| import OAI.Probability.ContinuumTransition.BlockEnergy |
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| namespace OAI |
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| section |
| open scoped BigOperators |
| open MeasureTheory Set |
| namespace ContinuumTemperature |
| open Classical |
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| theorem gridBox_subset_translate_cube {u : ℝ} (_hu : 0 < u) (k : Lattice) : |
| gridBox u k ⊆ (fun x : Space => x+gridCenter u k) '' cube u := by |
| intro x hx |
| refine ⟨x-gridCenter u k, ?_, sub_add_cancel _ _⟩ |
| intro i |
| have hxi := hx i |
| change -u/2 ≤ x i-u*((k i : ℝ)+1/2) ∧ x i-u*((k i : ℝ)+1/2) ≤ u/2 |
| constructor <;> linarith [hxi.1, hxi.2] |
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| theorem volume_translate (A : Set Space) (a : Space) : |
| volume ((fun x : Space => x+a) '' A) = volume A := by |
| have he : (fun x : Space => x+a) '' A = (fun x : Space => x-a) ⁻¹' A := by |
| ext x |
| constructor |
| · rintro ⟨y, hy, rfl⟩ |
| simpa only [mem_preimage, add_sub_cancel_right] using hy |
| · intro hx |
| exact ⟨x-a, hx, sub_add_cancel _ _⟩ |
| rw [he, show (fun x : Space => x-a) = (fun x => x+(-a)) by funext; rw [sub_eq_add_neg], |
| measure_preimage_add_right] |
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| theorem setPartition_mono {φ : ℝ → ℝ} (hφ : Measurable (radialPotential φ)) |
| {β B : ℝ} (hβ : 0 ≤ β) |
| (hst : ∀ (n : ℕ) (x : Fin n → Space), ((-B*n : ℝ) : EReal) ≤ energy φ x) |
| {A D : Set Space} (hD : volume D ≠ ⊤) (hAD : A ⊆ D) (n : ℕ) : |
| setPartition φ β A n ≤ setPartition φ β D n := by |
| unfold setPartition generalPartition |
| apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _) |
| refine setIntegral_mono_set (integrableOn_symmetricWeight hφ hβ hst ?_) |
| (Filter.Eventually.of_forall (symmetricWeight_nonneg φ β)) ?_ |
| · rw [volume_generalConfigurations] |
| exact ENNReal.pow_ne_top hD |
| · exact Filter.Eventually.of_forall (fun _ hx i => hAD (hx i)) |
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| theorem setPartition_gridBox_le {φ : ℝ → ℝ} (hφ : Measurable (radialPotential φ)) |
| {β B u : ℝ} (hβ : 0 ≤ β) (hu : 0 < u) |
| (hst : ∀ (n : ℕ) (x : Fin n → Space), ((-B*n : ℝ) : EReal) ≤ energy φ x) |
| (k : Lattice) (n : ℕ) : setPartition φ β (gridBox u k) n ≤ partition φ β u n := by |
| have hfin : volume ((fun x : Space => x+gridCenter u k) '' cube u) ≠ ⊤ := by |
| rw [volume_translate, volume_cube] |
| exact ENNReal.pow_ne_top ENNReal.ofReal_ne_top |
| have hh := setPartition_mono hφ hβ hst hfin (gridBox_subset_translate_cube hu k) n |
| rw [setPartition_translate] at hh |
| simp only [setPartition, generalPartition_eq_partition] at hh |
| exact hh |
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|
| theorem physicalProfile_ge_constant {t : ℝ} (ht : 0 ≤ t) {σ R : ℕ → ℝ} |
| (hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) (r : ℝ) : |
| -(t*(∑' n, σ n)*kernelHeight) ≤ physicalProfile t σ R r := by |
| have ha := attraction_le ht hσ hR hs r |
| have hc := coreProfile_nonneg t r |
| change _ ≤ coreProfile t r-attraction t σ R r |
| linarith |
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| theorem quadraticBlock_lower {φ : ℝ → ℝ} {C γ : ℝ} (hC : 0 ≤ C) (hγ : 0 ≤ γ) |
| (hφ : ∀ r, -C ≤ φ r) (q : ℕ) (x y : Space) : |
| -(C+γ) ≤ quadraticBlock q φ γ x y := by |
| unfold quadraticBlock blockCut |
| split_ifs |
| · linarith [hφ ‖x-y‖] |
| · linarith |
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| theorem volume_gridBlocks_ne_top {u : ℝ} (hu : 0 ≤ u) (s : Finset Lattice) : |
| volume (⋃ k ∈ s, gridBox u k) ≠ ⊤ := by |
| apply ne_top_of_le_ne_top ?_ (measure_biUnion_finset_le s (gridBox u)) |
| apply ENNReal.sum_ne_top.mpr |
| intro k _ |
| rw [volume_gridBox hu] |
| exact ENNReal.ofReal_ne_top |
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|
| theorem quadraticBlock_familyTerm_le {t β μ γ : ℝ} (ht : 0 ≤ 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) |
| {q : ℕ} (hq : 0 < q) (k : Lattice) (n : ℕ) : |
| familyTerm (gibbsFamily (quadraticBlock q (physicalProfile t σ R) γ) β (μ+β*γ/2)) |
| (gridBox ((q : ℝ)/4) k) n ≤ |
| quadraticGrandTerm (physicalProfile t σ R) β μ (β*γ/2*((q : ℝ)/4)^3) ((q : ℝ)/4) n := by |
| have hu : 0 < (q : ℝ)/4 := by positivity |
| rw [familyTerm_quadraticBlock hq] |
| unfold quadraticGrandTerm |
| rw [show β*γ/2*((q : ℝ)/4)^3*(n : ℝ)^2/((q : ℝ)/4)^3 = β*γ/2*(n : ℝ)^2 by field_simp] |
| exact mul_le_mul_of_nonneg_left |
| (setPartition_gridBox_le (measurable_radialPotential (measurable_physicalProfile t hσ hR hs)) |
| hβ hu (physicalProfile_stable ht hσ hR hs hsmall) k n) (Real.exp_pos _).le |
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| theorem grandFinite_quadraticBlock_box {t β μ γ : ℝ} (ht : 0 ≤ t) (hβ : 0 ≤ β) (hγ : 0 ≤ γ) |
| {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) {q : ℕ} (hq : 0 < q) |
| (hcoeff : β*γ/2*((q : ℝ)/4)^3 ≤ |
| β*(t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)/(4+2/((q : ℝ)/4))^3) (k : Lattice) : |
| GrandFinite (gibbsFamily (quadraticBlock q (physicalProfile t σ R) γ) β (μ+β*γ/2)) |
| (gridBox ((q : ℝ)/4) k) := by |
| have hu : 0 < (q : ℝ)/4 := by positivity |
| have hC : 0 ≤ t*(∑' n, σ n)*kernelHeight := |
| mul_nonneg (mul_nonneg ht (tsum_nonneg hσ)) kernelHeight_pos.le |
| refine ⟨fun ι _ => integrableOn_gibbs_of_lower |
| (measurable_quadraticBlock q (measurable_physicalProfile t hσ hR hs) γ) |
| (add_nonneg hC hγ) hβ |
| (quadraticBlock_lower hC hγ (physicalProfile_ge_constant ht hσ hR hs) q) ?_ ι, ?_⟩ |
| · rw [volume_gridBox hu.le]; exact ENNReal.ofReal_ne_top |
| · exact Summable.of_nonneg_of_le (familyTerm_nonneg _ _ (gibbsFamily_nonneg _ _ _)) |
| (quadraticBlock_familyTerm_le ht hβ hσ hR hs hsmall hq k) |
| (summable_quadraticGrandTerm ht hβ hu hσ hR hs hsmall hcoeff) |
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| end ContinuumTemperature |
| end |
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| end OAI |
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