| import Mathlib |
| import OAI.Analysis.HardSphere.Thermodynamics.BridgeScaleTail |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Bridge |
| open Wiener GaussianPacking PathCutoff |
|
|
| lemma leg_window_coord {t S C : ℝ} (hS : 0< S) (hC : 0≤ C) (y u : Point) |
| (hdis : ‖u-y‖≤ C) (β : Noise) (w q : Time) (hq : (q:ℝ)≤(w:ℝ)) (i : Fin 3) : |
| |leg t y u β q i-leg t y u β 0 i|≤ |
| (w:ℝ)*C+S*incrementFluctuation t S 0 w β := by |
| have he : suffixTime 0 q=q := by apply Subtype.ext; simp [suffixTime] |
| have hd : |(q:ℝ)*(u i-y i)|≤(w:ℝ)*C := by |
| rw [abs_mul,abs_of_nonneg q.property.1] |
| exact (mul_le_mul_of_nonneg_left ((norm_le_pi_norm (u-y) i).trans hdis) q.property.1).trans |
| (mul_le_mul_of_nonneg_right hq hC) |
| have hn := incrementFluctuation_eval (t:=t) hS 0 w q hq β i |
| rw [he] at hn |
| have hl := leg_segment_increment t y u β 0 q i |
| rw [he] at hl |
| simp only [show ((0:Time):ℝ)=0 from rfl,sub_zero,one_mul] at hl |
| rw [hl,show leg t y u β 0 i+(q:ℝ)*(u i-y i)+Real.sqrt (2*t)*(β i q-β i 0)-leg t y u β 0 i=(q:ℝ)*(u i-y i)+Real.sqrt (2*t)*(β i q-β i 0) by ring] |
| exact (abs_add_le _ _).trans (add_le_add hd hn) |
|
|
| lemma leg_window_diameter {t S C : ℝ} (hS : 0< S) (hC : 0≤ C) (y u : Point) |
| (hdis : ‖u-y‖≤ C) (β : Noise) (w : Time) : |
| liftedDiameter 0 w (pathCoordinates (leg t y u β))≤ |
| 6*((w:ℝ)*C+S*incrementFluctuation t S 0 w β) := by |
| have hR : 0≤(w:ℝ)*C+S*incrementFluctuation t S 0 w β := |
| add_nonneg (mul_nonneg w.property.1 hC) (mul_nonneg hS.le (incrementFluctuation_nonneg hS.le 0 w β)) |
| have hw (q : Time) : (intervalTime 0 w q:ℝ)≤(w:ℝ) := by |
| change (1-(q:ℝ))*0+(q:ℝ)*(w:ℝ)≤(w:ℝ) |
| nlinarith [q.property.2,w.property.1] |
| have h := liftedDiameter_coord_le (pathCoordinates (leg t y u β)) 0 w |
| (mul_nonneg (by norm_num : (0:ℝ)≤2) hR) (fun i s q ↦ by |
| have hs := leg_window_coord (t:=t) hS hC y u hdis β w (intervalTime 0 w s) (hw s) i |
| have hq := leg_window_coord (t:=t) hS hC y u hdis β w (intervalTime 0 w q) (hw q) i |
| change |leg t y u β (intervalTime 0 w s) i-leg t y u β (intervalTime 0 w q) i|≤_ |
| have he : leg t y u β (intervalTime 0 w s) i-leg t y u β (intervalTime 0 w q) i= |
| (leg t y u β (intervalTime 0 w s) i-leg t y u β 0 i)-(leg t y u β (intervalTime 0 w q) i-leg t y u β 0 i) := by ring |
| rw [he] |
| exact (abs_sub _ _).trans (by linarith)) |
| linarith |
|
|
| lemma leg_window_tail {t S C : ℝ} (ht : 0≤ t) (hS : 0< S) (hC : 0≤ C) (y u : Point) |
| (hdis : ‖u-y‖≤ C) (w : Time) (hdrift : (w:ℝ)*C≤ S/12) (p : ℕ) (hp : p≠0) : |
| law {β | S≤ liftedDiameter 0 w (pathCoordinates (leg t y u β))}≤ |
| ENNReal.ofReal (scaleMoment p t S 0 w)/(ENNReal.ofReal (1/12:ℝ))^p := by |
| apply (measure_mono (show {β | S≤ liftedDiameter 0 w (pathCoordinates (leg t y u β))}⊆ |
| {β | (1/12:ℝ)≤ incrementFluctuation t S 0 w β} from ?_)).trans |
| (incrementFluctuation_scale_tail ht hS (by norm_num) 0 w p hp) |
| intro β hβ |
| have h := leg_window_diameter (t:=t) hS hC y u hdis β w |
| change S≤_ at hβ |
| change (1/12:ℝ)≤ incrementFluctuation t S 0 w β |
| nlinarith |
|
|
| end HardSphere.Bridge |
|
|
| end |
| end |
|
|
| section |
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Bridge |
| open Wiener PathCutoff GaussianPacking |
|
|
| lemma torusDistance_nonnegative (L : Length) (x y : Torus L) : 0≤torusDistance L x y := Real.sqrt_nonneg _ |
|
|
| abbrev projectPoint (L : Length) (y : Point) : Torus L := fun j ↦ (y j : AddCircle L.val) |
|
|
| def nearestLift (L : Length) (v : Point) (x : Torus L) : Point := fun j ↦ |
| (((AddCircle.equivIco L.val (-L.val/2)) (x j-(v j : AddCircle L.val))).val)+v j |
|
|
| lemma nearestLift_project (L : Length) (v : Point) (x : Torus L) : |
| projectPoint L (nearestLift L v x)=x := by |
| ext j |
| simp only [projectPoint,nearestLift,AddCircle.coe_add,AddCircle.coe_equivIco,sub_add_cancel] |
|
|
| lemma nearestLift_coord (L : Length) (v : Point) (x : Torus L) (j : Fin 3) : |
| |nearestLift L v x j-v j|=dist (x j) (projectPoint L v j) := by |
| let z := (AddCircle.equivIco L.val (-L.val/2)) (x j-(v j : AddCircle L.val)) |
| have hz : |z.val|≤|L.val|/2 := by |
| rw [abs_of_pos L.property,abs_le] |
| exact ⟨by linarith [z.property.1],by linarith [z.property.2]⟩ |
| have he := (AddCircle.norm_coe_eq_abs_iff L.val L.property.ne').mpr hz |
| rw [AddCircle.coe_equivIco,←dist_eq_norm] at he |
| simpa only [nearestLift,add_sub_cancel_right] using he.symm |
|
|
| lemma nearestLift_bound (L : Length) (v : Point) (x : Torus L) {R : ℝ} |
| (h : torusDistance L x (projectPoint L v)≤R) : ‖nearestLift L v x-v‖≤R := by |
| apply (pi_norm_le_iff_of_nonneg ((torusDistance_nonnegative L x _).trans h)).mpr |
| intro j |
| change |nearestLift L v x j-v j|≤R |
| rw [nearestLift_coord] |
| exact (CellOrbital.dist_coord_le_torusDistance L _ _ j).trans h |
|
|
| lemma exists_pathLift_at {L : Length} (ω : ConfigurationPath 1 L) (y : Point) |
| (hy : projectPoint L y=ω 0 0) : ∃ζ : EuclideanPath,IsPathLift ω ζ ∧ (fun j ↦ ζ j 0)=y := by |
| have hx (j : Fin 3) : ∃ ζ : C(Time,ℝ),(∀s,(ζ s : AddCircle L.val)=ω s 0 j) ∧ ζ 0=y j := by |
| let γ : C(Time,AddCircle L.val) := ⟨fun s ↦ ω s 0 j,by fun_prop⟩ |
| have h0 : γ 0=(y j : AddCircle L.val) := (congrFun hy j).symm |
| obtain ⟨ζ,hζ,hζ0⟩ := (AddCircle.isCoveringMap_coe L.val).exists_path_lifts γ (y j) h0 |
| exact ⟨ζ,(fun s ↦ congrFun hζ s),hζ0⟩ |
| choose ζ hζ using hx |
| exact ⟨ζ,fun j s ↦ (hζ j).1 s,funext (fun j ↦ (hζ j).2)⟩ |
|
|
| lemma local_collision_lifts {L : Length} {v x c c₀ : Point} |
| (hL : 50<L.val) (hx : ‖x-v‖≤10) (hc₀ : ‖c₀-v‖≤12) (hc : ‖c-c₀‖≤1) |
| {r : ℝ} (hh : torusDistance L (projectPoint L x) (projectPoint L c)≤r) : ‖x-c‖≤r := by |
| have hr : 0≤r := (torusDistance_nonnegative L _ _).trans hh |
| apply (pi_norm_le_iff_of_nonneg hr).mpr |
| intro j |
| have hnorm : |x j-c j|≤23 := by |
| have hxj := (norm_le_pi_norm (x-v) j).trans hx |
| have hcj := (norm_le_pi_norm (c-c₀) j).trans hc |
| have hc₀j := (norm_le_pi_norm (c₀-v) j).trans hc₀ |
| change |x j-v j|≤10 at hxj |
| change |c j-c₀ j|≤1 at hcj |
| change |c₀ j-v j|≤12 at hc₀j |
| have he : x j-c j=(x j-v j)-(c₀ j-v j)-(c j-c₀ j) := by ring |
| rw [he] |
| exact ((abs_sub _ _).trans (add_le_add ((abs_sub _ _).trans (add_le_add hxj hc₀j)) hcj)).trans (by norm_num) |
| have he := (AddCircle.norm_coe_eq_abs_iff L.val L.property.ne').mpr |
| (show |x j-c j|≤|L.val|/2 by rw [abs_of_pos L.property]; linarith) |
| rw [AddCircle.coe_sub,←dist_eq_norm] at he |
| change |x j-c j|≤r |
| rw [←he] |
| exact (CellOrbital.dist_coord_le_torusDistance L (projectPoint L x) (projectPoint L c) j).trans hh |
|
|
| end HardSphere.Bridge |
|
|
| end |
| end |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Bridge |
| open Wiener GaussianPacking PathCutoff |
|
|
| lemma bridge_start_zero : ∀ᵐβ ∂law,∀j : Fin 3,β j 0=0 := by |
| apply ae_all_iff.mpr |
| intro j |
| exact (measurePreserving_eval (fun _ : Fin 3↦bridgeLaw) j).quasiMeasurePreserving.ae |
| (bridge_endpoints.mono (fun _ h↦h.1)) |
|
|
| lemma leg_confined_of_fluctuation (L : Length) (t : ℝ) (y u v : Point) (β : Noise) |
| (hy : ‖y-v‖≤3) (hu : ‖u-v‖≤3) (hzero : ∀j,β j 0=0) |
| (hnoise : incrementFluctuation t 1 0 1 β≤1/3) (s : Time) : |
| ‖leg t y u β s-v‖≤10 ∧ |
| torusDistance L (projectPoint L (leg t y u β s)) (projectPoint L v)≤10 := by |
| have hc (j : Fin 3) : |leg t y u β s j-v j|≤10/3 := by |
| have hn := incrementFluctuation_eval (t:=t) (by norm_num : (0:ℝ)<1) 0 1 s s.property.2 β j |
| have hs : suffixTime 0 s=s := by apply Subtype.ext; simp [suffixTime] |
| rw [hs,hzero j,sub_zero,one_mul] at hn |
| have hyj : |y j-v j|≤3 := (norm_le_pi_norm (y-v) j).trans hy |
| have huj : |u j-v j|≤3 := (norm_le_pi_norm (u-v) j).trans hu |
| have hm : |(1-(s:ℝ))*(y j-v j)+(s:ℝ)*(u j-v j)|≤3 := by |
| apply (abs_add_le _ _).trans |
| rw [abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr s.property.2),abs_of_nonneg s.property.1] |
| have h₁ := mul_le_mul_of_nonneg_left hyj (sub_nonneg.mpr s.property.2) |
| have h₂ := mul_le_mul_of_nonneg_left huj s.property.1 |
| linarith |
| change |(1-(s:ℝ))*y j+(s:ℝ)*u j+Real.sqrt (2*t)*β j s-v j|≤10/3 |
| rw [show (1-(s:ℝ))*y j+(s:ℝ)*u j+Real.sqrt (2*t)*β j s-v j= |
| ((1-(s:ℝ))*(y j-v j)+(s:ℝ)*(u j-v j))+Real.sqrt (2*t)*β j s by ring] |
| exact (abs_add_le _ _).trans (by linarith) |
| constructor |
| · apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤10)).mpr |
| intro j |
| exact (hc j).trans (by norm_num) |
| · unfold torusDistance |
| apply (Real.sqrt_le_iff).mpr |
| refine ⟨by norm_num, ?_⟩ |
| calc |
| _ ≤ ∑ _j : Fin 3, (10/3:ℝ)^2 := by |
| apply Finset.sum_le_sum |
| intro j _ |
| apply (sq_le_sq₀ dist_nonneg (by norm_num : (0:ℝ)≤10/3)).mpr |
| rw [dist_eq_norm, ← AddCircle.coe_sub] |
| exact (QuotientAddGroup.norm_mk_le_norm (m := leg t y u β s j-v j)).trans |
| (by simpa only [Real.norm_eq_abs] using hc j) |
| _ ≤ (10:ℝ)^2 := by norm_num |
|
|
| lemma leg_confinement_tail (L : Length) {t : ℝ} (ht : 0≤ t) (y u v : Point) |
| (hy : ‖y-v‖≤3) (hu : ‖u-v‖≤3) (p : ℕ) (hp : p≠0) : |
| law {β | ∃s : Time,¬(‖leg t y u β s-v‖≤10 ∧ |
| torusDistance L (projectPoint L (leg t y u β s)) (projectPoint L v)≤10)}≤ |
| ENNReal.ofReal (scaleMoment p t 1 0 1)/(ENNReal.ofReal (1/3:ℝ))^p := by |
| apply (measure_mono_ae (show {β | ∃s : Time,¬(‖leg t y u β s-v‖≤10 ∧ |
| torusDistance L (projectPoint L (leg t y u β s)) (projectPoint L v)≤10)}≤ᵐ[law] |
| {β | (1/3:ℝ)≤ incrementFluctuation t 1 0 1 β} from ?_)).trans |
| (incrementFluctuation_scale_tail ht (by norm_num) (by norm_num) 0 1 p hp) |
| filter_upwards [bridge_start_zero] with β hβ |
| rintro ⟨s,hs⟩ |
| by_contra hn |
| exact hs (leg_confined_of_fluctuation L t y u v β hy hu hβ (le_of_not_ge hn) s) |
|
|
| end HardSphere.Bridge |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
