| import Mathlib |
| import OAI.Analysis.HardSphere.PathBounds.ShiftInteracting |
| import OAI.Analysis.HardSphere.PathBounds.PhysicalVisitTails |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.ShiftVisits |
| open Wiener PathCutoff BathSlice HardSphere.Lattice CellOrbital HardSphere.FreeVisits |
|
|
| theorem ground_excessVisitors_tail : |
| ∃ αstar : ℝ, 0<αstar ∧ ∀ αmax : ℝ, 0<αmax → |
| ∃ H : ℝ, 0<H ∧ ∀ (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 → D*r≤αmax → 0<r → |
| (n:ℝ)*(2*Real.pi*r)^3≤L.val^3/2 → |
| ∀ (g : Jet n L), IsGroundVector n L r g → ∀ (x₀ : Torus L) (A : Finset (DiscreteTorus K)) {t : ℝ}, |
| 0<t → t≤2 → |
| groundPathLaw n L r t g {ω | ∀v∈A,H*D≤(visitorCount L x₀ Set.univ v ω:ℝ)}≤ |
| ENNReal.ofReal (Real.exp (-(D*(A.card:ℝ)))) := by |
| obtain ⟨Cd,Cf,Cs,α,hCd,hCf,hCs,hα,hm⟩ := interacting_visit_mgf (η₀:=1) (by norm_num) |
| refine ⟨α,hα,?_⟩ |
| intro αmax hαmax |
| let B := Real.exp (2*Cd*αmax)*Cf |
| have hB : 0<B := by dsimp [B]; positivity |
| let η := min 1 B⁻¹ |
| have hη : 0<η := lt_min (by norm_num) (inv_pos.mpr hB) |
| have hη1 : η≤1 := min_le_left _ _ |
| have hb : B*η≤1 := by |
| calc |
| B*η≤B*B⁻¹ := mul_le_mul_of_nonneg_left (min_le_right _ _) hB.le |
| _ = 1 := mul_inv_cancel₀ hB.ne' |
| let H := (Cs+1)/η |
| have hH : 0<H := div_pos (by linarith) hη |
| refine ⟨H,hH,?_⟩ |
| intro n K inst L hL r D hDeq hD hαr hαup hr hs g hg x₀ A t ht ht2 |
| have hD0 : 0≤D := by linarith |
| have hxp : t*Cd*D*r≤2*Cd*αmax := by |
| calc |
| t*Cd*D*r = (t*Cd)*(D*r) := by ring |
| _ ≤ (t*Cd)*αmax := mul_le_mul_of_nonneg_left hαup (by positivity) |
| _ ≤ (2*Cd)*αmax := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right ht2 hCd.le) hαmax.le |
| have hz : Real.exp (t*Cd*D*r)*(Cf*η*(1:ℝ)^(1/3:ℝ))≤1 := by |
| simp only [Real.one_rpow,mul_one] |
| calc |
| _ ≤ Real.exp (2*Cd*αmax)*(Cf*η) := mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr hxp) (by positivity) |
| _ = B*η := by dsimp [B]; ring |
| _ ≤ 1 := hb |
| have hp (y : Torus L) : coordinateWiener 1 {w | freeTrajectory 1 L t w (oneConfiguration y)∈(Set.univ : Set (ConfigurationPath 1 L))}≤ENNReal.ofReal 1 := by simp |
| have hh := hm n K L hL r D hDeq hD hαr hr hs g hg x₀ A ht ht2 hη.le hη1 |
| (by norm_num : (0:ℝ)≤1) Set.univ MeasurableSet.univ (fun _ ↦ Iff.rfl) hp hz |
| have hj := joint_visitors_markov L x₀ A MeasurableSet.univ (groundPathLaw n L r t g) hη.le hh (H:=H*D) |
| apply hj.trans |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| rw [←neg_mul] |
| apply mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg A.card) |
| have hhH : η*H=Cs+1 := mul_div_cancel₀ _ hη.ne' |
| have he : Cs*(Real.exp (t*Cd*D*r)*(Cf*η*(1:ℝ)^(1/3:ℝ)))*D≤Cs*D := by |
| exact (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hz hCs.le) hD0).trans_eq (by ring) |
| change _≤ -D |
| nlinarith |
|
|
| theorem ground_longVisitors_tail : |
| ∃ αstar : ℝ, 0<αstar ∧ ∀ αmax θ : ℝ, 0<αmax → 0<θ → |
| ∃ t₀ : ℝ, 0<t₀ ∧ ∀ (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 → D*r≤αmax → 0<r → |
| (n:ℝ)*(2*Real.pi*r)^3≤L.val^3/2 → |
| ∀ (g : Jet n L), IsGroundVector n L r g → ∀ (x₀ : Torus L) (A : Finset (DiscreteTorus K)) {t : ℝ}, |
| 0<t → t ≤ min 2 t₀ → |
| groundPathLaw n L r t g {ω | ∀v∈A,θ*D≤ |
| (visitorCount L x₀ {p | 1 < intervalDiameter 0 1 p} v ω:ℝ)}≤ |
| ENNReal.ofReal (Real.exp (-(θ*D/2)*(A.card:ℝ))) := by |
| obtain ⟨Cd,Cf,Cs,α,hCd,hCf,hCs,hα,hm⟩ := interacting_visit_mgf (η₀:=1) (by norm_num) |
| refine ⟨α,hα,?_⟩ |
| intro αmax θ hαmax hθ |
| let B := Real.exp (2*Cd*αmax)*Cf |
| have hB : 0<B := by dsimp [B]; positivity |
| let δ := min 1 (θ/(2*Cs)) |
| have hδ : 0<δ := lt_min (by norm_num) (by positivity) |
| let ε := (δ/B)^3 |
| have hε : 0<ε := by dsimp [ε]; positivity |
| have hrp : ε^(1/3:ℝ)=δ/B := by |
| simpa [ε] using Real.pow_rpow_inv_natCast (n:=3) (div_pos hδ hB).le (by norm_num : (3:ℕ)≠0) |
| obtain ⟨t₀,ht₀,hsmall⟩ := free_long_probability_small hε |
| refine ⟨t₀,ht₀,?_⟩ |
| intro n K inst L hL r D hDeq hD hαr hαup hr hs g hg x₀ A t ht htt |
| have ht2 := (le_min_iff.mp htt).1 |
| have hD0 : 0≤D := by linarith |
| have hxp : t*Cd*D*r≤2*Cd*αmax := by |
| calc |
| t*Cd*D*r = (t*Cd)*(D*r) := by ring |
| _ ≤ (t*Cd)*αmax := mul_le_mul_of_nonneg_left hαup (by positivity) |
| _ ≤ (2*Cd)*αmax := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right ht2 hCd.le) hαmax.le |
| have hzδ : Real.exp (t*Cd*D*r)*(Cf*1*ε^(1/3:ℝ))≤δ := by |
| rw [hrp,mul_one] |
| calc |
| _ ≤ Real.exp (2*Cd*αmax)*(Cf*(δ/B)) := mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr hxp) (by positivity) |
| _ = B*(δ/B) := by dsimp [B]; ring |
| _ = δ := mul_div_cancel₀ _ hB.ne' |
| have hz1 := hzδ.trans (min_le_left 1 (θ/(2*Cs))) |
| let E : Set (ConfigurationPath 1 L) := {p | 1 < intervalDiameter 0 1 p} |
| have hE : MeasurableSet E := measurableSet_lt measurable_const (intervalDiameter_continuous L 0 1).measurable |
| have hrev (p : ConfigurationPath 1 L) : reverseConfigurationPath 1 L p∈E↔p∈E := by |
| simp only [E,Set.mem_ofPred_eq,intervalDiameter_reverse,reverseTime_zero,reverseTime_one] |
| have hp (y : Torus L) : coordinateWiener 1 {w | freeTrajectory 1 L t w (oneConfiguration y)∈E}≤ENNReal.ofReal ε := |
| hsmall L ht (le_min_iff.mp htt).2 y |
| have hh := hm n K L hL r D hDeq hD hαr hr hs g hg x₀ A ht ht2 (by norm_num : (0:ℝ)≤1) |
| (le_refl 1) hε.le E hE hrev hp hz1 |
| have hj := joint_visitors_markov L x₀ A hE (groundPathLaw n L r t g) (by norm_num : (0:ℝ)≤1) hh (H:=θ*D) |
| apply hj.trans |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| apply mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg A.card) |
| have hδθ : Cs*δ≤θ/2 := by |
| have hh := (le_div_iff₀ (by positivity : 0<(2:ℝ)*Cs)).mp (min_le_right 1 (θ/(2*Cs))) |
| dsimp [δ] |
| linarith |
| have hc := mul_le_mul_of_nonneg_right ((mul_le_mul_of_nonneg_left hzδ hCs.le).trans hδθ) hD0 |
| nlinarith |
|
|
| end HardSphere.ShiftVisits |
|
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| end |
| end |
|
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| end |
|
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| end OAI |
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