File size: 4,202 Bytes
a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Probability.RadialTransition.ParticleMeasure
namespace OAI
universe uX
noncomputable section
open scoped BigOperators NNReal ContDiff ENNReal
open Filter MeasureTheory Set
namespace RadialTransition
def boltzmannPartition {X : Type uX} [MeasurableSpace X] (μ : Measure X) (E : X → ℝ) : ℝ :=
∫ x, Real.exp (-E x) ∂μ
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)))
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]
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]
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
end RadialTransition
end
end OAI
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