| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.UnitCenter |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| noncomputable section |
| open scoped BigOperators ComplexConjugate |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology |
| noncomputable section |
| open MeasureTheory Set |
| open scoped ENNReal |
| namespace HardSphere |
| open MeasureTheory Set Filter SphericalCoordinates |
| open scoped ENNReal |
| noncomputable section |
|
|
| def latticeExponentialSum : ℝ := ∑' z : ℤ, Real.exp (-|(z:ℝ)|/3) |
|
|
| lemma latticeExponential_summable : Summable (fun z : ℤ ↦ Real.exp (-|(z:ℝ)|/3)) := by |
| have h : Summable (fun n : ℕ ↦ Real.exp (-(n:ℝ)/3)) := by |
| have hg := summable_geometric_of_abs_lt_one (r:=Real.exp (-1/3)) (by |
| rw [abs_of_pos (Real.exp_pos _),Real.exp_lt_one_iff]; norm_num) |
| convert hg using 1 |
| funext n |
| rw [←Real.exp_nat_mul] |
| congr 1 |
| ring |
| apply Summable.of_nat_of_neg |
| · simpa using h |
| · simpa using h |
|
|
| lemma latticeExponentialSum_pos : 0<latticeExponentialSum := by |
| have h := latticeExponential_summable.le_tsum 0 (fun _ _ ↦ (Real.exp_pos _).le) |
| simp only [Int.cast_zero,abs_zero,neg_zero,zero_div,Real.exp_zero] at h |
| exact lt_of_lt_of_le zero_lt_one h |
|
|
| def periodicLatticeIndex {K : ℕ} (L : Length) (x : AddCircle L.val) (v : Fin K) : ℤ := |
| (v.val:ℤ)-round (L.val⁻¹*((v.val:ℝ)+1/2-((AddCircle.equivIco L.val 0) x).val))*(K:ℤ)- |
| ⌊((AddCircle.equivIco L.val 0) x).val-1/2⌋ |
|
|
| lemma periodicLatticeIndex_injective {K : ℕ} {L : Length} (x : AddCircle L.val) : |
| Function.Injective (periodicLatticeIndex (K:=K) L x) := by |
| intro v w hvw |
| have h := congrArg (fun z : ℤ ↦ (z+⌊((AddCircle.equivIco L.val 0) x).val-1/2⌋)%(K:ℤ)) hvw |
| simp only [periodicLatticeIndex,sub_add_cancel,Int.sub_mul_emod_self_right] at h |
| rw [Int.emod_eq_of_lt (by omega) (by exact_mod_cast v.isLt), |
| Int.emod_eq_of_lt (by omega) (by exact_mod_cast w.isLt)] at h |
| exact Fin.ext (by exact_mod_cast h) |
|
|
| lemma periodicLatticeIndex_bound {K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (x : AddCircle L.val) (v : Fin K) : |
| |(periodicLatticeIndex L x v:ℝ)|≤dist (((v.val:ℝ)+1/2:ℝ):AddCircle L.val) x+1 := by |
| let t := ((AddCircle.equivIco L.val 0) x).val |
| let δ := t-1/2-(⌊t-1/2⌋:ℝ) |
| have hδ0 : 0≤δ := Int.fract_nonneg (t-1/2) |
| have hδ1 : δ<1 := Int.fract_lt_one (t-1/2) |
| have hδ : |δ|≤1 := by rw [abs_of_nonneg hδ0]; exact hδ1.le |
| have hd : dist (((v.val:ℝ)+1/2:ℝ):AddCircle L.val) x= |
| |(periodicLatticeIndex L x v:ℝ)-δ| := by |
| conv_lhs => rw [←(show (t:AddCircle L.val)=x from AddCircle.coe_equivIco)] |
| rw [dist_eq_norm,←QuotientAddGroup.mk_sub,AddCircle.norm_eq] |
| congr 1 |
| dsimp [periodicLatticeIndex,δ,t] |
| push_cast |
| rw [←hL] |
| simp only [Int.fract] |
| ring |
| rw [hd] |
| calc |
| _ = |((periodicLatticeIndex L x v:ℝ)-δ)+δ| := by congr 1; ring |
| _ ≤ |(periodicLatticeIndex L x v:ℝ)-δ|+|δ| := abs_add_le _ _ |
| _ ≤ _ := add_le_add le_rfl hδ |
|
|
| lemma circle_lattice_exponential_bound {K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (x : AddCircle L.val) : |
| (∑ v : Fin K, Real.exp (-dist (((v.val:ℝ)+1/2:ℝ):AddCircle L.val) x/3))≤ |
| Real.exp (1/3)*latticeExponentialSum := by |
| classical |
| calc |
| _ ≤ ∑ v : Fin K, Real.exp (1/3)*Real.exp (-|(periodicLatticeIndex L x v:ℝ)|/3) := by |
| apply Finset.sum_le_sum |
| intro v _ |
| rw [←Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| have h := periodicLatticeIndex_bound hL x v |
| linarith only [h] |
| _ = Real.exp (1/3)*∑ v : Fin K, Real.exp (-|(periodicLatticeIndex L x v:ℝ)|/3) := (Finset.mul_sum ..).symm |
| _ ≤ _ := by |
| apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le |
| rw [←Finset.sum_image (f:=fun z : ℤ ↦ Real.exp (-|(z:ℝ)|/3)) (s:=Finset.univ) (periodicLatticeIndex_injective x).injOn] |
| exact latticeExponential_summable.sum_le_tsum _ (fun _ _ ↦ (Real.exp_pos _).le) |
|
|
| lemma torus_exp_le_coordinate_exp (L : Length) (x y : Torus L) : |
| Real.exp (-torusDistance L x y)≤Real.exp (-dist (x 0) (y 0)/3)* |
| Real.exp (-dist (x 1) (y 1)/3)*Real.exp (-dist (x 2) (y 2)/3) := by |
| rw [←Real.exp_add,←Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| have h0 := coordinate_distance_le L x y 0 |
| have h1 := coordinate_distance_le L x y 1 |
| have h2 := coordinate_distance_le L x y 2 |
| linarith only [h0,h1,h2] |
|
|
| theorem unitGrid_exponential_bound : ∃ C : ℝ, 0<C ∧ |
| ∀ (K : ℕ) (L : Length), L.val=(K:ℝ) → ∀ x : Torus L, |
| (∑ v : CubeCell K, Real.exp (-torusDistance L (unitCenter L v) x))≤C := by |
| let B := Real.exp (1/3)*latticeExponentialSum |
| have hB : 0<B := mul_pos (Real.exp_pos _) latticeExponentialSum_pos |
| refine ⟨B^3,pow_pos hB _,?_⟩ |
| intro K L hL x |
| let f := fun (j : Fin 3) (v : Fin K) ↦ Real.exp (-dist (((v.val:ℝ)+1/2:ℝ):AddCircle L.val) (x j)/3) |
| have hb (j : Fin 3) : (∑ v, f j v)≤B := circle_lattice_exponential_bound hL (x j) |
| have hn (j : Fin 3) : 0≤∑ v, f j v := Finset.sum_nonneg (fun _ _ ↦ (Real.exp_pos _).le) |
| calc |
| _ ≤ ∑ v : CubeCell K, f 0 v.1*f 1 v.2.1*f 2 v.2.2 := by |
| apply Finset.sum_le_sum |
| intro v _ |
| exact torus_exp_le_coordinate_exp L (unitCenter L v) x |
| _ = (∑ v, f 0 v)*(∑ v, f 1 v)*(∑ v, f 2 v) := by |
| simp_rw [Fintype.sum_prod_type,←Finset.mul_sum,←Finset.sum_mul] |
| rw [←Fintype.sum_mul_sum] |
| _ ≤ B*B*B := mul_le_mul (mul_le_mul (hb 0) (hb 1) (hn 1) hB.le) (hb 2) (hn 2) (mul_nonneg hB.le hB.le) |
| _ = B^3 := by ring |
|
|
| end |
| end HardSphere |
|
|
| namespace HardSphere |
| open MeasureTheory Set Filter SphericalCoordinates |
| open scoped ENNReal |
| noncomputable section |
|
|
| lemma unitGrid_site_exponential_bound : ∃ C : ℝ, 0<C ∧ |
| ∀ (K : ℕ) (L : Length), L.val=(K:ℝ) → ∀ (A : Finset (Torus L)), A.Nonempty → |
| (∑ v : CubeCell K, Real.exp (-spatialSiteDistance L A (unitCenter L v)))≤C*(A.card:ℝ) := by |
| obtain ⟨C,hC,h⟩ := unitGrid_exponential_bound |
| refine ⟨C,hC,?_⟩ |
| intro K L hL A hA |
| calc |
| _ ≤ ∑ v : CubeCell K, ∑ w∈A, Real.exp (-torusDistance L (unitCenter L v) w) := by |
| apply Finset.sum_le_sum |
| intro v _ |
| obtain ⟨w,hw,he⟩ := spatialSiteDistance_attained L A hA (unitCenter L v) |
| rw [he] |
| exact Finset.single_le_sum (f:=fun w ↦ Real.exp (-torusDistance L (unitCenter L v) w)) (fun _ _ ↦ (Real.exp_pos _).le) hw |
| _ = ∑ w∈A, ∑ v : CubeCell K, Real.exp (-torusDistance L (unitCenter L v) w) := Finset.sum_comm |
| _ ≤ ∑ _w∈A, C := Finset.sum_le_sum (fun w _ ↦ h K L hL w) |
| _ = _ := by rw [Finset.sum_const, nsmul_eq_mul,mul_comm] |
|
|
| lemma unitOccupation_full_card {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (v : CubeCell K) (X : Configuration n L) : |
| unitOccupation hL (fun _ ↦ True) v X=Fintype.card {i : Fin n // unitSite hL (X i)=v} := by |
| classical |
| apply Fintype.card_congr |
| exact { |
| toFun := fun i ↦ ⟨i.val.val,i.property⟩ |
| invFun := fun i ↦ ⟨⟨i.val,trivial⟩,i.property⟩ |
| left_inv := fun _ ↦ rfl |
| right_inv := fun _ ↦ rfl } |
|
|
| lemma unitOccupation_sum_weight {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (a : CubeCell K → ℝ) (X : Configuration n L) : |
| (∑ v : CubeCell K, a v*(unitOccupation hL (fun _ ↦ True) v X:ℝ))= |
| ∑ i : Fin n, a (unitSite hL (X i)) := by |
| classical |
| rw [←Fintype.sum_fiberwise (fun i : Fin n ↦ unitSite hL (X i)) (fun i ↦ a (unitSite hL (X i)))] |
| apply Finset.sum_congr rfl |
| intro v _ |
| simp only [unitOccupation_full_card] |
| have he : (fun i : {i : Fin n // unitSite hL (X i)=v} ↦ a (unitSite hL (X i.val)))=fun _ ↦ a v := by |
| funext i |
| rw [i.property] |
| rw [he,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_comm] |
|
|
| lemma weightedSiteCount_le_cell {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (A : Finset (Torus L)) (hA : A.Nonempty) (X : Configuration n L) : |
| weightedSiteCount L A X≤Real.exp 1*(∑ v : CubeCell K, |
| Real.exp (-spatialSiteDistance L A (unitCenter L v))*(unitOccupation hL (fun _ ↦ True) v X:ℝ)) := by |
| rw [unitOccupation_sum_weight,Finset.mul_sum,weightedSiteCount,ite_eq_left hA] |
| apply Finset.sum_le_sum |
| intro i _ |
| rw [←Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| have hd := spatialSiteDistance_triangle L A (unitCenter L (unitSite hL (X i))) (X i) |
| have hc := unitSite_dist_center hL (X i) |
| rw [torusDistance_comm] at hd |
| linarith only [hd,hc] |
|
|
| end |
| end HardSphere |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
