| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.UnitPairCount |
|
|
| 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 SphericalCoordinates |
| open scoped ENNReal |
|
|
| lemma unitOccupation_le {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (p : Fin n → Prop) [DecidablePred p] (v : CubeCell K) (X : Configuration n L) : |
| unitOccupation hL p v X≤n := by |
| exact (Fintype.card_subtype_le _).trans ((Fintype.card_subtype_le _).trans (by simp)) |
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| lemma unitPairCount_le {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (p : Fin n → Prop) [DecidablePred p] (v : CubeCell K) (X : Configuration n L) : |
| unitPairCount hL p v X≤(n:ℝ)^2 := by |
| have hn : 0≤(unitOccupation hL p v X:ℝ) := Nat.cast_nonneg _ |
| have hb : (unitOccupation hL p v X:ℝ)≤n := by exact_mod_cast unitOccupation_le hL p v X |
| unfold unitPairCount |
| nlinarith only [hn, mul_self_le_mul_self hn hb] |
|
|
| lemma unitPairCount_integrable {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (p : Fin n → Prop) [DecidablePred p] (v : CubeCell K) (u : ScalarL2 n L) : |
| Integrable (fun X ↦ unitPairCount hL p v X*‖u X‖^2) volume := |
| L2_bounded_weight_integrable u (unitPairCount_measurable hL p v) |
| (fun X ↦ by rw [abs_of_nonneg (unitPairCount_nonneg hL p v X)]; exact unitPairCount_le hL p v X) |
|
|
| lemma localizedNearLabels_integrable {n : ℕ} (L : Length) (A : Finset (Torus L)) (s : ℝ) (u : ScalarL2 n L) : |
| Integrable (fun X ↦ ((localizedNearLabels L A s X).card:ℝ)*‖u X‖^2) volume := |
| L2_bounded_weight_integrable u ((measurable_of_countable (fun k : ℕ ↦ (k:ℝ))).comp (localizedNearLabels_card_measurable L A s)) |
| (B:=(n:ℝ)) (fun X ↦ by rw [abs_of_nonneg (Nat.cast_nonneg _)]; exact_mod_cast (show (localizedNearLabels L A s X).card≤n by simpa using (localizedNearLabels L A s X).card_le_univ)) |
|
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| def weightedUnitPairCount {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (A : Finset (Torus L)) (X : Configuration n L) : ℝ := |
| ∑ v : CubeCell K, Real.exp (-spatialSiteDistance L A (unitCenter L v))*unitPairCount hL (fun _ ↦ True) v X |
|
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| lemma weightedUnitPairCount_nonneg {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (A : Finset (Torus L)) (X : Configuration n L) : 0≤weightedUnitPairCount hL A X := |
| Finset.sum_nonneg (fun v _ ↦ mul_nonneg (Real.exp_pos _).le (unitPairCount_nonneg hL _ v X)) |
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| lemma weightedUnitPairCount_integral {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (A : Finset (Torus L)) (u : ScalarL2 n L) : |
| (∫ X, weightedUnitPairCount hL A X*‖u X‖^2)= |
| ∑ v : CubeCell K, Real.exp (-spatialSiteDistance L A (unitCenter L v))*(∫ X, unitPairCount hL (fun _ ↦ True) v X*‖u X‖^2) := by |
| simp only [weightedUnitPairCount,Finset.sum_mul,mul_assoc] |
| rw [integral_finsetSum] |
| · simp only [integral_const_mul] |
| · intro v _ |
| exact (unitPairCount_integrable hL _ v u).const_mul _ |
|
|
| lemma weightedUnitPairCount_integrable {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (A : Finset (Torus L)) (u : ScalarL2 n L) : |
| Integrable (fun X ↦ weightedUnitPairCount hL A X*‖u X‖^2) volume := by |
| simp only [weightedUnitPairCount,Finset.sum_mul,mul_assoc] |
| exact integrable_finsetSum _ (fun v _ ↦ (unitPairCount_integrable hL _ v u).const_mul _) |
|
|
| lemma spatial_cell_average_real {n K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (A : Finset (Torus L)) {r : ℝ} (hr : 0<r) (u : Jet n L) (hu : u∈dirichletDomain n L r) : |
| ENNReal.ofReal (Real.exp (-1)/(64*Real.sqrt 3)*r*(∫ X, weightedUnitPairCount hL A X*‖u.1 X‖^2))≤ |
| ∫⁻ H, ENNReal.ofReal (spatialPartitionEnergy L A H u) ∂radiusLaw := by |
| apply le_trans _ (spatial_cell_average hL A hr u hu) |
| apply le_of_eq |
| have hi (v : CubeCell K) : ENNReal.ofReal (∫ X, unitPairCount hL (fun _ ↦ True) v X*‖u.1 X‖^2)= |
| ∫⁻ X, ENNReal.ofReal (unitPairCount hL (fun _ ↦ True) v X)*ENNReal.ofReal (‖u.1 X‖^2) := by |
| rw [ofReal_integral_eq_lintegral_ofReal (unitPairCount_integrable hL _ v u.1) |
| (Eventually.of_forall (fun X ↦ mul_nonneg (unitPairCount_nonneg hL _ v X) (sq_nonneg _)))] |
| simp only [ENNReal.ofReal_mul (unitPairCount_nonneg hL _ v _)] |
| have hn (v : CubeCell K) : 0≤∫ X, unitPairCount hL (fun _ ↦ True) v X*‖u.1 X‖^2 := |
| integral_nonneg (fun X ↦ mul_nonneg (unitPairCount_nonneg hL _ v X) (sq_nonneg _)) |
| rw [weightedUnitPairCount_integral,Finset.mul_sum,ENNReal.ofReal_sum_of_nonneg] |
| · apply Finset.sum_congr rfl |
| intro v _ |
| have he : Real.exp (-1)/(64*Real.sqrt 3)*r*(Real.exp (-spatialSiteDistance L A (unitCenter L v))*(∫ X, unitPairCount hL (fun _ ↦ True) v X*‖u.1 X‖^2))= |
| Real.exp (-(spatialSiteDistance L A (unitCenter L v)+1))*(1/(64*Real.sqrt 3)*r)*(∫ X, unitPairCount hL (fun _ ↦ True) v X*‖u.1 X‖^2) := by |
| rw [neg_add,Real.exp_add] |
| ring |
| rw [he,ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_mul (Real.exp_pos _).le,hi] |
| exact mul_assoc _ _ _ |
| · intro v _ |
| exact mul_nonneg (by positivity) (mul_nonneg (Real.exp_pos _).le (hn v)) |
|
|
| lemma spatial_near_average_real {n : ℕ} (L : Length) (A : Finset (Torus L)) |
| {r s : ℝ} (hs : 0<s) (hs' : 2*s≤1) (hL : 10*s≤L.val) (hr : 0<r) |
| (u : Jet n L) (hu : u∈dirichletDomain n L r) : |
| ENNReal.ofReal (Real.exp (-8)*(27/(262144*Real.sqrt 3))*r/s^3* |
| (∫ X, ((localizedNearLabels L A s X).card:ℝ)*‖u.1 X‖^2))≤ |
| ∫⁻ H, ENNReal.ofReal (spatialPartitionEnergy L A H u) ∂radiusLaw := by |
| have hi : ENNReal.ofReal (∫ X, ((localizedNearLabels L A s X).card:ℝ)*‖u.1 X‖^2)= |
| ∫⁻ X, ((localizedNearLabels L A s X).card:ℝ≥0∞)*ENNReal.ofReal (‖u.1 X‖^2) := by |
| rw [ofReal_integral_eq_lintegral_ofReal (localizedNearLabels_integrable L A s u.1) |
| (Eventually.of_forall (fun X ↦ mul_nonneg (Nat.cast_nonneg _) (sq_nonneg _)))] |
| simp only [ENNReal.ofReal_mul (Nat.cast_nonneg _),ENNReal.ofReal_natCast] |
| rw [ENNReal.ofReal_mul (by positivity),hi] |
| exact spatial_near_average L A hs hs' hL hr u hu |
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|
| theorem spatial_before_square : ∃ C D : ℝ, 0<C ∧ 0<D ∧ |
| ∀ (n K : ℕ) (L : Length) (hL : L.val=(K:ℝ)) (r s : ℝ), |
| 0<r → 0<s → 2*s≤1 → 10*s≤L.val → (n:ℝ)*(2*Real.pi*r)^3≤L.val^3/2 → |
| ∀ (g : Jet n L), IsGroundVector n L r g → |
| ∀ (A : Finset (Torus L)), A.Nonempty → ∀ (u : Jet n L), u∈dirichletDomain n L r → |
| Real.exp (-1)/(128*Real.sqrt 3)*r*(∫ X, weightedUnitPairCount hL A X*‖u.1 X‖^2)+ |
| Real.exp (-8)*(27/(524288*Real.sqrt 3))*r/s^3*(∫ X, ((localizedNearLabels L A s X).card:ℝ)*‖u.1 X‖^2)- |
| (C*(n:ℝ)*r/L.val^3+D)*(∫ X, weightedSiteCount L A X*‖u.1 X‖^2)≤energy u-energy g*‖u.1‖^2 := by |
| obtain ⟨C,D,hC,hD,hsp⟩ := spatial_spectral_average |
| refine ⟨C,D,hC,hD,?_⟩ |
| intro n K L hL r s hr hs hs' hsL hsmall g hg A hA u hu |
| have hF : 0≤∫ X, weightedSiteCount L A X*‖u.1 X‖^2 := |
| integral_nonneg (fun X ↦ mul_nonneg (weightedSiteCount_nonneg L A X) (sq_nonneg _)) |
| have hLpos : 0<L.val := L.property |
| have hB : 0≤energy u-energy g*‖u.1‖^2+(C*(n:ℝ)*r/L.val^3+D)*(∫ X, weightedSiteCount L A X*‖u.1 X‖^2) := |
| add_nonneg (sub_nonneg.mpr (hg.rayleigh_all u hu)) (mul_nonneg (by positivity) hF) |
| have hc := (ENNReal.ofReal_le_ofReal_iff hB).mp ((spatial_cell_average_real hL A hr u hu).trans |
| (hsp n L r hr hsmall g hg A hA u hu)) |
| have hk := (ENNReal.ofReal_le_ofReal_iff hB).mp ((spatial_near_average_real L A hs hs' hsL hr u hu).trans |
| (hsp n L r hr hsmall g hg A hA u hu)) |
| have he : Real.exp (-1)/(128*Real.sqrt 3)*r=Real.exp (-1)/(64*Real.sqrt 3)*r/2 := by ring |
| have he' : Real.exp (-8)*(27/(524288*Real.sqrt 3))*r/s^3=Real.exp (-8)*(27/(262144*Real.sqrt 3))*r/s^3/2 := by ring |
| rw [he,he'] |
| linarith only [hc,hk] |
|
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| end HardSphere |
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| end |
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| end OAI |
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