File size: 3,121 Bytes
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 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 | import OAI.Probability.ContinuumTransition.ModifiedEnergy
namespace OAI
section
open scoped BigOperators
open MeasureTheory Set
namespace ContinuumTemperature
open Classical
theorem gibbsEnergy_radial (φ : ℝ → ℝ) {ι : Type} [Fintype ι] (x : ι → Space) :
gibbsEnergy (fun x y => φ ‖x-y‖) x = realPairEnergy φ x := by
unfold gibbsEnergy realPairEnergy
ring
theorem gibbsFamily_radial (φ : ℝ → ℝ) (β h : ℝ) (ι : Type) [Fintype ι] (x : ι → Space) :
gibbsFamily (fun x y => φ ‖x-y‖) β h ι x =
Real.exp (h*Fintype.card ι)*symmetricWeight φ β x := by
unfold gibbsFamily symmetricWeight
split_ifs
· rw [gibbsEnergy_radial, ← Real.exp_add]
congr 1
ring
· simp
theorem familyTerm_radial (φ : ℝ → ℝ) (β h L : ℝ) (n : ℕ) :
familyTerm (gibbsFamily (fun x y => φ ‖x-y‖) β h) (cube L) n =
Real.exp (h*n)*partition φ β L n := by
unfold familyTerm familyIntegral
simp_rw [gibbsFamily_radial, Fintype.card_fin]
rw [integral_const_mul, mul_div_assoc]
simpa only [generalPartition, Fintype.card_fin] using
congrArg (fun a : ℝ => Real.exp (h*n)*a) (generalPartition_eq_partition φ β L n)
theorem grandIntegral_radial (φ : ℝ → ℝ) (t β h L : ℝ) :
grandIntegral (gibbsFamily (fun x y => φ ‖x-y‖) β (t*h)) (cube L) =
grandPartition φ t β h L := by
unfold grandIntegral grandPartition
simp_rw [familyTerm_radial]
theorem CellLowerBound.mono {t : ℝ} {σ R : ℕ → ℝ} {U V : Space → Space → ℝ}
(hU : CellLowerBound t σ R U) (hUV : ∀ x y, U x y ≤ V x y) : CellLowerBound t σ R V :=
fun x y => (hU x y).trans (hUV x y)
theorem physical_cellLowerBound {t : ℝ} (ht : 0 ≤ t) {σ R : ℕ → ℝ}
(hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) :
CellLowerBound t σ R (fun x y => physicalProfile t σ R ‖x-y‖) := by
intro x y
have hb := attraction_le_mixedEnvelope ht hσ hR hs x y
change attraction t σ R ‖x-y‖ ≤ physicalCoupling t σ R (cellIndex x) (cellIndex y) at hb
dsimp only [physicalProfile]
split_ifs
· linarith only [hb]
· linarith only [hb, coreProfile_nonneg t ‖x-y‖]
theorem masked_physical_cellLowerBound {t : ℝ} (ht : 0 ≤ t) {σ R : ℕ → ℝ}
(hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ)
(p q : Space → Space → ℝ) (hp : ∀ x y, 0 ≤ p x y)
(hpc : ∀ x y, cellIndex x = cellIndex y → p x y = 1) (hq : ∀ x y, q x y ≤ 1) :
CellLowerBound t σ R (fun x y => p x y*coreProfile t ‖x-y‖-q x y*attraction t σ R ‖x-y‖) := by
intro x y
have hb := attraction_le_mixedEnvelope ht hσ hR hs x y
have ha := attraction_nonneg ht hσ hR ‖x-y‖
have hc := coreProfile_nonneg t ‖x-y‖
have hq' := mul_le_mul_of_nonneg_right (hq x y) ha
change attraction t σ R ‖x-y‖ ≤ physicalCoupling t σ R (cellIndex x) (cellIndex y) at hb
dsimp only
split_ifs with hxy
· rw [hpc x y hxy]
nlinarith only [hb, hq']
· nlinarith only [hb, hq', mul_nonneg (hp x y) hc]
end ContinuumTemperature
end
end OAI
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