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67aee1d | 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 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.MollifiedCloseCountMultiplier
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
open scoped Topology
attribute [local irreducible] torusDistance
lemma coordDerivative_zero_of_particle_locality {n : ℕ} (L : Length) (f : Configuration n L→ℂ)
(hf : Differentiable ℝ (fun x : LiftedConfiguration n ↦ f (project n L x)))
(v : Torus L) (r : ℝ)
(hloc : ∀ (X Y : Configuration n L) (k : Fin n), (∀ i, i≠k → X i=Y i) → r≤torusDistance L (X k) v →
r≤torusDistance L (Y k) v → f X=f Y)
(p : Direction n) (X : Configuration n L) (hfar : r<torusDistance L (X p.1) v) :
coordDerivative n L f p X=0 := by
let w : LiftedConfiguration n := Pi.single p.1 (Pi.single p.2 1)
let Y : ℝ → Configuration n L := fun t ↦ project n L (representative n L X+t • w)
have hY0 : Y 0=X := by simp only [Y,zero_smul,add_zero,project_representative]
have hY : Continuous Y := by
apply (continuous_project n L).comp
exact continuous_const.add (continuous_id.smul continuous_const)
have hdcont : Continuous (fun t : ℝ ↦ torusDistance L (Y t p.1) v) :=
(torusDistance_continuous_pair L).comp (((continuous_apply p.1).comp hY).prodMk continuous_const)
have hevent : ∀ᶠ t in 𝓝 (0:ℝ), r<torusDistance L (Y t p.1) v :=
hdcont.continuousAt.eventually_const_lt (by simpa only [hY0] using hfar)
have hrest (t : ℝ) (i : Fin n) (hi : i≠p.1) : X i=Y t i := by
have he : Y t i=project n L (representative n L X) i := by
ext j
simp only [Y,project,Pi.add_apply,Pi.smul_apply,w,Pi.single_eq_of_ne hi,Pi.zero_apply,
smul_zero,add_zero]
rw [he,project_representative]
have heq : (fun t : ℝ ↦ f (Y t))=ᶠ[𝓝 (0:ℝ)] (fun _ ↦ f X) := by
filter_upwards [hevent] with t ht
exact (hloc X (Y t) p.1 (hrest t) hfar.le ht.le).symm
have hz : HasDerivAt (fun t : ℝ ↦ f (Y t)) 0 0 :=
(hasDerivAt_const 0 (f X)).congr_of_eventuallyEq heq
have hline := (hasFDerivAt_const (𝕜:=ℝ) (representative n L X) (0:ℝ)).add
((hasFDerivAt_id (𝕜:=ℝ) (0:ℝ)).smul_const w)
have hf0 : HasFDerivAt (fun x : LiftedConfiguration n ↦ f (project n L x))
(fderiv ℝ (fun x : LiftedConfiguration n ↦ f (project n L x)) (representative n L X))
(representative n L X+(0:ℝ) • w) := by
simpa only [zero_smul,add_zero] using (hf (representative n L X)).hasFDerivAt
have hd := (hf0.comp 0 hline).hasDerivAt
have he := hd.unique hz
simpa only [coordDerivative,w,ContinuousLinearMap.comp_apply,add_apply,zero_apply,
ContinuousLinearMap.smulRight_apply,ContinuousLinearMap.id_apply,one_smul,zero_add] using he
variable {n : ℕ} [Nontrivial (Fin n)]
lemma mollifiedCloseCountMultiplier_partial_zero (L : Length) {s : ℝ} (hs : 0<s) (hs1 : 2*s≤1)
(v : Torus L) (φ : ContDiffBump (0 : EuclideanConfiguration (Fin n))) (hφ : φ.rOut≤1)
(p : Direction n) (X : Configuration n L) (hfar : 8<torusDistance L (X p.1) v) :
coordDerivative n L (mollifiedCloseCountMultiplier L s v φ).toFun p X=0 := by
apply coordDerivative_zero_of_particle_locality L _
((mollifiedCloseCountMultiplier L s v φ).smooth.differentiable (by simp)) v 8 _ p X hfar
intro X Y k hr hx hy
simp only [mollifiedCloseCountMultiplier_apply,
mollifiedCloseCount_away_eq L hs hs1 v φ hφ X Y k hr hx hy]
lemma mollifiedCloseCutoff_partial_zero (L : Length) {s : ℝ} (hs : 0<s) (hs1 : 2*s≤1)
(v : Torus L) (φ : ContDiffBump (0 : EuclideanConfiguration (Fin n))) (hφ : φ.rOut≤1)
(h : ℝ) (p : Direction n) (X : Configuration n L) (hfar : 8<torusDistance L (X p.1) v) :
coordDerivative n L (mollifiedCloseCutoff L s v φ h).toFun p X=0 := by
rw [mollifiedCloseCutoff_partial,mollifiedCloseCountMultiplier_partial_zero L hs hs1 v φ hφ p X hfar,mul_zero]
end HardSphere
end
end OAI
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