| import OAI.Probability.RadialTransition.MacroLower |
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| namespace OAI |
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| universe uX uY |
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| noncomputable section |
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| open Function MeasureTheory Set Filter |
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| open scoped ContDiff BigOperators ENNReal NNReal |
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| namespace RadialTransition |
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| 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))) |
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| 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 |
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| 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] |
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| 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] |
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| 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 |
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| end MapWeight |
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| 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] |
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| 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 |
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| 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 _ _ _)))] |
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| end RadialTransition |
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| end |
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| end OAI |
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