openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /FiniteDepthKernelOrder.lean
| import OAI.MathematicalPhysics.Boltzmann.Collision.KernelInvariantOrder | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.EntropyIntegralBalance | |
| import Mathlib.MeasureTheory.Integral.Prod | |
| namespace OAI | |
| section | |
| /-! # Recovering the finite absolute-entropy datum class from relative entropy -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness | |
| theorem FiniteEntropyDatum.of_relativeEntropy_integrable {f : Density} | |
| (hn : ∀ᵐ z, 0 ≤ f z) (hf : Integrable f) | |
| (hE : Integrable (fun z => ‖z.2‖^2*f z)) | |
| (hH : Integrable (fun z => scalarRelativeEntropy (maxwellian z.2) (f z))) : | |
| FiniteEntropyDatum f := by | |
| have hlog : Integrable (fun z => f z*Real.log (f z)) := by | |
| have h := ((hH.add (hf.const_mul (1-(3/2:ℝ)*Real.log (2*Real.pi)))).sub | |
| (hE.const_mul (1/2:ℝ))).sub integrable_maxwellian_phase | |
| apply h.congr | |
| filter_upwards [] with z | |
| simp only [Pi.sub_apply,Pi.add_apply,scalarRelativeEntropy,log_maxwellian] | |
| ring | |
| refine ⟨hn,hf.aestronglyMeasurable,?_⟩ | |
| apply ((hf.add hE).add hlog.norm).congr | |
| filter_upwards [hn] with z hz | |
| simp only [Pi.add_apply,Real.norm_eq_abs,abs_mul,abs_of_nonneg hz,entropyMoment] | |
| ring | |
| theorem bounded_finiteEntropyDatum {q A p : ℝ} {f : Density} | |
| (hq : q < 2) (hA : 0 ≤ A) (hp : 1 < p) | |
| (hpi : Integrable (fun v : Velocity => H q v^p)) | |
| (hfm : Measurable f) (hfn : ∀ z, 0 ≤ f z) (hfb : ∀ z, f z ≤ A*H q z.2) : | |
| FiniteEntropyDatum f := by | |
| have hfi : Integrable f := by | |
| apply (((integrable_H hq).const_mul A).comp_snd (volume : Measure Position)).mono' | |
| hfm.aestronglyMeasurable | |
| filter_upwards [] with z | |
| rw [Real.norm_of_nonneg (hfn z)] | |
| exact hfb z | |
| have hEi : Integrable (fun z => ‖z.2‖^2*f z) := by | |
| have hmajor : Integrable (fun z : Phase => A*(‖z.2‖^2*H q z.2)) := by | |
| rw [Measure.volume_eq_prod] | |
| simpa only [Nat.reduceMul] using | |
| ((integrable_even_moment_H hq 1).const_mul A).comp_snd (volume : Measure Position) | |
| apply hmajor.mono' (((continuous_snd.norm.pow 2).measurable).mul hfm).aestronglyMeasurable | |
| filter_upwards [] with z | |
| simp only [Pi.mul_apply,Pi.pow_apply] | |
| rw [Real.norm_of_nonneg (mul_nonneg (sq_nonneg _) (hfn z))] | |
| exact (mul_le_mul_of_nonneg_left (hfb z) (sq_nonneg _)).trans_eq (by ring) | |
| have hHi : Integrable (fun z => scalarRelativeEntropy (maxwellian z.2) (f z)) := by | |
| rw [Measure.volume_eq_prod] | |
| exact bounded_relative_entropy_integrable hq hA hp hpi hfm hfn hfb | |
| exact FiniteEntropyDatum.of_relativeEntropy_integrable (Filter.Eventually.of_forall hfn) hfi hEi hHi | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| section | |
| /-! # Finite sums preserve convergence in integral L1 -/ | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology | |
| namespace BoltzmannNonuniqueness.Compactness | |
| theorem finite_integral_L1_zero {X ι : Type*} [MeasurableSpace X] | |
| {μ : Measure X} (s : Finset ι) {f : ι → ℕ → X → ℝ} | |
| (hi : ∀ i ∈ s, ∀ n, Integrable (f i n) μ) | |
| (hl : ∀ i ∈ s, Tendsto (fun n => ∫ x, |f i n x| ∂μ) atTop (𝓝 0)) : | |
| Tendsto (fun n => ∫ x, |∑ i ∈ s, f i n x| ∂μ) atTop (𝓝 0) := by | |
| have hb := tendsto_finsetSum s hl | |
| simp only [Finset.sum_const_zero] at hb | |
| apply squeeze_zero (fun _ => integral_nonneg (fun _ => abs_nonneg _)) _ hb | |
| intro n | |
| calc | |
| _ ≤ ∫ x, ∑ i ∈ s, |f i n x| ∂μ := by | |
| apply integral_mono_of_nonneg (ae_of_all _ (fun _ => abs_nonneg _)) | |
| (integrable_finsetSum s (fun i hi' => (hi i hi' n).abs)) | |
| exact ae_of_all _ (fun x => Finset.abs_sum_le_sum_abs (fun i => f i n x) s) | |
| _ = _ := integral_finsetSum s (fun i hi' => (hi i hi' n).abs) | |
| end BoltzmannNonuniqueness.Compactness | |
| end | |
| end | |
| section | |
| /-! # Finite history comparison on a depth-dependent reachable region -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ProbabilityTheory | |
| namespace BoltzmannNonuniqueness | |
| variable {X : Type*} [MeasurableSpace X] | |
| theorem kernel_pow_le_on_depth (K L : Kernel X X) (D : ℕ → Set X) (N : ℕ) | |
| (hstep : ∀ n < N, ∀ x ∈ D (n+1), ∀ᵐ y ∂K x, y ∈ D n) | |
| (hle : ∀ n < N, ∀ x ∈ D (n+1), K x ≤ L x) : | |
| ∀ n ≤ N, ∀ x ∈ D n, (K^n) x ≤ (L^n) x := by | |
| intro n hn | |
| induction n with | |
| | zero => intro x hx; exact le_rfl | |
| | succ n ih => | |
| intro x hx | |
| have hnN : n < N := Nat.lt_of_succ_le hn | |
| rw [show K^(n+1) = (K^n) ∘ₖ K by simpa only [pow_one] using Kernel.pow_add K n 1, | |
| show L^(n+1) = (L^n) ∘ₖ L by simpa only [pow_one] using Kernel.pow_add L n 1] | |
| apply KernelHistoryOrder.measure_comp_mono_ae _ (hle n hnN x hx) | |
| filter_upwards [hstep n hnN x hx] with y hy | |
| exact ih (Nat.le_of_lt hnN) y hy | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| end OAI | |