| 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 |
|
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| end OAI |
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