| import Mathlib |
| import OAI.Analysis.HardSphere.FormDomain.RawCutoffGradientIntegral |
| import OAI.Analysis.HardSphere.FormDomain.RmsPermuteFunBosonic |
|
|
| namespace OAI |
|
|
| 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 |
|
|
| lemma iterated_insertion (C : ℝ) (_hC : 0<C) |
| (hins : ∀ (n : ℕ) (L : Length) (r : ℝ), 0<r → |
| (n:ℝ)*(2*Real.pi*r)^3≤L.val^3/2 → |
| ∀ u : Jet n L, u∈dirichletDomain n L r → |
| ∃ v : Jet (n+1) L, v∈dirichletDomain (n+1) L r ∧ |
| ‖v.1‖=‖u.1‖ ∧ energy v≤energy u+(C*(n:ℝ)*r/L.val^3)*‖u.1‖^2) |
| (n m : ℕ) (L : Length) {r : ℝ} (hr : 0<r) |
| (hs : ((n+m:ℕ):ℝ)*(2*Real.pi*r)^3≤L.val^3/2) |
| (u : Jet n L) (hu : u∈dirichletDomain n L r) : |
| ∃ v : Jet (n+m) L, v∈dirichletDomain (n+m) L r ∧ ‖v.1‖=‖u.1‖ ∧ |
| energy v≤energy u+(∑ j∈Finset.range m,C*((n+j:ℕ):ℝ)*r/L.val^3)*‖u.1‖^2 := by |
| induction m with |
| | zero => exact ⟨u,hu,rfl,by simp⟩ |
| | succ m ih => |
| have hs' : ((n+m:ℕ):ℝ)*(2*Real.pi*r)^3≤L.val^3/2 := by |
| exact (mul_le_mul_of_nonneg_right |
| (show ((n+m:ℕ):ℝ)≤((n+(m+1):ℕ):ℝ) by exact_mod_cast Nat.le_succ (n+m)) |
| (by positivity)).trans hs |
| obtain ⟨v,hv,hn,he⟩ := ih hs' |
| obtain ⟨w,hw,hnw,hew⟩ := hins (n+m) L r hr hs' v hv |
| refine ⟨w,hw,hnw.trans hn,?_⟩ |
| rw [hn] at hew |
| rw [Finset.sum_range_succ] |
| nlinarith |
|
|
| lemma ground_energy_removal : ∃ C : ℝ, 0<C ∧ |
| ∀ (n m : ℕ) (L : Length) (r : ℝ), 0<r → |
| ((n+m:ℕ):ℝ)*(2*Real.pi*r)^3≤L.val^3/2 → |
| ∀ u : Jet (n+m) L, IsGroundVector (n+m) L r u → |
| ∀ v : Jet n L, v∈dirichletDomain n L r → |
| (energy u-C*(m:ℝ)*((n+m:ℕ):ℝ)*r/L.val^3)*‖v.1‖^2≤energy v := by |
| obtain ⟨C,hC,hins⟩ := uniform_insertion |
| refine ⟨C,hC,fun n m L r hr hs u hu v hv ↦ ?_⟩ |
| have hL : 0<L.val := L.property |
| obtain ⟨w,hw,hn,he⟩ := iterated_insertion C hC hins n m L hr hs v hv |
| have hmin := hu.rayleigh_all w hw |
| rw [hn] at hmin |
| have hsum : (∑ j∈Finset.range m,C*((n+j:ℕ):ℝ)*r/L.val^3)≤ |
| C*(m:ℝ)*((n+m:ℕ):ℝ)*r/L.val^3 := by |
| calc |
| _ ≤ ∑ _j∈Finset.range m,C*((n+m:ℕ):ℝ)*r/L.val^3 := by |
| apply Finset.sum_le_sum |
| intro j hj |
| apply div_le_div_of_nonneg_right _ (by positivity) |
| apply mul_le_mul_of_nonneg_right _ hr.le |
| apply mul_le_mul_of_nonneg_left _ hC.le |
| exact_mod_cast Nat.add_le_add_left (Nat.le_of_lt (Finset.mem_range.mp hj)) n |
| _ = _ := by simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]; ring |
| have := mul_le_mul_of_nonneg_right hsum (sq_nonneg ‖v.1‖) |
| nlinarith |
| end HardSphere |
|
|
| end |
|
|
| end OAI |
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|
