| import Mathlib | |
| import OAI.Analysis.HardSphere.Geometry.FirstVariation | |
| import OAI.Analysis.HardSphere.FormDomain.EnergyParallelogram | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped ENNReal Topology BigOperators | |
| noncomputable section | |
| open scoped BigOperators ComplexConjugate | |
| namespace HardSphere | |
| lemma permutationPullback_comp {N : ℕ} {L : Length} | |
| (σ τ : Equiv.Perm (Fin N)) (ψ : ScalarL2 N L) : | |
| permutationPullback σ (permutationPullback τ ψ) = permutationPullback (σ * τ) ψ := by | |
| change Lp.compMeasurePreserving _ (measurePreserving_permute σ) | |
| (Lp.compMeasurePreserving _ (measurePreserving_permute τ) ψ) = _ | |
| rw [← Lp.compMeasurePreserving_comp_apply] | |
| rfl | |
| lemma inner_permutationPullback {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N)) | |
| (ψ φ : ScalarL2 N L) : | |
| inner ℂ (permutationPullback σ ψ) (permutationPullback σ φ) = inner ℂ ψ φ := | |
| (Lp.compMeasurePreservingₗᵢ ℂ _ (measurePreserving_permute σ)).inner_map_map ψ φ | |
| lemma gradientPairing_permutationJet {N : ℕ} {L : Length} | |
| (σ : Equiv.Perm (Fin N)) (u v : Jet N L) : | |
| Variational.gradientPairing (permutationJet σ u) (permutationJet σ v) = | |
| Variational.gradientPairing u v := by | |
| change (∑ p : Direction N, inner ℂ (permutationPullback σ (u.2 (σ.symm p.1, p.2))) | |
| (permutationPullback σ (v.2 (σ.symm p.1, p.2)))) = _ | |
| simp only [inner_permutationPullback] | |
| exact Equiv.sum_comp (Equiv.prodCongr σ.symm (Equiv.refl (Fin 3))) | |
| (fun p ↦ inner ℂ (u.2 p) (v.2 p)) | |
| theorem IsGroundVector.permutationInvariant {N : ℕ} {L : Length} {a : ℝ} | |
| {u : Jet N L} (hu : IsGroundVector N L a u) (σ : Equiv.Perm (Fin N)) : | |
| permutationJet σ u = u := by | |
| have hs : (permutationJet σ u).1 = u.1 := (bosonic_iff_pullback u.1).mp hu.2.1 σ | |
| have hv : permutationJet σ u ∈ groundFormSubmodule N L a := by | |
| rw [mem_groundFormSubmodule] | |
| exact ⟨permutationJet_mem_domain σ hu.1, hs ▸ hu.2.1⟩ | |
| exact Variational.minimizer_eq_of_scalar_eq (groundFormSubmodule N L a) u | |
| (mem_groundFormSubmodule.mpr ⟨hu.1, hu.2.1⟩) hu.2.2.1 | |
| (fun w hw hn ↦ hu.2.2.2 w (mem_groundFormSubmodule.mp hw).1 | |
| (mem_groundFormSubmodule.mp hw).2 hn) | |
| (permutationJet σ u) hv hs (le_of_eq (energy_permutationJet σ u)) | |
| lemma bosonic_sum_permutationPullback {N : ℕ} {L : Length} (ψ : ScalarL2 N L) : | |
| bosonic (∑ σ : Equiv.Perm (Fin N), permutationPullback σ ψ) := by | |
| rw [bosonic_iff_pullback] | |
| intro τ | |
| simp only [map_sum, permutationPullback_comp] | |
| exact Equiv.sum_comp (Equiv.mulLeft τ) (fun σ ↦ permutationPullback σ ψ) | |
| lemma gradientPairing_sum_right {N : ℕ} {L : Length} {I : Type*} [Fintype I] | |
| (u : Jet N L) (v : I → Jet N L) : | |
| Variational.gradientPairing u (∑ i, v i) = | |
| ∑ i, Variational.gradientPairing u (v i) := by | |
| simp only [Variational.gradientPairing, Prod.snd_sum, Finset.sum_apply, inner_sum] | |
| exact Finset.sum_comm | |
| theorem IsGroundVector.full_first_variation {N : ℕ} {L : Length} {a : ℝ} | |
| {u : Jet N L} (hu : IsGroundVector N L a u) (v : Jet N L) | |
| (hv : v ∈ dirichletDomain N L a) : | |
| (∑ p : Direction N, inner ℂ (u.2 p) (v.2 p)) = | |
| (energy u : ℂ) * inner ℂ u.1 v.1 := by | |
| let w : Jet N L := ∑ σ : Equiv.Perm (Fin N), permutationJet σ v | |
| have hwd : w ∈ dirichletDomain N L a := by | |
| change w ∈ dirichletSubmodule N L a | |
| exact Submodule.sum_mem _ fun σ _ ↦ permutationJet_mem_domain σ hv | |
| have hwb : bosonic w.1 := by | |
| dsimp only [w] | |
| rw [Prod.fst_sum] | |
| exact bosonic_sum_permutationPullback v.1 | |
| have hgp (σ : Equiv.Perm (Fin N)) : | |
| Variational.gradientPairing u (permutationJet σ v) = Variational.gradientPairing u v := by | |
| calc | |
| _ = Variational.gradientPairing (permutationJet σ u) (permutationJet σ v) := by | |
| rw [hu.permutationInvariant σ] | |
| _ = _ := gradientPairing_permutationJet σ u v | |
| have hip (σ : Equiv.Perm (Fin N)) : | |
| inner ℂ u.1 (permutationJet σ v).1 = inner ℂ u.1 v.1 := by | |
| calc | |
| _ = inner ℂ (permutationPullback σ u.1) (permutationPullback σ v.1) := by | |
| rw [(bosonic_iff_pullback u.1).mp hu.2.1 σ]; rfl | |
| _ = _ := inner_permutationPullback σ u.1 v.1 | |
| have he := hu.first_variation w hwd hwb | |
| change Variational.gradientPairing u w = _ at he | |
| simp only [w, gradientPairing_sum_right, hgp, Prod.fst_sum, inner_sum, hip, | |
| Finset.sum_const, Finset.card_univ, nsmul_eq_mul] at he | |
| have hc : (Fintype.card (Equiv.Perm (Fin N)) : ℂ) ≠ 0 := by | |
| exact_mod_cast Fintype.card_ne_zero | |
| apply mul_left_cancel₀ hc | |
| change (Fintype.card (Equiv.Perm (Fin N)) : ℂ) * Variational.gradientPairing u v = _ | |
| rw [he] | |
| ring | |
| end HardSphere | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |