File size: 3,236 Bytes
a89d282 | 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 | import OAI.Probability.InvariantIsing.Pressure.GroundStateFinite
namespace OAI
/-! Integrable physical energies and the expected soft-maximum inequality. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped Classical BigOperators
namespace InvariantIsing
lemma integrable_physicalPressure {Ω : Type*} [MeasurableSpace Ω]
{N : ℕ} (hN : 0 < N) (P : Measure Ω) [IsProbabilityMeasure P]
(U : Ω → Orthogonal N) (hU : Measurable U) (eig c : Fin N → ℝ) :
Integrable (fun ω => rotatedPressure eig (matrixRotation (U ω)⁻¹) c) P := by
let K := ∑ i, |eig i|
have he i : |eig i| ≤ K := Finset.single_le_sum (f := fun i => |eig i|) (fun _ _ => abs_nonneg _) (Finset.mem_univ i)
apply (integrable_const (K/2+|noninteractingPressure c|)).mono'
((measurable_physicalPressure hN eig c).comp hU).aestronglyMeasurable
filter_upwards [] with ω
rw [Real.norm_eq_abs]
have hb := abs_pressure_sub_noninteracting_le hN eig c (matrixRotation (U ω)⁻¹) K he
have ha := abs_add_le (rotatedPressure eig (matrixRotation (U ω)⁻¹) c-noninteractingPressure c)
(noninteractingPressure c)
rw [sub_add_cancel] at ha
exact ha.trans (add_le_add hb le_rfl)
lemma integrable_physicalGroundState {Ω : Type*} [MeasurableSpace Ω]
{N : ℕ} (hN : 0 < N) (P : Measure Ω) [IsProbabilityMeasure P]
(U : Ω → Orthogonal N) (hU : Measurable U) (eig : Fin N → ℝ) :
Integrable (fun ω => groundStateEnergy eig (matrixRotation (U ω)⁻¹)) P := by
let K := ∑ i, |eig i|
have he i : |eig i| ≤ K := Finset.single_le_sum (f := fun i => |eig i|) (fun _ _ => abs_nonneg _) (Finset.mem_univ i)
exact (integrable_const (K/2)).mono'
((measurable_physical_groundStateEnergy hN eig).comp hU).aestronglyMeasurable
(ae_of_all _ fun ω => by simpa only [Real.norm_eq_abs] using
abs_groundStateEnergy_le hN eig (matrixRotation (U ω)⁻¹) K he)
lemma mean_groundState_squeeze {Ω : Type*} [MeasurableSpace Ω]
{N : ℕ} (hN : 0 < N) (P : Measure Ω) [IsProbabilityMeasure P]
(U : Ω → Orthogonal N) (hU : Measurable U) (eig : Fin N → ℝ)
(β : ℝ) (hβ : 0 < β) :
(∫ ω, rotatedPressure (fun i => β*eig i) (matrixRotation (U ω)⁻¹) (fun _ => 0) ∂P)/β ≤
(∫ ω, groundStateEnergy eig (matrixRotation (U ω)⁻¹) ∂P) ∧
(∫ ω, groundStateEnergy eig (matrixRotation (U ω)⁻¹) ∂P) ≤
(∫ ω, rotatedPressure (fun i => β*eig i) (matrixRotation (U ω)⁻¹) (fun _ => 0) ∂P)/β+
Real.log 2/β := by
have hp := integrable_physicalPressure hN P U hU (fun i => β*eig i) (fun _ => 0)
have hg := integrable_physicalGroundState hN P U hU eig
have hl := integral_mono (hp.div_const β) hg (fun ω =>
(groundStateEnergy_squeeze hN eig (matrixRotation (U ω)⁻¹) β hβ).1)
have hu := integral_mono hg ((hp.div_const β).add (integrable_const (Real.log 2/β))) (fun ω =>
(groundStateEnergy_squeeze hN eig (matrixRotation (U ω)⁻¹) β hβ).2)
rw [integral_div] at hl
simp only [Pi.add_apply] at hu
rw [integral_add (hp.div_const β) (integrable_const _),integral_div,integral_const,
probReal_univ,one_smul] at hu
exact ⟨hl,hu⟩
end InvariantIsing
end
end OAI
|