openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Collision /CollisionComparisonError.lean
| import OAI.MathematicalPhysics.Boltzmann.Compactness.PeriodicCharacteristicGainUnit | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.PeriodicCharacteristicGainUnit | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.L1VelocityTranslationApproximation | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.PhysicalFrequencyWindow | |
| namespace OAI | |
| /-! # The four physical errors in positive characteristic comparison -/ | |
| noncomputable section | |
| open MeasureTheory | |
| namespace BoltzmannNonuniqueness.PeriodicApproximation | |
| open CollisionCompactness | |
| variable {X : Type*} [MeasurableSpace X] | |
| def collisionComparisonError (N M : ℝ) (F H : X × Velocity → ℝ) (p : X × Velocity) : ℝ := | |
| |(gain N F p-gain M H p)/(1+F p+H p)|+ | |
| |(lossDensity F p-loss N F p)/(1+F p)|+ | |
| |(lossDensity H p-loss M H p)/(1+H p)|+ | |
| |kineticFrequency F p-kineticFrequency H p| | |
| theorem collisionComparisonError_measurable {N M : ℝ} {F H : X × Velocity → ℝ} | |
| (hF : Measurable F) (hH : Measurable H) : Measurable (collisionComparisonError N M F H) := by | |
| exact (((((gain_measurable N hF).sub (gain_measurable M hH)).div | |
| ((measurable_const.add hF).add hH)).abs.add | |
| (((lossDensity_measurable hF).sub (loss_measurable N hF)).div (measurable_const.add hF)).abs).add | |
| (((lossDensity_measurable hH).sub (loss_measurable M hH)).div (measurable_const.add hH)).abs).add | |
| ((kineticFrequency_measurable hF).sub (kineticFrequency_measurable hH)).abs | |
| theorem collisionComparisonError_nonneg (N M : ℝ) (F H : X × Velocity → ℝ) (p : X × Velocity) : | |
| 0 ≤ collisionComparisonError N M F H p := by | |
| unfold collisionComparisonError | |
| positivity | |
| end BoltzmannNonuniqueness.PeriodicApproximation | |
| end | |
| end OAI | |