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a2ee19a | 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 | import OAI.Probability.RadialTransition.ConvolutionIntegral
namespace OAI
universe uX
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
section Fock
variable {X : Type uX} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
[SecondCountableTopology X]
lemma particleBase_convolution {F G : Multiset X → ℝ≥0∞}
(hF : MeasurableConfiguration F) (hG : MeasurableConfiguration G)
(μ : Measure X) [SigmaFinite μ] (z : ℝ) :
(∫⁻ x, clusterConvolution F G (particleMultiset x) ∂particleBase μ z) =
(∫⁻ x, F (particleMultiset x) ∂particleBase μ z)*
(∫⁻ x, G (particleMultiset x) ∂particleBase μ z) := by
classical
have hmF := measurable_particleObservable F hF 0
have hmG := measurable_particleObservable G hG 0
have hmC := measurable_particleObservable _ (measurable_clusterConvolution hF hG) 0
simp only [zero_add] at hmF hmG hmC
rw [lintegral_particleBase μ z hmC, lintegral_particleBase μ z hmF,
lintegral_particleBase μ z hmG]
let A (n : ℕ) := ∫⁻ x : Fin n → X, F (configurationMultiset x) ∂Measure.pi (fun _ => μ)
let B (n : ℕ) := ∫⁻ x : Fin n → X, G (configurationMultiset x) ∂Measure.pi (fun _ => μ)
have hN (N : ℕ) :
(∫⁻ x : Fin N → X, clusterConvolution F G (configurationMultiset x) ∂Measure.pi (fun _ => μ)) =
∑ k ∈ Finset.range (N+1), (N.choose k : ℝ≥0∞)* (A (N-k)*B k) := by
simp_rw [clusterConvolution_configuration]
rw [lintegral_finsetSum]
· have hi (T : Finset (Fin N)) :
(∫⁻ x : Fin N → X,
F (configurationMultiset (fun i : {i // i ∉ T} => x i))*
G (configurationMultiset (fun i : T => x i)) ∂Measure.pi (fun _ => μ)) = A (N-T.card)*B T.card := by
rw [configuration_split_integral hF hG μ T,
configuration_integral_card F μ, configuration_integral_card G μ]
have hcardA : Fintype.card {i : Fin N // i ∈ T} = T.card := Fintype.card_coe T
have hcardB : Fintype.card {i : Fin N // i ∉ T} = N-T.card := by
rw [Fintype.card_subtype_compl, Fintype.card_fin, hcardA]
rw [hcardB, Fintype.card_coe T]
simp_rw [hi]
exact ennreal_subsets_card N (fun k => A (N-k)*B k)
· intro T _
have hFa : Measurable (fun x : Fin N → X => F (configurationMultiset (fun i : {i // i ∉ T} => x i))) := by
have hh := (hF 0 {i // i ∉ T}).comp (show Measurable (fun x : Fin N → X => fun i : {i // i ∉ T} => x i) from measurable_pi_iff.mpr (fun i => measurable_pi_apply i.val))
simpa only [zero_add, Function.comp_def] using hh
have hGa : Measurable (fun x : Fin N → X => G (configurationMultiset (fun i : T => x i))) := by
have hh := (hG 0 T).comp (show Measurable (fun x : Fin N → X => fun i : T => x i) from measurable_pi_iff.mpr (fun i => measurable_pi_apply i.val))
simpa only [zero_add, Function.comp_def] using hh
exact hFa.mul hGa
simp only [particleMultiset]
simp_rw [hN, Finset.mul_sum]
have hcoeff (N k : ℕ) (hk : k ∈ Finset.range (N+1)) :
fockCoeff z N*((N.choose k : ℝ≥0∞)*(A (N-k)*B k)) =
(fockCoeff z k*B k)*(fockCoeff z (N-k)*A (N-k)) := by
rw [← mul_assoc, fockCoeff_choose z (Nat.le_of_lt_succ (Finset.mem_range.mp hk))]
ring
have hsum (N : ℕ) :
(∑ k ∈ Finset.range (N+1), fockCoeff z N*((N.choose k : ℝ≥0∞)*(A (N-k)*B k))) =
∑ k ∈ Finset.range (N+1), (fockCoeff z k*B k)*(fockCoeff z (N-k)*A (N-k)) := by
exact Finset.sum_congr rfl (fun k hk => hcoeff N k hk)
change (∑' N, ∑ k ∈ Finset.range (N+1), fockCoeff z N*((N.choose k : ℝ≥0∞)*(A (N-k)*B k))) = _
simp_rw [hsum]
rw [ennreal_tsum_triangle (fun k n => (fockCoeff z k*B k)*(fockCoeff z n*A n))]
simp_rw [ENNReal.tsum_mul_left]
rw [ENNReal.tsum_mul_right, mul_comm]
rfl
end Fock
end RadialTransition
end
end OAI
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