| import Mathlib |
| import OAI.Analysis.HardSphere.Paths.Admissible |
| import OAI.Analysis.HardSphere.Allocation.PhysicalGroupComparison |
| import OAI.Analysis.HardSphere.Orbitals.PhysicalOrbitalReference |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital _root_.OAI.HardSphere.Lattice HardSphere.DyadicSkeleton |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma measureAffinity_nsmul {X : Type*} [MeasurableSpace X] |
| (P Q : Measure X) [IsFiniteMeasure P] [IsFiniteMeasure Q] (n : ℕ) : |
| measureAffinity (n • P) (n • Q)=(n:ℝ)*measureAffinity P Q := by |
| have h := measureAffinity_smul P Q (n:ℝ) (Nat.cast_nonneg n) |
| rw [ENNReal.ofReal_natCast] at h |
| simpa only [Nat.cast_smul_eq_nsmul] using h |
|
|
| lemma affinity_double_sum_bound {X I J : Type*} [MeasurableSpace X] [Fintype I] [Fintype J] |
| (A : I→J→Bool→Measure X) (Q : Bool→Measure X) (n : ℕ) |
| [∀i j b,IsFiniteMeasure (A i j b)] [∀b,IsFiniteMeasure (Q b)] |
| (h : ∀b,(∑i,∑j,A i j b)≤n • Q b) : |
| (∑i,∑j,measureAffinity (A i j true) (A i j false))≤(n:ℝ)*measureAffinity (Q true) (Q false) := by |
| have hb : (∑i,∑j,measureAffinity (A i j true) (A i j false))≤ |
| measureAffinity (∑i,∑j,A i j true) (∑i,∑j,A i j false) := by |
| refine (Finset.sum_le_sum (fun i _ ↦ measureAffinity_finset_sum Finset.univ (fun j ↦ A i j true) (fun j ↦ A i j false))).trans ?_ |
| exact measureAffinity_finset_sum Finset.univ (fun i ↦ ∑j,A i j true) (fun i ↦ ∑j,A i j false) |
| exact hb.trans ((measureAffinity_mono _ _ _ _ (h true) (h false)).trans_eq (measureAffinity_nsmul _ _ n)) |
|
|
| lemma physical_total_posts {N K : ℕ} [NeZero K] {L : Length} |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (b : Bool) |
| (P : Measure (Torus L×Configuration N L)) [SFinite P] : |
| (∑ij : DiscreteTorus K×DiscreteTorus K, |
| physicalCellVolumePost L (siteIndex (if b then ij.2 else ij.1)) |
| (siteIndex (if b then ij.1 else ij.2)) b ∘ₘ P)= |
| (P.prod volume).map (cellObservation b) := by |
| rw [Fintype.sum_prod_type] |
| cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true] |
| · rw [Finset.sum_comm] |
| have he := physical_cell_posts_sum hL hK false P |
| rw [←he] |
| exact (siteIndexEquiv.sum_comp (fun tag : Fin 3→Fin K ↦ |
| ∑extra : DiscreteTorus K,physicalCellVolumePost L (siteIndex extra) tag false ∘ₘ P)).trans |
| (Finset.sum_congr rfl (fun tag _ ↦ siteIndexEquiv.sum_comp |
| (fun extra : Fin 3→Fin K ↦ physicalCellVolumePost L extra tag false ∘ₘ P))) |
| · rw [←physical_cell_posts_sum hL hK true P] |
| exact (siteIndexEquiv.sum_comp (fun tag : Fin 3→Fin K ↦ |
| ∑extra : DiscreteTorus K,physicalCellVolumePost L (siteIndex extra) tag true ∘ₘ P)).trans |
| (Finset.sum_congr rfl (fun tag _ ↦ siteIndexEquiv.sum_comp |
| (fun extra : Fin 3→Fin K ↦ physicalCellVolumePost L extra tag true ∘ₘ P))) |
|
|
| lemma physicalGroupReference_finite {N K M G : ℕ} [NeZero K] {L : Length} {r : ℝ} |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) {ψ : Jet (N+1) L} (hψ : IsGroundVector (N+1) L r ψ) |
| (m : ℕ) (t R C S s₀ : ℝ) (ht : 0≤t) (u : Time) |
| (x y : DiscreteTorus K) (g : Fin G) (b : Bool) : |
| IsFiniteMeasure (physicalGroupReference (M:=M) (x:=x) (y:=y) hL ψ g m t r R C S s₀ u b) := by |
| let (flags : Fin (N+1)→Bool) (a : GridAllocation (N+1) K (G*M)) := |
| physicalBranchLaw_finite hL hK hψ g flags a m R C S s₀ (2*t) (by linarith) u |
| unfold physicalGroupReference |
| infer_instance |
|
|
| lemma physical_group_references_le {N K M G : ℕ} [NeZero K] {L : Length} {r : ℝ} |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val) |
| {ψ : Jet (N+1) L} (hψ : IsGroundVector (N+1) L r ψ) |
| (m : ℕ) (t R C S s₀ : ℝ) (ht : 0≤t) (u : Time) |
| (x y : DiscreteTorus K) (b : Bool) : |
| (∑g : Fin G,physicalGroupReference (M:=M) (x:=x) (y:=y) hL ψ g m t r R C S s₀ u b)≤ |
| (N+1) • (physicalCellVolumePost L (siteIndex (if b then y else x)) |
| (siteIndex (if b then x else y)) b ∘ₘ |
| (((volume : Measure (Torus L)).prod volume).withDensity |
| (fun p : Torus L×Configuration N L ↦ ENNReal.ofReal (‖ψ.1 (Fin.cons p.1 p.2)‖^2)))) := by |
| simpa only [physicalGroupReference,physicalObservedReference,physicalTaggedReference, |
| branchParticles,Function.Embedding.subtype,Function.Embedding.coeFn_mk] using |
| physical_all_references_le (M:=M) (G:=G) hL hK h50 hψ m R C S s₀ t ht u |
| (siteIndex (if b then y else x)) (siteIndex (if b then x else y)) b |
|
|
| lemma physical_reference_sum_bound {N K M G : ℕ} [NeZero K] {L : Length} {r : ℝ} |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val) |
| {ψ : Jet (N+1) L} (hψ : IsGroundVector (N+1) L r ψ) |
| (m : ℕ) (t R C S s₀ : ℝ) (ht : 0≤t) (u : Time) (b : Bool) : |
| (∑ij : DiscreteTorus K×DiscreteTorus K,∑g : Fin G, |
| physicalGroupReference (M:=M) (x:=ij.1) (y:=ij.2) hL ψ g m t r R C S s₀ u b)≤ |
| (N+1) • ((((volume : Measure (Torus L)).prod volume).withDensity |
| (fun p : Torus L×Configuration N L ↦ ENNReal.ofReal (‖ψ.1 (Fin.cons p.1 p.2)‖^2))).prod volume).map (cellObservation b) := by |
| let P : Measure (Torus L×Configuration N L) := (volume.prod volume).withDensity |
| (fun p ↦ ENNReal.ofReal (‖ψ.1 (Fin.cons p.1 p.2)‖^2)) |
| have hf := ((Lp.memLp ψ.1).comp_measurePreserving (cons_preserving (n:=N) L)).norm |
| have hf' := hf.comp_measurePreserving (Measure.measurePreserving_swap (μ:=volume) (ν:=volume)) |
| let : IsFiniteMeasure P := isFiniteMeasure_withDensity_ofReal |
| ((memLp_two_iff_integrable_sq hf'.aestronglyMeasurable).1 hf').2 |
| change _≤(N+1) • (P.prod volume).map (cellObservation b) |
| rw [←physical_total_posts hL hK b P,Finset.smul_sum] |
| exact Finset.sum_le_sum (fun ij _ ↦ physical_group_references_le hL hK h50 hψ m t R C S s₀ ht u ij.1 ij.2 b) |
| |
| attribute [local irreducible] physicalGroupReference physicalCellVolumePost |
| |
| theorem physical_affinity_sum_bound {N K M G : ℕ} [NeZero K] {L : Length} {r : ℝ} |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val) |
| {ψ : Jet (N+1) L} (hψ : IsGroundVector (N+1) L r ψ) |
| (m : ℕ) (t R C S s₀ : ℝ) (ht : 0≤t) (u : Time) : |
| (∑ij : DiscreteTorus K×DiscreteTorus K,∑g : Fin G, |
| measureAffinity (physicalGroupReference (M:=M) (x:=ij.1) (y:=ij.2) hL ψ g m t r R C S s₀ u true) |
| (physicalGroupReference (M:=M) (x:=ij.1) (y:=ij.2) hL ψ g m t r R C S s₀ u false)) ≤ |
| (N+1:ℝ)*(∫Y : Configuration N L,(∫x : Torus L,‖ψ.1 (Fin.cons x Y)‖)^2) := by |
| classical |
| let P : Measure (Torus L×Configuration N L) := (volume.prod volume).withDensity |
| (fun p ↦ ENNReal.ofReal (‖ψ.1 (Fin.cons p.1 p.2)‖^2)) |
| let Q := fun b ↦ (P.prod volume).map (cellObservation b) |
| let A := fun ij : DiscreteTorus K×DiscreteTorus K ↦ fun g : Fin G ↦ fun b ↦ |
| physicalGroupReference (M:=M) (x:=ij.1) (y:=ij.2) hL ψ g m t r R C S s₀ u b |
| have hm : Measurable (fun p : Configuration N L×Torus L ↦ ‖ψ.1 (Fin.cons p.2 p.1)‖) := |
| ((Lp.stronglyMeasurable ψ.1).measurable.comp (cons_preserving (n:=N) L).measurable).norm |
| have hf := ((Lp.memLp ψ.1).comp_measurePreserving (cons_preserving (n:=N) L)).norm |
| have hf' := hf.comp_measurePreserving (Measure.measurePreserving_swap (μ:=volume) (ν:=volume)) |
| let : IsFiniteMeasure P := isFiniteMeasure_withDensity_ofReal |
| ((memLp_two_iff_integrable_sq hf'.aestronglyMeasurable).1 hf').2 |
| let (ij : DiscreteTorus K×DiscreteTorus K) (g : Fin G) (b : Bool) : IsFiniteMeasure (A ij g b) := |
| physicalGroupReference_finite hL hK hψ m t R C S s₀ ht u ij.1 ij.2 g b |
| have hsall (b : Bool) : (∑ij,∑g,A ij g b)≤(N+1) • Q b := |
| physical_reference_sum_bound hL hK h50 hψ m t R C S s₀ ht u b |
| have hnorm : measureAffinity (Q true) (Q false)= |
| ∫Y : Configuration N L,(∫x : Torus L,‖ψ.1 (Fin.cons x Y)‖)^2 := |
| physical_unrestricted_affinity (n:=N) (volume : Measure (Torus L)) (volume : Measure (Configuration N L)) |
| (fun p : Configuration N L×Torus L ↦ ‖ψ.1 (Fin.cons p.2 p.1)‖) hm hf (fun _ ↦ norm_nonneg _) |
| have hb := affinity_double_sum_bound A Q (N+1) hsall |
| simpa only [hnorm,Nat.cast_add,Nat.cast_one] using hb |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal BigOperators Topology |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital _root_.OAI.HardSphere.Lattice HardSphere.DyadicSkeleton |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma pair_group_sum_lower {K G : ℕ} (hK : 400≤K) [NeZero K] |
| (f : (DiscreteTorus K×DiscreteTorus K)→Fin G→ℝ) {B : ℝ} (hB : 0<B) |
| (hf : ∀i g,0≤f i g) |
| (hgain : ∀i∈currentAdmissiblePairs K,∀g,1/(8*B)≤f i g) : |
| (K:ℝ)^6*(G:ℝ)/(16000*B)≤∑i,∑g,f i g := by |
| have hc : (K:ℝ)^6≤2000*((currentAdmissiblePairs K).card:ℝ) := by |
| exact_mod_cast current_admissible_card hK |
| have hs : ((currentAdmissiblePairs K).card:ℝ)*(G:ℝ)/(8*B)≤ |
| ∑i∈currentAdmissiblePairs K,∑g,f i g := by |
| calc |
| _ = ∑i∈currentAdmissiblePairs K,∑g : Fin G,1/(8*B) := by |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] |
| ring |
| _ ≤ _ := Finset.sum_le_sum (fun i hi ↦ Finset.sum_le_sum (fun g _ ↦ hgain i hi g)) |
| refine le_trans ?_ (hs.trans (Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) |
| (fun i _ _ ↦ Finset.sum_nonneg (fun g _ ↦ hf i g)))) |
| apply (div_le_div_iff₀ (by positivity : 0<16000*B) (by positivity : 0<8*B)).mpr |
| have hc' := mul_le_mul_of_nonneg_right hc (by positivity : 0≤(G:ℝ)) |
| nlinarith |
| |
| lemma modulus_numeric_gain {V D I ell M B G : ℝ} |
| (hV : 0<V) (hD : 0<D) (hell : 0<ell) (hM : 0<M) (hB : 0<B) |
| (hG : ell/(8*M)≤G/D) (hbound : V^2*G/(16000*B)≤(D*V)*I) : |
| ell/(128000*M*B)≤I/V := by |
| calc |
| _ = (ell/(8*M))/(16000*B) := by field_simp; ring |
| _ ≤ (G/D)/(16000*B) := div_le_div_of_nonneg_right hG (by positivity) |
| _ = (V^2*G/(16000*B))/((D*V)*V) := by field_simp |
| _ ≤ ((D*V)*I)/((D*V)*V) := div_le_div_of_nonneg_right hbound (by positivity) |
| _ = I/V := by field_simp |
| |
| theorem physical_modulus_from_gains {N K M G : ℕ} [NeZero K] {L : Length} {r : ℝ} |
| (hL : L.val=(K:ℝ)) (hK : 400≤K) |
| {ψ : Jet (N+1) L} (hψ : IsGroundVector (N+1) L r ψ) |
| (m : ℕ) (t R C S s₀ : ℝ) (ht : 0≤t) (u : Time) |
| (D ell B : ℝ) (hDeq : D=(N+1:ℝ)/L.val^3) (hell : 0<ell) (hM : 0<M) (hB : 1≤B) |
| (hG : ell/(8*M)≤(G:ℝ)/D) |
| (hgain : ∀ij∈currentAdmissiblePairs K,∀g : Fin G, |
| 1/(8*B)≤ measureAffinity |
| (physicalGroupReference (M:=M) (x:=ij.1) (y:=ij.2) hL ψ g m t r R C S s₀ u true) |
| (physicalGroupReference (M:=M) (x:=ij.1) (y:=ij.2) hL ψ g m t r R C S s₀ u false)) : |
| ell/(128000*M*B)≤Phase.modulusOccupation |
| (fun p : Configuration N L×Torus L ↦ ψ.1 (Fin.cons p.2 p.1)) volume volume := by |
| have h1 : 1≤K := by omega |
| have h50 : 50<L.val := by rw [hL]; exact_mod_cast (show 50<K by omega) |
| have hDp : 0<D := by rw [hDeq]; positivity |
| let f := fun ij : DiscreteTorus K×DiscreteTorus K ↦ fun g : Fin G ↦ |
| measureAffinity |
| (physicalGroupReference (M:=M) (x:=ij.1) (y:=ij.2) hL ψ g m t r R C S s₀ u true) |
| (physicalGroupReference (M:=M) (x:=ij.1) (y:=ij.2) hL ψ g m t r R C S s₀ u false) |
| have hl := pair_group_sum_lower hK f (by linarith : 0<B) |
| (fun _ _ ↦ affinity_nonneg _ _) hgain |
| have hu := physical_affinity_sum_bound (M:=M) (G:=G) hL h1 h50 hψ m t R C S s₀ ht u |
| have hbound : (L.val^3)^2*(G:ℝ)/(16000*B)≤ |
| (D*L.val^3)*(∫Y : Configuration N L,(∫x : Torus L,‖ψ.1 (Fin.cons x Y)‖)^2) := by |
| have he : D*L.val^3=(N+1:ℝ) := by rw [hDeq]; field_simp |
| rw [he] |
| calc |
| _ = (K:ℝ)^6*(G:ℝ)/(16000*B) := by rw [hL,←pow_mul] |
| _ ≤ _ := hl.trans hu |
| have H := modulus_numeric_gain (by positivity : 0<L.val^3) hDp hell |
| (by exact_mod_cast hM) (by linarith : 0<B) hG hbound |
| simpa only [Phase.modulusOccupation,Measure.real,BathSlice.volume_torus, |
| ENNReal.toReal_ofReal (by positivity : 0≤L.val^3)] using H |
| |
| end HardSphere.Groups |
| end |
| end |
| |
| end OAI |
| |
