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9b6e014 | 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 | import OAI.Probability.ContinuumTransition.ActiveSupports
namespace OAI
section
open scoped Topology
open Set Filter MeasureTheory
noncomputable section
namespace ContinuumTemperature
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]
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
end ContinuumTemperature
end
end
end OAI
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