| import OAI.Probability.RadialTransition.MixtureBounds | |
| namespace OAI | |
| noncomputable section | |
| open Function MeasureTheory Set Filter | |
| open scoped ContDiff BigOperators ENNReal NNReal | |
| namespace RadialTransition | |
| def macroMeasure (m q : ℕ) (z : ℝ) : Measure (MacroState m q) := | |
| Measure.pi (fun i : GridIndex m => particleBase (regionMeasure (macroRegion q (gridEmbed i))) z) | |
| def macroProduct (φ : Potential) (m q : ℕ) (β : ℝ) (F : ℕ → ℝ) (x : MacroState m q) : ℝ := | |
| ∏ i : GridIndex m, (F (particleCount (x i))* | |
| Real.exp (-β*pairWithin (fun (y z : macroRegion q (gridEmbed i)) => radialPair φ y z) (x i))) | |
| lemma macroProduct_nonneg (φ : Potential) (m q : ℕ) (β : ℝ) (F : ℕ → ℝ) | |
| (hF : ∀ n, 0 ≤ F n) (x : MacroState m q) : 0 ≤ macroProduct φ m q β F x := | |
| Finset.prod_nonneg (fun _ _ => mul_nonneg (hF _) (Real.exp_pos _).le) | |
| lemma integrable_macroProduct {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ) | |
| {q : ℕ} (hq : 0 < q) (m : ℕ) (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 (macroProduct φ m q β F) (macroMeasure m q z) := by | |
| let (i : GridIndex m) : IsFiniteMeasure (regionMeasure (macroRegion q (gridEmbed i))) := | |
| regionMeasure_finite (measurableSet_macroRegion _ _) (by rw [volume_macroRegion hq]; finiteness) | |
| exact Integrable.fintype_prod_dep (fun i => integrable_macroWeight hφ hb hq (gridEmbed i) β z F hF hs) | |
| lemma integral_macroProduct {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ) | |
| {q : ℕ} (hq : 0 < q) (m : ℕ) (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)) : | |
| (∫ x, macroProduct φ m q β F x ∂macroMeasure m q z) = | |
| (∑' n, F n*Real.exp (z*n)*partition φ ((q:ℝ)/4) n β)^(m^3) := by | |
| let (i : GridIndex m) : IsFiniteMeasure (regionMeasure (macroRegion q (gridEmbed i))) := | |
| regionMeasure_finite (measurableSet_macroRegion _ _) (by rw [volume_macroRegion hq]; finiteness) | |
| unfold macroProduct macroMeasure | |
| rw [integral_fintype_prod_eq_prod (fun i : GridIndex m => fun s : ParticleState (macroRegion q (gridEmbed i)) => F (particleCount s)*Real.exp (-β*pairWithin (fun (y w : macroRegion q (gridEmbed i)) => radialPair φ y w) s))] | |
| simp only [integral_macroWeight hφ hb hq _ β z F hF hs, Finset.prod_const, Finset.card_univ, | |
| Fintype.card_fun, Fintype.card_fin] | |
| lemma macroQuad_upper {φ : Potential} {m q : ℕ} (hq : 0 < q) {β R lam : ℝ} | |
| (hβ : 0 ≤ β) (hR : (q:ℝ)/4 ≤ R) (hlam : 0 ≤ lam) (x : MacroState m q) : | |
| Real.exp (-β*macroKacEnergy φ m q R lam x) ≤ | |
| macroProduct φ m q β (fun n => Real.exp (β*lam*(1+10000*(bumpLip:ℝ)*q/R)/(2*((q:ℝ)/4)^3)*(n:ℝ)^2)) x := by | |
| have hh := mul_le_mul_of_nonpos_left (macroKacEnergy_lower (φ := φ) hq hR hlam x) (neg_nonpos.mpr hβ) | |
| apply (Real.exp_le_exp.mpr hh).trans_eq | |
| simp only [macroProduct, ← Real.exp_add, ← Real.exp_sum] | |
| congr 1 | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| ring | |
| lemma integrable_macroKac {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ) | |
| {m q : ℕ} (hq : 0 < q) {β R lam z : ℝ} (hβ : 0 ≤ β) (hR : (q:ℝ)/4 ≤ R) (hlam : 0 ≤ lam) | |
| (hs : Summable (quadraticTerm φ ((q:ℝ)/4) β z (β*lam*(1+10000*(bumpLip:ℝ)*q/R)/2))) : | |
| Integrable (fun x => Real.exp (-β*macroKacEnergy φ m q R lam x)) (macroMeasure m q z) := by | |
| have hs' : Summable (fun n : ℕ => Real.exp (β*lam*(1+10000*(bumpLip:ℝ)*q/R)/(2*((q:ℝ)/4)^3)*(n:ℝ)^2)*Real.exp (z*n)*partition φ ((q:ℝ)/4) n β) := by | |
| convert hs using 1 | |
| funext n | |
| dsimp only [quadraticTerm] | |
| have he : β*lam*(1+10000*(bumpLip:ℝ)*q/R)/(2*((q:ℝ)/4)^3)*(n:ℝ)^2 = β*lam*(1+10000*(bumpLip:ℝ)*q/R)/2*(n:ℝ)^2/((q:ℝ)/4)^3 := by ring | |
| rw [he]; ring | |
| apply (integrable_macroProduct hφ hb hq m β z _ (fun _ => (Real.exp_pos _).le) hs').mono' | |
| (Real.measurable_exp.comp (measurable_const.mul (measurable_macroKacEnergy hφ m q R lam))).aestronglyMeasurable | |
| exact Eventually.of_forall (fun x => by change |Real.exp (-β*macroKacEnergy φ m q R lam x)| ≤ _; rw [abs_of_pos (Real.exp_pos _)]; exact macroQuad_upper hq hβ hR hlam x) | |
| lemma macroKac_partition_upper {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ) | |
| {m q : ℕ} (hq : 0 < q) {β R lam z : ℝ} (hβ : 0 ≤ β) (hR : (q:ℝ)/4 ≤ R) (hlam : 0 ≤ lam) | |
| (hs : Summable (quadraticTerm φ ((q:ℝ)/4) β z (β*lam*(1+10000*(bumpLip:ℝ)*q/R)/2))) : | |
| boltzmannPartition (macroMeasure m q z) (fun x => β*macroKacEnergy φ m q R lam x) ≤ | |
| (∑' n, quadraticTerm φ ((q:ℝ)/4) β z (β*lam*(1+10000*(bumpLip:ℝ)*q/R)/2) n)^(m^3) := by | |
| have he (n : ℕ) : Real.exp (β*lam*(1+10000*(bumpLip:ℝ)*q/R)/(2*((q:ℝ)/4)^3)*(n:ℝ)^2)*Real.exp (z*n)*partition φ ((q:ℝ)/4) n β = | |
| quadraticTerm φ ((q:ℝ)/4) β z (β*lam*(1+10000*(bumpLip:ℝ)*q/R)/2) n := by | |
| dsimp only [quadraticTerm] | |
| have h : β*lam*(1+10000*(bumpLip:ℝ)*q/R)/(2*((q:ℝ)/4)^3)*(n:ℝ)^2 = β*lam*(1+10000*(bumpLip:ℝ)*q/R)/2*(n:ℝ)^2/((q:ℝ)/4)^3 := by ring | |
| rw [h]; ring | |
| have hs' : Summable (fun n : ℕ => Real.exp (β*lam*(1+10000*(bumpLip:ℝ)*q/R)/(2*((q:ℝ)/4)^3)*(n:ℝ)^2)*Real.exp (z*n)*partition φ ((q:ℝ)/4) n β) := by simpa only [he] using hs | |
| have hh := integral_mono (integrable_macroKac hφ hb hq hβ hR hlam hs) | |
| (integrable_macroProduct hφ hb hq m β z _ (fun _ => (Real.exp_pos _).le) hs') | |
| (macroQuad_upper hq hβ hR hlam) | |
| rw [integral_macroProduct hφ hb hq m β z _ (fun _ => (Real.exp_pos _).le) hs'] at hh | |
| simpa only [he, boltzmannPartition, neg_mul] using hh | |
| end RadialTransition | |
| end | |
| end OAI | |