| import OAI.Probability.ContinuumTransition.CubeCoercivity |
|
|
| namespace OAI |
|
|
| universe u_1 |
|
|
| section |
| open scoped Topology BigOperators ENNReal |
| open Filter MeasureTheory Set |
|
|
| namespace ContinuumTemperature |
|
|
| @[simp] theorem radialPotential_ne_bot (φ : ℝ → ℝ) (x : Space) : |
| radialPotential φ x ≠ ⊥ := by |
| unfold radialPotential |
| split_ifs |
| · exact top_ne_bot |
| · exact EReal.coe_ne_bot _ |
|
|
| theorem ereal_sum_ne_bot {α : Type u_1} (s : Finset α) (f : α → EReal) |
| (hf : ∀ i ∈ s, f i ≠ ⊥) : ∑ i ∈ s, f i ≠ ⊥ := by |
| classical |
| induction s using Finset.induction_on with |
| | empty => simp |
| | @insert i s hi ih => |
| rw [Finset.sum_insert hi, EReal.add_ne_bot_iff] |
| exact ⟨hf i (Finset.mem_insert_self i s), ih (fun j hj => hf j (Finset.mem_insert_of_mem hj))⟩ |
|
|
| theorem measurable_energy {φ : ℝ → ℝ} (hφ : Measurable (radialPotential φ)) |
| (N : ℕ) : Measurable (fun x : Fin N → Space => energy φ x) := by |
| classical |
| unfold energy |
| fun_prop |
|
|
| theorem measurable_boltzmann {φ : ℝ → ℝ} (hφ : Measurable (radialPotential φ)) |
| (β : ℝ) (N : ℕ) : Measurable (fun x : Fin N → Space => boltzmann φ β x) := by |
| have hE := measurable_energy hφ N |
| exact Measurable.ite (hE (measurableSet_singleton ⊤)) measurable_const |
| ((measurable_const.mul hE.ereal_toReal).exp) |
|
|
| theorem boltzmann_nonneg (φ : ℝ → ℝ) (β : ℝ) {N : ℕ} (x : Fin N → Space) : |
| 0 ≤ boltzmann φ β x := by |
| unfold boltzmann |
| split_ifs |
| · rfl |
| · exact (Real.exp_pos _).le |
|
|
| theorem boltzmann_le_of_stable {φ : ℝ → ℝ} {β B : ℝ} (hβ : 0 ≤ β) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| {N : ℕ} (x : Fin N → Space) : |
| boltzmann φ β x ≤ Real.exp (β * B * N) := by |
| unfold boltzmann |
| split_ifs with htop |
| · exact (Real.exp_pos _).le |
| · apply Real.exp_le_exp.mpr |
| have he := EReal.toReal_le_toReal (hstable N x) (EReal.coe_ne_bot _) htop |
| simp only [EReal.toReal_coe] at he |
| nlinarith [mul_le_mul_of_nonneg_left he hβ] |
|
|
| theorem isCompact_cube (L : ℝ) : IsCompact (cube L) := by |
| have h := isCompact_pi_infinite (fun _ : Fin 3 => isCompact_Icc (a := -L / 2) (b := L / 2)) |
| exact (PiLp.homeomorph 2 (fun _ : Fin 3 => ℝ)).isCompact_preimage.mpr h |
|
|
| theorem measurableSet_cube (L : ℝ) : MeasurableSet (cube L) := |
| (isCompact_cube L).measurableSet |
|
|
| theorem isCompact_configurations (L : ℝ) (N : ℕ) : |
| IsCompact {x : Fin N → Space | ∀ i, x i ∈ cube L} := |
| isCompact_pi_infinite (fun _ => isCompact_cube L) |
|
|
| theorem integrableOn_boltzmann {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) |
| (L : ℝ) (N : ℕ) : |
| IntegrableOn (fun x : Fin N → Space => boltzmann φ β x) |
| {x : Fin N → Space | ∀ i, x i ∈ cube L} := by |
| have hK := isCompact_configurations L N |
| have : IsFiniteMeasure (volume.restrict {x : Fin N → Space | ∀ i, x i ∈ cube L}) := |
| ⟨by simpa only [Measure.restrict_apply_univ] using hK.measure_lt_top (μ := volume)⟩ |
| apply (integrable_const (Real.exp (β * B * N))).mono' |
| (measurable_boltzmann hφ β N).aestronglyMeasurable |
| filter_upwards [] with x |
| rw [Real.norm_eq_abs, abs_of_nonneg (boltzmann_nonneg φ β x)] |
| exact boltzmann_le_of_stable hβ hstable x |
|
|
| theorem energy_eq_real_of_injective (φ : ℝ → ℝ) {N : ℕ} {x : Fin N → Space} |
| (hx : Function.Injective x) : |
| energy φ x = ((∑ i : Fin N, ∑ j ∈ Finset.univ.filter (fun j : Fin N => i < j), |
| φ ‖x i - x j‖ : ℝ) : EReal) := by |
| classical |
| unfold energy |
| simp only [ereal_coe_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j hj |
| have hij : i ≠ j := ne_of_lt (Finset.mem_filter.mp hj).2 |
| exact ite_eq_right (sub_ne_zero.mpr (hx.ne hij)) |
|
|
| theorem boltzmann_pos_of_injective (φ : ℝ → ℝ) (β : ℝ) {N : ℕ} |
| {x : Fin N → Space} (hx : Function.Injective x) : 0 < boltzmann φ β x := by |
| unfold boltzmann |
| rw [ite_eq_right (by rw [energy_eq_real_of_injective φ hx]; exact EReal.coe_ne_top _)] |
| exact Real.exp_pos _ |
|
|
| def openConfigurations (L : ℝ) (N : ℕ) : Set (Fin N → Space) := |
| {x | (∀ i k, x i k ∈ Ioo (-L / 2) (L / 2)) ∧ Function.Injective x} |
|
|
| theorem isOpen_openConfigurations (L : ℝ) (N : ℕ) : |
| IsOpen (openConfigurations L N) := by |
| have hbox : IsOpen {x : Fin N → Space | ∀ i k, x i k ∈ Ioo (-L / 2) (L / 2)} := by |
| simp only [ofPred_forall] |
| apply isOpen_iInter_of_finite |
| intro i |
| apply isOpen_iInter_of_finite |
| intro k |
| exact isOpen_Ioo.preimage ((PiLp.continuous_apply 2 (fun _ : Fin 3 => ℝ) k).comp |
| (continuous_apply i)) |
| have hinj : IsOpen {x : Fin N → Space | Function.Injective x} := by |
| have heq : {x : Fin N → Space | Function.Injective x} = |
| ⋂ i, ⋂ j, ⋂ (_ : i ≠ j), {x | x i ≠ x j} := by |
| ext x |
| simp only [mem_ofPred_eq, mem_iInter] |
| exact ⟨fun h i j hij => h.ne hij, fun h i j hij => by_contra fun hij' => h i j hij' hij⟩ |
| rw [heq] |
| apply isOpen_iInter_of_finite |
| intro i |
| apply isOpen_iInter_of_finite |
| intro j |
| apply isOpen_iInter_of_finite |
| intro _ |
| exact (isClosed_eq (continuous_apply i) (continuous_apply j)).isOpen_compl |
| exact hbox.inter hinj |
|
|
| theorem openConfigurations_nonempty {L : ℝ} (hL : 0 < L) (N : ℕ) : |
| (openConfigurations L N).Nonempty := by |
| let c : ℝ := L / (4 * ((N : ℝ) + 1)) |
| have hc : 0 < c := by dsimp [c]; positivity |
| let x : Fin N → Space := fun i => WithLp.toLp 2 (fun _ : Fin 3 => (i : ℝ) * c) |
| refine ⟨x, ?_, ?_⟩ |
| · intro i k |
| change -L / 2 < (i : ℝ) * c ∧ (i : ℝ) * c < L / 2 |
| have hi : (i : ℝ) < N := by exact_mod_cast i.isLt |
| have hinonneg : 0 ≤ (i : ℝ) := by positivity |
| constructor |
| · nlinarith [mul_nonneg hinonneg hc.le] |
| · dsimp [c] |
| rw [← mul_div_assoc, div_lt_iff₀ (by positivity)] |
| nlinarith [mul_pos hL (by exact_mod_cast Nat.zero_lt_succ N : 0 < (N : ℝ) + 1)] |
| · intro i j he |
| have hij := congrArg (fun y : Space => y 0) he |
| change (i : ℝ) * c = (j : ℝ) * c at hij |
| have hij' := mul_right_cancel₀ hc.ne' hij |
| exact Fin.ext (by exact_mod_cast hij') |
|
|
| theorem partition_pos {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) : 0 < partition φ β L N := by |
| classical |
| unfold partition |
| split_ifs |
| · norm_num |
| · apply mul_pos (by positivity) |
| rw [setIntegral_pos_iff_support_of_nonneg_ae |
| (ae_of_all _ (boltzmann_nonneg φ β)) (integrableOn_boltzmann hφ hβ hstable L N)] |
| apply lt_of_lt_of_le ((isOpen_openConfigurations L N).measure_pos volume |
| (openConfigurations_nonempty hL N)) |
| apply measure_mono |
| intro x hx |
| exact ⟨(boltzmann_pos_of_injective φ β hx.2).ne', |
| fun i k => ⟨(hx.1 i k).1.le, (hx.1 i k).2.le⟩⟩ |
|
|
| theorem boltzmann_add (φ : ℝ → ℝ) (β γ : ℝ) {N : ℕ} (x : Fin N → Space) : |
| boltzmann φ (β + γ) x = boltzmann φ β x * boltzmann φ γ x := by |
| unfold boltzmann |
| split_ifs |
| · ring |
| · rw [← Real.exp_add] |
| congr 1 |
| ring |
|
|
| theorem boltzmann_rpow (φ : ℝ → ℝ) (β : ℝ) {p : ℝ} (hp : 0 < p) |
| {N : ℕ} (x : Fin N → Space) : |
| boltzmann φ β x ^ p = boltzmann φ (p * β) x := by |
| unfold boltzmann |
| split_ifs |
| · exact Real.zero_rpow hp.ne' |
| · rw [← Real.exp_mul] |
| congr 1 |
| ring |
|
|
| theorem integral_boltzmann_logconvex {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {B : ℝ} |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (L : ℝ) (N : ℕ) {β γ a b : ℝ} |
| (hβ : 0 < β) (hγ : 0 < γ) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) : |
| (∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ (a * β + b * γ) x) ≤ |
| (∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ β x) ^ a * |
| (∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ γ x) ^ b := by |
| let μ : Measure (Fin N → Space) := |
| volume.restrict {x : Fin N → Space | ∀ i, x i ∈ cube L} |
| have hp (c u : ℝ) (hc : 0 < c) (x : Fin N → Space) : |
| ‖boltzmann φ (c * u) x‖ ^ (1 / c) = boltzmann φ u x := by |
| rw [Real.norm_eq_abs, abs_of_nonneg (boltzmann_nonneg φ (c * u) x), |
| boltzmann_rpow φ (c * u) (one_div_pos.mpr hc)] |
| congr 1 |
| field_simp |
| have hLp (c u : ℝ) (hc : 0 < c) (hu : 0 < u) : |
| MemLp (fun x : Fin N → Space => boltzmann φ (c * u) x) (ENNReal.ofReal (1 / c)) μ := by |
| have A : ENNReal.ofReal (1 / c) ≠ 0 := by positivity |
| have B' : ENNReal.ofReal (1 / c) ≠ ∞ := ENNReal.ofReal_ne_top |
| rw [← memLp_norm_rpow_iff _ A B', ENNReal.toReal_ofReal (by positivity : 0 ≤ 1 / c), |
| ENNReal.div_self A B', memLp_one_iff_integrable] |
| · apply (integrableOn_boltzmann hφ hu.le hstable L N).congr |
| exact ae_of_all _ (fun x => (hp c u hc x).symm) |
| · exact (measurable_boltzmann hφ (c * u) N).aestronglyMeasurable |
| have hHolder := integral_mul_le_Lp_mul_Lq_of_nonneg |
| (Real.holderConjugate_one_div ha hb hab) |
| (ae_of_all μ (boltzmann_nonneg φ (a * β))) |
| (ae_of_all μ (boltzmann_nonneg φ (b * γ))) |
| (hLp a β ha hβ) (hLp b γ hb hγ) |
| simp only [one_div_one_div] at hHolder |
| convert hHolder using 1 |
| · exact integral_congr_ae (ae_of_all μ (boltzmann_add φ (a * β) (b * γ))) |
| · congr 2 |
| · apply integral_congr_ae |
| filter_upwards [] with x |
| simpa only [Real.norm_eq_abs, abs_of_nonneg (boltzmann_nonneg φ (a * β) x)] |
| using (hp a β ha x).symm |
| · apply integral_congr_ae |
| filter_upwards [] with x |
| simpa only [Real.norm_eq_abs, abs_of_nonneg (boltzmann_nonneg φ (b * γ) x)] |
| using (hp b γ hb x).symm |
|
|
| theorem integral_boltzmann_pos {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) : |
| 0 < ∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ β x := by |
| rw [setIntegral_pos_iff_support_of_nonneg_ae |
| (ae_of_all _ (boltzmann_nonneg φ β)) (integrableOn_boltzmann hφ hβ hstable L N)] |
| apply lt_of_lt_of_le ((isOpen_openConfigurations L N).measure_pos volume |
| (openConfigurations_nonempty hL N)) |
| apply measure_mono |
| intro x hx |
| exact ⟨(boltzmann_pos_of_injective φ β hx.2).ne', |
| fun i k => ⟨(hx.1 i k).1.le, (hx.1 i k).2.le⟩⟩ |
|
|
| theorem convexOn_log_integral_boltzmann {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {B L : ℝ} (hL : 0 < L) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (N : ℕ) : ConvexOn ℝ (Ioi 0) (fun β => Real.log |
| (∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ β x)) := by |
| refine convexOn_iff_forall_pos.mpr ⟨convex_Ioi _, ?_⟩ |
| intro β hβ γ hγ a b ha hb hab |
| simp only [smul_eq_mul] |
| have hβ' := integral_boltzmann_pos hφ hβ.le hL hstable N |
| have hγ' := integral_boltzmann_pos hφ hγ.le hL hstable N |
| have hac : 0 < a * β + b * γ := add_pos (mul_pos ha hβ) (mul_pos hb hγ) |
| have hc := integral_boltzmann_pos hφ hac.le hL hstable N |
| have hHolder := integral_boltzmann_logconvex hφ hstable L N hβ hγ ha hb hab |
| calc |
| _ ≤ Real.log ((∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ β x) ^ a * |
| (∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ γ x) ^ b) := |
| Real.log_le_log hc hHolder |
| _ = _ := by rw [Real.log_mul (Real.rpow_pos_of_pos hβ' a).ne' (Real.rpow_pos_of_pos hγ' b).ne', |
| Real.log_rpow hβ', Real.log_rpow hγ'] |
|
|
| theorem convexOn_log_partition {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {B L : ℝ} (hL : 0 < L) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) |
| (N : ℕ) : ConvexOn ℝ (Ioi 0) (fun β => Real.log (partition φ β L N)) := by |
| by_cases hN : N = 0 |
| · simp only [hN, partition, ↓reduceIte, Real.log_one] |
| exact convexOn_const _ (convex_Ioi _) |
| · have hlog (β : ℝ) (hβ : 0 < β) : Real.log (partition φ β L N) = |
| Real.log (1 / (N.factorial : ℝ)) + Real.log |
| (∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ β x) := by |
| unfold partition |
| rw [ite_eq_right hN, Real.log_mul (one_div_ne_zero (Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero N))) |
| (integral_boltzmann_pos hφ hβ.le hL hstable N).ne'] |
| have hc := convexOn_log_integral_boltzmann hφ hL hstable N |
| refine convexOn_iff_forall_pos.mpr ⟨convex_Ioi _, ?_⟩ |
| intro β hβ γ hγ a b ha hb hab |
| simp only [smul_eq_mul] |
| rw [hlog _ (add_pos (mul_pos ha hβ) (mul_pos hb hγ)), hlog β hβ, hlog γ hγ] |
| have hi := hc.2 hβ hγ ha.le hb.le hab |
| simp only [smul_eq_mul] at hi |
| have hcst : a * Real.log (1 / (N.factorial : ℝ)) + |
| b * Real.log (1 / (N.factorial : ℝ)) = Real.log (1 / (N.factorial : ℝ)) := by |
| rw [← add_mul, hab, one_mul] |
| nlinarith only [hi, hcst] |
|
|
| theorem volume_cube (L : ℝ) : volume (cube L) = (ENNReal.ofReal L) ^ 3 := by |
| have heq : cube L = (@WithLp.ofLp 2 (Fin 3 → ℝ)) ⁻¹' |
| (Set.pi Set.univ (fun _ : Fin 3 => Icc (-L / 2) (L / 2))) := by |
| ext x |
| simp only [cube, mem_preimage, mem_ofPred_eq, Set.mem_pi, mem_univ, forall_const] |
| rw [heq, (PiLp.volume_preserving_ofLp (Fin 3)).measure_preimage |
| ((MeasurableSet.pi Set.countable_univ (fun _ _ => measurableSet_Icc)).nullMeasurableSet)] |
| change (Measure.pi (fun _ : Fin 3 => (volume : Measure ℝ))) _ = _ |
| rw [Measure.pi_pi] |
| simp only [Real.volume_Icc] |
| have ha : L / 2 - -L / 2 = L := by ring |
| simp [ha] |
|
|
| theorem volume_configurations (L : ℝ) (N : ℕ) : |
| volume {x : Fin N → Space | ∀ i, x i ∈ cube L} = (ENNReal.ofReal L) ^ (3 * N) := by |
| have heq : {x : Fin N → Space | ∀ i, x i ∈ cube L} = |
| Set.pi Set.univ (fun _ : Fin N => cube L) := by ext x; simp |
| rw [heq] |
| change (Measure.pi (fun _ : Fin N => (volume : Measure Space))) _ = _ |
| rw [Measure.pi_pi] |
| simp only [volume_cube, Finset.prod_const, Finset.card_univ, Fintype.card_fin, ← pow_mul] |
|
|
| theorem partition_nonneg (φ : ℝ → ℝ) (β L : ℝ) (N : ℕ) : |
| 0 ≤ partition φ β L N := by |
| unfold partition |
| split_ifs |
| · norm_num |
| · apply mul_nonneg (by positivity) |
| exact integral_nonneg (boltzmann_nonneg φ β) |
|
|
| theorem partition_le_exp_of_stable {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) : |
| partition φ β L N ≤ (L ^ 3 * Real.exp (β * B)) ^ N / (N.factorial : ℝ) := by |
| by_cases hN : N = 0 |
| · simp [hN, partition] |
| · have : IsFiniteMeasure (volume.restrict {x : Fin N → Space | ∀ i, x i ∈ cube L}) := |
| ⟨by simpa only [Measure.restrict_apply_univ] using |
| (isCompact_configurations L N).measure_lt_top (μ := volume)⟩ |
| have hi := integral_mono_ae (integrableOn_boltzmann hφ hβ hstable L N) |
| (integrable_const (Real.exp (β * B * N))) |
| (ae_of_all _ (boltzmann_le_of_stable hβ hstable (N := N))) |
| rw [integral_const] at hi |
| simp only [Measure.real, Measure.restrict_apply_univ, volume_configurations, |
| ENNReal.toReal_pow, ENNReal.toReal_ofReal hL, smul_eq_mul] at hi |
| unfold partition |
| rw [ite_eq_right hN] |
| have hfac : 0 ≤ (1 / (N.factorial : ℝ)) := by positivity |
| apply (mul_le_mul_of_nonneg_left hi hfac).trans_eq |
| rw [show β * B * (N : ℝ) = (N : ℝ) * (β * B) by ring, Real.exp_nat_mul, |
| mul_pow, pow_mul] |
| ring |
|
|
| end ContinuumTemperature |
| end |
|
|
| end OAI |
|
|
