File size: 5,837 Bytes
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 | 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
|