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f18e7de | 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 | 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
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