| 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 | |