| import OAI.Probability.RadialTransition.ParticleMeasure |
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| namespace OAI |
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| universe uX |
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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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| def boltzmannPartition {X : Type uX} [MeasurableSpace X] (μ : Measure X) (E : X → ℝ) : ℝ := |
| ∫ x, Real.exp (-E x) ∂μ |
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| def boltzmannMeasure {X : Type uX} [MeasurableSpace X] (μ : Measure X) (E : X → ℝ) : Measure X := |
| (ENNReal.ofReal (boltzmannPartition μ E))⁻¹ • μ.withDensity (fun x => ENNReal.ofReal (Real.exp (-E x))) |
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| theorem boltzmannMeasure_probability {X : Type uX} [MeasurableSpace X] {μ : Measure X} |
| {E : X → ℝ} (hi : Integrable (fun x => Real.exp (-E x)) μ) |
| (hp : 0 < boltzmannPartition μ E) : IsProbabilityMeasure (boltzmannMeasure μ E) := by |
| constructor |
| rw [boltzmannMeasure, Measure.smul_apply, withDensity_apply _ MeasurableSet.univ, |
| Measure.restrict_univ, ← ofReal_integral_eq_lintegral_ofReal hi |
| (Eventually.of_forall (fun x => (Real.exp_pos _).le))] |
| change (ENNReal.ofReal (boltzmannPartition μ E))⁻¹ * ENNReal.ofReal (boltzmannPartition μ E) = 1 |
| rw [ENNReal.inv_mul_cancel (ENNReal.ofReal_ne_zero_iff.mpr hp) ENNReal.ofReal_ne_top] |
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| theorem integral_boltzmann {X : Type uX} [MeasurableSpace X] {μ : Measure X} |
| {E : X → ℝ} (hE : Measurable E) (f : X → ℝ) : |
| (∫ x, f x ∂boltzmannMeasure μ E) = |
| (boltzmannPartition μ E)⁻¹ * ∫ x, Real.exp (-E x) * f x ∂μ := by |
| have hw : Measurable (fun x => ENNReal.ofReal (Real.exp (-E x))) := |
| (Real.measurable_exp.comp hE.neg).ennreal_ofReal |
| rw [boltzmannMeasure, integral_smul_measure, |
| integral_withDensity_eq_integral_toReal_smul |
| hw (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))] |
| have hZ : 0 ≤ boltzmannPartition μ E := integral_nonneg (fun _ => (Real.exp_pos _).le) |
| simp only [ENNReal.toReal_inv, ENNReal.toReal_ofReal hZ, |
| ENNReal.toReal_ofReal (Real.exp_pos _).le, smul_eq_mul] |
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| theorem integrable_boltzmann {X : Type uX} [MeasurableSpace X] {μ : Measure X} |
| {E f : X → ℝ} (hE : Measurable E) |
| (hi : Integrable (fun x => Real.exp (-E x) * f x) μ) |
| (hp : 0 < boltzmannPartition μ E) : Integrable f (boltzmannMeasure μ E) := by |
| have hw : Measurable (fun x => ENNReal.ofReal (Real.exp (-E x))) := |
| (Real.measurable_exp.comp hE.neg).ennreal_ofReal |
| apply Integrable.smul_measure |
| · apply (integrable_withDensity_iff_integrable_smul' |
| hw |
| (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))).mpr |
| simpa only [ENNReal.toReal_ofReal (Real.exp_pos _).le, smul_eq_mul] using hi |
| · exact ENNReal.inv_ne_top.mpr (ENNReal.ofReal_ne_zero_iff.mpr hp) |
| |
| theorem log_boltzmann_difference_le {X : Type uX} [MeasurableSpace X] {μ : Measure X} |
| {E F : X → ℝ} (hE : Measurable E) |
| (hiE : Integrable (fun x => Real.exp (-E x)) μ) |
| (hiF : Integrable (fun x => Real.exp (-F x)) μ) |
| (hpE : 0 < boltzmannPartition μ E) (hpF : 0 < boltzmannPartition μ F) |
| (hi : Integrable (fun x => F x - E x) (boltzmannMeasure μ E)) : |
| Real.log (boltzmannPartition μ E) - Real.log (boltzmannPartition μ F) ≤ |
| ∫ x, F x - E x ∂boltzmannMeasure μ E := by |
| let := boltzmannMeasure_probability hiE hpE |
| have hexp : Integrable (fun x => Real.exp (-(F x - E x))) (boltzmannMeasure μ E) := by |
| apply integrable_boltzmann hE _ hpE |
| convert hiF using 1 |
| ext x |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
| have hj := convexOn_exp.map_integral_le Real.continuous_exp.continuousOn isClosed_univ |
| (Eventually.of_forall (fun x => mem_univ (-(F x - E x)))) hi.neg hexp |
| rw [integral_neg, integral_boltzmann hE (fun x => Real.exp (-(F x - E x)))] at hj |
| have he (x : X) : Real.exp (-E x) * Real.exp (-(F x - E x)) = Real.exp (-F x) := by |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
| simp_rw [he] at hj |
| have hl := Real.log_le_log (Real.exp_pos _) hj |
| change Real.log (Real.exp (-∫ x, F x - E x ∂boltzmannMeasure μ E)) ≤ |
| Real.log ((boltzmannPartition μ E)⁻¹ * boltzmannPartition μ F) at hl |
| rw [Real.log_exp, Real.log_mul (inv_ne_zero hpE.ne') hpF.ne', Real.log_inv] at hl |
| linarith |
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| end RadialTransition |
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| end |
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| end OAI |
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