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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import OAI.Probability.ContinuumTransition.GibbsIntegrability
namespace OAI
section
open scoped BigOperators
open MeasureTheory Set
namespace ContinuumTemperature
open Classical
theorem idealFamily_finite (d : ℝ) {D : Set Space} (hD : volume D ≠ ⊤) :
GrandFinite (idealFamily d) D :=
GrandFinite.of_bound _ hD (fun _ _ => measurable_const)
(idealFamily_nonneg d) (fun _ _ _ => le_refl _)
theorem grandIntegral_ideal (d : ℝ) (D : Set Space) :
grandIntegral (idealFamily d) D = Real.exp (volume.real D*Real.exp d) := by
have he (n : ℕ) : familyTerm (idealFamily d) D n =
(volume.real D*Real.exp d)^n/(n.factorial : ℝ) := by
simp only [familyTerm, familyIntegral, idealFamily, Fintype.card_fin,
setIntegral_const, smul_eq_mul, Measure.real, volume_generalConfigurations,
ENNReal.toReal_pow, mul_pow, mul_comm d, Real.exp_nat_mul]
simp_rw [grandIntegral, he]
simpa only [Real.exp_eq_exp_ℝ] using
(NormedSpace.expSeries_div_hasSum_exp (volume.real D*Real.exp d)).tsum_eq
theorem core_gibbs_finite (t d : ℝ) {β : ℝ} (hβ : 0 ≤ β)
{D : Set Space} (hD : volume D ≠ ⊤) :
GrandFinite (gibbsFamily (fun x y => coreProfile t ‖x-y‖) β d) D := by
apply GrandFinite.of_bound _ hD
(fun ι _ => measurable_gibbsFamily (V := fun x y : Space => coreProfile t ‖x-y‖) ((measurable_coreProfile t).comp
((measurable_fst.sub measurable_snd).norm)) β d ι)
(gibbsFamily_nonneg _ β d)
intro ι _ x
unfold gibbsFamily
split_ifs
· apply Real.exp_le_exp.mpr
have he : 0 ≤ gibbsEnergy (fun x y => coreProfile t ‖x-y‖) x := by
unfold gibbsEnergy
apply div_nonneg _ (by norm_num)
apply Finset.sum_nonneg
intro i _
apply Finset.sum_nonneg
intro j _
split_ifs
· rfl
· exact coreProfile_nonneg _ _
exact sub_le_self _ (mul_nonneg hβ he)
· exact (Real.exp_pos _).le
theorem core_grand_bad_bound (t d L : ℝ) {β : ℝ} (hβ : 0 ≤ β) :
grandIntegral (gibbsFamily (fun x y => coreProfile t ‖x-y‖) β d) (cube L) ≤
Real.exp (volume.real (cube L)*Real.exp (d-β*t^2/2))*
grandIntegral (freeFamily t d) (cube L) := by
rw [← grandIntegral_ideal]
exact grandIntegral_label_convolution _ _ _ (idealFamily_relabel _) (freeFamily_relabel t d)
(measurableSet_cube L) (core_gibbs_finite t d hβ (isCompact_cube L).measure_lt_top.ne)
(idealFamily_finite _ (isCompact_cube L).measure_lt_top.ne)
(GrandFinite.of_bound _ (isCompact_cube L).measure_lt_top.ne
(freeFamily_measurable t d) (freeFamily_nonneg t d) (freeFamily_le_exp t d))
(fun _ x _ => core_gibbs_bad_endpoints t d hβ x)
lemma truncated_exp_poly_bound {V θ : ℝ} (hV : 0 ≤ V) (hθ : 0 ≤ θ) (M : ℕ) :
(∑ k ∈ Finset.range (M+1), (V*θ)^k/(k.factorial : ℝ)) ≤
(M+1)*Real.exp V*(max 1 θ)^M := by
have hs : HasSum (fun k : ℕ => V^k/(k.factorial : ℝ)) (Real.exp V) := by
simpa only [Real.exp_eq_exp_ℝ] using NormedSpace.expSeries_div_hasSum_exp V
calc
_ ≤ ∑ _k ∈ Finset.range (M+1), Real.exp V*(max 1 θ)^M := by
apply Finset.sum_le_sum
intro k hk
have hkn : k ≤ M := by have := Finset.mem_range.mp hk; omega
have hp : V^k/(k.factorial : ℝ) ≤ Real.exp V :=
hs.summable.le_tsum k (fun j _ => by positivity) |>.trans_eq hs.tsum_eq
rw [mul_pow, mul_div_right_comm]
exact mul_le_mul hp
((pow_le_pow_left₀ hθ (le_max_right 1 θ) k).trans
(pow_le_pow_right₀ (le_max_left 1 θ) hkn))
(pow_nonneg hθ k) (Real.exp_pos V).le
_ = _ := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_add,
Nat.cast_one]; ring
noncomputable def clusterActivity (t d : ℝ) : ℝ :=
Real.exp d*(max 1 (volume.real (Metric.closedBall (0 : Space)
((1+shellDelta)*Real.exp (-t)))*Real.exp d))^3
theorem core_grand_upper {t : ℝ} (ht : 2*(1+shellDelta)*Real.exp (-t) < 1)
(d L : ℝ) {β : ℝ} (hβ : 0 ≤ β) :
grandIntegral (gibbsFamily (fun x y => coreProfile t ‖x-y‖) β d) (cube L) ≤
Real.exp (volume.real (cube L)*(1+Real.exp (d-β*t^2/2))+(packingNumber L : ℝ))*
((packingNumber L : ℝ)+1)*(max 1 (clusterActivity t d))^(packingNumber L) := by
have h1 := core_grand_bad_bound t d L hβ
have h2 := free_grand_upper ht d L
have h3 := truncated_exp_poly_bound (V := volume.real (cube L)) ENNReal.toReal_nonneg
(θ := clusterActivity t d) (by unfold clusterActivity; positivity) (packingNumber L)
have he : volume.real (cube L)*Real.exp d*
(max 1 (volume.real (Metric.closedBall (0 : Space) ((1+shellDelta)*Real.exp (-t)))*Real.exp d))^3 =
volume.real (cube L)*clusterActivity t d := by unfold clusterActivity; ring
rw [he] at h2
apply h1.trans
apply (mul_le_mul_of_nonneg_left h2 (Real.exp_pos _).le).trans
apply (mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right h3 (Real.exp_pos _).le)
(Real.exp_pos _).le).trans_eq
rw [show volume.real (cube L)*(1+Real.exp (d-β*t^2/2))+(packingNumber L : ℝ) =
volume.real (cube L)*Real.exp (d-β*t^2/2)+volume.real (cube L)+(packingNumber L : ℝ) by ring,
Real.exp_add, Real.exp_add]
ring
end ContinuumTemperature
end
end OAI
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