| import OAI.Probability.ContinuumTransition.ActiveSupports |
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| namespace OAI |
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| section |
| open scoped Topology |
| open Set Filter MeasureTheory |
| noncomputable section |
| namespace ContinuumTemperature |
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| lemma physicalCanonical_quadratic_lower {t β d : ℝ} (ht : 1 < t) (hβ : 0 ≤ β) |
| {σ R : ℕ → ℝ} (hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) |
| (hsmall : 2197*t*kernelHeight*(∑' n, σ n)/2 ≤ t^2/8) (hd : 0 ≤ d) : |
| (β*((t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)/64)/t)*d^2- |
| (β*t/2)*d-1/t ≤ physicalCanonical (physicalProfile t σ R) t β d := by |
| have hh := canonicalDensity_lower_coercive ht hβ hσ hR hs hsmall hd |
| have he := entropy_dual_bound 0 hd |
| norm_num only [zero_mul,Real.exp_zero,zero_sub] at he |
| apply (le_div_iff₀ (lt_trans zero_lt_one ht)).mpr |
| have hne : t ≠ 0 := (lt_trans zero_lt_one ht).ne' |
| field_simp [hne] |
| nlinarith only [he,hh] |
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| lemma physical_meanField_attained {t a : ℝ} (ht : 1 < t) (ha : 0 < a) |
| {σ R : ℕ → ℝ} (hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) |
| (hsmall : 2197*t*kernelHeight*(∑' n, σ n)/2 ≤ t^2/8) |
| (hgap : t*a/2 < (t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)/64) |
| {x : Parameter} (hx : x ∈ pressureDomain) : |
| ∃ d, quadraticCost (physicalPressure (physicalProfile t σ R) t) a x d = |
| meanField (physicalPressure (physicalProfile t σ R) t) a x := by |
| have ht0 : 0 < t := lt_trans zero_lt_one ht |
| have hφ := measurable_radialPotential (measurable_physicalProfile t hσ hR hs) |
| have hst := physicalProfile_stable ht0.le hσ hR hs hsmall |
| have hcore := fun r (_ : 0 < r) => physicalProfile_le_core ht0.le hσ hR r |
| have hP := isMeanField_meanField (physical_meanField ht ha.le hσ hR hs hsmall hgap.le) |
| apply meanField_attained ha (continuousOn_physicalPressure hφ ht hst hcore) hP hx |
| (fun h => physicalPressure_isLUB hφ ht (le_of_lt hx) hst hcore (h := h)) |
| (c := x.1*((t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)/64)/t) |
| (b := x.1*t/2) (k := 1/t) |
| · apply (lt_div_iff₀ ht0).mpr |
| have hh := mul_lt_mul_of_pos_left hgap hx |
| nlinarith only [hh] |
| · exact fun d hd => physicalCanonical_quadratic_lower ht (le_of_lt hx) hσ hR hs hsmall hd |
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| end ContinuumTemperature |
| end |
| end |
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| end OAI |
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