| import OAI.Analysis.HardSphere.CondensationDefinitions |
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| namespace OAI |
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| section |
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| section |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
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| namespace HardSphere |
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| abbrev HilbertJet (N : ℕ) (L : Length) := |
| PiLp 2 (fun _ : Option (Direction N) ↦ ScalarL2 N L) |
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| def Condensation : Prop := |
| ∃ ε₀ c₀ : ℝ, 0 < ε₀ ∧ 0 < c₀ ∧ |
| ∀ (a ρ : ℝ), 0 < a → 0 < ρ → ρ * a ^ 3 < ε₀ → |
| ∀ (N : ℕ → ℕ) (L : ℕ → Length) (hN : ∀ k, 0 < N k), |
| Tendsto N atTop atTop → |
| Tendsto (fun k ↦ (L k).val) atTop atTop → |
| Tendsto (fun k ↦ (N k : ℝ) / (L k).val ^ 3) atTop (𝓝 ρ) → |
| ∀ (Ψ : (k : ℕ) → Jet (N k) (L k)), |
| (∀ k, IsGroundVector (N k) (L k) a (Ψ k)) → |
| c₀ ≤ Filter.liminf (fun k ↦ occupation (hN k) (Ψ k).1) atTop |
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| lemma scalarL2_norm_exponent : Fact (1 ≤ (2 : ℝ≥0∞)) := inferInstance |
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| attribute [local instance] scalarL2_norm_exponent |
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| abbrev scalarL2_normedAddCommGroup (N : ℕ) (L : Length) : |
| NormedAddCommGroup (ScalarL2 N L) := inferInstance |
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| attribute [local instance] scalarL2_normedAddCommGroup |
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| abbrev scalarL2_addCommMonoid (N : ℕ) (L : Length) : AddCommMonoid (ScalarL2 N L) := |
| (scalarL2_normedAddCommGroup N L).toAddCommMonoid |
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| abbrev scalarL2_innerProductSpace (N : ℕ) (L : Length) : |
| InnerProductSpace ℂ (ScalarL2 N L) := inferInstance |
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| attribute [local instance] scalarL2_innerProductSpace |
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| abbrev scalarL2_module (N : ℕ) (L : Length) : Module ℂ (ScalarL2 N L) := |
| (scalarL2_innerProductSpace N L).toNormedSpace.toModule |
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| attribute [local instance] scalarL2_module |
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| lemma scalarL2_completeSpace (N : ℕ) (L : Length) : CompleteSpace (ScalarL2 N L) := |
| inferInstance |
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| end HardSphere |
| end |
| end |
| end |
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| end OAI |
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