File size: 11,829 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 | import Mathlib
import OAI.Analysis.CoulombIonization.Fermionic.QuantumEventTilt
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Set Finset
open scoped ENNReal NNReal BigOperators
namespace CoulombObservation
def dyadicObservationWidth (r : ℝ) (k : ℕ) : ℝ := ((2:ℝ)^k * r)^(1.01:ℝ)
lemma dyadicObservationWidth_pos {r : ℝ} (hr : 0 < r) (k : ℕ) :
0 < dyadicObservationWidth r k := Real.rpow_pos_of_pos (by positivity) _
lemma dyadicObservationWidth_inv_sq {r : ℝ} (hr : 0 < r) (k : ℕ) :
(dyadicObservationWidth r k)⁻¹^2 =
r^(-2.02:ℝ) * ((2:ℝ)^(-2.02:ℝ))^k := by
unfold dyadicObservationWidth
rw [← Real.rpow_natCast, ← Real.rpow_neg (by positivity), ← Real.rpow_mul (by positivity)]
norm_num
rw [Real.mul_rpow (by positivity) hr.le, ← Real.rpow_pow_comm (by norm_num)]
ring
lemma dyadicObservationRatio_nonneg : 0 ≤ (2:ℝ)^(-2.02:ℝ) := Real.rpow_nonneg (by norm_num) _
lemma dyadicObservationRatio_le_half : (2:ℝ)^(-2.02:ℝ) ≤ 1/2 := by
calc
_ ≤ (2:ℝ)^(-1:ℝ) := Real.rpow_le_rpow_of_exponent_le (by norm_num) (by norm_num)
_ = _ := by norm_num
lemma geometric_sum_le_two {q : ℝ} (hq : 0 ≤ q) (hq2 : q ≤ 1/2) (K : ℕ) :
(∑ k ∈ range K, q^k) ≤ 2 := by
have hh (n : ℕ) : (∑ k ∈ range n, q^k) ≤ 2 * (1-q^n) := by
induction n with
| zero => simp
| succ n ih =>
rw [sum_range_succ, pow_succ]
nlinarith [pow_nonneg hq n, mul_le_mul_of_nonneg_right hq2 (pow_nonneg hq n)]
exact (hh K).trans (by nlinarith [pow_nonneg hq K])
lemma dyadicObservation_inv_sq_sum {r : ℝ} (hr : 0 < r) (K : ℕ) :
(∑ k : Fin K, (dyadicObservationWidth r k)⁻¹^2) ≤ 2*r^(-2.02:ℝ) := by
simp only [dyadicObservationWidth_inv_sq hr]
rw [← mul_sum, Fin.sum_univ_eq_sum_range]
have hh := mul_le_mul_of_nonneg_left
(geometric_sum_le_two dyadicObservationRatio_nonneg dyadicObservationRatio_le_half K)
(Real.rpow_nonneg hr.le (-2.02))
simpa only [mul_comm] using hh
end CoulombObservation
namespace CoulombAtom
open CoulombObservation
theorem quantum_dyadic_observation_event_tilt {Z : ℝ} (hZ : 0 ≤ Z) {N : ℕ}
(F : fermionGraph N) (hn : ‖fermionGraphValue N F‖^2 = 1)
(hF : formEnergy Z (graphFormVector F) = energy Z N)
{r : ℝ} (hr : 0 < r) (K : ℕ)
{s : Set (Fin K × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s)
(hsy : QuantumEventSymmetric s)
(hp : 0 < quantumEventProbability F (fun k : Fin K => (dyadicObservationWidth r k)⁻¹) s) :
∃ G : fermionGraph N,
‖fermionGraphValue N G‖^2 = 1 ∧
graphRawLaw G =
(ENNReal.ofReal (quantumEventProbability F
(fun k : Fin K => (dyadicObservationWidth r k)⁻¹) s))⁻¹ •
Measure.map Prod.fst
(((graphRawLaw F).prod
(Measure.pi (fun _ : Fin K × (Fin N × Fin 3) => compactNoiseLaw))).restrict
(quantumObservationEvent (fun k : Fin K => (dyadicObservationWidth r k)⁻¹) s)) ∧
formEnergy Z (graphFormVector G) ≤ energy Z N +
observationFisherConstant * r^(-2.02:ℝ) *
(Real.log (Real.exp 1/quantumEventProbability F
(fun k : Fin K => (dyadicObservationWidth r k)⁻¹) s))^5 := by
let b : Fin K → ℝ := fun k => (dyadicObservationWidth r k)⁻¹
obtain ⟨G,hnG,hlaw,hE⟩ := quantum_observation_event_tilt hZ F hn hF b hs hsy hp
refine ⟨G,hnG,hlaw,hE.trans ?_⟩
have hp1 : quantumEventProbability F b s ≤ 1 := by
let := graphRawLaw_probability F hn
have hi := integrable_bounded_probability (μ := graphRawLaw F)
(quantumEventLikelihood_measurable b hs) (quantumEventLikelihood_nonneg b hs)
(quantumEventLikelihood_le_one b hs)
have hh := integral_mono hi (integrable_const (1:ℝ)) (quantumEventLikelihood_le_one b hs)
simpa [quantumEventProbability] using hh
have hlog : 0 ≤ (Real.log (Real.exp 1/quantumEventProbability F b s))^5 := by
apply pow_nonneg
rw [eventLog_eq hp]
have h := Real.log_nonpos hp.le hp1
linarith
have hsum := dyadicObservation_inv_sq_sum hr K
have hC := observationFisherConstant_pos
have hh := mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hsum (show 0 ≤ observationFisherConstant/2 by positivity)) hlog
apply add_le_add_right
calc
_ ≤ _ := hh
_ = _ := by ring
end CoulombAtom
open MeasureTheory Set Finset
open scoped ENNReal NNReal BigOperators
namespace CoulombObservation
def scaleObservationArray {J I : Type*} [Fintype J] [Fintype I]
(ell : J → ℝ) : (J × I → ℝ) →L[ℝ] (J × I → ℝ) :=
ContinuousLinearMap.pi (fun q => ell q.1 • ContinuousLinearMap.proj q)
@[simp] lemma scaleObservationArray_apply {J I : Type*} [Fintype J] [Fintype I]
(ell : J → ℝ) (u : J × I → ℝ) (q : J × I) :
scaleObservationArray ell u q = ell q.1 * u q := rfl
lemma scaleObservationArray_reindex {J I : Type*} [Fintype J] [Fintype I]
(ell : J → ℝ) (e : Equiv.Perm I) (u : J × I → ℝ) :
scaleObservationArray ell (arrayReindex e u) = arrayReindex e (scaleObservationArray ell u) := by
ext q
simp only [scaleObservationArray_apply, arrayReindex_apply]
end CoulombObservation
namespace CoulombAtom
open CoulombObservation
def physicalObservationArray {N : ℕ} {J : Type*} [Fintype J]
(ell : J → ℝ) (z : Configuration N × (J × (Fin N × Fin 3) → ℝ)) :
J × (Fin N × Fin 3) → ℝ := fun q => z.1 q.2.1 q.2.2 + ell q.1 * z.2 q
def physicalObservationEvent {N : ℕ} {J : Type*} [Fintype J]
(ell : J → ℝ) (s : Set (J × (Fin N × Fin 3) → ℝ)) :
Set (Configuration N × (J × (Fin N × Fin 3) → ℝ)) :=
physicalObservationArray ell ⁻¹' s
def physicalObservationProbability {N : ℕ} {J : Type*} [Fintype J]
(F : fermionGraph N) (ell : J → ℝ) (s : Set (J × (Fin N × Fin 3) → ℝ)) : ℝ :=
(((graphRawLaw F).prod (Measure.pi (fun _ : J × (Fin N × Fin 3) => compactNoiseLaw)))
(physicalObservationEvent ell s)).toReal
lemma physicalObservationEvent_eq {N : ℕ} {J : Type*} [Fintype J]
(ell : J → ℝ) (hell : ∀ j, 0 < ell j) (s : Set (J × (Fin N × Fin 3) → ℝ)) :
quantumObservationEvent (fun j => (ell j)⁻¹) (scaleObservationArray ell ⁻¹' s) =
physicalObservationEvent ell s := by
have he (z : Configuration N × (J × (Fin N × Fin 3) → ℝ)) :
scaleObservationArray ell
(observationPlacement (fun j => (ell j)⁻¹) (flattenConfiguration N z.1) + z.2) =
physicalObservationArray ell z := by
ext q
simp only [scaleObservationArray_apply, Pi.add_apply, observationPlacement_apply,
flattenConfiguration_apply, physicalObservationArray]
rw [mul_add, ← mul_assoc, mul_inv_cancel₀ (hell q.1).ne', one_mul]
ext z
exact congrArg (· ∈ s) (he z) |>.to_iff
lemma scaledObservationEvent_symmetric {N : ℕ} {J : Type*} [Fintype J]
(ell : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hsy : QuantumEventSymmetric s) :
QuantumEventSymmetric (scaleObservationArray ell ⁻¹' s) := by
intro π u
change scaleObservationArray ell (arrayReindex (π.prodCongr (Equiv.refl (Fin 3))) u) ∈ s ↔
scaleObservationArray ell u ∈ s
rw [scaleObservationArray_reindex]
exact hsy π _
lemma quantumEventProbability_eq_physical {N : ℕ} {J : Type*} [Fintype J]
(F : fermionGraph N) (hn : ‖fermionGraphValue N F‖^2 = 1)
(ell : J → ℝ) (hell : ∀ j, 0 < ell j)
{s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) :
quantumEventProbability F (fun j => (ell j)⁻¹) (scaleObservationArray ell ⁻¹' s) =
physicalObservationProbability F ell s := by
have he := quantumEventProbability_ofReal F hn (fun j => (ell j)⁻¹)
(hs.preimage (scaleObservationArray ell).measurable)
rw [physicalObservationEvent_eq ell hell] at he
have hp : 0 ≤ quantumEventProbability F (fun j => (ell j)⁻¹)
(scaleObservationArray ell ⁻¹' s) :=
integral_nonneg (quantumEventLikelihood_nonneg _ (hs.preimage (scaleObservationArray (I := Fin N × Fin 3) ell).measurable))
exact (ENNReal.toReal_ofReal hp).symm.trans (congrArg ENNReal.toReal he)
theorem quantum_physical_observation_event_tilt {Z : ℝ} (hZ : 0 ≤ Z) {N : ℕ}
(F : fermionGraph N) (hn : ‖fermionGraphValue N F‖^2 = 1)
(hF : formEnergy Z (graphFormVector F) = energy Z N)
{J : Type*} [Fintype J] (ell : J → ℝ) (hell : ∀ j, 0 < ell j)
{s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s)
(hsy : QuantumEventSymmetric s) (hp : 0 < physicalObservationProbability F ell s) :
∃ G : fermionGraph N,
‖fermionGraphValue N G‖^2 = 1 ∧
graphRawLaw G =
(ENNReal.ofReal (physicalObservationProbability F ell s))⁻¹ •
Measure.map Prod.fst
(((graphRawLaw F).prod
(Measure.pi (fun _ : J × (Fin N × Fin 3) => compactNoiseLaw))).restrict
(physicalObservationEvent ell s)) ∧
formEnergy Z (graphFormVector G) ≤ energy Z N +
(observationFisherConstant/2) * (∑ j, ((ell j)⁻¹)^2) *
(Real.log (Real.exp 1/physicalObservationProbability F ell s))^5 := by
have hm := hs.preimage (scaleObservationArray ell).measurable
have hp' : 0 < quantumEventProbability F (fun j => (ell j)⁻¹)
(scaleObservationArray ell ⁻¹' s) := by
rwa [quantumEventProbability_eq_physical F hn ell hell hs]
obtain ⟨G, hG, hlaw, hE⟩ := quantum_observation_event_tilt hZ F hn hF
(fun j => (ell j)⁻¹) hm (scaledObservationEvent_symmetric ell hsy) hp'
rw [quantumEventProbability_eq_physical F hn ell hell hs,
physicalObservationEvent_eq ell hell] at hlaw
rw [quantumEventProbability_eq_physical F hn ell hell hs] at hE
exact ⟨G,hG,hlaw,hE⟩
theorem quantum_physical_dyadic_event_tilt {Z : ℝ} (hZ : 0 ≤ Z) {N : ℕ}
(F : fermionGraph N) (hn : ‖fermionGraphValue N F‖^2 = 1)
(hF : formEnergy Z (graphFormVector F) = energy Z N)
{r : ℝ} (hr : 0 < r) (K : ℕ)
{s : Set (Fin K × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s)
(hsy : QuantumEventSymmetric s)
(hp : 0 < physicalObservationProbability F (fun k : Fin K => dyadicObservationWidth r k) s) :
∃ G : fermionGraph N,
‖fermionGraphValue N G‖^2 = 1 ∧
graphRawLaw G =
(ENNReal.ofReal (physicalObservationProbability F
(fun k : Fin K => dyadicObservationWidth r k) s))⁻¹ •
Measure.map Prod.fst
(((graphRawLaw F).prod
(Measure.pi (fun _ : Fin K × (Fin N × Fin 3) => compactNoiseLaw))).restrict
(physicalObservationEvent (fun k : Fin K => dyadicObservationWidth r k) s)) ∧
formEnergy Z (graphFormVector G) ≤ energy Z N +
observationFisherConstant * r^(-2.02:ℝ) *
(Real.log (Real.exp 1/physicalObservationProbability F
(fun k : Fin K => dyadicObservationWidth r k) s))^5 := by
let ell : Fin K → ℝ := fun k => dyadicObservationWidth r k
have hell : ∀ k, 0 < ell k := fun k => dyadicObservationWidth_pos hr k
have hm := hs.preimage (scaleObservationArray ell).measurable
have hp' : 0 < quantumEventProbability F (fun k : Fin K => (dyadicObservationWidth r k)⁻¹)
(scaleObservationArray ell ⁻¹' s) := by
rwa [quantumEventProbability_eq_physical F hn ell hell hs]
obtain ⟨G,hG,hlaw,hE⟩ := quantum_dyadic_observation_event_tilt hZ F hn hF hr K hm
(scaledObservationEvent_symmetric ell hsy) hp'
rw [quantumEventProbability_eq_physical F hn ell hell hs,
physicalObservationEvent_eq ell hell] at hlaw
rw [quantumEventProbability_eq_physical F hn ell hell hs] at hE
exact ⟨G,hG,hlaw,hE⟩
end CoulombAtom
end
end OAI
|