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159357f | 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 | import Mathlib
import OAI.Analysis.CoulombIonization.Fermionic.Multiplier
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff InnerProductSpace Convolution
open Filter
open scoped Topology InnerProductSpace
open MeasureTheory Complex Filter
open scoped Topology InnerProductSpace
open MeasureTheory Complex Filter
open scoped Topology InnerProductSpace ContDiff
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff InnerProductSpace Convolution
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff InnerProductSpace
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff InnerProductSpace ENNReal
open MeasureTheory Filter
open scoped Topology ContDiff BigOperators
open Set Filter Topology InnerProductSpace Laplacian
open MeasureTheory Filter
open scoped Topology
open MeasureTheory Filter
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter
open scoped Topology BigOperators InnerProductSpace
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter
open scoped Topology BigOperators Pointwise
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian
open scoped Convolution
open scoped RealInnerProductSpace
open MeasureTheory Filter Set Metric
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian
open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian
open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian
open Filter Set Metric Topology InnerProductSpace Laplacian
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology
open Filter
open scoped Topology
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Filter Set Metric Topology
open MeasureTheory Complex Filter
open scoped Topology InnerProductSpace ContDiff BigOperators
namespace CoulombAtom
lemma graph_value_components_ext {N : ℕ} {F G : fermionGraph N}
(h : ∀ s, graphComponent s none F = graphComponent s none G) : F = G := by
apply fermionGraphValue_injective N
apply Subtype.ext
apply PiLp.ext
intro s
exact h s
lemma complex_square_partition (p q : ℝ) (h : p ^ 2 + q ^ 2 = 1) (z : ℂ) :
(p : ℂ) * ((p : ℂ) * z) + (q : ℂ) * ((q : ℂ) * z) = z := by
calc
_ = ((p ^ 2 + q ^ 2 : ℝ) : ℂ) * z := by push_cast; ring
_ = z := by rw [h]; simp
attribute [local irreducible] FermionMultiplier.apply graphComponent
fermionGraphValue coulombFormOperator graphFormVector formEnergy
lemma FermionMultiplier.partition_apply {N : ℕ} (p q : FermionMultiplier N)
(h : ∀ x, p.value x ^ 2 + q.value x ^ 2 = 1) (F : fermionGraph N) :
p.apply (p.apply F) + q.apply (q.apply F) = F := by
apply graph_value_components_ext
intro s
rw [map_add]
apply Lp.ext
filter_upwards [Lp.coeFn_add (graphComponent s none (p.apply (p.apply F)))
(graphComponent s none (q.apply (q.apply F))),
p.apply_value (p.apply F) s, p.apply_value F s,
q.apply_value (q.apply F) s, q.apply_value F s] with x ha hb hc hd he
exact ha.trans ((congrArg₂ (fun a b : ℂ => a + b)
(hb.trans (congrArg (fun z : ℂ => (p.value x : ℂ) * z) hc))
(hd.trans (congrArg (fun z : ℂ => (q.value x : ℂ) * z) he))).trans
(complex_square_partition (p.value x) (q.value x) (h x) _))
lemma FermionMultiplier.partition_mass {N : ℕ} (p q : FermionMultiplier N)
(h : ∀ x, p.value x ^ 2 + q.value x ^ 2 = 1) (F : fermionGraph N) :
‖fermionGraphValue N (p.apply F)‖ ^ 2 +
‖fermionGraphValue N (q.apply F)‖ ^ 2 = ‖fermionGraphValue N F‖ ^ 2 := by
have he := congrArg (fermionGraphValue N) (p.partition_apply q h F)
have hi := congrArg (fun U => (⟪U, fermionGraphValue N F⟫_ℂ).re) he
simp only [map_add, inner_add_left, Complex.add_re] at hi
calc
_ = _ := congrArg₂ (fun a b : ℝ => a + b)
(p.value_pairing F).symm (q.value_pairing F).symm
_ = _ := hi
_ = _ := (norm_sq_eq_re_inner (𝕜 := ℂ) (fermionGraphValue N F)).symm
def multiplierError {N : ℕ} (p : FermionMultiplier N) (F : fermionGraph N) : ℝ :=
(1 / 2 : ℝ) * (∑ s : Spins N, ∑ i : Fin N, ∑ a : Fin 3,
∫ x, (lineDeriv ℝ p.value x (direction i a)) ^ 2 *
‖graphComponent s none F x‖ ^ 2)
lemma multiplierError_nonneg {N : ℕ} (p : FermionMultiplier N) (F : fermionGraph N) :
0 ≤ multiplierError p F := by
apply mul_nonneg (by norm_num)
apply Finset.sum_nonneg; intro s _
apply Finset.sum_nonneg; intro i _
apply Finset.sum_nonneg; intro a _
exact integral_nonneg fun x => mul_nonneg (sq_nonneg _) (sq_nonneg _)
theorem quantum_partition_ims {N : ℕ} (Z : ℝ) (p q : FermionMultiplier N)
(h : ∀ x, p.value x ^ 2 + q.value x ^ 2 = 1) (F : fermionGraph N) :
formEnergy Z (graphFormVector (p.apply F)) +
formEnergy Z (graphFormVector (q.apply F)) =
formEnergy Z (graphFormVector F) + multiplierError p F + multiplierError q F := by
have hp := p.form_ims Z F
have hq := q.form_ims Z F
have hi : (⟪p.apply (p.apply F), coulombFormOperator Z N F⟫_ℂ).re +
(⟪q.apply (q.apply F), coulombFormOperator Z N F⟫_ℂ).re =
formEnergy Z (graphFormVector F) := by
rw [← Complex.add_re, ← inner_add_left, p.partition_apply q h F,
coulombFormOperator_inner]
change _ = multiplierError p F at hp
change _ = multiplierError q F at hq
linarith
lemma quantum_form_above_sector {Z : ℝ} (hZ : 0 ≤ Z) {N : ℕ} (F : fermionGraph N) :
energy Z N * ‖fermionGraphValue N F‖ ^ 2 ≤ formEnergy Z (graphFormVector F) := by
rw [← coulombFormOperator_inner]
exact form_lower_of_unit (fermionGraphValue N) (fermionGraphValue_injective N)
(coulombFormOperator Z N) (fun U hU => energy_le_graphForm hZ U hU) F
theorem quantum_partition_excess {Z b : ℝ} (hZ : 0 ≤ Z) {N : ℕ}
(hb : b ≤ energy Z N) (p q : FermionMultiplier N)
(h : ∀ x, p.value x ^ 2 + q.value x ^ 2 = 1) (F : fermionGraph N) :
formEnergy Z (graphFormVector (p.apply F)) -
b * ‖fermionGraphValue N (p.apply F)‖ ^ 2 ≤
formEnergy Z (graphFormVector F) - b * ‖fermionGraphValue N F‖ ^ 2 +
multiplierError p F + multiplierError q F := by
have hl := quantum_form_above_sector hZ (q.apply F)
have hbq := mul_le_mul_of_nonneg_right hb (sq_nonneg ‖fermionGraphValue N (q.apply F)‖)
have hmass := p.partition_mass q h F
have henergy := quantum_partition_ims Z p q h F
have hm := congrArg (fun x : ℝ => b * x) hmass
simp only [mul_add] at hm
linarith
end CoulombAtom
open MeasureTheory Filter Set
open scoped Topology BigOperators
end
end OAI
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