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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 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.UnitCenter
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators
open scoped BigOperators ComplexConjugate
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Real Complex
open scoped ENNReal ComplexConjugate
open ProbabilityTheory
open MeasureTheory Filter Set
open scoped ENNReal Topology
open MeasureTheory Set
open scoped ENNReal
namespace HardSphere
open MeasureTheory Set Filter SphericalCoordinates
open scoped ENNReal
def unitPairCount {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(p : Fin n → Prop) [DecidablePred p] (v : CubeCell K) (X : Configuration n L) : ℝ :=
(unitOccupation hL p v X:ℝ)*((unitOccupation hL p v X:ℝ)-1)
lemma unitPairCount_nonneg {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(p : Fin n → Prop) [DecidablePred p] (v : CubeCell K) (X : Configuration n L) :
0≤unitPairCount hL p v X := by
unfold unitPairCount
by_cases h : unitOccupation hL p v X=0
· simp only [h,Nat.cast_zero,zero_mul,le_refl]
· have hp : (1:ℝ)≤unitOccupation hL p v X := by exact_mod_cast Nat.one_le_iff_ne_zero.mpr h
exact mul_nonneg (Nat.cast_nonneg _) (sub_nonneg.mpr hp)
lemma unitPairCount_measurable {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(p : Fin n → Prop) [DecidablePred p] (v : CubeCell K) : Measurable (unitPairCount hL p v) := by
have hm : Measurable (fun X : Configuration n L ↦ fun i ↦ unitSite hL (X i)) :=
Measurable.of_eval (fun i ↦ (measurable_unitSite hL).comp (measurable_pi_apply i))
let W : (Fin n → CubeCell K) → ℝ := fun σ ↦
(Fintype.card {i : {i // p i} // σ i.val=v}:ℝ)*((Fintype.card {i : {i // p i} // σ i.val=v}:ℝ)-1)
have hW : Measurable W := measurable_of_finite W
change Measurable (W ∘ fun X : Configuration n L ↦ fun i ↦ unitSite hL (X i))
exact hW.comp hm
lemma spatialPartition_cell_weight {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(A : Finset (Torus L)) (H : ℝ) (X : Configuration n L) :
(∑ v : CubeCell K, if spatialSiteDistance L A (unitCenter L v)+1≤H then
unitPairCount hL (fun _ ↦ True) v X else 0)≤
∑ σ : Fin n → Bool, (∑ v : CubeCell K, unitPairCount hL (fun i ↦ σ i=true) v X)*
‖(productPartition n L (spatialInside L A H) (spatialOutside L A H) σ).toFun X‖^2 := by
classical
simp_rw [Finset.sum_mul]
rw [Finset.sum_comm]
apply Finset.sum_le_sum
intro v _
by_cases hv : spatialSiteDistance L A (unitCenter L v)+1≤H
· rw [ite_eq_left hv]
apply le_of_eq
have hp := productPartition_norm_unity (spatialInside L A H) (spatialOutside L A H)
(spatialInside_real L A H) (spatialOutside_real L A H) (spatialPartition_unity L A H) X
calc
_ = unitPairCount hL (fun _ ↦ True) v X*
(∑ σ : Fin n → Bool, ‖(productPartition n L (spatialInside L A H) (spatialOutside L A H) σ).toFun X‖^2) := by rw [hp,mul_one]
_ = ∑ σ : Fin n → Bool, unitPairCount hL (fun _ ↦ True) v X*
‖(productPartition n L (spatialInside L A H) (spatialOutside L A H) σ).toFun X‖^2 := Finset.mul_sum ..
_ = _ := by
apply Finset.sum_congr rfl
intro σ _
by_cases hz : (productPartition n L (spatialInside L A H) (spatialOutside L A H) σ).toFun X=0
· simp only [hz,norm_zero,zero_pow (by norm_num : 2≠0),mul_zero]
· unfold unitPairCount
rw [spatialPartition_cell_occupation hL A H σ X hz v hv]
· rw [ite_eq_right hv]
exact Finset.sum_nonneg (fun σ _ ↦ mul_nonneg (unitPairCount_nonneg hL _ v X) (sq_nonneg _))
lemma spatialPartition_cell_integral {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(A : Finset (Torus L)) (H : ℝ) {r : ℝ} (hr : 0<r) (u : Jet n L)
(hu : u∈dirichletDomain n L r) :
(∑ v : CubeCell K, (Ici (spatialSiteDistance L A (unitCenter L v)+1)).indicator
(fun _ : ℝ ↦ ENNReal.ofReal (1/(64*Real.sqrt 3)*r)*
(∫⁻ X, ENNReal.ofReal (unitPairCount hL (fun _ ↦ True) v X)*ENNReal.ofReal (‖u.1 X‖^2))) H)≤
ENNReal.ofReal (spatialPartitionEnergy L A H u) := by
classical
let w := fun σ : Fin n → Bool ↦ productPartition n L (spatialInside L A H) (spatialOutside L A H) σ
let c : ℝ := 1/(64*Real.sqrt 3)*r
have hc : 0≤c := by dsimp [c]; positivity
let q := fun (p : Fin n → Prop) [DecidablePred p] (v : CubeCell K) (X : Configuration n L) ↦ unitPairCount hL p v X
have hcoe : ∀ᵐ X ∂volume, ∀ σ, (w σ).onScalar u.1 X=(w σ).toFun X*u.1 X :=
ae_all_iff.mpr (fun σ ↦ (w σ).coe_onScalar u.1)
have hm (σ : Fin n → Bool) : Measurable (fun X : Configuration n L ↦
ENNReal.ofReal (c*∑ v : CubeCell K, q (fun i ↦ σ i=true) v X)*ENNReal.ofReal (‖(w σ).onScalar u.1 X‖^2)) :=
((Finset.measurable_sum _ (fun v _ ↦ unitPairCount_measurable hL _ v)).const_mul c).ennreal_ofReal.mul
((Lp.stronglyMeasurable ((w σ).onScalar u.1)).measurable.norm.pow_const 2).ennreal_ofReal
have hn (σ : Fin n → Bool) (X : Configuration n L) : 0≤∑ v : CubeCell K, q (fun i ↦ σ i=true) v X :=
Finset.sum_nonneg (fun v _ ↦ unitPairCount_nonneg hL _ v X)
have havg : (∫⁻ X, ENNReal.ofReal (c*(∑ v : CubeCell K, if spatialSiteDistance L A (unitCenter L v)+1≤H then q (fun _ ↦ True) v X else 0))*ENNReal.ofReal (‖u.1 X‖^2))≤
ENNReal.ofReal (spatialPartitionEnergy L A H u) := by
calc
_ ≤ ∫⁻ X, ∑ σ : Fin n → Bool, ENNReal.ofReal (c*∑ v : CubeCell K, q (fun i ↦ σ i=true) v X)*ENNReal.ofReal (‖(w σ).onScalar u.1 X‖^2) := by
apply lintegral_mono_ae
filter_upwards [hcoe] with X hX
have hh := ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left (spatialPartition_cell_weight hL A H X) hc)
conv_rhs at hh => rw [Finset.mul_sum]
have hp (σ : Fin n → Bool) : 0≤c*((∑ v : CubeCell K, q (fun i ↦ σ i=true) v X)*‖(w σ).toFun X‖^2) :=
mul_nonneg hc (mul_nonneg (hn σ X) (sq_nonneg _))
rw [ENNReal.ofReal_sum_of_nonneg (fun σ _ ↦ hp σ)] at hh
simp_rw [←mul_assoc,ENNReal.ofReal_mul (mul_nonneg hc (hn _ X))] at hh
simp_rw [hX,norm_mul,mul_pow,ENNReal.ofReal_mul (sq_nonneg _),←mul_assoc,←Finset.sum_mul]
exact mul_le_mul_of_nonneg_right hh zero_le
_ = ∑ σ : Fin n → Bool, ∫⁻ X, ENNReal.ofReal (c*∑ v : CubeCell K, q (fun i ↦ σ i=true) v X)*ENNReal.ofReal (‖(w σ).onScalar u.1 X‖^2) := by
rw [lintegral_finsetSum _ (fun σ _ ↦ hm σ)]
_ ≤ ∑ σ : Fin n → Bool, ENNReal.ofReal (selectedEnergy (fun i ↦ σ i=true) ((w σ).onJet u)) := by
apply Finset.sum_le_sum
intro σ _
have hh := dirichlet_physical_cell_energy hL (fun i ↦ σ i=true) hr ((w σ).onJet u) ((w σ).onJet_mem_domain hu)
refine (le_of_eq ?_).trans hh
apply lintegral_congr
intro X
congr 2
dsimp [c,q,unitPairCount]
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro v _
ring
_ = _ := by rw [spatialPartitionEnergy,ENNReal.ofReal_sum_of_nonneg (fun σ _ ↦ selectedEnergy_nonneg _ _)]
apply le_trans _ havg
apply le_of_eq
have hp (X : Configuration n L) :
ENNReal.ofReal (c*(∑ v : CubeCell K, if spatialSiteDistance L A (unitCenter L v)+1≤H then q (fun _ ↦ True) v X else 0))*ENNReal.ofReal (‖u.1 X‖^2)=
∑ v : CubeCell K, ENNReal.ofReal (c*(if spatialSiteDistance L A (unitCenter L v)+1≤H then q (fun _ ↦ True) v X else 0))*ENNReal.ofReal (‖u.1 X‖^2) := by
have hpv (v : CubeCell K) : 0≤c*(if spatialSiteDistance L A (unitCenter L v)+1≤H then q (fun _ ↦ True) v X else 0) := by
apply mul_nonneg hc
split_ifs
· exact unitPairCount_nonneg hL _ v X
· exact le_refl _
rw [Finset.mul_sum,ENNReal.ofReal_sum_of_nonneg (fun v _ ↦ hpv v),Finset.sum_mul]
simp_rw [hp]
rw [lintegral_finsetSum]
· apply Finset.sum_congr rfl
intro v _
by_cases hv : spatialSiteDistance L A (unitCenter L v)+1≤H
· rw [indicator_of_mem (show H∈Ici (spatialSiteDistance L A (unitCenter L v)+1) from hv)]
simp only [ite_eq_left hv,ENNReal.ofReal_mul hc,mul_assoc]
rw [lintegral_const_mul' _ _ (ENNReal.ofReal_ne_top)]
· rw [indicator_of_notMem (show H∉Ici (spatialSiteDistance L A (unitCenter L v)+1) from hv)]
simp only [ite_eq_right hv,mul_zero,ENNReal.ofReal_zero,zero_mul,lintegral_zero]
· intro v _
by_cases hv : spatialSiteDistance L A (unitCenter L v)+1≤H
· simp only [ite_eq_left hv]
exact (((unitPairCount_measurable hL _ v).const_mul c).ennreal_ofReal.mul
((Lp.stronglyMeasurable u.1).measurable.norm.pow_const 2).ennreal_ofReal)
· simp only [ite_eq_right hv,mul_zero,ENNReal.ofReal_zero,zero_mul]
exact measurable_const
theorem spatial_cell_average {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(A : Finset (Torus L)) {r : ℝ} (hr : 0<r) (u : Jet n L) (hu : u∈dirichletDomain n L r) :
(∑ v : CubeCell K, ENNReal.ofReal (Real.exp (-(spatialSiteDistance L A (unitCenter L v)+1)))*
(ENNReal.ofReal (1/(64*Real.sqrt 3)*r)*
(∫⁻ X, ENNReal.ofReal (unitPairCount hL (fun _ ↦ True) v X)*ENNReal.ofReal (‖u.1 X‖^2))))≤
∫⁻ H, ENNReal.ofReal (spatialPartitionEnergy L A H u) ∂radiusLaw := by
have hb := lintegral_mono (μ:=radiusLaw) (fun H ↦ spatialPartition_cell_integral hL A H hr u hu)
rw [lintegral_finsetSum] at hb
· apply le_trans _ hb
apply Finset.sum_le_sum
intro v _
rw [lintegral_indicator_const measurableSet_Ici]
have hd : 0≤ spatialSiteDistance L A (unitCenter L v)+1 := by
have := spatialSiteDistance_nonneg L A (unitCenter L v)
linarith
have ht : radiusLaw (Ici (spatialSiteDistance L A (unitCenter L v)+1))=
ENNReal.ofReal (Real.exp (-(spatialSiteDistance L A (unitCenter L v)+1))) := by
rw [←radiusLaw_tail_eq hd]
exact (ENNReal.ofReal_toReal (measure_ne_top radiusLaw _)).symm
rw [ht,mul_comm]
· intro v _
exact measurable_const.indicator measurableSet_Ici
end HardSphere
end
end OAI
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