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d5b562f | 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 | import OAI.MathematicalPhysics.CriticalMixing.Comparison
namespace OAI
noncomputable section
open scoped BigOperators Topology NNReal ENNReal
open MeasureTheory ProbabilityTheory Filter
namespace CriticalSK
theorem critical_two_clock_mixing (ε : ℝ) (hε : 0 < ε) :
Tendsto (fun n : ℕ => (disorderLaw n).real (twoClockMixingEvent n ε))
atTop (𝓝 1) := by
obtain ⟨β, hβ0, hβ1, hsmall⟩ := temperature_close_to_one hε
obtain ⟨C, hC, hP⟩ := fixed_temperature_poincare β hβ0 hβ1
have hl : Tendsto (fun n : ℕ => (disorderLaw n).real (poincareEvent n β C) -
mixingReductionError n ε) atTop (𝓝 1) := by
simpa only [sub_zero] using hP.sub (mixingReductionError_tendsto hε)
exact tendsto_of_tendsto_of_tendsto_of_le_of_le' hl tendsto_const_nhds
(mixingGapReduction_lower hε hβ1.le hsmall hC)
(Filter.Eventually.of_forall (fun _ => measureReal_le_one))
theorem critical_mixing_bounds (ε : ℝ) (hε : 0 < ε) :
Tendsto (fun n : ℕ => (disorderLaw n {W |
(n : ℝ) ^ ((2 : ℝ) / 3 - ε) ≤ continuousMixingTime W ∧
continuousMixingTime W ≤ Real.exp (ε * n)}).toReal) atTop (𝓝 1) ∧
Tendsto (fun n : ℕ => (disorderLaw n {W |
(n : ℝ) ^ ((5 : ℝ) / 3 - ε) ≤ (discreteMixingTime W : ℝ) ∧
(discreteMixingTime W : ℝ) ≤ Real.exp (ε * n)}).toReal) atTop (𝓝 1) := by
have hp := critical_two_clock_mixing ε hε
constructor
· apply tendsto_of_tendsto_of_tendsto_of_le_of_le hp tendsto_const_nhds
· intro n
refine measureReal_mono ?_
intro W hW
exact ⟨hW.1, hW.2.1⟩
· intro n
exact measureReal_le_one
· apply tendsto_of_tendsto_of_tendsto_of_le_of_le hp tendsto_const_nhds
· intro n
refine measureReal_mono ?_
intro W hW
exact ⟨hW.2.2.1, hW.2.2.2⟩
· intro n
exact measureReal_le_one
end CriticalSK
end
end OAI
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