File size: 3,857 Bytes
159357f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | import Mathlib
import OAI.Analysis.CoulombIonization.Variational.BlockSwap
import OAI.Analysis.CoulombIonization.FieldAnalysis.ConditionalFieldIdentity
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Filter Set
open scoped BigOperators
namespace CoulombAtom
open CoulombNeumann
lemma weighted_coreSlice_integrable {N M : ℕ} (ψ : FormVector (N+M)) (s : Spins M)
(F : Configuration M → ℝ)
(hF : ∀ t : Spins N, Integrable (fun p : Configuration M × Configuration N =>
F p.1*‖(swapBlocks ψ).value (joinLists s t) (joinLists p.1 p.2)‖^2)) :
Integrable (fun x => F x*formMass (coreSlice ψ s x)) := by
have hi (t : Spins N) : Integrable (fun x : Configuration M =>
F x*(∫ y : Configuration N, ‖(coreSlice ψ s x).value t y‖^2)) := by
simpa only [swapBlocks_value,coreSlice,integral_const_mul] using (hF t).integral_prod_left
simpa only [formMass,Finset.mul_sum] using integrable_finsetSum Finset.univ (fun t _ => hi t)
lemma retained_pressure_weighted_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) {b : ℝ} (hb : 0 < b) (s : Spins M)
(S : Configuration M → Finset (Fin M)) (hS : ∀ i, MeasurableSet {x | i ∈ S x}) :
Integrable (fun x => retainedPressure b (S x) x*formMass (coreSlice ψ s x)) :=
weighted_coreSlice_integrable ψ s _ (fun t => retained_pressure_join_integrable
(swapBlocks_sobolev hψ) hb (fun p => S p.1) (fun i => measurable_fst (hS i)) s t)
lemma retained_direct_weighted_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) {b : ℝ} (hb : 0 < b) (s : Spins M)
(S : Configuration M → Finset (Fin M)) (hS : ∀ i, MeasurableSet {x | i ∈ S x}) :
Integrable (fun x => retainedDirect b (S x) x*formMass (coreSlice ψ s x)) :=
weighted_coreSlice_integrable ψ s _ (fun t => retained_direct_join_integrable
(swapBlocks_sobolev hψ) hb (fun p => S p.1) (fun i => measurable_fst (hS i)) s t)
lemma conditionalFieldParticle_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (Z lam : ℝ) (s : Spins M) (i : Fin M) :
Integrable (fun x => formMass (coreSlice ψ s x)*normalizedCoreField Z lam (coreSlice ψ s x) (x i)) := by
have hh := (((coreSlice_mass_div_integrable hψ s i).const_mul Z).sub
(coreSlice_cross_integrable hψ s i)).sub ((hψ.coreSlice_mass_integrable s).const_mul lam)
apply hh.congr
filter_upwards [hψ.ae_coreSlice s] with x hx
simp only [Pi.sub_apply]
rw [mul_comm (formMass _),normalizedCoreField_mul_mass hx]
ring
def conditionalRetainedFieldSum {N M : ℕ} (Z lam : ℝ) (ψ : FormVector (N+M))
(s : Spins M) (S : Configuration M → Finset (Fin M)) (x : Configuration M) : ℝ :=
formMass (coreSlice ψ s x)*∑ i ∈ S x, normalizedCoreField Z lam (coreSlice ψ s x) (x i)
lemma conditionalRetainedFieldSum_integrable {N M : ℕ} {ψ : FormVector (N+M)}
(hψ : SobolevVector ψ) (Z lam : ℝ) (s : Spins M)
(S : Configuration M → Finset (Fin M)) (hS : ∀ i, MeasurableSet {x | i ∈ S x}) :
Integrable (conditionalRetainedFieldSum Z lam ψ s S) := by
classical
have hi (i : Fin M) := (conditionalFieldParticle_integrable hψ Z lam s i).indicator (hS i)
have he (x : Configuration M) : conditionalRetainedFieldSum Z lam ψ s S x =
∑ i : Fin M, if i ∈ S x then
formMass (coreSlice ψ s x)*normalizedCoreField Z lam (coreSlice ψ s x) (x i) else 0 := by
simp only [conditionalRetainedFieldSum,Finset.mul_sum,Finset.sum_ite_mem,Finset.univ_inter]
change Integrable (fun x => conditionalRetainedFieldSum Z lam ψ s S x)
simp_rw [he]
exact integrable_finsetSum _ (fun i _ => by
convert hi i using 1
funext x
simp only [Set.indicator_apply,Set.mem_ofPred_eq])
end CoulombAtom
end
end OAI
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