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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | 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
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