| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.HeadLift |
| import OAI.Analysis.HardSphere.Cutoffs.RawAngleLift |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| namespace HardSphere.TightCutoff |
| open PeriodicCutoff MeasureTheory Set Filter |
| open scoped ComplexConjugate |
|
|
| def closeLabels {n : ℕ} {L : Length} (r : ℝ) (w : SmoothMultiplier 1 L) |
| (Y : Configuration n L) : Finset (Fin n) := by |
| classical |
| exact Finset.univ.filter (fun i ↦ ∃ y∈tsupport w.orbital, torusDistance L y (Y i)≤2*r) |
|
|
| lemma mem_closeLabels_of_near {n : ℕ} {L : Length} {r : ℝ} (hr : 0<r) |
| (w : SmoothMultiplier 1 L) (Y : Configuration n L) {y : Torus L} (hy : w.orbital y≠0) |
| (i : Fin n) (hi : y∈near L.val r (Y i)) : i∈closeLabels r w Y := by |
| classical |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,y,subset_tsupport w.orbital hy,near_distance hr hi⟩ |
|
|
| def orbitalCutoff {L : Length} (n : ℕ) (r : ℝ) (w : SmoothMultiplier 1 L) : SmoothMultiplier (n+1) L := |
| (w.liftHead n).mul (rawCutoff n L r) |
|
|
| lemma orbitalCutoff_real {L : Length} (n : ℕ) (r : ℝ) (w : SmoothMultiplier 1 L) |
| (hw : ∀ X, conj (w.toFun X)=w.toFun X) (X : Configuration (n+1) L) : |
| conj ((orbitalCutoff n r w).toFun X)=(orbitalCutoff n r w).toFun X := by |
| change conj (w.orbital (X 0)*(rawCutoff n L r).toFun X)=_ |
| rw [map_mul,rawCutoff_real] |
| change conj (w.toFun (fun _ ↦ X 0)) * _ = _ |
| rw [hw] |
| rfl |
|
|
| lemma orbitalCutoff_clearance {L : Length} (n : ℕ) {r : ℝ} (hr : 0<r) (w : SmoothMultiplier 1 L) : |
| InsertionClearance r (orbitalCutoff n r w) := |
| Set.Subset.trans tsupport_mul_subset_right (rawCutoff_clearance n L hr) |
|
|
| lemma weighted_near_le {n : ℕ} {L : Length} {r : ℝ} (hr : 0<r) |
| (w : SmoothMultiplier 1 L) (Y : Configuration n L) (y : Torus L) : |
| ‖w.orbital y‖^2 * ∑ i : Fin n, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y ≤ |
| w.orbitalBound * ∑ i ∈ closeLabels r w Y, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y := by |
| classical |
| have hb (i : Fin n) : 0≤(near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y := |
| Set.indicator_nonneg (fun _ _ ↦ zero_le_one) _ |
| by_cases hy : w.orbital y=0 |
| · simp only [hy,norm_zero,zero_pow (by decide : 2≠0),zero_mul] |
| exact mul_nonneg (sq_nonneg _) (Finset.sum_nonneg fun i _ ↦ hb i) |
| · have he : (∑ i : Fin n, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y)= |
| ∑ i ∈ closeLabels r w Y, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y := by |
| symm |
| apply Finset.sum_subset (Finset.subset_univ _) |
| intro i _ hi |
| exact Set.indicator_of_notMem (fun hh ↦ hi (mem_closeLabels_of_near hr w Y hy i hh)) _ |
| rw [he] |
| exact mul_le_mul_of_nonneg_right (w.orbital_le_bound y) (Finset.sum_nonneg fun i _ ↦ hb i) |
|
|
| lemma rawCutoff_mass_pointwise (n : ℕ) (L : Length) {r : ℝ} (hr : 0<r) |
| (y : Torus L) (Y : Configuration n L) : |
| 1≤‖(rawCutoff n L r).toFun (Fin.cons y Y)‖^2+ |
| ∑ i : Fin n, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y := by |
| rw [rawCutoff_norm_sq] |
| have hb (i : Fin n) : 0≤(near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y := |
| Set.indicator_nonneg (fun _ _ ↦ zero_le_one) _ |
| by_cases hh : ∃ i, y∈near L.val r (Y i) |
| · obtain ⟨i,hi⟩ := hh |
| have hs := Finset.single_le_sum (f := fun i : Fin n ↦ |
| (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y) (fun i _ ↦ hb i) (Finset.mem_univ i) |
| rw [Set.indicator_of_mem hi] at hs |
| exact hs.trans (le_add_of_nonneg_left (sq_nonneg _)) |
| · have ha (i : Fin n) : angle L.val r y (Y i)=Real.pi/2 := |
| angle_one_outside L.val hr (fun hi ↦ hh ⟨i,hi⟩) |
| have he : rawCutoffReal n L r (Fin.cons y Y)=1 := by |
| simp only [rawCutoffReal,Fin.cons_zero,Fin.cons_succ,ha,Real.sin_pi_div_two,Finset.prod_const_one] |
| rw [he,one_pow] |
| exact le_add_of_nonneg_right (Finset.sum_nonneg (fun i _ ↦ hb i)) |
|
|
| lemma localized_mass_pointwise {n : ℕ} {L : Length} {r : ℝ} (hr : 0<r) |
| (w : SmoothMultiplier 1 L) (Y : Configuration n L) (y : Torus L) : |
| ‖w.orbital y‖^2≤‖(orbitalCutoff n r w).toFun (Fin.cons y Y)‖^2+ |
| w.orbitalBound * ∑ i ∈ closeLabels r w Y, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y := by |
| have h := mul_le_mul_of_nonneg_left (rawCutoff_mass_pointwise n L hr y Y) (sq_nonneg ‖w.orbital y‖) |
| simp only [mul_one,mul_add] at h |
| have he : ‖(orbitalCutoff n r w).toFun (Fin.cons y Y)‖^2= |
| ‖w.orbital y‖^2 * ‖(rawCutoff n L r).toFun (Fin.cons y Y)‖^2 := by |
| change ‖w.orbital y*(rawCutoff n L r).toFun (Fin.cons y Y)‖^2=_ |
| rw [norm_mul,mul_pow] |
| rw [he] |
| exact h.trans (add_le_add_right (weighted_near_le hr w Y y) _) |
|
|
| lemma near_sum_integral_le {n : ℕ} {L : Length} {r : ℝ} (hr : 0<r) |
| (Y : Configuration n L) (J : Finset (Fin n)) : |
| (∫ y : Torus L, ∑ i ∈ J, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y)≤ |
| (J.card:ℝ)*(4*r)^3 := by |
| rw [integral_finsetSum _ (fun i _ ↦ (integrable_const (1:ℝ)).indicator (near_measurable L.val r (Y i)))] |
| calc |
| _ = ∑ i ∈ J, volume.real (near L.val r (Y i)) := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact integral_indicator_one (near_measurable L.val _ _) |
| _ ≤ ∑ i ∈ J, (4*r)^3 := Finset.sum_le_sum fun i _ ↦ near_volume_le L.val hr (Y i) |
| _ = _ := by simp |
|
|
| lemma localized_mass_lower {n : ℕ} {L : Length} {r : ℝ} (hr : 0<r) |
| (w : SmoothMultiplier 1 L) (hw : ∫ y : Torus L, ‖w.orbital y‖^2=1) |
| (Y : Configuration n L) : |
| 1-64*w.orbitalBound*(closeLabels r w Y).card*r^3≤(orbitalCutoff n r w).fiberMass Y := by |
| have hf : Integrable (fun y : Torus L ↦ ‖(orbitalCutoff n r w).toFun (Fin.cons y Y)‖^2) volume := |
| (((continuous_of_continuous_lift (orbitalCutoff n r w).smooth.continuous).comp |
| ((continuous_id : Continuous (id : Torus L → Torus L)).finCons |
| (continuous_const : Continuous (fun _ : Torus L ↦ Y)))).norm.pow 2).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _) |
| have hw' : Integrable (fun y : Torus L ↦ ‖w.orbital y‖^2) volume := |
| (w.orbital_continuous.norm.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hi : Integrable (fun y : Torus L ↦ ∑ i ∈ closeLabels r w Y, |
| (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y) volume := |
| integrable_finsetSum _ (fun i _ ↦ (integrable_const (1:ℝ)).indicator (near_measurable L.val r (Y i))) |
| have h := integral_mono hw' (hf.add (hi.const_mul w.orbitalBound)) (localized_mass_pointwise hr w Y) |
| change (∫ y : Torus L, ‖w.orbital y‖^2) ≤ ∫ y : Torus L, |
| ‖(orbitalCutoff n r w).toFun (Fin.cons y Y)‖^2+ |
| w.orbitalBound * ∑ i ∈ closeLabels r w Y, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y at h |
| rw [hw,integral_add hf (hi.const_mul w.orbitalBound),integral_const_mul] at h |
| have hh := mul_le_mul_of_nonneg_left (near_sum_integral_le hr Y (closeLabels r w Y)) (show 0≤w.orbitalBound from sq_nonneg _) |
| change 1-64*w.orbitalBound*(closeLabels r w Y).card*r^3≤∫ y : Torus L, ‖(orbitalCutoff n r w).toFun (Fin.cons y Y)‖^2 |
| nlinarith |
|
|
| end HardSphere.TightCutoff |
|
|
| namespace HardSphere.TightCutoff |
| open PeriodicCutoff MeasureTheory Set Filter |
|
|
| lemma norm_add_sq_le_two (z v : ℂ) : ‖z+v‖^2≤2*‖z‖^2+2*‖v‖^2 := by |
| have h := norm_add_le z v |
| have h' := sq_nonneg (‖z‖-‖v‖) |
| nlinarith [norm_nonneg (z+v),norm_nonneg z,norm_nonneg v] |
|
|
| lemma orbitalCutoff_derivative_le {n : ℕ} {L : Length} (r : ℝ) |
| (w : SmoothMultiplier 1 L) (X : Configuration (n+1) L) (p : Direction (n+1)) : |
| ‖coordDerivative (n+1) L (orbitalCutoff n r w).toFun p X‖^2≤ |
| 2*‖w.orbital (X 0)‖^2*‖coordDerivative (n+1) L (rawCutoff n L r).toFun p X‖^2+ |
| 2*‖coordDerivative (n+1) L (w.liftHead n).toFun p X‖^2 := by |
| change ‖coordDerivative (n+1) L ((w.liftHead n).toFun*(rawCutoff n L r).toFun) p X‖^2≤_ |
| rw [coordDerivative_mul ((w.liftHead n).smooth.differentiable (by simp)) |
| ((rawCutoff n L r).smooth.differentiable (by simp))] |
| simp only [Pi.add_apply,Pi.mul_apply] |
| have hh := norm_add_sq_le_two |
| ((w.liftHead n).toFun X*coordDerivative (n+1) L (rawCutoff n L r).toFun p X) |
| ((rawCutoff n L r).toFun X*coordDerivative (n+1) L (w.liftHead n).toFun p X) |
| simp only [norm_mul,mul_pow] at hh |
| have hb : ‖(rawCutoff n L r).toFun X‖^2≤1 := by |
| rw [rawCutoff_norm_sq] |
| nlinarith [(rawCutoffReal_bounds n L r X).1,(rawCutoffReal_bounds n L r X).2] |
| have hc := mul_le_mul_of_nonneg_left hb |
| (sq_nonneg ‖coordDerivative (n+1) L (w.liftHead n).toFun p X‖) |
| change ‖coordDerivative (n+1) L (w.liftHead n).toFun p X‖^2 * ‖(rawCutoff n L r).toFun X‖^2≤_ at hc |
| change _≤2*‖(w.liftHead n).toFun X‖^2*‖coordDerivative (n+1) L (rawCutoff n L r).toFun p X‖^2+_ |
| nlinarith |
|
|
| lemma localized_gradient_pointwise {n : ℕ} {L : Length} {r M : ℝ} (hr : 0<r) (hM : 0≤M) |
| (hb : ∀ t : ℝ, |deriv Real.smoothTransition t|≤M) |
| (w : SmoothMultiplier 1 L) (Y : Configuration n L) (y : Torus L) : |
| (∑ p : Direction (n+1), ‖coordDerivative (n+1) L (orbitalCutoff n r w).toFun p (Fin.cons y Y)‖^2)≤ |
| 2*(∑ j : Fin 3, ‖coordDerivative 1 L w.toFun (0,j) (fun _ ↦ y)‖^2)+ |
| (288*Real.pi^2*M^2/r^2)*w.orbitalBound * |
| ∑ i ∈ closeLabels r w Y, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y := by |
| have hs := Finset.sum_le_sum (s := (Finset.univ : Finset (Direction (n+1)))) |
| (fun p _ ↦ orbitalCutoff_derivative_le r w (Fin.cons y Y) p) |
| simp only [Finset.sum_add_distrib,← Finset.mul_sum,w.liftHead_gradient_sum,Fin.cons_zero] at hs |
| have hg := rawCutoff_gradient_le n L hr hM hb (Fin.cons y Y) |
| simp only [Fin.cons_zero,Fin.cons_succ] at hg |
| have h1 := mul_le_mul_of_nonneg_left hg (show 0≤2*‖w.orbital y‖^2 by positivity) |
| have h2 := mul_le_mul_of_nonneg_left (weighted_near_le hr w Y y) |
| (show 0≤288*Real.pi^2*M^2/r^2 by positivity) |
| have he : 2*‖w.orbital y‖^2 * (144*Real.pi^2*M^2/r^2 * |
| ∑ i : Fin n, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y)= |
| (288*Real.pi^2*M^2/r^2) * (‖w.orbital y‖^2 * |
| ∑ i : Fin n, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y) := by ring |
| rw [he] at h1 |
| have hh := h1.trans h2 |
| rw [← mul_assoc] at hh |
| linarith |
|
|
| lemma derivative_fiber_integrable {n : ℕ} {L : Length} (f : SmoothMultiplier (n+1) L) |
| (Y : Configuration n L) (p : Direction (n+1)) : |
| Integrable (fun y : Torus L ↦ ‖coordDerivative (n+1) L f.toFun p (Fin.cons y Y)‖^2) volume := by |
| exact (((continuous_coordDerivative f.smooth p).comp |
| ((continuous_id : Continuous (id : Torus L → Torus L)).finCons |
| (continuous_const : Continuous (fun _ : Torus L ↦ Y)))).norm.pow 2).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _) |
|
|
| lemma orbital_derivative_integrable {L : Length} (w : SmoothMultiplier 1 L) (j : Fin 3) : |
| Integrable (fun y : Torus L ↦ ‖coordDerivative 1 L w.toFun (0,j) (fun _ ↦ y)‖^2) volume := |
| (((continuous_coordDerivative w.smooth (0,j)).comp (continuous_pi fun _ ↦ continuous_id)).norm.pow 2 |
| ).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
|
|
| lemma localized_gradient_integral_le {n : ℕ} {L : Length} {r M : ℝ} (hr : 0<r) (hM : 0≤M) |
| (hb : ∀ t : ℝ, |deriv Real.smoothTransition t|≤M) |
| (w : SmoothMultiplier 1 L) (Y : Configuration n L) : |
| (∑ p : Direction (n+1), ∫ y : Torus L, |
| ‖coordDerivative (n+1) L (orbitalCutoff n r w).toFun p (Fin.cons y Y)‖^2)≤ |
| 2*w.orbitalEnergy+18432*Real.pi^2*M^2*w.orbitalBound*(closeLabels r w Y).card*r := by |
| have hf := derivative_fiber_integrable (orbitalCutoff n r w) Y |
| have hw := orbital_derivative_integrable w |
| have hi : Integrable (fun y : Torus L ↦ ∑ i ∈ closeLabels r w Y, |
| (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y) volume := |
| integrable_finsetSum _ (fun i _ ↦ (integrable_const (1:ℝ)).indicator (near_measurable L.val r (Y i))) |
| have h := integral_mono (integrable_finsetSum Finset.univ (fun p _ ↦ hf p)) |
| (((integrable_finsetSum Finset.univ (fun j _ ↦ hw j)).const_mul 2).add |
| (hi.const_mul ((288*Real.pi^2*M^2/r^2)*w.orbitalBound))) |
| (localized_gradient_pointwise hr hM hb w Y) |
| change (∫ y : Torus L, ∑ p : Direction (n+1), |
| ‖coordDerivative (n+1) L (orbitalCutoff n r w).toFun p (Fin.cons y Y)‖^2)≤ |
| ∫ y : Torus L, 2*(∑ j : Fin 3, ‖coordDerivative 1 L w.toFun (0,j) (fun _ ↦ y)‖^2)+ |
| (288*Real.pi^2*M^2/r^2)*w.orbitalBound * |
| ∑ i ∈ closeLabels r w Y, (near L.val r (Y i)).indicator (fun _ ↦ (1:ℝ)) y at h |
| rw [integral_finsetSum _ (fun p _ ↦ hf p),integral_add |
| ((integrable_finsetSum Finset.univ (fun j _ ↦ hw j)).const_mul 2) |
| (hi.const_mul ((288*Real.pi^2*M^2/r^2)*w.orbitalBound)), |
| integral_const_mul,integral_const_mul,integral_finsetSum _ (fun j _ ↦ hw j)] at h |
| have hh := mul_le_mul_of_nonneg_left (near_sum_integral_le hr Y (closeLabels r w Y)) |
| (show 0≤(288*Real.pi^2*M^2/r^2)*w.orbitalBound from mul_nonneg (by positivity) (sq_nonneg _)) |
| have he : (288*Real.pi^2*M^2/r^2)*w.orbitalBound*((closeLabels r w Y).card*(4*r)^3)= |
| 18432*Real.pi^2*M^2*w.orbitalBound*(closeLabels r w Y).card*r := by |
| field_simp [hr.ne'] |
| ring |
| rw [he] at hh |
| exact h.trans (add_le_add_right hh _) |
|
|
| end HardSphere.TightCutoff |
|
|
| end |
|
|
| end OAI |
|
|
