| import OAI.Probability.RadialTransition.CanonicalLimit |
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| namespace OAI |
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| universe uG |
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| noncomputable section |
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| open scoped BigOperators NNReal ContDiff ENNReal |
| open Filter MeasureTheory Set |
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| namespace RadialTransition |
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| lemma sum_fin_two {G : Type uG} [AddCommMonoid G] (k r : ℕ) (hr : r ≤ k) (u v : G) : |
| (∑ i : Fin k, if (i : ℕ) < r then u else v) = r • u + (k - r) • v := by |
| rw [Finset.sum_ite] |
| simp only [Finset.sum_const, Fin.card_filter_val_lt, min_eq_right hr] |
| congr 1 |
| have he : (Finset.univ.filter (fun i : Fin k => ¬(i : ℕ) < r)).card = k - r := by |
| rw [Finset.filter_not, Finset.card_sdiff_of_subset (Finset.filter_subset _ _)] |
| simp only [Finset.card_univ, Fintype.card_fin, Fin.card_filter_val_lt, min_eq_right hr] |
| rw [he] |
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| theorem boxCost_limit_convex {H : ℝ → ℕ → ℝ} {A : ℝ → ℝ} |
| (hrep : HasBoxReplication H) |
| (hlim : ∀ d : ℝ, 0 ≤ d → Tendsto (fun z : ℝ × ℕ => H z.1 z.2) |
| (thermodynamicFilter d) (nhds (A d))) : ConvexOn ℝ (Ici 0) A := by |
| refine ⟨convex_Ici _, ?_⟩ |
| intro d₁ hd₁ d₂ hd₂ a b ha hb hab |
| simp only [smul_eq_mul] |
| have ha1 : a ≤ 1 := by linarith |
| let B (j : ℕ) : ℕ := j + 1 |
| let s (j : ℕ) : ℝ := B j |
| let k (j : ℕ) : ℕ := B j ^ 3 |
| let r (j : ℕ) : ℕ := ⌊a * (k j : ℝ)⌋₊ |
| let m₁ (j : ℕ) : ℕ := ⌊d₁ * s j ^ 3⌋₊ |
| let m₂ (j : ℕ) : ℕ := ⌊d₂ * s j ^ 3⌋₊ |
| let N (j : ℕ) : ℕ := r j * m₁ j + (k j - r j) * m₂ j |
| let L (j : ℕ) : ℝ := (B j : ℝ) * (s j + 2) |
| have hspos (j : ℕ) : 0 < s j := by dsimp [s, B]; positivity |
| have hkpos (j : ℕ) : 0 < (k j : ℝ) := by dsimp [k, B]; positivity |
| have hkr (j : ℕ) : r j ≤ k j := by |
| apply Nat.floor_le_of_le |
| exact mul_le_of_le_one_left (by positivity) ha1 |
| have hs : Tendsto s atTop atTop := by |
| have hh := tendsto_atTop_add_const_right atTop (1 : ℝ) |
| (tendsto_natCast_atTop_atTop : Tendsto (fun j : ℕ => (j : ℝ)) atTop atTop) |
| simpa only [s, B, Nat.cast_add, Nat.cast_one] using hh |
| have hk : Tendsto (fun j => (k j : ℝ)) atTop atTop := by |
| simpa only [k, s, Nat.cast_pow, Function.comp_def] using (tendsto_pow_atTop (by norm_num : 3 ≠ 0)).comp hs |
| have hr : Tendsto (fun j => (r j : ℝ) / k j) atTop (nhds a) := |
| (tendsto_nat_floor_mul_div_atTop ha).comp hk |
| have hrem : Tendsto (fun j => ((k j - r j : ℕ) : ℝ) / k j) atTop (nhds b) := by |
| have hh := hr.const_sub 1 |
| rw [show 1 - a = b by linarith] at hh |
| apply hh.congr' |
| exact Eventually.of_forall (fun j => by dsimp only; rw [Nat.cast_sub (hkr j)]; field_simp [(hkpos j).ne']) |
| have hm₁ : Tendsto (fun j => (m₁ j : ℝ) / s j ^ 3) atTop (nhds d₁) := |
| (floor_density_tendsto hd₁).comp hs |
| have hm₂ : Tendsto (fun j => (m₂ j : ℝ) / s j ^ 3) atTop (nhds d₂) := |
| (floor_density_tendsto hd₂).comp hs |
| have hratio : Tendsto (fun j => (s j / (s j + 2)) ^ 3) atTop (nhds 1) := by |
| simpa only [one_pow, Function.comp_def] using (tendsto_side_ratio.comp hs).pow 3 |
| have hL : Tendsto L atTop atTop := hs.atTop_mul_atTop₀ |
| (tendsto_atTop_add_const_right atTop (2 : ℝ) hs) |
| have hden : Tendsto (fun j => (N j : ℝ) / L j ^ 3) atTop (nhds (a * d₁ + b * d₂)) := by |
| have hh := hratio.mul ((hr.mul hm₁).add (hrem.mul hm₂)) |
| simp only [one_mul] at hh |
| apply hh.congr' |
| refine Eventually.of_forall (fun j => ?_) |
| have hs0 := (hspos j).ne' |
| have hk0 := (hkpos j).ne' |
| have hsum0 : s j + 2 ≠ 0 := by linarith [hspos j] |
| dsimp only [N, L, Function.comp_def] |
| rw [Nat.cast_add, Nat.cast_mul, Nat.cast_mul] |
| have hkEq : (B j : ℝ) ^ 3 = k j := by simp [k] |
| rw [mul_pow, hkEq] |
| have hsk : s j ^ 3 = (k j : ℝ) := hkEq |
| field_simp |
| rw [hsk] |
| have hnl : Tendsto (fun j => (L j, N j)) atTop (thermodynamicFilter (a * d₁ + b * d₂)) := by |
| apply tendsto_inf.mpr |
| exact ⟨tendsto_comap_iff.mpr hL, tendsto_comap_iff.mpr hden⟩ |
| have hH : Tendsto (fun j => H (L j) (N j)) atTop (nhds (A (a * d₁ + b * d₂))) := |
| (hlim _ (add_nonneg (mul_nonneg ha hd₁) (mul_nonneg hb hd₂))).comp hnl |
| have hH₁ : Tendsto (fun j => H (s j) (m₁ j)) atTop (nhds (A d₁)) := |
| (hlim d₁ hd₁).comp ((floor_tendsto_thermodynamicFilter hd₁).comp hs) |
| have hH₂ : Tendsto (fun j => H (s j) (m₂ j)) atTop (nhds (A d₂)) := |
| (hlim d₂ hd₂).comp ((floor_tendsto_thermodynamicFilter hd₂).comp hs) |
| have hupper : Tendsto (fun j => (s j / (s j + 2)) ^ 3 * |
| ((r j : ℝ) / k j * H (s j) (m₁ j) + ((k j - r j : ℕ) : ℝ) / k j * H (s j) (m₂ j))) |
| atTop (nhds (a * A d₁ + b * A d₂)) := by |
| simpa only [one_mul] using hratio.mul ((hr.mul hH₁).add (hrem.mul hH₂)) |
| apply le_of_tendsto_of_tendsto hH hupper |
| refine Eventually.of_forall (fun j => ?_) |
| let f : Fin (k j) → ℕ := fun i => if (i : ℕ) < r j then m₁ j else m₂ j |
| have hsum : (∑ i, f i) = N j := by |
| simpa only [f, N, nsmul_eq_mul, smul_eq_mul] using sum_fin_two (k j) (r j) (hkr j) (m₁ j) (m₂ j) |
| have hcost : (∑ i, H (s j) (f i)) = (r j : ℝ) * H (s j) (m₁ j) + |
| ((k j - r j : ℕ) : ℝ) * H (s j) (m₂ j) := by |
| dsimp only [f] |
| simp only [apply_ite] |
| simpa only [nsmul_eq_mul] using sum_fin_two (k j) (r j) (hkr j) |
| (H (s j) (m₁ j)) (H (s j) (m₂ j)) |
| have hh := hrep (s j) (L j) (B j) (hspos j) |
| (by dsimp [L] |
| rw [hsum, hcost] at hh |
| apply hh.trans_eq |
| have hs0 := (hspos j).ne' |
| have hk0 := (hkpos j).ne' |
| have hsum0 : s j + 2 ≠ 0 := by linarith [hspos j] |
| dsimp only [L] |
| have hkEq : (B j : ℝ) ^ 3 = k j := by simp [k] |
| rw [mul_pow, hkEq] |
| have hsk : s j ^ 3 = (k j : ℝ) := hkEq |
| field_simp |
| rw [hsk] |
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| def finiteGrandPressure (φ : Potential) (t L β h : ℝ) : ℝ := |
| Real.log (grandPartition φ t L β h) / (t * L ^ 3) |
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| def grandPressure (φ : Potential) (t : ℝ) : Pressure := |
| fun β h => limsup (fun L : ℝ => finiteGrandPressure φ t L β h) atTop |
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| theorem grandPressure_spec {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ) |
| {B t β : ℝ} (hstable : StableWith φ B) (ht : 0 < t) (hβ : 0 ≤ β) |
| (hc : ∀ r : ℝ≥0, 2 ≤ (r : ℝ) → φ r ≤ 0) (h : ℝ) : |
| Tendsto (fun L : ℝ => finiteGrandPressure φ t L β h) atTop (nhds (grandPressure φ t β h)) ∧ |
| 0 ≤ grandPressure φ t β h ∧ grandPressure φ t β h ≤ Real.exp (β * B + t * h) / t := by |
| obtain ⟨p, hp0, hpB, hlim⟩ := grandPressure_exists hφ hb hstable ht hβ hc h |
| have he : grandPressure φ t β h = p := hlim.limsup_eq |
| rw [he] |
| exact ⟨hlim, hp0, hpB⟩ |
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| theorem finiteGrandPressure_convex_mix {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ) |
| {B t L β γ a b : ℝ} (hstable : StableWith φ B) (ht : 0 < t) (hL : 0 < L) |
| (hβ : 0 ≤ β) (hγ : 0 ≤ γ) (h j : ℝ) (ha : 0 ≤ a) (hbb : 0 ≤ b) (hab : a + b = 1) : |
| finiteGrandPressure φ t L (a * β + b * γ) (a * h + b * j) ≤ |
| a * finiteGrandPressure φ t L β h + b * finiteGrandPressure φ t L γ j := by |
| have hmix : 0 ≤ a * β + b * γ := by positivity |
| have hh := log_tsum_convex_mix ha hbb hab |
| (grandPartition_summable hφ hb hstable hL.le hmix t (a * h + b * j)) |
| (grandPartition_summable hφ hb hstable hL.le hβ t h) |
| (grandPartition_summable hφ hb hstable hL.le hγ t j) |
| (fun n => mul_pos (Real.exp_pos _) (partition_pos hφ hb hL n _)) |
| (fun n => mul_pos (Real.exp_pos _) (partition_pos hφ hb hL n _)) |
| (fun n => mul_pos (Real.exp_pos _) (partition_pos hφ hb hL n _)) |
| (fun n => by |
| simp only [Real.log_mul (Real.exp_ne_zero _) (partition_pos hφ hb hL n _).ne', Real.log_exp] |
| have hh := log_partition_convex_mix hφ hb hL n β γ ha hbb hab |
| nlinarith) |
| dsimp only [finiteGrandPressure] |
| have hh' := div_le_div_of_nonneg_right hh (show 0 ≤ t * L ^ 3 by positivity) |
| simpa only [grandPartition, add_div, mul_div_assoc] using hh' |
| |
| theorem grandPressure_convex_mix {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ) |
| {B t β γ a b : ℝ} (hstable : StableWith φ B) (ht : 0 < t) |
| (hc : ∀ r : ℝ≥0, 2 ≤ (r : ℝ) → φ r ≤ 0) |
| (hβ : 0 ≤ β) (hγ : 0 ≤ γ) (h j : ℝ) (ha : 0 ≤ a) (hbb : 0 ≤ b) (hab : a + b = 1) : |
| grandPressure φ t (a * β + b * γ) (a * h + b * j) ≤ |
| a * grandPressure φ t β h + b * grandPressure φ t γ j := by |
| have hl := (grandPressure_spec hφ hb hstable ht (β := a * β + b * γ) (by positivity) hc (a * h + b * j)).1 |
| have hr := ((grandPressure_spec hφ hb hstable ht hβ hc h).1.const_mul a).add |
| ((grandPressure_spec hφ hb hstable ht hγ hc j).1.const_mul b) |
| apply le_of_tendsto_of_tendsto hl hr |
| filter_upwards [eventually_gt_atTop (0 : ℝ)] with L hL |
| exact finiteGrandPressure_convex_mix hφ hb hstable ht hL hβ hγ h j ha hbb hab |
| |
| end RadialTransition |
| |
| end |
| |
| end OAI |
| |
