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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 | import Mathlib
import OAI.Analysis.HardSphere.Model
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators
namespace HardSphere
lemma coordDerivative_add {N : ℕ} {L : Length} {a : ℝ}
(f g : SmoothTest N L a) (p : Direction N) :
coordDerivative N L (f.toFun + g.toFun) p =
coordDerivative N L f.toFun p + coordDerivative N L g.toFun p := by
funext X
simp only [coordDerivative, Pi.add_apply]
rw [fderiv_fun_add (f.smooth.differentiable (by simp)).differentiableAt
(g.smooth.differentiable (by simp)).differentiableAt]
rfl
lemma coordDerivative_smul {N : ℕ} {L : Length} {a : ℝ}
(c : ℂ) (f : SmoothTest N L a) (p : Direction N) :
coordDerivative N L (c • f.toFun) p = c • coordDerivative N L f.toFun p := by
funext X
simp only [coordDerivative, Pi.smul_apply]
rw [fderiv_fun_const_smul (f.smooth.differentiable (by simp)).differentiableAt c]
rfl
lemma coordDerivative_zero (N : ℕ) (L : Length) (p : Direction N) :
coordDerivative N L 0 p = 0 := by
funext X
simp [coordDerivative]
def SmoothTest.zero (N : ℕ) (L : Length) (a : ℝ) : SmoothTest N L a where
toFun := 0
smooth := contDiff_const
compact := HasCompactSupport.zero
support_allowed := by simp
memLp := MemLp.zero
partial_memLp := by
intro p
rw [coordDerivative_zero]
exact MemLp.zero
def SmoothTest.add {N : ℕ} {L : Length} {a : ℝ}
(f g : SmoothTest N L a) : SmoothTest N L a where
toFun := f.toFun + g.toFun
smooth := f.smooth.add g.smooth
compact := f.compact.add g.compact
support_allowed := (tsupport_add _ _).trans (union_subset f.support_allowed g.support_allowed)
memLp := f.memLp.add g.memLp
partial_memLp := by
intro p
rw [coordDerivative_add]
exact (f.partial_memLp p).add (g.partial_memLp p)
def SmoothTest.smul {N : ℕ} {L : Length} {a : ℝ}
(c : ℂ) (f : SmoothTest N L a) : SmoothTest N L a where
toFun := c • f.toFun
smooth := f.smooth.const_smul c
compact := f.compact.smul_left
support_allowed := (tsupport_smul_subset_right (fun _ ↦ c) f.toFun).trans f.support_allowed
memLp := f.memLp.const_smul c
partial_memLp := by
intro p
rw [coordDerivative_smul]
exact (f.partial_memLp p).const_smul c
@[simp] lemma SmoothTest.jet_zero (N : ℕ) (L : Length) (a : ℝ) :
(SmoothTest.zero N L a).jet = 0 := by
apply Prod.ext
· rfl
· funext p
change MemLp.toLp (coordDerivative N L 0 p) ((SmoothTest.zero N L a).partial_memLp p) = 0
simp only [coordDerivative_zero, MemLp.toLp_zero]
@[simp] lemma SmoothTest.jet_add {N : ℕ} {L : Length} {a : ℝ}
(f g : SmoothTest N L a) : (f.add g).jet = f.jet + g.jet := by
apply Prod.ext
· rfl
· funext p
change _ = (f.partial_memLp p).toLp _ + (g.partial_memLp p).toLp _
rw [← MemLp.toLp_add]
refine MemLp.toLp_congr ((f.add g).partial_memLp p)
((f.partial_memLp p).add (g.partial_memLp p)) ?_
exact Filter.Eventually.of_forall (congrFun (coordDerivative_add f g p))
@[simp] lemma SmoothTest.jet_smul {N : ℕ} {L : Length} {a : ℝ}
(c : ℂ) (f : SmoothTest N L a) : (f.smul c).jet = c • f.jet := by
apply Prod.ext
· rfl
· funext p
change _ = c • (f.partial_memLp p).toLp _
rw [← MemLp.toLp_const_smul]
refine MemLp.toLp_congr ((f.smul c).partial_memLp p)
((f.partial_memLp p).const_smul c) ?_
exact Filter.Eventually.of_forall (congrFun (coordDerivative_smul c f p))
def smoothGraph (N : ℕ) (L : Length) (a : ℝ) : Submodule ℂ (Jet N L) where
carrier := Set.range (SmoothTest.jet (N := N) (L := L) (a := a))
zero_mem' := ⟨SmoothTest.zero N L a, SmoothTest.jet_zero N L a⟩
add_mem' := by
rintro _ _ ⟨f, rfl⟩ ⟨g, rfl⟩
exact ⟨f.add g, f.jet_add g⟩
smul_mem' := by
rintro c _ ⟨f, rfl⟩
exact ⟨f.smul c, f.jet_smul c⟩
def dirichletSubmodule (N : ℕ) (L : Length) (a : ℝ) : Submodule ℂ (Jet N L) :=
(smoothGraph N L a).topologicalClosure
@[simp] lemma mem_dirichletSubmodule {N : ℕ} {L : Length} {a : ℝ} {u : Jet N L} :
u ∈ dirichletSubmodule N L a ↔ u ∈ dirichletDomain N L a := Iff.rfl
def permuteConfiguration {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N))
(X : Configuration N L) : Configuration N L := fun i ↦ X (σ i)
lemma measurePreserving_permute {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N)) :
MeasurePreserving (permuteConfiguration (L := L) σ) volume volume := by
have h := volume_measurePreserving_piCongrLeft (fun _ : Fin N ↦ Torus L) σ.symm
convert h using 1
funext X i
simp [permuteConfiguration, MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply]
section
attribute [local instance] scalarL2_addCommMonoid scalarL2_module
def permutationPullback {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N)) :
ScalarL2 N L →ₗ[ℂ] ScalarL2 N L :=
Lp.compMeasurePreservingₗ ℂ _ (measurePreserving_permute σ)
end
lemma bosonic_iff_pullback {N : ℕ} {L : Length} (ψ : ScalarL2 N L) :
bosonic ψ ↔ ∀ σ, permutationPullback σ ψ = ψ := by
have hae (σ : Equiv.Perm (Fin N)) :
(permutationPullback σ ψ : Configuration N L → ℂ) =ᵐ[volume]
fun X ↦ ψ (permuteConfiguration σ X) :=
Lp.coeFn_compMeasurePreserving ψ (measurePreserving_permute σ)
constructor
· intro h σ
exact Lp.ext ((hae σ).trans (h σ))
· intro h σ
have hh := hae σ
rw [h σ] at hh
exact hh.symm
section
attribute [local instance] scalarL2_normedAddCommGroup scalarL2_module
def bosonicSubmodule (N : ℕ) (L : Length) : Submodule ℂ (ScalarL2 N L) :=
⨅ σ : Equiv.Perm (Fin N),
@LinearMap.ker ℂ ℂ (ScalarL2 N L) (ScalarL2 N L)
Complex.instSemiring Complex.instSemiring
(scalarL2_normedAddCommGroup N L).toAddCommMonoid
(scalarL2_normedAddCommGroup N L).toAddCommMonoid
(scalarL2_module N L) (scalarL2_module N L) (RingHom.id ℂ)
(@HSub.hSub _ _ _
(@instHSub _ (@LinearMap.instSub ℂ ℂ (ScalarL2 N L) (ScalarL2 N L)
Complex.instSemiring Complex.instSemiring
(scalarL2_normedAddCommGroup N L).toAddCommMonoid
(scalarL2_normedAddCommGroup N L).toAddCommGroup
(scalarL2_module N L) (scalarL2_module N L) (RingHom.id ℂ)))
(permutationPullback σ)
(@LinearMap.id ℂ (ScalarL2 N L) Complex.instSemiring
(scalarL2_normedAddCommGroup N L).toAddCommMonoid (scalarL2_module N L)))
@[simp] lemma mem_bosonicSubmodule {N : ℕ} {L : Length} {ψ : ScalarL2 N L} :
ψ ∈ bosonicSubmodule N L ↔ bosonic ψ := by
simp only [bosonicSubmodule, Submodule.mem_iInf, LinearMap.mem_ker,
LinearMap.sub_apply, LinearMap.id_apply, sub_eq_zero]
exact (bosonic_iff_pullback ψ).symm
end
def groundFormSubmodule (N : ℕ) (L : Length) (a : ℝ) : Submodule ℂ (Jet N L) :=
dirichletSubmodule N L a ⊓
(bosonicSubmodule N L).comap (LinearMap.fst ℂ (ScalarL2 N L) (Direction N → ScalarL2 N L))
@[simp] lemma mem_groundFormSubmodule {N : ℕ} {L : Length} {a : ℝ} {u : Jet N L} :
u ∈ groundFormSubmodule N L a ↔ u ∈ dirichletDomain N L a ∧ bosonic u.1 := by
simp [groundFormSubmodule]
end HardSphere
end
end OAI
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