| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.SmoothMultiplier |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| namespace HardSphere |
| open MeasureTheory Set Filter |
| open scoped ComplexConjugate |
|
|
| lemma coordinateLine_cons_succ {n : ℕ} {L : Length} (x : Torus L) |
| (Y : Configuration n L) (i : Fin n) (j : Fin 3) (t : AddCircle L.val) : |
| coordinateLine (Fin.cons x Y) (i.succ,j) t=Fin.cons x (coordinateLine Y (i,j) t) := by |
| funext k q |
| refine Fin.cases ?_ (fun k ↦ ?_) k |
| · simp [coordinateLine] |
| · by_cases h : k=i |
| · subst k; simp [coordinateLine] |
| · simp [coordinateLine,Pi.single_eq_of_ne h,Pi.single_eq_of_ne (fun e ↦ h (Fin.succ_injective _ e))] |
|
|
| lemma SmoothMultiplier.hasDerivAt_fiberIntegral {n : ℕ} {L : Length} |
| (f : SmoothMultiplier (n+1) L) (Y : Configuration n L) (i : Fin n) (j : Fin 3) : |
| HasDerivAt (fun t : ℝ ↦ ∫ x : Torus L, |
| f.toFun (Fin.cons x (coordinateLine Y (i,j) (t : AddCircle L.val)))) |
| (∫ x : Torus L, coordDerivative (n+1) L f.toFun (i.succ,j) (Fin.cons x Y)) 0 := by |
| let F : ℝ → Torus L → ℂ := fun t x ↦ |
| f.toFun (Fin.cons x (coordinateLine Y (i,j) (t : AddCircle L.val))) |
| let D : ℝ → Torus L → ℂ := fun t x ↦ |
| coordDerivative (n+1) L f.toFun (i.succ,j) |
| (Fin.cons x (coordinateLine Y (i,j) (t : AddCircle L.val))) |
| have cF (t) : Continuous (F t) := |
| (continuous_of_continuous_lift f.smooth.continuous).comp |
| ((continuous_id : Continuous (id : Torus L → Torus L)).finCons |
| (continuous_const : Continuous (fun _ : Torus L ↦ coordinateLine Y (i,j) (t : AddCircle L.val)))) |
| have cD (t) : Continuous (D t) := |
| (continuous_coordDerivative f.smooth (i.succ,j)).comp |
| ((continuous_id : Continuous (id : Torus L → Torus L)).finCons |
| (continuous_const : Continuous (fun _ : Torus L ↦ coordinateLine Y (i,j) (t : AddCircle L.val)))) |
| let d : C(Configuration (n+1) L,ℂ) := |
| ⟨_,continuous_coordDerivative f.smooth (i.succ,j)⟩ |
| have h := hasDerivAt_integral_of_dominated_loc_of_deriv_le |
| (F := F) (F' := D) (μ := (volume : Measure (Torus L))) (s := Set.univ) (x₀ := 0) |
| (bound := fun _ ↦ ‖d‖) univ_mem |
| (Eventually.of_forall fun t ↦ (cF t).aestronglyMeasurable) |
| ((cF 0).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| (cD 0).aestronglyMeasurable |
| (Eventually.of_forall fun x t _ ↦ d.norm_coe_le_norm _) |
| (integrable_const _) (Eventually.of_forall fun x t _ ↦ by |
| have hh := hasDerivAt_coordinateLine (f.smooth.differentiable (by simp)) |
| (Fin.cons x Y) (i.succ,j) t |
| simpa only [coordinateLine_cons_succ] using hh) |
| simpa only [F,D,Complex.ofReal_zero,AddCircle.coe_zero,coordinateLine,Pi.single_zero,add_zero] using h.2 |
| |
| lemma SmoothMultiplier.insertion_orthogonal {n : ℕ} {L : Length} |
| (f : SmoothMultiplier (n+1) L) |
| (hr : ∀ X, conj (f.toFun X)=f.toFun X) |
| (hn : ∀ Y : Configuration n L, ∫ x : Torus L, ‖f.toFun (Fin.cons x Y)‖^2=1) |
| (i : Fin n) (j : Fin 3) (Y : Configuration n L) : |
| ∫ x : Torus L, conj (f.toFun (Fin.cons x Y))* |
| coordDerivative (n+1) L f.toFun (i.succ,j) (Fin.cons x Y)=0 := by |
| have he (X) : f.toFun X*f.toFun X = ((‖f.toFun X‖^2 : ℝ) : ℂ) := by |
| calc |
| _ = f.toFun X*conj (f.toFun X) := by rw [hr] |
| _ = _ := by rw [Complex.mul_conj,Complex.sq_norm] |
| have hc (Y) : (∫ x : Torus L, (f.mul f).toFun (Fin.cons x Y))=1 := by |
| simp only [SmoothMultiplier.mul,Pi.mul_apply,he] |
| rw [integral_complex_ofReal,hn,Complex.ofReal_one] |
| have h := (f.mul f).hasDerivAt_fiberIntegral Y i j |
| simp only [hc] at h |
| have hz := h.unique (hasDerivAt_const (0:ℝ) (1:ℂ)) |
| have hd := coordDerivative_mul (f.smooth.differentiable (by simp)) |
| (f.smooth.differentiable (by simp)) (i.succ,j) |
| change (∫ x : Torus L, coordDerivative (n+1) L (f.toFun*f.toFun) (i.succ,j) (Fin.cons x Y))=0 at hz |
| rw [hd] at hz |
| have hfactor : (∫ x : Torus L, |
| (f.toFun * coordDerivative (n+1) L f.toFun (i.succ,j) + |
| f.toFun * coordDerivative (n+1) L f.toFun (i.succ,j)) (Fin.cons x Y))= |
| (2:ℂ)*(∫ x : Torus L, conj (f.toFun (Fin.cons x Y))* |
| coordDerivative (n+1) L f.toFun (i.succ,j) (Fin.cons x Y)) := by |
| rw [← integral_const_mul] |
| apply integral_congr_ae |
| exact Eventually.of_forall fun x ↦ by simp only [Pi.add_apply,Pi.mul_apply,hr]; ring |
| rw [hfactor] at hz |
| exact (mul_eq_zero.mp hz).resolve_left (by norm_num) |
| |
| end HardSphere |
| |
| end |
| |
| end OAI |
| |
