| import OAI.Probability.RadialTransition.BoltzmannMeasure |
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| namespace OAI |
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| universe uX uI |
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| noncomputable section |
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| open MeasureTheory Filter Set |
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| open scoped ContDiff BigOperators NNReal ENNReal |
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| namespace RadialTransition |
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| lemma cell_young {D G n : ℝ} (hD : 0 < D) : n * G ≤ D / 4 * n^2 + G^2 / D := by |
| have hh := sq_nonneg (D*n - 2*G) |
| have he : G^2 / D * D = G^2 := div_mul_cancel₀ _ hD.ne' |
| nlinarith |
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| theorem one_cell_exponential_moment {X : Type uX} [TopologicalSpace X] [MeasurableSpace X] |
| [BorelSpace X] [SecondCountableTopology X] (ν : Measure X) [IsFiniteMeasure ν] |
| (z : ℝ) {β D B J G : ℝ} (hβ : 0 ≤ β) (hD : 0 < D) (hJ : J ≤ D) |
| {H : ParticleState X → ℝ} (hH : Measurable H) |
| (henergy : ∀ x, (D-J/2) * (particleCount x : ℝ)^2 - B*particleCount x - |
| (particleCount x : ℝ)*G ≤ H x - H particleVacuum) : |
| (∫⁻ x, ENNReal.ofReal (Real.exp (β*D/4*(particleCount x : ℝ)^2 - β*H x)) ∂particleBase ν z) ≤ |
| ENNReal.ofReal (Real.exp ((ν univ).toReal * Real.exp (z+β*B))) * |
| ENNReal.ofReal (Real.exp (β*G^2/D)) * |
| ∫⁻ x, ENNReal.ofReal (Real.exp (-β*H x)) ∂particleBase ν z := by |
| let A := Real.exp (β*G^2/D - β*H particleVacuum) |
| have hm : Measurable (fun x => ENNReal.ofReal (Real.exp (-β*H x))) := |
| (Real.measurable_exp.comp (measurable_const.mul hH)).ennreal_ofReal |
| have hlower : ENNReal.ofReal (Real.exp (-β*H particleVacuum)) ≤ |
| ∫⁻ x, ENNReal.ofReal (Real.exp (-β*H x)) ∂particleBase ν z := by |
| calc |
| _ = ∫⁻ x, ENNReal.ofReal (Real.exp (-β*H x)) ∂Measure.dirac particleVacuum := |
| (lintegral_dirac' _ hm).symm |
| _ ≤ _ := lintegral_mono' (dirac_vacuum_le_particleBase ν z) le_rfl |
| have hupper : (∫⁻ x, ENNReal.ofReal (Real.exp (β*D/4*(particleCount x : ℝ)^2 - β*H x)) ∂particleBase ν z) ≤ |
| ENNReal.ofReal A * ENNReal.ofReal (Real.exp ((ν univ).toReal * Real.exp (z+β*B))) := by |
| calc |
| _ ≤ ∫⁻ x, ENNReal.ofReal A * ENNReal.ofReal (Real.exp (β*B*particleCount x)) ∂particleBase ν z := by |
| apply lintegral_mono |
| intro x |
| dsimp only [A] |
| rw [← ENNReal.ofReal_mul (Real.exp_pos _).le, ← Real.exp_add] |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| have hy := cell_young (G := G) (n := (particleCount x : ℝ)) hD |
| have hj := mul_nonneg (sub_nonneg.mpr hJ) (sq_nonneg (particleCount x : ℝ)) |
| have he := mul_le_mul_of_nonneg_left (henergy x) hβ |
| have hy' := mul_le_mul_of_nonneg_left hy hβ |
| have hj' := mul_nonneg hβ hj |
| have heq : β * (G^2/D) = β*G^2/D := by ring |
| nlinarith |
| _ = _ := by rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top, lintegral_exp_particleCount] |
| calc |
| _ ≤ _ := hupper |
| _ = ENNReal.ofReal (Real.exp ((ν univ).toReal * Real.exp (z+β*B))) * |
| ENNReal.ofReal (Real.exp (β*G^2/D)) * ENNReal.ofReal (Real.exp (-β*H particleVacuum)) := by |
| dsimp [A] |
| rw [sub_eq_add_neg, Real.exp_add, ENNReal.ofReal_mul (Real.exp_pos _).le] |
| rw [neg_mul] |
| ac_rfl |
| _ ≤ _ := mul_le_mul' le_rfl hlower |
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| section ProductCells |
| variable {I : Type uI} [Fintype I] {X : I → Type uX} |
| [∀ i, TopologicalSpace (X i)] [∀ i, MeasurableSpace (X i)] |
| [∀ i, BorelSpace (X i)] [∀ i, SecondCountableTopology (X i)] |
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| def cellN (i : I) (x : ∀ j, ParticleState (X j)) : ℝ := particleCount (x i) |
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| omit [Fintype I] [∀ i, BorelSpace (X i)] [∀ i, SecondCountableTopology (X i)] |
| [∀ i, TopologicalSpace (X i)] [∀ i, MeasurableSpace (X i)] in |
| theorem cellN_nonneg (i : I) (x : ∀ j, ParticleState (X j)) : 0 ≤ cellN i x := Nat.cast_nonneg _ |
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| omit [Fintype I] [∀ i, BorelSpace (X i)] [∀ i, SecondCountableTopology (X i)] |
| [∀ i, MeasurableSpace (X i)] in |
| theorem measurable_cellN (i : I) : Measurable (cellN (X := X) i) := |
| ((show Measurable (fun n : ℕ => (n : ℝ)) from continuous_of_discreteTopology.measurable).comp |
| continuous_particleCount.measurable).comp (measurable_pi_apply i) |
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| theorem integrable_exp_cell_total (ν : ∀ i, Measure (X i)) [∀ i, IsFiniteMeasure (ν i)] |
| (z c : ℝ) : Integrable (fun x => Real.exp (c * ∑ i, cellN i x)) |
| (Measure.pi (fun i => particleBase (ν i) z)) := by |
| have hh := Integrable.fintype_prod_dep (fun i => integrable_exp_particleCount (ν i) z c) |
| convert hh using 1 |
| ext x |
| rw [Finset.mul_sum, Real.exp_sum] |
| rfl |
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| theorem cell_partition_pos (ν : ∀ i, Measure (X i)) [∀ i, IsFiniteMeasure (ν i)] |
| (z : ℝ) {E : (∀ i, ParticleState (X i)) → ℝ} |
| (hi : Integrable (fun x => Real.exp (-E x)) (Measure.pi (fun i => particleBase (ν i) z))) : |
| 0 < boltzmannPartition (Measure.pi (fun i => particleBase (ν i) z)) E := by |
| apply (integral_pos_iff_support_of_nonneg (fun _ => (Real.exp_pos _).le) hi).mpr |
| have hs : Function.support (fun x => Real.exp (-E x)) = univ := by |
| ext x |
| simp [Function.support, Real.exp_ne_zero] |
| rw [hs] |
| rw [Measure.pi_univ] |
| apply pos_iff_ne_zero.mpr |
| apply Finset.prod_ne_zero_iff.mpr |
| intro i _ |
| rw [particleBase_univ (measure_ne_top (ν i) univ)] |
| exact ENNReal.ofReal_ne_zero_iff.mpr (Real.exp_pos _) |
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| theorem weighted_square_le {I : Type uI} [Fintype I] (w n : I → ℝ) |
| (hw : ∀ i, 0 ≤ w i) (hrow : ∑ i, w i ≤ 1) : |
| (∑ i, w i * n i)^2 ≤ ∑ i, w i * (n i)^2 := by |
| have hh := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ |
| (fun i _ => hw i) (fun i _ => mul_nonneg (hw i) (sq_nonneg (n i))) |
| (fun i (_ : i ∈ Finset.univ) => (show (w i * n i)^2 ≤ w i * (w i * (n i)^2) by nlinarith)) |
| exact hh.trans (mul_le_of_le_one_left (Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (sq_nonneg (n i)))) hrow) |
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| theorem integrable_cell_numerator (ν : ∀ i, Measure (X i)) [∀ i, IsFiniteMeasure (ν i)] |
| (z : ℝ) {D B : ℝ} {E f : (∀ i, ParticleState (X i)) → ℝ} |
| (hE : Measurable E) (hf : Measurable f) |
| (hstable : ∀ x, D/2 * ∑ i, (cellN i x)^2 - B * ∑ i, cellN i x ≤ E x) |
| (hfbound : ∀ x, f x ≤ D/2 * ∑ i, (cellN i x)^2) : |
| Integrable (fun x => Real.exp (f x - E x)) (Measure.pi (fun i => particleBase (ν i) z)) := by |
| apply (integrable_exp_cell_total ν z B).mono' (Real.measurable_exp.comp (hf.sub hE)).aestronglyMeasurable |
| filter_upwards [] with x |
| change |Real.exp (f x - E x)| ≤ Real.exp (B * ∑ i, cellN i x) |
| rw [abs_of_pos (Real.exp_pos _)] |
| exact Real.exp_le_exp.mpr (by linarith [hstable x, hfbound x]) |
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| theorem uniform_cell_exponential_moment [Nonempty I] [DecidableEq I] |
| (ν : ∀ i, Measure (X i)) [∀ i, IsFiniteMeasure (ν i)] |
| (z : ℝ) {D B V : ℝ} (hD : 0 < D) (hV : 0 ≤ V) |
| (hvol : ∀ i, (ν i univ).toReal ≤ V) |
| (θ : I → I → ℝ) (hθ : ∀ i j, 0 ≤ θ i j) (hθdiag : ∀ i, θ i i = 0) |
| (hrow : ∀ i, ∑ j, θ i j ≤ 1) |
| {E : (∀ i, ParticleState (X i)) → ℝ} (hE : Measurable E) |
| (hstable : ∀ x, D/2 * ∑ i, (cellN i x)^2 - B * ∑ i, cellN i x ≤ E x) |
| (hconditional : ∀ i x s, D/2 * (particleCount s : ℝ)^2 - B * particleCount s - |
| (particleCount s : ℝ) * (D/8 * ∑ j, θ i j * cellN j x) ≤ |
| E (Function.update x i s) - E (Function.update x i particleVacuum)) : |
| ∀ i, (∫ x, Real.exp (D/4 * (cellN i x)^2) |
| ∂boltzmannMeasure (Measure.pi (fun j => particleBase (ν j) z)) E) ≤ |
| Real.exp (16/15 * V * Real.exp (z+B)) := by |
| classical |
| let μ := Measure.pi (fun j => particleBase (ν j) z) |
| let F (i : I) (x : ∀ j, ParticleState (X j)) := D/4 * (cellN i x)^2 |
| let g (i : I) (x : ∀ j, ParticleState (X j)) := (1/16:ℝ) * ∑ j, θ i j * F j x |
| let C := Real.exp (V * Real.exp (z+B)) |
| have hFmeas (i : I) : Measurable (F i) := measurable_const.mul ((measurable_cellN i).pow_const 2) |
| have hF0 (i : I) (x : ∀ j, ParticleState (X j)) : 0 ≤ F i x := mul_nonneg (by positivity) (sq_nonneg _) |
| have hθ1 (i j : I) : θ i j ≤ 1 := |
| (Finset.single_le_sum (fun k _ => hθ i k) (Finset.mem_univ j)).trans (hrow i) |
| have hFbound (i : I) (x : ∀ j, ParticleState (X j)) : F i x ≤ D/2 * ∑ j, (cellN j x)^2 := by |
| have hh := Finset.single_le_sum (s := Finset.univ) (f := fun j => (cellN j x)^2) |
| (fun _ _ => sq_nonneg _) (Finset.mem_univ i) |
| dsimp [F] |
| nlinarith [sq_nonneg (cellN i x)] |
| have hgmeas (i : I) : Measurable (g i) := |
| measurable_const.mul (Finset.measurable_sum _ (fun j _ => measurable_const.mul (hFmeas j))) |
| have hgBound (i : I) (x : ∀ j, ParticleState (X j)) : g i x ≤ D/2 * ∑ j, (cellN j x)^2 := by |
| have hh := Finset.sum_le_sum (s := Finset.univ) |
| (fun j (_ : j ∈ Finset.univ) => mul_le_of_le_one_left (hF0 j x) (hθ1 i j)) |
| have hsum : ∑ j, F j x = D/4 * ∑ j, (cellN j x)^2 := by simp only [F, Finset.mul_sum] |
| have hn : 0 ≤ ∑ j, (cellN j x)^2 := Finset.sum_nonneg (fun _ _ => sq_nonneg _) |
| dsimp only [g] |
| rw [hsum] at hh |
| nlinarith |
| have hiE : Integrable (fun x => Real.exp (-E x)) μ := by |
| simpa using integrable_cell_numerator ν z hE measurable_const hstable |
| (f := fun _ => 0) (fun x => mul_nonneg (by positivity) (Finset.sum_nonneg (fun _ _ => sq_nonneg _))) |
| have hp := cell_partition_pos ν z hiE |
| let := boltzmannMeasure_probability hiE hp |
| have hiF (i : I) := integrable_cell_numerator ν z hE (hFmeas i) hstable (hFbound i) |
| have hig (i : I) := integrable_cell_numerator ν z hE (hgmeas i) hstable (hgBound i) |
| have hif (i : I) : Integrable (fun x => Real.exp (F i x)) (boltzmannMeasure μ E) := by |
| apply integrable_boltzmann hE _ hp |
| convert hiF i using 1 |
| ext x |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
| have hC1 : 1 ≤ C := Real.one_le_exp_iff.mpr (mul_nonneg hV (Real.exp_pos _).le) |
| have hint (i : I) : (∫ x, Real.exp (F i x) ∂boltzmannMeasure μ E) ≤ |
| C*(∫ x, Real.exp (g i x) ∂boltzmannMeasure μ E) := by |
| have hfmeas : Measurable (fun x => ENNReal.ofReal (Real.exp (F i x - E x))) := |
| (Real.measurable_exp.comp ((hFmeas i).sub hE)).ennreal_ofReal |
| have hgmeas' : Measurable (fun x => ENNReal.ofReal C * ENNReal.ofReal (Real.exp (g i x - E x))) := |
| measurable_const.mul (Real.measurable_exp.comp ((hgmeas i).sub hE)).ennreal_ofReal |
| have hlocal : lmarginal (fun j => particleBase (ν j) z) {i} |
| (fun x => ENNReal.ofReal (Real.exp (F i x - E x))) ≤ |
| lmarginal (fun j => particleBase (ν j) z) {i} |
| (fun x => ENNReal.ofReal C * ENNReal.ofReal (Real.exp (g i x - E x))) := by |
| rw [lmarginal_singleton, lmarginal_singleton] |
| intro x |
| have hginv (s : ParticleState (X i)) : g i (Function.update x i s) = g i x := by |
| dsimp only [g, F, cellN] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro j _ |
| by_cases hj : j = i |
| · subst j |
| simp [hθdiag] |
| · simp [Function.update_of_ne hj] |
| have hcur := one_cell_exponential_moment (ν i) z (β := 1) (D := D) (B := B) (J := D) |
| (G := D/8 * ∑ j, θ i j * cellN j x) (by norm_num) hD le_rfl |
| (H := fun s => E (Function.update x i s)) |
| (hE.comp (measurable_update x (a := i))) (fun s => by |
| convert hconditional i x s using 1; ring) |
| have hG : (D/8 * ∑ j, θ i j * cellN j x)^2 / D ≤ g i x := by |
| have hh := weighted_square_le (θ i) (fun j => cellN j x) (hθ i) (hrow i) |
| have hs : ∑ j, θ i j * F j x = D/4 * ∑ j, θ i j * (cellN j x)^2 := by |
| dsimp only [F] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| dsimp only [g] |
| rw [hs] |
| apply (div_le_iff₀ hD).mpr |
| nlinarith [mul_nonneg (sq_nonneg D) (sub_nonneg.mpr hh)] |
| have hCi : Real.exp ((ν i univ).toReal * Real.exp (z+B)) ≤ C := |
| Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right (hvol i) (Real.exp_pos _).le) |
| simp only [one_mul, neg_one_mul] at hcur |
| calc |
| _ = ∫⁻ s, ENNReal.ofReal (Real.exp (D/4*(particleCount s : ℝ)^2 - |
| E (Function.update x i s))) ∂particleBase (ν i) z := by |
| apply lintegral_congr |
| intro s |
| simp only [F, cellN, Function.update_self] |
| _ ≤ _ := hcur |
| _ ≤ ENNReal.ofReal C * ENNReal.ofReal (Real.exp (g i x)) * |
| ∫⁻ s, ENNReal.ofReal (Real.exp (-E (Function.update x i s))) ∂particleBase (ν i) z := by |
| gcongr |
| _ = _ := by |
| rw [mul_assoc, ← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top, |
| ← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top] |
| apply lintegral_congr |
| intro s |
| rw [hginv, Real.exp_sub] |
| simp only [div_eq_mul_inv, Real.exp_neg, ENNReal.ofReal_mul (Real.exp_pos _).le] |
| have htotal := lintegral_le_of_lmarginal_le {i} hfmeas hgmeas' hlocal |
| rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top, |
| ← ofReal_integral_eq_lintegral_ofReal (hiF i) (Eventually.of_forall (fun _ => (Real.exp_pos _).le)), |
| ← ofReal_integral_eq_lintegral_ofReal (hig i) (Eventually.of_forall (fun _ => (Real.exp_pos _).le)), |
| ← ENNReal.ofReal_mul (by linarith [hC1])] at htotal |
| have hr := (ENNReal.ofReal_le_ofReal_iff (mul_nonneg (by linarith [hC1]) |
| (integral_nonneg (fun _ => (Real.exp_pos _).le)))).mp htotal |
| rw [integral_boltzmann hE, integral_boltzmann hE] |
| simp_rw [← Real.exp_add, neg_add_eq_sub] |
| have hh := mul_le_mul_of_nonneg_left hr (inv_nonneg.mpr hp.le) |
| nlinarith |
| have hc := occupation_moment_closure F θ hC1 (by norm_num : 0 ≤ (1/16:ℝ)) |
| (by norm_num : (1/16:ℝ) < 1) hFmeas hF0 hif hθ hrow hint |
| intro i |
| convert hc i using 1 |
| congr 1 |
| dsimp [C] |
| rw [Real.log_exp] |
| ring |
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| end ProductCells |
| end RadialTransition |
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| end |
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| end OAI |
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