| import Mathlib |
| import OAI.Analysis.HardSphere.Comparison.FiniteFiberSum |
|
|
| 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 |
|
|
| lemma exists_injective_near {ι E : Type*} [Fintype ι] [DecidableEq ι] |
| [MetricSpace E] [PerfectSpace E] (X : ι→E) {ε : ℝ} (hε : 0<ε) : |
| ∃ Y : ι→E, Function.Injective Y ∧ ∀ i, dist (Y i) (X i)<ε := by |
| classical |
| have aux (I : Finset ι) : ∃ Y : ι→E, Set.InjOn Y I ∧ ∀ i∈I, dist (Y i) (X i)<ε := by |
| induction I using Finset.induction_on with |
| | empty => exact ⟨X,by simp [Set.InjOn],by simp⟩ |
| | @insert a I ha ih => |
| obtain ⟨Y,hinj,hclose⟩ := ih |
| have hd := dense_univ.sdiff_finset (I.image Y) |
| obtain ⟨y,hy,hball⟩ := hd.exists_mem_open Metric.isOpen_ball (Metric.nonempty_ball.mpr hε) |
| refine ⟨Function.update Y a y,?_,?_⟩ |
| · intro i hi j hj he |
| rcases Finset.mem_insert.mp hi with hi | hi |
| · subst i |
| rcases Finset.mem_insert.mp hj with hj | hj |
| · exact hj.symm |
| · have hja : j≠a := fun h ↦ ha (h ▸ hj) |
| rw [Function.update_self,Function.update_of_ne hja] at he |
| exact (hy.2 (by rw [he]; exact Finset.mem_image.mpr ⟨j,hj,rfl⟩)).elim |
| · rcases Finset.mem_insert.mp hj with hj | hj |
| · subst j |
| have hia : i≠a := fun h ↦ ha (h ▸ hi) |
| rw [Function.update_self,Function.update_of_ne hia] at he |
| exact (hy.2 (Finset.mem_image.mpr ⟨i,hi,he⟩)).elim |
| · have hia : i≠a := fun h ↦ ha (h ▸ hi) |
| have hja : j≠a := fun h ↦ ha (h ▸ hj) |
| rw [Function.update_of_ne hia,Function.update_of_ne hja] at he |
| exact hinj hi hj he |
| · intro i hi |
| rcases Finset.mem_insert.mp hi with rfl | hi |
| · simpa using hball |
| · rw [Function.update_of_ne (show i≠a from fun h ↦ ha (h ▸ hi))] |
| exact hclose i hi |
| obtain ⟨Y,hi,hc⟩ := aux Finset.univ |
| exact ⟨Y,fun i j h ↦ hi (Finset.mem_univ _) (Finset.mem_univ _) h,fun i ↦ hc i (Finset.mem_univ _)⟩ |
|
|
| lemma dense_injective_configurations {ι E : Type*} [Fintype ι] [DecidableEq ι] |
| [MetricSpace E] [PerfectSpace E] : Dense {X : ι→E | Function.Injective X} := by |
| apply Metric.dense_iff.mpr |
| intro X ε hε |
| obtain ⟨Y,hY,hclose⟩ := exists_injective_near X hε |
| refine ⟨Y,?_,hY⟩ |
| exact (dist_pi_lt_iff hε).mpr hclose |
|
|
| lemma torus_nontrivial (L : Length) : Nontrivial (Torus L) := by |
| let hc : Nontrivial (AddCircle L.val) := by |
| refine ⟨⟨((L.val/2:ℝ):AddCircle L.val),0,?_⟩⟩ |
| intro he |
| have hh := AddCircle.norm_half_period_eq L.val |
| rw [he,norm_zero,abs_of_pos L.property] at hh |
| linarith only [L.property,hh] |
| let := hc |
| infer_instance |
|
|
| lemma torusDistance_continuous_pair (L : Length) : |
| Continuous (fun p : Torus L×Torus L ↦ torusDistance L p.1 p.2) := by |
| unfold torusDistance |
| fun_prop |
|
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| attribute [local irreducible] torusDistance nearestParticleDistance |
|
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| lemma nearestDifference_continuous {ι : Type*} [Fintype ι] [DecidableEq ι] [Nontrivial ι] |
| (L : Length) : Continuous (fun p : (ι→Torus L)×(ι→Torus L) ↦ |
| ∑ i, (nearestParticleDistance L p.2 i-nearestParticleDistance L p.1 i)^2) := by |
| apply continuous_finsetSum |
| intro i _ |
| exact (((nearestParticleDistance_continuous L i).comp continuous_snd).sub |
| ((nearestParticleDistance_continuous L i).comp continuous_fst)).pow 2 |
|
|
| lemma torusMotion_continuous {ι : Type*} [Fintype ι] (L : Length) : |
| Continuous (fun p : (ι→Torus L)×(ι→Torus L) ↦ 502*∑ i, torusDistance L (p.1 i) (p.2 i)^2) := by |
| apply continuous_const.mul |
| apply continuous_finsetSum |
| intro i _ |
| apply Continuous.pow |
| exact (torusDistance_continuous_pair L).comp |
| (f := fun p : (ι→Torus L)×(ι→Torus L) ↦ (p.1 i,p.2 i)) |
| (((continuous_apply i).comp continuous_fst).prodMk ((continuous_apply i).comp continuous_snd)) |
|
|
| theorem nearestParticleDistance_sum_sq_all {ι : Type*} [Fintype ι] [DecidableEq ι] [Nontrivial ι] |
| (L : Length) (X Y : ι→Torus L) : |
| (∑ i, (nearestParticleDistance L Y i-nearestParticleDistance L X i)^2)≤ |
| 502*∑ i, torusDistance L (X i) (Y i)^2 := by |
| let := torus_nontrivial L |
| let S : Set (ι→Torus L) := {X | Function.Injective X} |
| have hd : Dense (S×ˢS) := dense_injective_configurations.prod dense_injective_configurations |
| have hc : IsClosed {p : (ι→Torus L)×(ι→Torus L) | |
| (∑ i, (nearestParticleDistance L p.2 i-nearestParticleDistance L p.1 i)^2)≤ |
| 502*∑ i, torusDistance L (p.1 i) (p.2 i)^2} := |
| isClosed_le (nearestDifference_continuous L) (torusMotion_continuous L) |
| exact closure_minimal (fun p hp ↦ nearestParticleDistance_sum_sq L p.1 p.2 hp.1 hp.2) hc (hd (X,Y)) |
|
|
| end HardSphere |
|
|
| namespace HardSphere |
| open MeasureTheory Set Filter |
|
|
| def closeRamp (t : ℝ) : ℝ := max 0 (min 1 t) |
|
|
| lemma closeRamp_nonneg (t : ℝ) : 0≤closeRamp t := le_max_left _ _ |
| lemma closeRamp_le_one (t : ℝ) : closeRamp t≤1 := max_le zero_le_one (min_le_left _ _) |
| lemma closeRamp_one {t : ℝ} (ht : 1≤t) : closeRamp t=1 := by simp [closeRamp,min_eq_left ht] |
| lemma closeRamp_zero {t : ℝ} (ht : t≤0) : closeRamp t=0 := by |
| exact max_eq_left ((min_le_right 1 t).trans ht) |
| lemma closeRamp_lipschitz : LipschitzWith 1 closeRamp := |
| (LipschitzWith.id.const_min 1).const_max 0 |
|
|
| lemma closeRamp_sub_le (x y : ℝ) : |closeRamp x-closeRamp y|≤|x-y| := by |
| simpa only [Real.dist_eq,NNReal.coe_one,one_mul] using closeRamp_lipschitz.dist_le_mul x y |
|
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| variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nontrivial ι] |
|
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| def closeAmplitude (L : Length) (s : ℝ) (v : Torus L) (X : ι→Torus L) (i : ι) : ℝ := |
| closeRamp (6-torusDistance L (X i) v)*closeRamp (2-nearestParticleDistance L X i/s) |
|
|
| def localCloseRoot (L : Length) (s : ℝ) (v : Torus L) (X : ι→Torus L) : ℝ := |
| ‖(WithLp.toLp 2 (closeAmplitude L s v X) : EuclideanSpace ℝ ι)‖ |
|
|
| def localCloseSmoothCount (L : Length) (s : ℝ) (v : Torus L) (X : ι→Torus L) : ℝ := |
| ∑ i, closeAmplitude L s v X i^2 |
|
|
| lemma localCloseRoot_sq (L : Length) (s : ℝ) (v : Torus L) (X : ι→Torus L) : |
| localCloseRoot L s v X^2=localCloseSmoothCount L s v X := |
| EuclideanSpace.real_norm_sq_eq _ |
|
|
| lemma closeAmplitude_nonneg (L : Length) (s : ℝ) (v : Torus L) (X : ι→Torus L) (i : ι) : |
| 0≤closeAmplitude L s v X i := mul_nonneg (closeRamp_nonneg _) (closeRamp_nonneg _) |
| lemma closeAmplitude_le_one (L : Length) (s : ℝ) (v : Torus L) (X : ι→Torus L) (i : ι) : |
| closeAmplitude L s v X i≤1 := |
| (mul_le_of_le_one_left (closeRamp_nonneg _) (closeRamp_le_one _)).trans |
| (closeRamp_le_one _) |
|
|
| lemma closeAmplitude_one (L : Length) {s : ℝ} (hs : 0<s) (v : Torus L) (X : ι→Torus L) (i : ι) |
| (hv : torusDistance L (X i) v≤5) (hc : nearestParticleDistance L X i ≤ s) : |
| closeAmplitude L s v X i=1 := by |
| unfold closeAmplitude |
| have hd : nearestParticleDistance L X i/s≤1 := (div_le_one hs).mpr hc |
| rw [closeRamp_one (by linarith only [hv]),closeRamp_one (by linarith only [hd]),mul_one] |
|
|
| lemma closeAmplitude_zero_of_far (L : Length) (s : ℝ) (v : Torus L) (X : ι→Torus L) (i : ι) |
| (hv : 6≤torusDistance L (X i) v) : closeAmplitude L s v X i=0 := by |
| unfold closeAmplitude |
| rw [closeRamp_zero (by linarith only [hv]),zero_mul] |
|
|
| lemma closeAmplitude_zero_of_isolated (L : Length) {s : ℝ} (hs : 0<s) (v : Torus L) (X : ι→Torus L) (i : ι) |
| (hc : 2*s≤nearestParticleDistance L X i) : closeAmplitude L s v X i=0 := by |
| unfold closeAmplitude |
| have hd : 2≤nearestParticleDistance L X i/s := (le_div_iff₀ hs).mpr hc |
| rw [closeRamp_zero (t := 2-nearestParticleDistance L X i/s) (by linarith only [hd]),mul_zero] |
|
|
| lemma torusDistance_sub_le (L : Length) (x y v : Torus L) : |
| |torusDistance L x v-torusDistance L y v|≤torusDistance L x y := by |
| simpa only [euclideanTorusPoint_dist] using abs_dist_sub_le |
| (euclideanTorusPoint L x) (euclideanTorusPoint L y) (euclideanTorusPoint L v) |
|
|
| lemma unit_product_sub_le {a b c d : ℝ} (ha : 0≤a) (_hb : 0≤b) (_hc : 0≤c) (hd : 0≤d) |
| (hA : a≤1) (_hB : b≤1) (_hC : c≤1) (hD : d≤1) : |
| |a*b-c*d|≤|a-c|+|b-d| := by |
| calc |
| _ = |a*(b-d)+(a-c)*d| := by congr 1; ring |
| _ ≤ |a*(b-d)|+|(a-c)*d| := abs_add_le _ _ |
| _ = a*|b-d|+|a-c| *d := by rw [abs_mul,abs_mul,abs_of_nonneg ha,abs_of_nonneg hd] |
| _ ≤ 1*|b-d|+|a-c| *1 := add_le_add |
| (mul_le_mul_of_nonneg_right hA (abs_nonneg _)) (mul_le_mul_of_nonneg_left hD (abs_nonneg _)) |
| _ = _ := by ring |
|
|
| lemma closeAmplitude_sub_le (L : Length) {s : ℝ} (hs : 0<s) (v : Torus L) (X Y : ι→Torus L) (i : ι) : |
| |closeAmplitude L s v Y i-closeAmplitude L s v X i|≤ |
| torusDistance L (X i) (Y i)+|nearestParticleDistance L Y i-nearestParticleDistance L X i|/s := by |
| unfold closeAmplitude |
| apply (unit_product_sub_le (closeRamp_nonneg _) (closeRamp_nonneg _) (closeRamp_nonneg _) (closeRamp_nonneg _) |
| (closeRamp_le_one _) (closeRamp_le_one _) (closeRamp_le_one _) (closeRamp_le_one _)).trans |
| apply add_le_add |
| · apply (closeRamp_sub_le _ _).trans |
| have he : (6-torusDistance L (Y i) v)-(6-torusDistance L (X i) v)= |
| torusDistance L (X i) v-torusDistance L (Y i) v := by ring |
| rw [he] |
| exact torusDistance_sub_le L (X i) (Y i) v |
| · apply (closeRamp_sub_le _ _).trans_eq |
| rw [show (2-nearestParticleDistance L Y i/s)-(2-nearestParticleDistance L X i/s)= |
| -(nearestParticleDistance L Y i-nearestParticleDistance L X i)/s by ring, |
| abs_div,abs_neg,abs_of_pos hs] |
|
|
| end HardSphere |
|
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| end |
|
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| end OAI |
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