| import OAI.Probability.ContinuumTransition.DensityLimit |
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| namespace OAI |
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| section |
| open scoped Topology BigOperators ENNReal |
| open Filter MeasureTheory Set |
| namespace ContinuumTemperature |
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| noncomputable def finiteCanonical (φ : ℝ → ℝ) (β L : ℝ) (N : ℕ) : ℝ := |
| -Real.log (partition φ β L N)/L^3 |
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| @[simp] theorem finiteCanonical_zero (φ : ℝ → ℝ) (β L : ℝ) : |
| finiteCanonical φ β L 0 = 0 := by simp [finiteCanonical, partition] |
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| theorem finiteCanonical_lower_constant {φ : ℝ → ℝ} (hφ : Measurable (radialPotential φ)) |
| {β B L : ℝ} (hβ : 0 ≤ β) (hL : 0 < L) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B*(N : ℝ) : ℝ) : EReal) ≤ energy φ x) (N : ℕ) : |
| -Real.exp (β*B) ≤ finiteCanonical φ β L N := by |
| have hc := (summable_grandPartition hφ hβ hL.le hstable 1 0).le_tsum N |
| (fun n _ => mul_nonneg (Real.exp_pos _).le (partition_nonneg φ β L n)) |
| have hc' : partition φ β L N ≤ grandPartition φ 1 β 0 L := by |
| simpa only [mul_zero, zero_mul, Real.exp_zero, one_mul, grandPartition] using hc |
| have hZ := hc'.trans (grandPartition_le_exp hφ hβ hL.le hstable 1 0) |
| simp only [mul_zero, zero_add] at hZ |
| have hlog := Real.log_le_log (partition_pos hφ hβ hL hstable N) hZ |
| rw [Real.log_exp] at hlog |
| unfold finiteCanonical |
| apply (le_div_iff₀ (pow_pos hL 3)).2 |
| nlinarith only [hlog] |
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| theorem finiteCanonical_upper_trial {φ : ℝ → ℝ} (hφ : Measurable (radialPotential φ)) |
| {q t β B L D : ℝ} (hq : 0 < q) (ht : 1 < t) (hβ : 0 ≤ β) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B*(N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (hcore : ∀ r, 0 < r → φ r ≤ coreProfile t r) |
| (hL : 8*q ≤ L) (hD : D ≤ (2*q)⁻¹^3) {N : ℕ} (hN : (N : ℝ) ≤ D*L^3) : |
| finiteCanonical φ β L N ≤ (β*trialEnergyConstant t q-Real.log (trialVolume q)) * (N : ℝ)/L^3 := by |
| have hpos : 0 < trialVolume q ^ N * Real.exp (-β*(trialEnergyConstant t q*N)) := by |
| exact mul_pos (pow_pos (trialVolume_pos hq) N) (Real.exp_pos _) |
| have hlow := partition_trial_lower hφ hq ht hβ hstable hcore hL hD hN |
| have hlog := Real.log_le_log hpos hlow |
| rw [Real.log_mul (pow_pos (trialVolume_pos hq) N).ne' (Real.exp_pos _).ne', |
| Real.log_pow, Real.log_exp] at hlog |
| unfold finiteCanonical |
| apply div_le_div_of_nonneg_right _ (pow_nonneg (by linarith : 0 ≤ L) 3) |
| nlinarith only [hlog] |
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| theorem finiteCanonical_densityBounded {φ : ℝ → ℝ} (hφ : Measurable (radialPotential φ)) |
| {t β B : ℝ} (ht : 1 < t) (hβ : 0 ≤ β) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B*(N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (hcore : ∀ r, 0 < r → φ r ≤ coreProfile t r) : |
| DensityBounded (finiteCanonical φ β) := by |
| intro D hD |
| let q : ℝ := (2*(D+1))⁻¹ |
| have hq : 0 < q := by dsimp [q]; positivity |
| have hdmesh : D ≤ (2*q)⁻¹^3 := by |
| have hh : (2*q)⁻¹ = D+1 := by dsimp [q]; field_simp |
| rw [hh] |
| nlinarith [sq_nonneg D] |
| let C : ℝ := |β*trialEnergyConstant t q-Real.log (trialVolume q)| |
| refine ⟨Real.exp (β*B)+C*D+1, 8*q, by dsimp [C]; positivity, by positivity, ?_⟩ |
| intro L hL N hN |
| have hLpos : 0 < L := lt_of_lt_of_le (by positivity) hL |
| have hlo := finiteCanonical_lower_constant hφ hβ hLpos hstable N |
| have hup := finiteCanonical_upper_trial hφ hq ht hβ hstable hcore hL hdmesh hN |
| have hup' : finiteCanonical φ β L N ≤ C*D := by |
| calc |
| _ ≤ (β*trialEnergyConstant t q-Real.log (trialVolume q)) * ((N : ℝ)/L^3) := by |
| simpa only [mul_div_assoc] using hup |
| _ ≤ C*((N : ℝ)/L^3) := mul_le_mul_of_nonneg_right (le_abs_self _) |
| (div_nonneg (Nat.cast_nonneg _) (pow_nonneg hLpos.le 3)) |
| _ ≤ C*D := mul_le_mul_of_nonneg_left ((div_le_iff₀ (pow_pos hLpos 3)).2 hN) (abs_nonneg _) |
| apply abs_le.mpr |
| have hCD : 0 ≤ C*D := mul_nonneg (abs_nonneg _) hD.le |
| have he := (Real.exp_pos (β*B)).le |
| constructor <;> linarith |
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| theorem finiteCanonical_hasBoxInequality {φ : ℝ → ℝ} (hφ : Measurable (radialPotential φ)) |
| {β B : ℝ} (hβ : 0 ≤ β) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B*(N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (hsign : ∀ r, 1 ≤ r → φ r ≤ 0) : HasBoxInequality (finiteCanonical φ β) := by |
| intro ℓ L hℓ hL r |
| classical |
| let e : (Fin 3 → Fin ⌊L/(ℓ+2)⌋₊) ≃ Fin (tileCount ℓ L) := |
| Fintype.equivFinOfCardEq (by simp [tileCount]) |
| have htile := partition_tiling_lower hφ hβ hℓ.le hL.le hstable hsign (fun i => r (e i)) |
| rw [e.sum_comp] at htile |
| have hprod := e.prod_comp (fun i => partition φ β ℓ (r i)) |
| rw [hprod] at htile |
| have hlog := Real.log_le_log (Finset.prod_pos (fun i _ => partition_pos hφ hβ hℓ hstable (r i))) htile |
| rw [Real.log_prod (fun i _ => (partition_pos hφ hβ hℓ hstable (r i)).ne')] at hlog |
| unfold finiteCanonical |
| calc |
| -Real.log (partition φ β L (∑ i, r i))/L^3 ≤ -(∑ i, Real.log (partition φ β ℓ (r i)))/L^3 := |
| div_le_div_of_nonneg_right (neg_le_neg hlog) (pow_nonneg hL.le 3) |
| _ = ℓ^3/L^3 * ∑ i, -Real.log (partition φ β ℓ (r i))/ℓ^3 := by |
| rw [← Finset.sum_div, Finset.sum_neg_distrib] |
| field_simp |
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| theorem exists_canonical_density_limit {φ : ℝ → ℝ} (hφ : Measurable (radialPotential φ)) |
| {t β B : ℝ} (ht : 1 < t) (hβ : 0 ≤ β) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B*(N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (hcore : ∀ r, 0 < r → φ r ≤ coreProfile t r) {d : ℝ} (hd : 0 < d) : |
| ∃ I : ℝ, Tendsto (fun p : ℝ × ℕ => finiteCanonical φ β p.1 p.2) (densityFilter d) (𝓝 I) := by |
| apply exists_density_limit (finiteCanonical_densityBounded hφ ht hβ hstable hcore) |
| (finiteCanonical_hasBoxInequality hφ hβ hstable _) hd |
| intro r hr |
| exact (hcore r (lt_of_lt_of_le (by norm_num) hr)).trans_eq (coreProfile_eq_zero ht hr) |
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| theorem finiteCanonical_lower_entropy {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {β B L : ℝ} (hβ : 0 ≤ β) (hL : 0 < L) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) (N : ℕ) : |
| -β * B * ((N : ℝ) / L ^ 3) + ((N : ℝ) / L ^ 3) * Real.log ((N : ℝ) / L ^ 3) |
| - (N : ℝ) / L ^ 3 ≤ finiteCanonical φ β L N := by |
| by_cases hN : N = 0 |
| · simp [hN] |
| have hh := neg_log_partition_lower hφ hβ hL hstable hN |
| (fun x _ => hstable N x) |
| have hh' := div_le_div_of_nonneg_right hh (pow_pos hL 3).le |
| unfold finiteCanonical |
| calc |
| _ = (β * (-B * (N : ℝ)) + N * Real.log ((N : ℝ) / L ^ 3) - N) / L ^ 3 := by ring |
| _ ≤ _ := hh' |
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| theorem finiteCanonical_tendsto_zero_density {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {t β B : ℝ} (ht : 1 < t) (hβ : 0 ≤ β) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (hcore : ∀ r, 0 < r → φ r ≤ coreProfile t r) : |
| Tendsto (fun p : ℝ × ℕ => finiteCanonical φ β p.1 p.2) (densityFilter 0) (𝓝 0) := by |
| have hden := densityFilter_density 0 |
| have hlo : Tendsto (fun p : ℝ × ℕ => |
| -β * B * ((p.2 : ℝ) / p.1 ^ 3) + ((p.2 : ℝ) / p.1 ^ 3) * |
| Real.log ((p.2 : ℝ) / p.1 ^ 3) - (p.2 : ℝ) / p.1 ^ 3) |
| (densityFilter 0) (𝓝 0) := by |
| simpa only [mul_zero, Real.log_zero, add_zero, sub_zero, Function.comp_def] using |
| ((hden.const_mul (-β * B)).add (Real.continuous_mul_log.tendsto 0 |>.comp hden)).sub hden |
| let C := β * trialEnergyConstant t (1/2) - Real.log (trialVolume (1/2)) |
| have hup : Tendsto (fun p : ℝ × ℕ => C * ((p.2 : ℝ) / p.1 ^ 3)) |
| (densityFilter 0) (𝓝 0) := by |
| simpa only [mul_zero] using hden.const_mul C |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hlo hup |
| · filter_upwards [(densityFilter_length 0).eventually (eventually_gt_atTop 0)] with p hp |
| exact finiteCanonical_lower_entropy hφ hβ hp hstable p.2 |
| · filter_upwards [(densityFilter_length 0).eventually (eventually_ge_atTop 4), |
| hden.eventually (eventually_le_nhds (by norm_num : (0 : ℝ) < 1))] with p hp hpd |
| have hL : 0 < p.1 := by linarith |
| have hh := finiteCanonical_upper_trial hφ (by norm_num : (0 : ℝ) < 1 / 2) |
| ht hβ hstable hcore (by linarith : 8*(1/2 : ℝ) ≤ p.1) |
| (by norm_num : (1 : ℝ) ≤ (2*(1/2))⁻¹^3) |
| ((div_le_iff₀ (pow_pos hL 3)).1 hpd) |
| simpa only [C, mul_div_assoc] using hh |
|
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| noncomputable def canonicalDensity (φ : ℝ → ℝ) (β d : ℝ) : ℝ := |
| liminf (fun p : ℝ × ℕ => finiteCanonical φ β p.1 p.2) (densityFilter d) |
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| theorem finiteCanonical_tendsto {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {t β B d : ℝ} (ht : 1 < t) (hβ : 0 ≤ β) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (hcore : ∀ r, 0 < r → φ r ≤ coreProfile t r) (hd : 0 ≤ d) : |
| Tendsto (fun p : ℝ × ℕ => finiteCanonical φ β p.1 p.2) |
| (densityFilter d) (𝓝 (canonicalDensity φ β d)) := by |
| let : Fact (0 ≤ d) := ⟨hd⟩ |
| obtain hzero | hpos := hd.eq_or_lt |
| · subst d |
| have hz := finiteCanonical_tendsto_zero_density hφ ht hβ hstable hcore |
| simpa only [canonicalDensity, hz.liminf_eq] using hz |
| · obtain ⟨I, hI⟩ := exists_canonical_density_limit hφ ht hβ hstable hcore hpos |
| simpa only [canonicalDensity, hI.liminf_eq] using hI |
|
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| @[simp] theorem canonicalDensity_zero {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {t β B : ℝ} (ht : 1 < t) (hβ : 0 ≤ β) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (hcore : ∀ r, 0 < r → φ r ≤ coreProfile t r) : canonicalDensity φ β 0 = 0 := by |
| let : Fact (0 ≤ (0 : ℝ)) := ⟨le_rfl⟩ |
| exact (finiteCanonical_tendsto_zero_density hφ ht hβ hstable hcore).liminf_eq |
|
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| noncomputable def canonicalFreeEnergy (φ : ℝ → ℝ) (β d : ℝ) : ℝ := |
| canonicalDensity φ β d / β |
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| theorem hasCanonicalFreeEnergy_of_core {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {t B : ℝ} (ht : 1 < t) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (hcore : ∀ r, 0 < r → φ r ≤ coreProfile t r) : |
| HasCanonicalFreeEnergy φ (canonicalFreeEnergy φ) := by |
| intro β d hβ hd |
| have hlim := (finiteCanonical_tendsto hφ ht hβ.le hstable hcore hd.le).comp |
| (floor_density_tendsto hd.le) |
| have hneg := hlim.neg |
| convert hneg using 1 |
| · ext L |
| simp only [Function.comp_def, finiteCanonical, div_eq_mul_inv, inv_pow] |
| ring |
| · simp only [canonicalFreeEnergy] |
| field_simp |
|
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| theorem concaveOn_canonicalDensity {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {t B d : ℝ} (ht : 1 < t) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (hcore : ∀ r, 0 < r → φ r ≤ coreProfile t r) (hd : 0 ≤ d) : |
| ConcaveOn ℝ (Ioi 0) (fun β => canonicalDensity φ β d) := by |
| let : Fact (0 ≤ d) := ⟨hd⟩ |
| refine ⟨convex_Ioi _, ?_⟩ |
| intro β hβ γ hγ a b ha hb hab |
| simp only [smul_eq_mul] |
| have hc : 0 < a * β + b * γ := by |
| have hh := (convex_Ioi (0 : ℝ)) hβ hγ ha hb hab |
| exact hh |
| have hβlim := (finiteCanonical_tendsto hφ ht hβ.le hstable hcore hd) |
| have hγlim := (finiteCanonical_tendsto hφ ht hγ.le hstable hcore hd) |
| have hclim := (finiteCanonical_tendsto hφ ht hc.le hstable hcore hd) |
| apply le_of_tendsto_of_tendsto ((hβlim.const_mul a).add (hγlim.const_mul b)) hclim |
| filter_upwards [(densityFilter_length d).eventually (eventually_gt_atTop 0)] with p hp |
| have hh := (convexOn_log_partition hφ hp hstable p.2).2 hβ hγ ha hb hab |
| simp only [smul_eq_mul] at hh |
| dsimp [finiteCanonical] |
| have hden : 0 < p.1 ^ 3 := pow_pos hp 3 |
| rw [← mul_div_assoc, ← mul_div_assoc, ← add_div] |
| exact div_le_div_of_nonneg_right (by linarith only [hh]) hden.le |
|
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| end ContinuumTemperature |
| end |
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| end OAI |
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