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9eeb9c7 | 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 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.FreePathExpectation
import OAI.Analysis.HardSphere.PathBounds.FreeVisitTravel
namespace OAI
section
section
noncomputable section
open MeasureTheory Set Filter
open scoped ENNReal NNReal Topology BigOperators
namespace HardSphere.FreeVisits
open HardSphere.Wiener HardSphere.BathSlice HardSphere.Lattice HardSphere.CellOrbital
attribute [local irreducible] freeTrajectory pathVisitedSites torusDistance
def singleVisitDefect {K : ℕ} (L : Length) (A : Finset (DiscreteTorus K))
(η : ℝ) (D : Set (ConfigurationPath 1 L)) (ω : ConfigurationPath 1 L) : ℝ≥0∞ := by
classical
exact ENNReal.ofReal (Real.exp (η*(if ω∈D then ((pathVisitedSites L (siteCenter L) A 0 ω).card:ℝ) else 0))-1)
lemma singleVisitDefect_measurable {K : ℕ} (L : Length) (A : Finset (DiscreteTorus K))
(η : ℝ) {D : Set (ConfigurationPath 1 L)} (hD : MeasurableSet D) :
Measurable (singleVisitDefect L A η D) := by
classical
exact ((measurable_const.mul ((pathVisitedSites_measurable L (siteCenter L) A 0).ite hD measurable_const)).exp.sub
measurable_const).ennreal_ofReal
lemma singleVisitDefect_reverse {K : ℕ} (L : Length) (A : Finset (DiscreteTorus K))
(η : ℝ) {D : Set (ConfigurationPath 1 L)}
(hD : ∀ω,reverseConfigurationPath 1 L ω∈D ↔ ω∈D) (ω : ConfigurationPath 1 L) :
singleVisitDefect L A η D (reverseConfigurationPath 1 L ω)=singleVisitDefect L A η D ω := by
classical
simp only [singleVisitDefect,hD,pathVisitedSites_reverse]
lemma free_single_defect_bound {η₀ : ℝ} (hη₀ : 0≤η₀) : ∃ C : ℝ, 0<C ∧
∀ {K : ℕ} [NeZero K] (L : Length), L.val=(K:ℝ) → ∀ A : Finset (DiscreteTorus K),
∀ {t η ε : ℝ}, 0<t → t≤2 → 0≤η → η≤η₀ → 0≤ε →
∀ D : Set (ConfigurationPath 1 L), MeasurableSet D →
(∀ y : Torus L, coordinateWiener 1 {w | freeTrajectory 1 L t w (oneConfiguration y)∈D}≤ENNReal.ofReal ε) →
∀ y : Torus L,
oneFreeExpectation t (singleVisitDefect L A η D) y ≤
ENNReal.ofReal (C*η*ε^(1/3:ℝ)*Real.exp (-spatialSiteDistance L (A.image (siteCenter L)) y)) := by
classical
obtain ⟨C,hC,hbound⟩ := free_visit_bound hη₀
refine ⟨C.toReal+1,by positivity,?_⟩
intro K _ L hL A t η ε ht ht2 hη hηη hε D hD hprob y
let E := {w : CoordinatePaths 1 | freeTrajectory 1 L t w (oneConfiguration y)∈D}
have hE : MeasurableSet E := hD.preimage
((freeTrajectory_continuous 1 L t).measurable.comp (measurable_id.prodMk measurable_const))
have H := hbound L hL A 0 ht ht2 hη hηη (oneConfiguration y) E hE (ENNReal.ofReal ε) (hprob y)
have heq : oneFreeExpectation t (singleVisitDefect L A η D) y =
∫⁻ w,visitDefect L A 0 t η (oneConfiguration y) E w ∂coordinateWiener 1 := by
rfl
rw [heq]
apply H.trans
have hC' : C≤ENNReal.ofReal (C.toReal+1) := by
calc
C = ENNReal.ofReal C.toReal := (ENNReal.ofReal_toReal hC.ne).symm
_ ≤ ENNReal.ofReal (C.toReal+1) := ENNReal.ofReal_le_ofReal (by linarith)
calc
C*ENNReal.ofReal η*ENNReal.ofReal ε^(1/3:ℝ)*
ENNReal.ofReal (Real.exp (-visitStartDistance L A 0 (oneConfiguration y))) ≤
ENNReal.ofReal (C.toReal+1)*ENNReal.ofReal η*ENNReal.ofReal ε^(1/3:ℝ)*
ENNReal.ofReal (Real.exp (-visitStartDistance L A 0 (oneConfiguration y))) := by gcongr
_ = _ := by
rw [ENNReal.ofReal_rpow_of_nonneg hε (show 0≤(1/3:ℝ) by norm_num),
←ENNReal.ofReal_mul (show 0≤C.toReal+1 by positivity),
←ENNReal.ofReal_mul (show 0≤(C.toReal+1)*η by positivity),
←ENNReal.ofReal_mul (show 0≤(C.toReal+1)*η*ε^(1/3:ℝ) by positivity)]
rfl
end HardSphere.FreeVisits
end
end
end
end OAI
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