File size: 3,451 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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import OAI.Probability.InvariantIsing.Pressure.PhysicalPressureMean
import OAI.Probability.InvariantIsing.Pressure.ThermalPressure
namespace OAI
/-! The actual maximum Ising energy and its uniform finite-temperature squeeze. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators Classical
namespace InvariantIsing
def groundStateEnergy {N : ℕ} (eig : Fin N → ℝ) (U : Rotation N) : ℝ :=
(N : ℝ)⁻¹ * Finset.univ.sup' Finset.univ_nonempty (rotatedEnergy eig U)
lemma groundStateEnergy_attained {N : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) :
(∀ σ, rotatedEnergy eig U σ ≤ N*groundStateEnergy eig U) ∧
∃ σ, rotatedEnergy eig U σ=N*groundStateEnergy eig U := by
have hn : (N : ℝ) ≠ 0 := by exact_mod_cast hN.ne'
have he : (N : ℝ)*groundStateEnergy eig U=
Finset.univ.sup' Finset.univ_nonempty (rotatedEnergy eig U) := by
unfold groundStateEnergy
field_simp
rw [he]
constructor
· intro σ
exact Finset.le_sup' _ (Finset.mem_univ σ)
· obtain ⟨σ,_,hσ⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (rotatedEnergy eig U)
exact ⟨σ,hσ.symm⟩
lemma groundStateEnergy_squeeze {N : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (β : ℝ) (hβ : 0 < β) :
rotatedPressure (fun i => β*eig i) U (fun _ => 0)/β ≤ groundStateEnergy eig U ∧
groundStateEnergy eig U ≤
rotatedPressure (fun i => β*eig i) U (fun _ => 0)/β+Real.log 2/β := by
have he := groundStateEnergy_attained hN eig U
have hh := finite_groundState_squeeze hN (rotatedEnergy eig U) β
(groundStateEnergy eig U) hβ he.1 he.2
have hp : rotatedPressure (fun i => β*eig i) U (fun _ => 0)=
(N : ℝ)⁻¹*logPartition (fun σ => β*rotatedEnergy eig U σ) := by
simp only [rotatedPressure,rotatedEnergy_scale,fieldEnergy,zero_mul,Finset.sum_const_zero,add_zero]
rw [hp]
exact ⟨hh.2,by linarith [hh.1]⟩
lemma abs_groundStateEnergy_le {N : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (K : ℝ) (hK : ∀ i, |eig i| ≤ K) :
|groundStateEnergy eig U| ≤ K/2 := by
obtain ⟨σ,hσ⟩ := (groundStateEnergy_attained hN eig U).2
have hh := abs_rotatedEnergy_le eig U K hK σ
rw [hσ,abs_mul,abs_of_nonneg (Nat.cast_nonneg N)] at hh
have hn : (0 : ℝ) < N := by exact_mod_cast hN
nlinarith
lemma measurable_groundStateEnergy {N : ℕ} (eig : Fin N → ℝ) :
Measurable (fun V : SpecialOrthogonal N => groundStateEnergy eig (specialRotation V)) := by
have hE (σ : Spin N) : Measurable (fun V : SpecialOrthogonal N =>
rotatedEnergy eig (specialRotation V) σ) :=
(Finset.measurable_sum _ fun i _ =>
((measurable_specialRotation_eval (spinVector σ) i).pow_const 2).const_mul (eig i)).const_mul _
have hh := Finset.measurable_sup' Finset.univ_nonempty (fun σ _ => hE σ)
convert hh.const_mul (N : ℝ)⁻¹ using 1
funext V
simp only [groundStateEnergy,Finset.sup'_apply]
lemma measurable_physical_groundStateEnergy {N : ℕ} (hN : 0 < N) (eig : Fin N → ℝ) :
Measurable (fun V : Orthogonal N => groundStateEnergy eig (matrixRotation V⁻¹)) := by
have hm := (measurable_groundStateEnergy eig).comp
((measurable_cavityOrientationLift hN).comp measurable_inv)
convert hm using 1
funext V
simp only [Function.comp_apply,groundStateEnergy,cavityOrientationLift_energy]
end InvariantIsing
end
end OAI
|