| import Mathlib |
| import OAI.Analysis.HardSphere.Operators.KilledTransitionFormHeat |
| import OAI.Analysis.HardSphere.Collisions.IteratedInsertion |
|
|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| namespace HardSphere |
| open ContinuousLinearMap |
|
|
| lemma formResolvent_norm_le_of_energy {N : ℕ} {L : Length} {a b : ℝ} |
| (hb : 0≤b) (henergy : ∀ u : Jet N L, u∈dirichletDomain N L a → b*‖u.1‖^2≤energy u) : |
| ‖formResolvent N L a‖≤(1+b)⁻¹ := by |
| have hp : 0<1+b := by linarith |
| apply opNorm_le_bound _ (inv_nonneg.mpr hp.le) |
| intro f |
| let g := (scalarEmbedding N L a).adjoint f |
| let u := hilbertJetEquiv N L (g : HilbertJet N L) |
| have hu : u∈dirichletDomain N L a := g.property |
| have he := henergy u hu |
| have hn : ‖g‖^2=‖u.1‖^2+energy u := hilbertJet_norm_sq (g : HilbertJet N L) |
| have hinner : ‖g‖^2=(inner ℂ f u.1).re := by |
| rw [norm_sq_eq_re_inner (𝕜 := ℂ)] |
| exact congrArg Complex.re (adjoint_inner_left (scalarEmbedding N L a) g f) |
| have hbound : (1+b)*‖u.1‖^2≤‖f‖*‖u.1‖ := by |
| have hi := re_inner_le_norm (𝕜 := ℂ) f u.1 |
| change (inner ℂ f u.1).re≤‖f‖*‖u.1‖ at hi |
| nlinarith |
| change ‖u.1‖≤(1+b)⁻¹*‖f‖ |
| by_cases hz : ‖u.1‖=0 |
| · rw [hz]; positivity |
| · have hupos : 0<‖u.1‖ := lt_of_le_of_ne (norm_nonneg _) (Ne.symm hz) |
| rw [← div_eq_inv_mul,le_div_iff₀ hp] |
| nlinarith |
|
|
| local instance scalarOperatorRealAlgebraCoercive (N : ℕ) (L : Length) : |
| Algebra ℝ (ScalarL2 N L →L[ℂ] ScalarL2 N L) := Algebra.restrictScalars ℝ ℂ _ |
|
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| lemma formHeat_norm_le_of_energy {N : ℕ} {L : Length} {a b t : ℝ} (ht : 0<t) |
| (henergy : ∀ u : Jet N L, u∈dirichletDomain N L a → b*‖u.1‖^2≤energy u) : |
| ‖formHeat N L a t‖≤Real.exp (-t*b) := by |
| cases subsingleton_or_nontrivial (ScalarL2 N L) with |
| | inl h => |
| let := h |
| rw [show formHeat N L a t=0 from Subsingleton.elim _ _,norm_zero] |
| exact (Real.exp_pos _).le |
| | inr h => |
| let := h |
| by_cases hb : 0≤b |
| · have hp : 0<1+b := by linarith |
| have hR := formResolvent_norm_le_of_energy hb henergy |
| unfold formHeat |
| apply norm_cfc_le (Real.exp_pos _).le |
| intro x hx |
| have hx0 : 0≤x := spectrum_nonneg_of_nonneg |
| (ContinuousLinearMap.nonneg_iff_isPositive.mpr (formResolvent_isPositive N L a)) hx |
| have hxb : x≤(1+b)⁻¹ := (Real.le_norm_self x).trans ((spectrum.norm_le_norm_of_mem hx).trans hR) |
| rw [Real.norm_eq_abs,abs_of_nonneg (heatScalar_nonneg t x)] |
| rcases eq_or_lt_of_le hx0 with rfl | hx0 |
| · rw [heatScalar_zero]; exact (Real.exp_pos _).le |
| · rw [heatScalar_of_pos ht hx0] |
| apply Real.exp_le_exp.mpr |
| have hxinv : 1+b≤x⁻¹ := by |
| simpa only [inv_inv] using (inv_le_inv₀ (inv_pos.mpr hp) hx0).mpr hxb |
| have hg := mul_le_mul_of_nonneg_left hxinv ht.le |
| rw [div_eq_mul_inv] |
| linarith |
| · have hnorm : ‖formHeat N L a t‖≤1 := by |
| rw [formHeat,← regularHeat_pos (formResolvent N L a) ht] |
| exact regularHeat_contract (formResolvent_isPositive N L a) (formResolvent_norm_le_one N L a) t |
| exact hnorm.trans (Real.one_le_exp (mul_nonneg_of_nonpos_of_nonpos (by linarith) (le_of_not_ge hb))) |
| lemma killedTransition_norm_le_of_energy {N : ℕ} {L : Length} {a b t : ℝ} (ht : 0<t) |
| (henergy : ∀ u : Jet N L, u∈dirichletDomain N L a → b*‖u.1‖^2≤energy u) : |
| ‖killedTransition N L a t‖≤Real.exp (-t*b) := by |
| rw [killedTransition_eq_formHeat N L a ht] |
| exact formHeat_norm_le_of_energy ht henergy |
|
|
| lemma bath_heat_bound : ∃ C : ℝ, 0<C ∧ |
| ∀ (n m : ℕ) (L : Length) (r t : ℝ), 0<r → 0<t → |
| ((n+m:ℕ):ℝ)*(2*Real.pi*r)^3≤L.val^3/2 → |
| ∀ u : Jet (n+m) L, IsGroundVector (n+m) L r u → |
| ‖killedTransition n L r t‖≤ |
| Real.exp (-t*(energy u-C*(m:ℝ)*((n+m:ℕ):ℝ)*r/L.val^3)) := by |
| obtain ⟨C,hC,hrem⟩ := ground_energy_removal |
| exact ⟨C,hC,fun n m L r t hr ht hs u hu ↦ |
| killedTransition_norm_le_of_energy ht (hrem n m L r hr hs u hu)⟩ |
| end HardSphere |
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| end |
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| end OAI |
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