| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.SmoothMultiplier |
| import OAI.Analysis.HardSphere.Geometry.TorusLiftBox |
|
|
| 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 |
| open MeasureTheory Set Filter |
| open scoped ComplexConjugate |
|
|
| def headLift (n : ℕ) : LiftedConfiguration (n+1) →L[ℝ] LiftedConfiguration 1 := |
| .pi fun _ ↦ .proj 0 |
|
|
| lemma headLift_direction_zero (n : ℕ) (j : Fin 3) : |
| headLift n (Pi.single 0 (Pi.single j 1))=Pi.single 0 (Pi.single j 1) := by |
| ext i k |
| have hi : i=0 := Subsingleton.elim _ _ |
| subst i |
| simp [headLift] |
|
|
| lemma headLift_direction_succ (n : ℕ) (i : Fin n) (j : Fin 3) : |
| headLift n (Pi.single i.succ (Pi.single j 1))=0 := by |
| ext k q |
| simp [headLift] |
|
|
| def SmoothMultiplier.orbital {L : Length} (w : SmoothMultiplier 1 L) (x : Torus L) : ℂ := |
| w.toFun (fun _ ↦ x) |
|
|
| lemma SmoothMultiplier.orbital_continuous {L : Length} (w : SmoothMultiplier 1 L) : |
| Continuous w.orbital := |
| (continuous_of_continuous_lift w.smooth.continuous).comp (continuous_pi fun _ ↦ continuous_id) |
|
|
| def SmoothMultiplier.orbitalBound {L : Length} (w : SmoothMultiplier 1 L) : ℝ := |
| ‖(⟨w.orbital,w.orbital_continuous⟩ : C(Torus L,ℂ))‖^2 |
|
|
| lemma SmoothMultiplier.orbital_le_bound {L : Length} (w : SmoothMultiplier 1 L) (x : Torus L) : |
| ‖w.orbital x‖^2≤w.orbitalBound := by |
| exact pow_le_pow_left₀ (norm_nonneg _) ((⟨w.orbital,w.orbital_continuous⟩ : C(Torus L,ℂ)).norm_coe_le_norm x) 2 |
|
|
| def SmoothMultiplier.orbitalEnergy {L : Length} (w : SmoothMultiplier 1 L) : ℝ := |
| ∑ j : Fin 3, ∫ x : Torus L, ‖coordDerivative 1 L w.toFun (0,j) (fun _ ↦ x)‖^2 |
|
|
| def SmoothMultiplier.liftHead {L : Length} (w : SmoothMultiplier 1 L) (n : ℕ) : |
| SmoothMultiplier (n+1) L := |
| .ofSmooth (fun X ↦ w.orbital (X 0)) (w.smooth.comp (headLift n).contDiff) |
|
|
| lemma SmoothMultiplier.coordDerivative_liftHead_zero {L : Length} |
| (w : SmoothMultiplier 1 L) (n : ℕ) (j : Fin 3) : |
| coordDerivative (n+1) L (w.liftHead n).toFun (0,j)= |
| fun X ↦ coordDerivative 1 L w.toFun (0,j) (fun _ ↦ X 0) := by |
| funext X |
| have h := (w.smooth.differentiable (by simp) |
| (headLift n (representative (n+1) L X))).hasFDerivAt.comp |
| (representative (n+1) L X) (headLift n).hasFDerivAt |
| change (fderiv ℝ ((fun Y : LiftedConfiguration 1 ↦ w.toFun (project 1 L Y)) ∘ headLift n) |
| (representative (n+1) L X)) (Pi.single 0 (Pi.single j 1))=_ |
| rw [h.fderiv,ContinuousLinearMap.comp_apply,headLift_direction_zero] |
| rfl |
|
|
| lemma SmoothMultiplier.coordDerivative_liftHead_succ {L : Length} |
| (w : SmoothMultiplier 1 L) (n : ℕ) (i : Fin n) (j : Fin 3) : |
| coordDerivative (n+1) L (w.liftHead n).toFun (i.succ,j)=0 := by |
| funext X |
| have h := (w.smooth.differentiable (by simp) |
| (headLift n (representative (n+1) L X))).hasFDerivAt.comp |
| (representative (n+1) L X) (headLift n).hasFDerivAt |
| change (fderiv ℝ ((fun Y : LiftedConfiguration 1 ↦ w.toFun (project 1 L Y)) ∘ headLift n) |
| (representative (n+1) L X)) (Pi.single i.succ (Pi.single j 1))=0 |
| rw [h.fderiv,ContinuousLinearMap.comp_apply,headLift_direction_succ,map_zero] |
|
|
| lemma SmoothMultiplier.liftHead_gradient_sum {L : Length} |
| (w : SmoothMultiplier 1 L) (n : ℕ) (X : Configuration (n+1) L) : |
| (∑ p : Direction (n+1), ‖coordDerivative (n+1) L (w.liftHead n).toFun p X‖^2)= |
| ∑ j : Fin 3, ‖coordDerivative 1 L w.toFun (0,j) (fun _ ↦ X 0)‖^2 := by |
| rw [Fintype.sum_prod_type,Fin.sum_univ_succ] |
| simp only [w.coordDerivative_liftHead_zero,w.coordDerivative_liftHead_succ, |
| Pi.zero_apply,norm_zero,zero_pow (by decide : 2≠0),Finset.sum_const_zero,add_zero] |
|
|
| end HardSphere |
|
|
| end |
|
|
| end OAI |
|
|
