openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Approximation /CutoffEntropyMeasure.lean
| import OAI.MathematicalPhysics.Boltzmann.Approximation.PositiveComparisonBound | |
| namespace OAI | |
| /-! # The extended nonnegative cutoff dissipation agrees with positive production -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped ENNReal | |
| namespace BoltzmannNonuniqueness.CollisionCompactness | |
| variable {X : Type*} [MeasurableSpace X] | |
| def cutoffExtendedProduction (b : ℝ) (f : X × Velocity → ℝ) | |
| (z : (X × (Velocity × Velocity)) × Sphere) : ℝ≥0∞ := | |
| if ‖z.1.2.1-z.1.2.2‖ < b then | |
| ENNReal.ofReal (collisionKernel z.1.2.1 z.1.2.2 z.2)* | |
| extendedEntropyProduction | |
| (f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2)) | |
| (f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2)) | |
| else 0 | |
| omit [MeasurableSpace X] in | |
| theorem positiveCutoffProduction_formula (b : ℝ) (f : X × Velocity → ℝ) | |
| (hf : ∀ z, 0 < f z) (z : (X × (Velocity × Velocity)) × Sphere) : | |
| positiveCutoffProduction b f z = | |
| if ‖z.1.2.1-z.1.2.2‖ < b then | |
| (1/4:ℝ)*collisionKernel z.1.2.1 z.1.2.2 z.2* | |
| ((f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2)- | |
| f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2))* | |
| Real.log ((f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2))/ | |
| (f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2)))) | |
| else 0 := by | |
| unfold positiveCutoffProduction | |
| rw [collisionJump_log_relative f hf,collisionJump_log_density f hf] | |
| simp only [truncatedPreRate,truncatedPostRate,Set.indicator_apply,Set.mem_ofPred_eq] | |
| by_cases hc : ‖z.1.2.1-z.1.2.2‖ < b | |
| · simp only [hc,↓reduceIte,postRate_formula,preRate] | |
| ring | |
| · simp only [hc,↓reduceIte,sub_self,mul_zero] | |
| omit [MeasurableSpace X] in | |
| theorem ofReal_positiveCutoffProduction (b : ℝ) (f : X × Velocity → ℝ) | |
| (hf : ∀ z, 0 < f z) (z : (X × (Velocity × Velocity)) × Sphere) : | |
| ENNReal.ofReal (positiveCutoffProduction b f z) = (4:ℝ≥0∞)⁻¹*cutoffExtendedProduction b f z := by | |
| rw [positiveCutoffProduction_formula b f hf] | |
| have hA := mul_pos (hf (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)) | |
| (hf (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2)) | |
| have hB := mul_pos (hf (z.1.1,z.1.2.1)) (hf (z.1.1,z.1.2.2)) | |
| by_cases hc : ‖z.1.2.1-z.1.2.2‖ < b | |
| · simp only [cutoffExtendedProduction,hc,↓reduceIte] | |
| have he : extendedEntropyProduction | |
| (f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2)) | |
| (f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2)) = | |
| ENNReal.ofReal ((f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2)- | |
| f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2))* | |
| Real.log ((f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2))/ | |
| (f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2)))) := by | |
| simp [extendedEntropyProduction,hA,hB,hA.ne',hB.ne'] | |
| rw [he] | |
| rw [ENNReal.ofReal_mul (mul_nonneg (by norm_num) (collisionKernel_nonneg _ _ _)), | |
| ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 1/4)] | |
| rw [ENNReal.ofReal_div_of_pos (by norm_num : (0:ℝ) < 4)] | |
| norm_num [mul_assoc] | |
| · simp [hc,cutoffExtendedProduction] | |
| def cutoffExtendedDissipation (μ : Measure X) (b : ℝ) (f : X × Velocity → ℝ) : ℝ≥0∞ := | |
| (4:ℝ≥0∞)⁻¹*∫⁻ z, cutoffExtendedProduction b f z | |
| ∂(μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea | |
| theorem ofReal_integral_positiveCutoffProduction {μ : Measure X} (b : ℝ) | |
| (f : X × Velocity → ℝ) (hf : ∀ z, 0 < f z) | |
| (hi : Integrable (positiveCutoffProduction b f) | |
| ((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea)) : | |
| ENNReal.ofReal (∫ z, positiveCutoffProduction b f z | |
| ∂(μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea) = | |
| cutoffExtendedDissipation μ b f := by | |
| rw [ofReal_integral_eq_lintegral_ofReal hi | |
| (Filter.Eventually.of_forall (positiveCutoffProduction_nonneg b f hf))] | |
| simp_rw [ofReal_positiveCutoffProduction b f hf] | |
| exact lintegral_const_mul' _ _ (by norm_num) | |
| end BoltzmannNonuniqueness.CollisionCompactness | |
| end | |
| end OAI | |