File size: 9,858 Bytes
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 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 | import Mathlib
import OAI.Analysis.HardSphere.PathBounds.InteractingVisits
namespace OAI
section
section
noncomputable section
open MeasureTheory Set Filter
open scoped ENNReal NNReal Topology BigOperators
namespace HardSphere.ShiftVisits
open HardSphere.Wiener HardSphere.Lattice HardSphere.CellOrbital HardSphere.BathSlice HardSphere.FreeVisits
attribute [local irreducible] freeTrajectory
def defectiveTotalVisits {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K))
(D : Set (ConfigurationPath 1 L)) (ω : ConfigurationPath n L) : ℕ := by
classical
exact ∑ i, if onePathAt i ω∈D then (pathVisitedSites L (center L c) A i ω).card else 0
lemma pathVisitedSites_onePathAt {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K))
(i : Fin n) (ω : ConfigurationPath n L) :
pathVisitedSites L (center L c) A 0 (onePathAt i ω)=pathVisitedSites L (center L c) A i ω := by
simp only [pathVisitedSites,pathVisitsBall,onePathAt,oneConfiguration,ContinuousMap.coe_mk]
rfl
lemma exp_defectiveTotalVisits {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K))
(D : Set (ConfigurationPath 1 L)) {η : ℝ} (hη : 0≤η) (ω : ConfigurationPath n L) :
ENNReal.ofReal (Real.exp (η*(defectiveTotalVisits L c A D ω : ℝ)))=
∏ i, (1+singleDefect L c A η D (onePathAt i ω)) := by
classical
simp only [defectiveTotalVisits,Nat.cast_sum,Finset.mul_sum,Real.exp_sum,
ENNReal.ofReal_prod_of_nonneg (fun _ _ ↦ (Real.exp_pos _).le)]
apply Finset.prod_congr rfl
intro i _
rw [singleDefect,pathVisitedSites_onePathAt]
have hx : 0≤η*(if onePathAt i ω∈D then ((pathVisitedSites L (center L c) A i ω).card:ℝ) else 0) := by positivity
have he : 0≤Real.exp (η*(if onePathAt i ω∈D then ((pathVisitedSites L (center L c) A i ω).card:ℝ) else 0))-1 := sub_nonneg.mpr (Real.one_le_exp_iff.mpr hx)
simp only [Nat.cast_ite,Nat.cast_zero]
rw [←ENNReal.ofReal_one,←ENNReal.ofReal_add zero_le_one he]
congr 1
ring
theorem interacting_visit_mgf {η₀ : ℝ} (hη₀ : 0≤η₀) :
∃ Cdel Cfree Csp αstar : ℝ,
0<Cdel ∧ 0<Cfree ∧ 0<Csp ∧ 0<αstar ∧
∀ (n K : ℕ) [NeZero K] (L : Length), L.val=(K:ℝ) → ∀ r D : ℝ,
D=(n:ℝ)/L.val^3 → 1≤D → αstar≤D*r → 0<r →
(n:ℝ)*(2*Real.pi*r)^3≤L.val^3/2 →
∀ (g : Jet n L), IsGroundVector n L r g →
∀ (c : Torus L) (A : Finset (DiscreteTorus K)) {t η ε : ℝ},
0<t → t≤2 → 0≤η → η≤η₀ → 0≤ε →
∀ E : Set (ConfigurationPath 1 L), MeasurableSet E →
(∀ω,reverseConfigurationPath 1 L ω∈E ↔ ω∈E) →
(∀ y : Torus L, coordinateWiener 1 {w | freeTrajectory 1 L t w (oneConfiguration y)∈E}≤ENNReal.ofReal ε) →
Real.exp (t*Cdel*D*r)*(Cfree*η*ε^(1/3:ℝ))≤1 →
(∫⁻ ω, ENNReal.ofReal (Real.exp (η*(defectiveTotalVisits L c A E ω:ℝ)))
∂groundPathLaw n L r t g) ≤
ENNReal.ofReal (Real.exp (Csp*(Real.exp (t*Cdel*D*r)*(Cfree*η*ε^(1/3:ℝ)))*D*(A.card:ℝ))) := by
classical
obtain ⟨Cdel,hCdel,hdel⟩ := ground_product_one_add
obtain ⟨Cfree,hCfree,hfree⟩ := free_single_defect_bound hη₀
obtain ⟨Csp,αstar,hCsp,hα,hsp⟩ := spatial_count_mgf_enn
refine ⟨Cdel,Cfree,Csp,αstar,hCdel,hCfree,hCsp,hα,?_⟩
intro n K inst L hL r D hDeq hD hαr hr hs g hg c A t η ε ht ht2 hη hη0 hε E hE hrev hprob hz1
let z := Real.exp (t*Cdel*D*r)*(Cfree*η*ε^(1/3:ℝ))
have hz : 0≤z := by positivity
have hsize : 1≤L.val := by rw [hL]; exact_mod_cast (NeZero.pos K)
let := hg.groundPathLaw_probability ht.le
by_cases hA : A.Nonempty
· let B := A.image (center L c)
have hB : B.Nonempty := hA.image _
have hDexp : t*Cdel*(n:ℝ)*r/L.val^3=t*Cdel*D*r := by rw [hDeq]; ring
calc
_ = ∫⁻ ω,∏i,(1+singleDefect L c A η E (onePathAt i ω)) ∂groundPathLaw n L r t g :=
lintegral_congr (exp_defectiveTotalVisits L c A E hη)
_ ≤ ∫⁻ X,∏i,(1+ENNReal.ofReal (Real.exp (t*Cdel*(n:ℝ)*r/L.val^3))*
oneFreeExpectation t (singleDefect L c A η E) (X i)) ∂groundPositionLaw n L g :=
hdel n L r t hr ht hs g hg _ (singleDefect_measurable L c A η hE) (singleDefect_reverse L c A η hrev)
_ ≤ ∫⁻ X,ENNReal.ofReal (Real.exp (z*weightedSiteCount L B X)) ∂groundPositionLaw n L g := by
apply lintegral_mono
intro X
apply le_trans _ (product_one_add_exp_le L B hB hz X)
apply Finset.prod_le_prod
intro i _
gcongr
have H := hfree L hL c A ht ht2 hη hη0 hε E hE hprob (X i)
calc
_ ≤ ENNReal.ofReal (Real.exp (t*Cdel*(n:ℝ)*r/L.val^3))*
ENNReal.ofReal (Cfree*η*ε^(1/3:ℝ)*Real.exp (-spatialSiteDistance L B (X i))) := by gcongr
_ = _ := by rw [←ENNReal.ofReal_mul (Real.exp_nonneg _),hDexp]; congr 1; dsimp [z]; ring
_ ≤ ENNReal.ofReal (Real.exp (Csp*z*D*(B.card:ℝ))) :=
hsp n K L hL r D hDeq hD hαr hr hsize hs g hg B z hz hz1
_ ≤ _ := by
apply ENNReal.ofReal_le_ofReal
apply Real.exp_le_exp.mpr
have hc : (B.card:ℝ)≤(A.card:ℝ) := by exact_mod_cast Finset.card_image_le
exact mul_le_mul_of_nonneg_left hc (by positivity)
· have he : A=∅ := Finset.not_nonempty_iff_eq_empty.mp hA
subst A
simp [defectiveTotalVisits,pathVisitedSites]
end HardSphere.ShiftVisits
end
end
section
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal Topology BigOperators
namespace HardSphere.ShiftVisits
open HardSphere.Wiener HardSphere.BathSlice HardSphere.Lattice HardSphere.CellOrbital HardSphere.FreeVisits
def visitorCount {n K : ℕ} (L : Length) (c : Torus L) (E : Set (ConfigurationPath 1 L))
(v : DiscreteTorus K) (ω : ConfigurationPath n L) : ℕ := by
classical
exact (Finset.univ.filter (fun i : Fin n ↦ onePathAt i ω∈E ∧ pathVisitsBall L 12 (center L c v) i ω)).card
lemma visitorCount_sum {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K))
(E : Set (ConfigurationPath 1 L)) (ω : ConfigurationPath n L) :
(∑v∈A,visitorCount L c E v ω)=defectiveTotalVisits L c A E ω := by
classical
simp only [visitorCount,Finset.card_filter]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro i _
by_cases hi : onePathAt i ω∈E
· simp only [hi,true_and,pathVisitedSites,ite_eq_left,Finset.card_filter]
· simp only [hi,false_and,ite_false,Finset.sum_const_zero,pathVisitedSites]
lemma visitorCount_measurable {n K : ℕ} (L : Length) (c : Torus L) {E : Set (ConfigurationPath 1 L)}
(hE : MeasurableSet E) (v : DiscreteTorus K) : Measurable (visitorCount (n:=n) L c E v) := by
classical
have he : visitorCount (n:=n) L c E v=(fun ω ↦ ∑i : Fin n, if onePathAt i ω∈E ∧ pathVisitsBall L 12 (center L c v) i ω then (1:ℕ) else 0) := by
funext ω
simp only [visitorCount,Finset.card_filter]
rw [he]
apply Finset.measurable_fun_sum
intro i _
have hi : MeasurableSet {ω : ConfigurationPath n L | onePathAt i ω∈E} := hE.preimage (onePathAt_continuous i).measurable
exact measurable_const.ite (hi.inter (pathVisitsBall_isClosed L 12 (center L c v) i).measurableSet) measurable_const
lemma defectiveTotalVisits_measurable {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K))
{E : Set (ConfigurationPath 1 L)} (hE : MeasurableSet E) : Measurable (defectiveTotalVisits (n:=n) L c A E) := by
have he : defectiveTotalVisits (n:=n) L c A E=fun ω ↦ ∑v∈A,visitorCount L c E v ω := by
funext ω; exact (visitorCount_sum L c A E ω).symm
rw [he]
exact Finset.measurable_fun_sum _ (fun v _ ↦ visitorCount_measurable L c hE v)
lemma joint_visitor_sum {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K))
(E : Set (ConfigurationPath 1 L)) {H : ℝ} {ω : ConfigurationPath n L}
(h : ∀v∈A,H≤(visitorCount L c E v ω:ℝ)) :
H*(A.card:ℝ)≤(defectiveTotalVisits L c A E ω:ℝ) := by
rw [←visitorCount_sum L c A E ω,Nat.cast_sum]
have hs := Finset.sum_le_sum h
simpa only [Finset.sum_const,nsmul_eq_mul,mul_comm] using hs
lemma joint_visitors_markov {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K))
{E : Set (ConfigurationPath 1 L)} (hE : MeasurableSet E)
(μ : Measure (ConfigurationPath n L)) {η H Q : ℝ} (hη : 0≤η)
(hmgf : (∫⁻ω,ENNReal.ofReal (Real.exp (η*(defectiveTotalVisits L c A E ω:ℝ))) ∂μ)≤
ENNReal.ofReal (Real.exp (Q*(A.card:ℝ)))) :
μ {ω | ∀v∈A,H≤(visitorCount L c E v ω:ℝ)}≤ENNReal.ofReal (Real.exp ((Q-η*H)*(A.card:ℝ))) := by
let F := fun ω : ConfigurationPath n L ↦ ENNReal.ofReal (Real.exp (η*(defectiveTotalVisits L c A E ω:ℝ)))
have hc : Measurable (fun k : ℕ ↦ (k:ℝ)) := measurable_of_countable _
have hF : Measurable F := ((measurable_const.mul (hc.comp (defectiveTotalVisits_measurable L c A hE))).exp).ennreal_ofReal
have hp : ENNReal.ofReal (Real.exp (η*(H*(A.card:ℝ))))≠0 := ne_of_gt (ENNReal.ofReal_pos.mpr (Real.exp_pos _))
have hm := meas_ge_le_lintegral_div (μ:=μ) hF.aemeasurable hp ENNReal.ofReal_ne_top
calc
_ ≤ μ {ω | ENNReal.ofReal (Real.exp (η*(H*(A.card:ℝ))))≤F ω} := by
apply measure_mono
intro ω hω
apply ENNReal.ofReal_le_ofReal
apply Real.exp_le_exp.mpr
exact mul_le_mul_of_nonneg_left (joint_visitor_sum L c A E hω) hη
_ ≤ _ := hm
_ ≤ ENNReal.ofReal (Real.exp (Q*(A.card:ℝ)))/ENNReal.ofReal (Real.exp (η*(H*(A.card:ℝ)))) := ENNReal.div_le_div_right hmgf _
_ = _ := by
rw [←ENNReal.ofReal_div_of_pos (Real.exp_pos _),←Real.exp_sub]
congr 2
ring
end HardSphere.ShiftVisits
end
end
end
end OAI
|