| 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 L2Multiplier | |
| open MeasureTheory | |
| open scoped ComplexConjugate ENNReal | |
| variable {α : Type*} [MeasurableSpace α] {μ : Measure α} | |
| lemma inner_clm_real_right (f : α → ℂ) (hf : MemLp f ∞ μ) | |
| (hreal : ∀ x, conj (f x) = f x) (u v : Lp ℂ 2 μ) : | |
| inner ℂ u (clm f hf v) = inner ℂ (clm f hf u) v := by | |
| simpa only [show (fun x ↦ conj (f x)) = f from funext hreal] using | |
| inner_clm_right f hf u v | |
| lemma clm_comm (f g : α → ℂ) (hf : MemLp f ∞ μ) (hg : MemLp g ∞ μ) | |
| (u : Lp ℂ 2 μ) : clm f hf (clm g hg u) = clm g hg (clm f hf u) := by | |
| rw [← clm_mul, ← clm_mul] | |
| congr 2 | |
| exact mul_comm f g | |
| end L2Multiplier | |
| namespace HardSphere | |
| open MeasureTheory Set Filter | |
| open scoped ComplexConjugate ENNReal | |
| lemma SmoothMultiplier.partial_real {N : ℕ} {L : Length} (f : SmoothMultiplier N L) | |
| (hf : ∀ X, conj (f.toFun X) = f.toFun X) (p : Direction N) (X : Configuration N L) : | |
| conj (coordDerivative N L f.toFun p X) = coordDerivative N L f.toFun p X := by | |
| have he : (fun X ↦ conj (f.toFun X)) = f.toFun := funext hf | |
| rw [← congrFun (coordDerivative_conj (f.smooth.differentiable (by simp)) p) X, he] | |
| def SmoothMultiplier.derivativeScalar {N : ℕ} {L : Length} (f : SmoothMultiplier N L) | |
| (p : Direction N) : ScalarL2 N L →L[ℂ] ScalarL2 N L := | |
| L2Multiplier.clm _ (f.partial_memLp_top p) | |
| @[simp] lemma SmoothMultiplier.onJet_fst {N : ℕ} {L : Length} | |
| (f : SmoothMultiplier N L) (u : Jet N L) : (f.onJet u).1 = f.onScalar u.1 := rfl | |
| @[simp] lemma SmoothMultiplier.onJet_snd {N : ℕ} {L : Length} | |
| (f : SmoothMultiplier N L) (u : Jet N L) (p : Direction N) : | |
| (f.onJet u).2 p = f.onScalar (u.2 p) + f.derivativeScalar p u.1 := rfl | |
| lemma SmoothMultiplier.inner_onScalar_right {N : ℕ} {L : Length} | |
| (f : SmoothMultiplier N L) (hf : ∀ X, conj (f.toFun X) = f.toFun X) | |
| (u v : ScalarL2 N L) : inner ℂ u (f.onScalar v) = inner ℂ (f.onScalar u) v := | |
| L2Multiplier.inner_clm_real_right _ _ hf u v | |
| lemma SmoothMultiplier.inner_derivativeScalar_right {N : ℕ} {L : Length} | |
| (f : SmoothMultiplier N L) (hf : ∀ X, conj (f.toFun X) = f.toFun X) | |
| (p : Direction N) (u v : ScalarL2 N L) : | |
| inner ℂ u (f.derivativeScalar p v) = inner ℂ (f.derivativeScalar p u) v := | |
| L2Multiplier.inner_clm_real_right _ _ (f.partial_real hf p) u v | |
| lemma SmoothMultiplier.onScalar_derivativeScalar_comm {N : ℕ} {L : Length} | |
| (f : SmoothMultiplier N L) (p : Direction N) (u : ScalarL2 N L) : | |
| f.onScalar (f.derivativeScalar p u) = f.derivativeScalar p (f.onScalar u) := | |
| L2Multiplier.clm_comm _ _ _ _ u | |
| theorem IsGroundVector.smooth_multiplier_identity {N : ℕ} {L : Length} {a : ℝ} | |
| {u : Jet N L} (hu : IsGroundVector N L a u) (f : SmoothMultiplier N L) | |
| (hf : ∀ X, conj (f.toFun X) = f.toFun X) : | |
| energy (f.onJet u) - energy u * ‖f.onScalar u.1‖ ^ 2 = | |
| ∑ p : Direction N, ‖f.derivativeScalar p u.1‖ ^ 2 := by | |
| have hv := f.onJet_mem_domain (f.onJet_mem_domain hu.1) | |
| have he := hu.full_first_variation (f.onJet (f.onJet u)) hv | |
| have hnorm (x : ScalarL2 N L) : (inner ℂ x x).re = ‖x‖ ^ 2 := | |
| (norm_sq_eq_re_inner (𝕜 := ℂ) x).symm | |
| have hpair (p : Direction N) : | |
| (inner ℂ (u.2 p) ((f.onJet (f.onJet u)).2 p)).re = | |
| ‖f.onScalar (u.2 p)‖ ^ 2 + | |
| 2 * (inner ℂ (f.onScalar (u.2 p)) (f.derivativeScalar p u.1)).re := by | |
| rw [f.onJet_snd, f.onJet_snd, f.onJet_fst, map_add, | |
| inner_add_right, inner_add_right, ← f.onScalar_derivativeScalar_comm, | |
| f.inner_onScalar_right hf (u.2 p) (f.onScalar (u.2 p)), | |
| f.inner_onScalar_right hf (u.2 p) (f.derivativeScalar p u.1)] | |
| simp only [Complex.add_re, hnorm] | |
| ring | |
| have hs : (inner ℂ u.1 (f.onJet (f.onJet u)).1).re = ‖f.onScalar u.1‖ ^ 2 := by | |
| rw [f.onJet_fst, f.onJet_fst, f.inner_onScalar_right hf, hnorm] | |
| have her := congrArg Complex.re he | |
| simp only [Complex.re_sum, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, | |
| zero_mul, sub_zero, hs, hpair] at her | |
| change (∑ p, ‖(f.onJet u).2 p‖ ^ 2) - energy u * ‖f.onScalar u.1‖ ^ 2 = _ | |
| simp only [f.onJet_snd, norm_add_sq (𝕜 := ℂ)] | |
| rw [← her] | |
| change (∑ p, (‖f.onScalar (u.2 p)‖ ^ 2 + | |
| 2 * (inner ℂ (f.onScalar (u.2 p)) (f.derivativeScalar p u.1)).re + | |
| ‖f.derivativeScalar p u.1‖ ^ 2)) - | |
| (∑ p, (‖f.onScalar (u.2 p)‖ ^ 2 + | |
| 2 * (inner ℂ (f.onScalar (u.2 p)) (f.derivativeScalar p u.1)).re)) = _ | |
| simp only [Finset.sum_add_distrib, ← Finset.mul_sum] | |
| ring | |
| end HardSphere | |
| end | |
| end OAI | |