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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 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 81 | import OAI.Probability.ContinuumTransition.GibbsQuadratic
namespace OAI
section
open scoped BigOperators
open MeasureTheory Set
namespace ContinuumTemperature
open Classical
theorem gibbsEnergy_mul_left (V : Space → Space → ℝ) (a : ℝ) {ι : Type} [Fintype ι] (x : ι → Space) :
gibbsEnergy (fun x y => a*V x y) x = a*gibbsEnergy V x := by
have he (i j : ι) : (if i = j then 0 else a*V (x i) (x j)) =
a*(if i = j then 0 else V (x i) (x j)) := by split_ifs <;> ring
simp only [gibbsEnergy, he, ← Finset.mul_sum]
ring
theorem gibbsEnergy_block_constant {q : ℕ} (hq : 0 < q) (γ : ℝ)
{ι : Type} [Fintype ι] (s : Finset Lattice) (c : ι → s) (x : ι → Space)
(hx : ∀ i, x i ∈ gridBox ((q : ℝ)/4) (c i)) :
gibbsEnergy (blockCut q (fun _ _ => γ)) x =
γ*((∑ k : s, (Fintype.card {i // c i = k} : ℝ)^2)-Fintype.card ι)/2 := by
have hu : 0 < (q : ℝ)/4 := by positivity
have he (i j : ι) : blockCut q (fun _ _ => γ) (x i) (x j) = if c i = c j then γ else 0 := by
simp only [blockCut, (mem_gridBox_iff hu _ _).1 (hx i),
(mem_gridBox_iff hu _ _).1 (hx j), Subtype.coe_inj]
rw [gibbsEnergy_eq_sum_sub_diag]
simp only [he, ite_true, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
rw [double_sum_comp_eq c (fun k l => if k = l then γ else 0)]
have hs : (∑ k : s, ∑ l : s, (if k = l then γ else 0)*
(Fintype.card {i // c i = k} : ℝ)*(Fintype.card {i // c i = l} : ℝ)) =
γ*∑ k : s, (Fintype.card {i // c i = k} : ℝ)^2 := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro k _
simp only [ite_mul, zero_mul, Finset.sum_ite_eq, Finset.mem_univ, ite_true]
ring
rw [hs]
ring
theorem quadraticBlock_eq_sub (q : ℕ) (φ : ℝ → ℝ) (γ : ℝ) :
quadraticBlock q φ γ = fun x y => blockCut q (fun x y => φ ‖x-y‖) x y-
blockCut q (fun _ _ => γ) x y := by
funext x y
unfold quadraticBlock blockCut
split_ifs <;> ring
theorem kac_gibbsFamily_le_quadratic {C : NNReal} (hC : LipschitzWith C kernel)
{R β a μ : ℝ} (hR : 0 < R) (hβ : 0 ≤ β) (ha : 0 ≤ a)
{q : ℕ} (hq : 0 < q) (φ : ℝ → ℝ)
{ι : Type} [Fintype ι] (s : Finset Lattice) (c : ι → s) (x : ι → Space)
(hx : ∀ i, x i ∈ gridBox ((q : ℝ)/4) (c i)) :
let u := (q : ℝ)/4
let γ := a*(1+(6*(C : ℝ)*u/R^4)*(2*(R+6*u))^3)/u^3
gibbsFamily (fun x y => blockCut q (fun x y => φ ‖x-y‖) x y-a*scaledKernel R (x-y)) β μ ι x ≤
gibbsFamily (quadraticBlock q φ γ) β (μ+β*γ/2) ι x := by
dsimp only
let u := (q : ℝ)/4
let γ := a*(1+(6*(C : ℝ)*u/R^4)*(2*(R+6*u))^3)/u^3
have hu : 0 < u := by dsimp only [u]; positivity
have he := kac_energy_upper hC hR hu s c x hx
have hb : a*gibbsEnergy (fun x y => scaledKernel R (x-y)) x ≤
γ/2*∑ k : s, (Fintype.card {i // c i = k} : ℝ)^2 := by
apply (mul_le_mul_of_nonneg_left he ha).trans_eq
dsimp only [γ]
ring
unfold gibbsFamily
split_ifs
· rw [gibbsEnergy_sub, gibbsEnergy_mul_left, quadraticBlock_eq_sub,
gibbsEnergy_sub, gibbsEnergy_block_constant hq γ s c x hx]
apply Real.exp_le_exp.mpr
change μ*(Fintype.card ι : ℝ)-β*(_-a*_) ≤
(μ+β*γ/2)*(Fintype.card ι : ℝ)-β*(_-γ*(_-Fintype.card ι)/2)
nlinarith only [mul_le_mul_of_nonneg_left hb hβ]
· rfl
end ContinuumTemperature
end
end OAI
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