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9eeb9c7 | 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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | import OAI.Analysis.HardSphere.CondensationDefinitions
namespace OAI
section
section
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators
namespace HardSphere
abbrev HilbertJet (N : ℕ) (L : Length) :=
PiLp 2 (fun _ : Option (Direction N) ↦ ScalarL2 N L)
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
lemma scalarL2_norm_exponent : Fact (1 ≤ (2 : ℝ≥0∞)) := inferInstance
attribute [local instance] scalarL2_norm_exponent
abbrev scalarL2_normedAddCommGroup (N : ℕ) (L : Length) :
NormedAddCommGroup (ScalarL2 N L) := inferInstance
attribute [local instance] scalarL2_normedAddCommGroup
abbrev scalarL2_addCommMonoid (N : ℕ) (L : Length) : AddCommMonoid (ScalarL2 N L) :=
(scalarL2_normedAddCommGroup N L).toAddCommMonoid
abbrev scalarL2_innerProductSpace (N : ℕ) (L : Length) :
InnerProductSpace ℂ (ScalarL2 N L) := inferInstance
attribute [local instance] scalarL2_innerProductSpace
abbrev scalarL2_module (N : ℕ) (L : Length) : Module ℂ (ScalarL2 N L) :=
(scalarL2_innerProductSpace N L).toNormedSpace.toModule
attribute [local instance] scalarL2_module
lemma scalarL2_completeSpace (N : ℕ) (L : Length) : CompleteSpace (ScalarL2 N L) :=
inferInstance
end HardSphere
end
end
end
end OAI
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