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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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Probability.RadialTransition.MacroLower
namespace OAI
universe uX uY
noncomputable section
open Function MeasureTheory Set Filter
open scoped ContDiff BigOperators ENNReal NNReal
namespace RadialTransition
lemma measurable_countWeight {φ : Potential} (hφ : Continuous φ) (β : ℝ) (F : ℕ → ℝ) :
Measurable (fun s : ParticleState Position => F (particleCount s)*Real.exp (-β*orderedEnergy φ s.2)) := by
apply Continuous.measurable
apply continuous_sigma
intro n
change Continuous (fun x : Fin n → Position => F n * Real.exp (-β*orderedEnergy φ x))
exact continuous_const.mul (Real.continuous_exp.comp (continuous_const.mul (continuous_orderedEnergy hφ)))
lemma lintegral_countWeight {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ)
(β z : ℝ) (F : ℕ → ℝ) (hF : ∀ n, 0 ≤ F n) {A : Set Position} (hAfin : volume A ≠ ⊤) :
(∫⁻ s : ParticleState Position,
ENNReal.ofReal (F (particleCount s)*Real.exp (-β*orderedEnergy φ s.2))
∂particleBase (volume.restrict A) z) =
∑' n : ℕ, ENNReal.ofReal (F n*Real.exp (z*n)*partitionOn φ A n β) := by
let : IsFiniteMeasure (volume.restrict A) := isFiniteMeasure_restrict.mpr hAfin
rw [lintegral_particleBase _ _ (measurable_countWeight hφ β F).ennreal_ofReal]
apply tsum_congr
intro n
dsimp only [particleCount]
have hi := (integrable_orderedBoltzmann (ι := Fin n) hφ hb β (Measure.pi (fun _ => volume.restrict A))).const_mul (F n)
rw [← ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall (fun x => mul_nonneg (hF n) (Real.exp_pos _).le)), integral_const_mul,
← ENNReal.ofReal_mul (by positivity)]
congr 1
simp only [partitionOn, rawPartition, div_eq_mul_inv]
ring
lemma partitionOn_macroRegion (φ : Potential) {q : ℕ} (hq : 0 < q) (i : Cell) (n : ℕ) (β : ℝ) :
partitionOn φ (macroRegion q i) n β = partition φ ((q:ℝ)/4) n β := by
rw [macroRegion_translate hq, partitionOn_translate]
have hh : partitionOn φ (halfCube ((q:ℝ)/4)) n β = partitionOn φ (cube ((q:ℝ)/4)) n β := by
simp only [partitionOn, rawPartition, restrict_halfCube]
rw [hh, partitionOn_cube]
section MapWeight
variable {X : Type uX} {Y : Type uY} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
[TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y]
[SecondCountableTopology X] [SecondCountableTopology Y]
lemma lintegral_pairWeight_map (v : Y → Y → ℝ) (hv : Measurable (Function.uncurry v))
(μ : Measure X) [IsFiniteMeasure μ] (f : X → Y) (hf : Measurable f) (β z : ℝ) (F : ℕ → ℝ) :
(∫⁻ s : ParticleState X, ENNReal.ofReal (F (particleCount s)*Real.exp (-β*pairWithin (fun x y => v (f x) (f y)) s)) ∂particleBase μ z) =
∫⁻ s : ParticleState Y, ENNReal.ofReal (F (particleCount s)*Real.exp (-β*pairWithin v s)) ∂particleBase (Measure.map f μ) z := by
have hfun : Measurable (fun s : ParticleState Y => ENNReal.ofReal (F (particleCount s)*Real.exp (-β*pairWithin v s))) := by
apply Measurable.ennreal_ofReal
exact (((continuous_of_discreteTopology : Continuous F).comp continuous_particleCount).measurable).mul
(Real.measurable_exp.comp (measurable_const.mul (measurable_pairWithin v hv)))
rw [← particleBase_map μ z hf, lintegral_map hfun (measurable_mapParticle hf)]
rfl
end MapWeight
lemma lintegral_macroWeight {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ)
{q : ℕ} (hq : 0 < q) (i : Cell) (β z : ℝ) (F : ℕ → ℝ) (hF : ∀ n, 0 ≤ F n) :
(∫⁻ s : ParticleState (macroRegion q i), ENNReal.ofReal (F (particleCount s)*
Real.exp (-β*pairWithin (fun (x y : macroRegion q i) => radialPair φ x y) s))
∂particleBase (regionMeasure (macroRegion q i)) z) =
∑' n : ℕ, ENNReal.ofReal (F n*Real.exp (z*n)*partition φ ((q:ℝ)/4) n β) := by
have hfin : volume (macroRegion q i) ≠ ⊤ := by rw [volume_macroRegion hq]; finiteness
let : IsFiniteMeasure (regionMeasure (macroRegion q i)) := regionMeasure_finite (measurableSet_macroRegion _ _) hfin
rw [lintegral_pairWeight_map (radialPair φ) (continuous_radialPair hφ).measurable _ Subtype.val measurable_subtype_coe,
regionMeasure_map (measurableSet_macroRegion _ _)]
simp only [pairWithin_radial]
rw [lintegral_countWeight hφ hb β z F hF hfin]
simp only [partitionOn_macroRegion φ hq]
lemma integrable_macroWeight {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ)
{q : ℕ} (hq : 0 < q) (i : Cell) (β z : ℝ) (F : ℕ → ℝ) (hF : ∀ n, 0 ≤ F n)
(hs : Summable (fun n => F n*Real.exp (z*n)*partition φ ((q:ℝ)/4) n β)) :
Integrable (fun s : ParticleState (macroRegion q i) => F (particleCount s)*
Real.exp (-β*pairWithin (fun (x y : macroRegion q i) => radialPair φ x y) s))
(particleBase (regionMeasure (macroRegion q i)) z) := by
have hm : Measurable (fun s : ParticleState (macroRegion q i) => F (particleCount s)*
Real.exp (-β*pairWithin (fun (x y : macroRegion q i) => radialPair φ x y) s)) := by
exact ((continuous_of_discreteTopology.comp continuous_particleCount).measurable).mul
(Real.measurable_exp.comp (measurable_const.mul (measurable_pairWithin _
(continuous_radialPair_comp hφ continuous_subtype_val continuous_subtype_val).measurable)))
apply (lintegral_ofReal_ne_top_iff_integrable hm.aestronglyMeasurable
(Eventually.of_forall (fun s => mul_nonneg (hF _) (Real.exp_pos _).le))).mp
rw [lintegral_macroWeight hφ hb hq i β z F hF,
← ENNReal.ofReal_tsum_of_nonneg (fun n => mul_nonneg (mul_nonneg (hF n) (Real.exp_pos _).le) (partition_nonneg _ _ _)) hs]
exact ENNReal.ofReal_ne_top
lemma integral_macroWeight {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ)
{q : ℕ} (hq : 0 < q) (i : Cell) (β z : ℝ) (F : ℕ → ℝ) (hF : ∀ n, 0 ≤ F n)
(hs : Summable (fun n => F n*Real.exp (z*n)*partition φ ((q:ℝ)/4) n β)) :
(∫ s : ParticleState (macroRegion q i), F (particleCount s)*
Real.exp (-β*pairWithin (fun (x y : macroRegion q i) => radialPair φ x y) s)
∂particleBase (regionMeasure (macroRegion q i)) z) =
∑' n, F n*Real.exp (z*n)*partition φ ((q:ℝ)/4) n β := by
rw [integral_eq_lintegral_of_nonneg_ae (Eventually.of_forall (fun s => mul_nonneg (hF _) (Real.exp_pos _).le))
(integrable_macroWeight hφ hb hq i β z F hF hs).aestronglyMeasurable,
lintegral_macroWeight hφ hb hq i β z F hF,
← ENNReal.ofReal_tsum_of_nonneg (fun n => mul_nonneg (mul_nonneg (hF n) (Real.exp_pos _).le) (partition_nonneg _ _ _)) hs,
ENNReal.toReal_ofReal (tsum_nonneg (fun n => mul_nonneg (mul_nonneg (hF n) (Real.exp_pos _).le) (partition_nonneg _ _ _)))]
end RadialTransition
end
end OAI
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