| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.LocalCloseLabels |
|
|
| 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 |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology |
| open MeasureTheory Set |
| open scoped ENNReal |
| namespace HardSphere |
| open MeasureTheory Set Filter |
|
|
| private def flattenLinear (n : ℕ) : LiftedConfiguration n→ₗ[ℝ] EuclideanConfiguration (Fin n) where |
| toFun X := WithLp.toLp 2 (fun p ↦ X p.1 p.2) |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| def euclideanFlatten (n : ℕ) : LiftedConfiguration n→L[ℝ] EuclideanConfiguration (Fin n) := |
| (flattenLinear n).toContinuousLinearMap |
|
|
| lemma euclideanFlatten_apply (n : ℕ) (X : LiftedConfiguration n) (p : Direction n) : |
| euclideanFlatten n X p=X p.1 p.2 := rfl |
|
|
| lemma euclideanFlatten_project (n : ℕ) (L : Length) (X : LiftedConfiguration n) : |
| euclideanProject L (euclideanFlatten n X)=project n L X := rfl |
|
|
| lemma euclideanFlatten_single (n : ℕ) (p : Direction n) : |
| euclideanFlatten n (Pi.single p.1 (Pi.single p.2 1))=EuclideanSpace.basisFun (Direction n) ℝ p := by |
| ext q |
| by_cases h1 : q.1=p.1 <;> by_cases h2 : q.2=p.2 <;> |
| simp [euclideanFlatten_apply,EuclideanSpace.basisFun_apply,Prod.ext_iff,h1,h2] |
|
|
| lemma realEuclidean_lift_smooth {n : ℕ} (L : Length) (f : Configuration n L→ℝ) |
| (hf : ContDiff ℝ ((⊤:ℕ∞):WithTop ℕ∞) (fun x : EuclideanConfiguration (Fin n) ↦ f (euclideanProject L x))) : |
| ContDiff ℝ ((⊤:ℕ∞):WithTop ℕ∞) (fun x : LiftedConfiguration n ↦ (f (project n L x):ℂ)) := |
| Complex.ofRealCLM.contDiff.comp (hf.comp (euclideanFlatten n).contDiff) |
|
|
| def realEuclideanMultiplier {n : ℕ} (L : Length) (f : Configuration n L→ℝ) |
| (hf : ContDiff ℝ ((⊤:ℕ∞):WithTop ℕ∞) (fun x : EuclideanConfiguration (Fin n) ↦ f (euclideanProject L x))) : |
| SmoothMultiplier n L := SmoothMultiplier.ofSmooth (fun X ↦ (f X:ℂ)) (realEuclidean_lift_smooth L f hf) |
|
|
| lemma coordDerivative_real_euclidean {n : ℕ} (L : Length) (f : Configuration n L→ℝ) |
| (hf : Differentiable ℝ (fun x : EuclideanConfiguration (Fin n) ↦ f (euclideanProject L x))) |
| (p : Direction n) (X : Configuration n L) : |
| coordDerivative n L (fun Y ↦ (f Y:ℂ)) p X= |
| ((fderiv ℝ (fun x : EuclideanConfiguration (Fin n) ↦ f (euclideanProject L x)) |
| (euclideanFlatten n (representative n L X)) (EuclideanSpace.basisFun (Direction n) ℝ p)):ℂ) := by |
| have hh := (Complex.ofRealCLM.hasFDerivAt.comp _ (hf _).hasFDerivAt).comp |
| (representative n L X) (euclideanFlatten n).hasFDerivAt |
| have he : ((Complex.ofRealCLM ∘ (fun x : EuclideanConfiguration (Fin n) ↦ f (euclideanProject L x))) ∘ euclideanFlatten n)= |
| (fun x : LiftedConfiguration n ↦ (f (project n L x):ℂ)) := by |
| funext x |
| rfl |
| rw [he] at hh |
| unfold coordDerivative |
| rw [hh.fderiv] |
| simp only [ContinuousLinearMap.comp_apply,euclideanFlatten_single,Complex.ofRealCLM_apply] |
|
|
| end HardSphere |
|
|
| namespace HardSphere |
| open MeasureTheory Set Filter |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nontrivial ι] |
|
|
| lemma clippedNearest_le (L : Length) (X Y : ι→Torus L) (i k : ι) (t : ℝ) |
| (hi : X i=Y i) (hrest : ∀ j, j≠k → X j=Y j) |
| (hfar : t≤torusDistance L (Y i) (Y k)) : |
| min (nearestParticleDistance L X i) t≤nearestParticleDistance L Y i := by |
| by_cases he : nearestLabel L Y i=k |
| · simpa only [nearestParticleDistance,he] using (min_le_right (nearestParticleDistance L X i) t).trans hfar |
| · have hn := nearestParticleDistance_le L X i (nearestLabel L Y i) (nearestLabel_ne L Y i) |
| rw [hi,hrest _ he] at hn |
| exact (min_le_left _ _).trans hn |
|
|
| lemma clippedNearest_eq (L : Length) (X Y : ι→Torus L) (i k : ι) (t : ℝ) |
| (hi : X i=Y i) (hrest : ∀ j, j≠k → X j=Y j) |
| (hXfar : t≤torusDistance L (X i) (X k)) (hYfar : t≤torusDistance L (Y i) (Y k)) : |
| min (nearestParticleDistance L X i) t=min (nearestParticleDistance L Y i) t := |
| le_antisymm (le_min (clippedNearest_le L X Y i k t hi hrest hYfar) (min_le_right _ _)) |
| (le_min (clippedNearest_le L Y X i k t hi.symm (fun j h ↦ (hrest j h).symm) hXfar) (min_le_right _ _)) |
|
|
| lemma closeRamp_clipped {s d : ℝ} (hs : 0<s) : |
| closeRamp (2-d/s)=closeRamp (2-min d (2*s)/s) := by |
| by_cases h : d≤2*s |
| · rw [min_eq_left h] |
| · rw [min_eq_right (le_of_not_ge h)] |
| have hd : 2≤d/s := (le_div_iff₀ hs).mpr (le_of_not_ge h) |
| rw [closeRamp_zero (t:=2-d/s) (by linarith only [hd])] |
| have he : 2-2*s/s=0 := by field_simp; ring |
| rw [he,closeRamp_zero le_rfl] |
|
|
| lemma far_from_near_separated (L : Length) (a b v : Torus L) (hnear : torusDistance L a v<6) |
| (hfar : 7≤torusDistance L b v) : 1≤torusDistance L a b := by |
| have ht := torusDistance_triangle' L b a v |
| have he : torusDistance L b a=torusDistance L a b := by simp only [torusDistance,dist_comm] |
| rw [he] at ht |
| linarith only [hnear,hfar,ht] |
|
|
| theorem closeAmplitude_away_eq (L : Length) {s : ℝ} (hs : 0<s) (hs1 : 2*s≤1) |
| (v : Torus L) (X Y : ι→Torus L) (k : ι) (hrest : ∀ i, i≠k → X i=Y i) |
| (hXfar : 7≤torusDistance L (X k) v) (hYfar : 7≤torusDistance L (Y k) v) (i : ι) : |
| closeAmplitude L s v X i=closeAmplitude L s v Y i := by |
| by_cases hik : i=k |
| · subst i |
| rw [closeAmplitude_zero_of_far L s v X k (by linarith only [hXfar]), |
| closeAmplitude_zero_of_far L s v Y k (by linarith only [hYfar])] |
| · have hi := hrest i hik |
| by_cases hv : torusDistance L (X i) v<6 |
| · have hxf := hs1.trans (far_from_near_separated L (X i) (X k) v hv hXfar) |
| have hyv : torusDistance L (Y i) v<6 := hi ▸ hv |
| have hyf := hs1.trans (far_from_near_separated L (Y i) (Y k) v hyv hYfar) |
| have he := clippedNearest_eq L X Y i k (2*s) hi hrest hxf hyf |
| unfold closeAmplitude |
| rw [hi,closeRamp_clipped (d:=nearestParticleDistance L X i) hs, |
| closeRamp_clipped (d:=nearestParticleDistance L Y i) hs,he] |
| · have hxv : 6≤torusDistance L (X i) v := le_of_not_gt hv |
| have hyv : 6≤torusDistance L (Y i) v := hi ▸ hxv |
| rw [closeAmplitude_zero_of_far L s v X i hxv,closeAmplitude_zero_of_far L s v Y i hyv] |
|
|
| lemma localCloseSmoothCount_away_eq (L : Length) {s : ℝ} (hs : 0<s) (hs1 : 2*s≤1) |
| (v : Torus L) (X Y : ι→Torus L) (k : ι) (hrest : ∀ i, i≠k → X i=Y i) |
| (hXfar : 7≤torusDistance L (X k) v) (hYfar : 7≤torusDistance L (Y k) v) : |
| localCloseSmoothCount L s v X=localCloseSmoothCount L s v Y := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [closeAmplitude_away_eq L hs hs1 v X Y k hrest hXfar hYfar i] |
|
|
| end HardSphere |
|
|
| namespace HardSphere |
| open MeasureTheory Set Filter |
| open scoped Convolution NNReal |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nontrivial ι] |
|
|
| def mollifiedCloseRoot (L : Length) (s : ℝ) (v : Torus L) |
| (φ : ContDiffBump (0 : EuclideanConfiguration ι)) : (ι→Torus L)→ℝ := |
| configurationMollify L φ (localCloseRoot L s v) |
|
|
| def mollifiedCloseCount (L : Length) (s : ℝ) (v : Torus L) |
| (φ : ContDiffBump (0 : EuclideanConfiguration ι)) (X : ι→Torus L) : ℝ := |
| mollifiedCloseRoot L s v φ X^2 |
|
|
| lemma mollifiedCloseRoot_nonneg (L : Length) (s : ℝ) (v : Torus L) |
| (φ : ContDiffBump (0 : EuclideanConfiguration ι)) (X : ι→Torus L) : |
| 0 ≤ mollifiedCloseRoot L s v φ X := |
| configurationMollify_nonneg L φ _ (fun _ ↦ norm_nonneg _) X |
|
|
| lemma mollifiedCloseRoot_smooth (L : Length) (s : ℝ) (v : Torus L) |
| (φ : ContDiffBump (0 : EuclideanConfiguration ι)) : |
| ContDiff ℝ ((⊤:ℕ∞):WithTop ℕ∞) (fun x : EuclideanConfiguration ι ↦ mollifiedCloseRoot L s v φ (euclideanProject L x)) := |
| configurationMollify_smooth L φ _ (localCloseRoot_continuous L s v) |
|
|
| lemma mollifiedCloseCount_smooth (L : Length) (s : ℝ) (v : Torus L) |
| (φ : ContDiffBump (0 : EuclideanConfiguration ι)) : |
| ContDiff ℝ ((⊤:ℕ∞):WithTop ℕ∞) (fun x : EuclideanConfiguration ι ↦ mollifiedCloseCount L s v φ (euclideanProject L x)) := |
| (mollifiedCloseRoot_smooth L s v φ).pow 2 |
|
|
| lemma mollifiedCloseRoot_lipschitz (L : Length) {s : ℝ} (hs : 0<s) (hs1 : s≤1) (v : Torus L) |
| (φ : ContDiffBump (0 : EuclideanConfiguration ι)) : |
| LipschitzWith ⟨32/s,by positivity⟩ (fun x : EuclideanConfiguration ι ↦ mollifiedCloseRoot L s v φ (euclideanProject L x)) := |
| configurationMollify_lipschitz L φ (localCloseRoot_continuous L s v) |
| (localCloseRoot_euclidean_lipschitz L hs hs1 v) |
|
|
| lemma mollifiedCloseRoot_approximation (L : Length) {s : ℝ} (hs : 0<s) (hs1 : s≤1) (v : Torus L) |
| (φ : ContDiffBump (0 : EuclideanConfiguration ι)) (X : ι→Torus L) : |
| |mollifiedCloseRoot L s v φ X-localCloseRoot L s v X|≤(32/s)*φ.rOut := by |
| exact configurationMollify_approximation L φ (localCloseRoot_continuous L s v) |
| (localCloseRoot_euclidean_lipschitz L hs hs1 v) X |
|
|
| lemma self_gradient_square_of_lipschitz {κ : Type*} [Fintype κ] {C : ℝ≥0} |
| {f : EuclideanSpace ℝ κ→ℝ} (hf : LipschitzWith C f) {x : EuclideanSpace ℝ κ} |
| (hx : DifferentiableAt ℝ f x) : |
| (∑ p, (fderiv ℝ (fun y ↦ f y^2) x (EuclideanSpace.basisFun κ ℝ p))^2)≤ |
| (4*(C:ℝ)^2)*f x^2 := by |
| rw [←(EuclideanSpace.basisFun κ ℝ).norm_dual,fderiv_fun_pow 2 hx] |
| simp only [Nat.reduceSub,pow_one,nsmul_eq_mul,Nat.cast_ofNat,norm_smul,Real.norm_eq_abs] |
| have hb := norm_fderiv_le_of_lipschitz ℝ (x₀:=x) hf |
| have hsq := (sq_le_sq₀ (norm_nonneg _) C.coe_nonneg).mpr hb |
| calc |
| _ = (4*f x^2)*‖fderiv ℝ f x‖^2 := by rw [mul_pow,sq_abs]; ring |
| _ ≤ (4*f x^2)*(C:ℝ)^2 := mul_le_mul_of_nonneg_left hsq (by positivity) |
| _ = _ := by ring |
|
|
| lemma mollifiedCloseCount_gradient (L : Length) {s : ℝ} (hs : 0<s) (hs1 : s≤1) (v : Torus L) |
| (φ : ContDiffBump (0 : EuclideanConfiguration ι)) (x : EuclideanConfiguration ι) : |
| (∑ p, (fderiv ℝ (fun y ↦ mollifiedCloseCount L s v φ (euclideanProject L y)) x |
| (EuclideanSpace.basisFun (ι×Fin 3) ℝ p))^2)≤ |
| (4096/s^2)*mollifiedCloseCount L s v φ (euclideanProject L x) := by |
| have h := self_gradient_square_of_lipschitz (mollifiedCloseRoot_lipschitz L hs hs1 v φ) |
| ((mollifiedCloseRoot_smooth L s v φ).differentiable (by simp) x) |
| have hc : (4:ℝ)*(32/s)^2=4096/s^2 := by ring |
| rw [←hc] |
| exact h |
|
|
| end HardSphere |
|
|
| end |
|
|
| end OAI |
|
|
