion i ψ ((x,s),y)‖^2) ∂ν ∂Measure.count := by |
| apply lintegral_congr |
| intro x |
| simp only [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,lintegral_fintype,Measure.count_singleton,mul_one,Finset.mul_sum] |
| rfl |
| _ = ∫⁻ p, ENNReal.ofReal (w p.1.1) * ENNReal.ofReal (‖splitFermion i ψ p‖^2) |
| ∂((spinSpaceMeasure (V := V)).prod ν) := by |
| rw [lintegral_prod _ hm,lintegral_prod _ hm.lintegral_prod_right'] |
| _ = ∫⁻ p, ENNReal.ofReal (w p.1.1) * ENNReal.ofReal (‖ψ ((splitAt i).symm p)‖^2) |
| ∂((spinSpaceMeasure (V := V)).prod ν) := by |
| apply lintegral_congr_ae |
| filter_upwards [ha] with p hp |
| rw [← hp] |
| rfl |
| _ = _ := by |
| have he := (hp.symm (splitAt i)).lintegral_comp_emb |
| (splitAt i).symm.measurableEmbedding |
| (fun q => ENNReal.ofReal (w (q i).1) * ENNReal.ofReal (‖ψ q‖^2)) |
| have hs (p : (V × Fin 2) × (Fin N → V × Fin 2)) : |
| (splitAt i).symm p i = p.1 := by |
| exact congrArg Prod.fst ((splitAt i).apply_symm_apply p) |
| simp_rw [hs] at he |
| exact he |
|
|
| lemma density_weighted_lintegral {N : ℕ} (ψ : fermionL2 (V := V) (N+1)) |
| (w : V → ℝ) (hw : Measurable w) : |
| (∫⁻ x, ENNReal.ofReal (w x) * fermionDensity ψ x) = |
| ∑ i : Fin (N+1), ∫⁻ q, ENNReal.ofReal (w (q i).1) * ENNReal.ofReal (‖ψ q‖^2) |
| ∂(configMeasure (N+1) (spinSpaceMeasure (V := V))) := by |
| change (∫⁻ x, ENNReal.ofReal (w x) * ∑ i : Fin (N+1), |
| ∑ s : Fin 2, ∫⁻ y, ENNReal.ofReal (‖splitFermion i ψ ((x,s),y)‖^2) |
| ∂(configMeasure N (spinSpaceMeasure (V := V)))) = _ |
| conv_lhs => arg 2; ext x; rw [Finset.mul_sum] |
| rw [lintegral_finsetSum' (f := fun i x => ENNReal.ofReal (w x) * |
| (∑ s : Fin 2, ∫⁻ y, ENNReal.ofReal (‖splitFermion i ψ ((x,s),y)‖^2) |
| ∂(configMeasure N (spinSpaceMeasure (V := V))))) Finset.univ (fun i _ => |
| hw.ennreal_ofReal.aemeasurable.mul (coordinateDensity_aemeasurable ψ i))] |
| exact Finset.sum_congr rfl (fun i _ => coordinate_density_weighted ψ i w hw) |
| end CoulombLT |
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| end |
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| end OAI |
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