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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 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.SmoothSiteWeight
import OAI.Analysis.HardSphere.Geometry.ScalarNormSqIntegral
import OAI.Analysis.HardSphere.Geometry.BoxWeight
import OAI.Analysis.HardSphere.Conditioning.BathPhase
namespace OAI
section
section
noncomputable section
open MeasureTheory MeasureTheory.Measure Set Filter
open scoped ENNReal NNReal Topology BigOperators ComplexConjugate
namespace HardSphere
lemma cons_measurableEmbedding {n : ℕ} (L : Length) :
MeasurableEmbedding (fun p : Configuration n L × Torus L ↦ Fin.cons (α:=fun _ ↦ Torus L) p.2 p.1) := by
have h := ((MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) 0).symm.measurableEmbedding).comp
(MeasurableEquiv.prodComm : Configuration n L × Torus L ≃ᵐ Torus L × Configuration n L).measurableEmbedding
convert h using 1
funext pair
change Fin.cons pair.2 pair.1 = Fin.insertNth (α:=fun _ : Fin (n+1) ↦ Torus L) 0 pair.2 pair.1
exact (Fin.insertNth_zero' _ _).symm
lemma ground_bath_integral {n : ℕ} {L : Length} {u : Jet (n+1) L} (hu : ‖u.1‖=1)
(F : Configuration n L → ℝ) (hF : Measurable F) {B : ℝ} (hB : ∀Y,|F Y|≤B) :
(∫Y,F Y*(∫x,‖u.1 (Fin.cons x Y)‖^2)) =
∫Y,F Y ∂(groundPositionLaw (n+1) L u).map (fun X ↦ fun i : Fin n ↦ X i.succ) := by
let μ := groundPositionLaw (n+1) L u
let := groundPositionLaw_probability u hu
have hm : Measurable (fun X : Configuration (n+1) L ↦ fun i : Fin n ↦ X i.succ) := by fun_prop
rw [integral_map hm.aemeasurable hF.aestronglyMeasurable,groundPositionLaw_integral]
let f : Configuration n L × Torus L → ℝ := fun p ↦ F p.1*‖u.1 (Fin.cons p.2 p.1)‖^2
have hf : Integrable f (volume.prod volume) := by
have hu' := (Lp.memLp u.1).comp_measurePreserving (cons_preserving (n:=n) L)
apply ((hu'.integrable_norm_pow (by norm_num : (2:ℕ)≠0)).const_mul B).mono'
· exact (hF.comp measurable_fst).aestronglyMeasurable.mul (hu'.aestronglyMeasurable.norm.pow 2)
· exact Eventually.of_forall fun p ↦ by
change ‖F p.1*‖u.1 (Fin.cons p.2 p.1)‖^2‖ ≤ B*‖u.1 (Fin.cons p.2 p.1)‖^2
simp only [Real.norm_eq_abs,abs_mul,abs_sq]
exact mul_le_mul_of_nonneg_right (hB _) (sq_nonneg _)
calc
_ = ∫p,f p ∂volume.prod volume := by rw [integral_prod _ hf]; simp only [f,integral_const_mul]
_ = _ := by
have he := (cons_preserving (n:=n) L).integral_comp (cons_measurableEmbedding (n:=n) L)
(fun X ↦ F (fun i : Fin n ↦ X i.succ)*‖u.1 X‖^2)
simpa only [Fin.cons_succ] using he
lemma ground_physical_phase_transfer {n : ℕ} {L : Length} {r h : ℝ}
{u : Jet (n+1) L} (hu : IsGroundVector (n+1) L r u) (hh : 0<h) :
Phase.modulusOccupation (fun p : Configuration n L × Torus L ↦ u.1 (Fin.cons p.2 p.1)) volume volume -
4*Real.sqrt ((∫Y,(volume : Measure (Torus L)).real (BathSlice.omitted h r Y)
∂(groundPositionLaw (n+1) L u).map (fun X ↦ fun i : Fin n ↦ X i.succ))/L.val^3) ≤
Phase.occupation (fun p : Configuration n L × Torus L ↦ u.1 (Fin.cons p.2 p.1)) volume volume := by
have hLp := L.property
let f : Configuration n L × Torus L → ℂ := fun p ↦ u.1 (Fin.cons p.2 p.1)
have hf : MemLp f 2 (volume.prod volume) :=
(Lp.memLp u.1).comp_measurePreserving (cons_preserving (n:=n) L)
have hn : ∫p,‖f p‖^2 ∂volume.prod volume=1 := by
rw [(cons_preserving (n:=n) L).integral_comp (cons_measurableEmbedding (n:=n) L)
(fun X ↦ ‖u.1 X‖^2),←scalar_norm_sq_integral,hu.2.2.1]
norm_num
have hV : (volume : Measure (Torus L)).real univ=L.val^3 := by
rw [Measure.real,BathSlice.volume_torus,ENNReal.toReal_ofReal (by positivity)]
have hU : MeasurableSet {p : Configuration n L × Torus L | p.2∈BathSlice.retained h r p.1} :=
(BathSlice.measurableSet_omitted_graph L h r).compl
have ht := Phase.phase_transfer hf hn (by rw [hV]; positivity) hU (hu.ae_retained_commonPhase hh)
have hl : Phase.geometricLoss f {p | p.2∈BathSlice.retained h r p.1} volume volume =
(∫Y,(volume : Measure (Torus L)).real (BathSlice.omitted h r Y)
∂(groundPositionLaw (n+1) L u).map (fun X ↦ fun i : Fin n ↦ X i.succ))/L.val^3 := by
unfold Phase.geometricLoss
simp only [hV]
congr 1
convert ground_bath_integral hu.2.2.1 (fun Y ↦ (volume : Measure (Torus L)).real (BathSlice.omitted h r Y))
(BathSlice.measurable_omitted_volume L h r).ennreal_toReal (B:=L.val^3) (fun Y ↦ by
rw [abs_of_nonneg measureReal_nonneg,←hV]
exact measureReal_mono (subset_univ _)) using 1
apply integral_congr_ae
exact Eventually.of_forall fun Y ↦ by
have he : Prod.mk Y ⁻¹' {p : Configuration n L × Torus L | p.2∈BathSlice.retained h r p.1}ᶜ = BathSlice.omitted h r Y := by
ext x
simp [BathSlice.retained]
dsimp only
rw [he]
simpa only [hl] using ht
end HardSphere
end
end
end
end OAI
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