File size: 3,280 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 | import OAI.Probability.RadialTransition.GridPressure
namespace OAI
noncomputable section
open Function MeasureTheory Set Filter
open scoped ContDiff BigOperators ENNReal NNReal
namespace RadialTransition
lemma summable_sectorWeight (φ : Potential) (L β z : ℝ) (n : ℕ) :
Summable (fun k : ℕ => (if k = n then (1:ℝ) else 0)*Real.exp (z*k)*partition φ L k β) :=
summable_of_ne_finset_zero (s := {n}) (by intro k hk; simp only [Finset.mem_singleton] at hk; simp [hk])
lemma sectorWeight_tsum (φ : Potential) (L β z : ℝ) (n : ℕ) :
(∑' k : ℕ, (if k = n then (1:ℝ) else 0)*Real.exp (z*k)*partition φ L k β) = Real.exp (z*n)*partition φ L n β := by
simp only [ite_mul, one_mul, zero_mul]
exact tsum_ite_eq n _
lemma macroProduct_sector (φ : Potential) (m q : ℕ) (β : ℝ) (n : ℕ) (x : MacroState m q) :
macroProduct φ m q β (fun k => if k = n then 1 else 0) x =
if (∀ i, particleCount (x i) = n) then
Real.exp (-β*∑ i : GridIndex m, pairWithin (fun (y z : macroRegion q (gridEmbed i)) => radialPair φ y z) (x i)) else 0 := by
classical
by_cases hh : ∀ i, particleCount (x i) = n
· rw [ite_eq_left hh]
simp only [macroProduct, hh, ite_true, one_mul, ← Real.exp_sum, ← Finset.mul_sum]
· simp only [hh, ite_false]
obtain ⟨i, hi⟩ := not_forall.mp hh
exact Finset.prod_eq_zero (Finset.mem_univ i) (by simp [hi])
lemma macroSector_lower {φ : Potential} {m q : ℕ} (hq : 0 < q) {β R lam : ℝ}
(hβ : 0 ≤ β) (hR : (q:ℝ)/4 ≤ R) (hlam : 0 ≤ lam) (n : ℕ) (x : MacroState m q) :
Real.exp (β*lam*((kernelInterior m q R).card*(1-kernelRowError q R)*(n:ℝ)^2/(2*((q:ℝ)/4)^3)))*
macroProduct φ m q β (fun k => if k = n then 1 else 0) x ≤
Real.exp (-β*macroKacEnergy φ m q R lam x) := by
classical
rw [macroProduct_sector]
split_ifs with hh
· rw [← Real.exp_add]
apply Real.exp_le_exp.mpr
have hb := mul_le_mul_of_nonneg_left (macro_attraction_lower hq hR x hh) (mul_nonneg hβ hlam)
unfold macroKacEnergy
nlinarith
· simpa only [mul_zero] using (Real.exp_pos _).le
lemma macroKac_partition_lower {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)
(hi : Integrable (fun x => Real.exp (-β*macroKacEnergy φ m q R lam x)) (macroMeasure m q z)) (n : ℕ) :
Real.exp (β*lam*((kernelInterior m q R).card*(1-kernelRowError q R)*(n:ℝ)^2/(2*((q:ℝ)/4)^3)))*
(Real.exp (z*n)*partition φ ((q:ℝ)/4) n β)^(m^3) ≤
boltzmannPartition (macroMeasure m q z) (fun x => β*macroKacEnergy φ m q R lam x) := by
have hs := summable_sectorWeight φ ((q:ℝ)/4) β z n
have hn (k : ℕ) : 0 ≤ (if k = n then (1:ℝ) else 0) := by split_ifs <;> norm_num
have hh := integral_mono ((integrable_macroProduct hφ hb hq m β z _ hn hs).const_mul
(Real.exp (β*lam*((kernelInterior m q R).card*(1-kernelRowError q R)*(n:ℝ)^2/(2*((q:ℝ)/4)^3))))) hi
(macroSector_lower hq hβ hR hlam n)
rw [integral_const_mul, integral_macroProduct hφ hb hq m β z _ hn hs, sectorWeight_tsum] at hh
simpa only [boltzmannPartition, neg_mul] using hh
end RadialTransition
end
end OAI
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