| import Mathlib |
| import OAI.Analysis.HardSphere.FormDomain.RmsPermuteFunBosonic |
| import OAI.Analysis.HardSphere.Collisions.IntegrableCompInsertNth |
|
|
| 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 |
| open MeasureTheory Filter Set |
|
|
| lemma rayleigh_larger {n m : ℕ} {L : Length} {a c : ℝ} (hn : 0 < n) (hnm : n ≤ m) |
| (h : ∀ u : Jet n L, u∈dirichletDomain n L a → c*(n:ℝ)*‖u.1‖^2≤energy u) : |
| ∀ v : Jet m L, v∈dirichletDomain m L a → c*(m:ℝ)*‖v.1‖^2≤energy v := by |
| induction m, hnm using Nat.le_induction with |
| | base => exact h |
| | succ m hnm ih => exact rayleigh_successor (hn.trans_le hnm) ih |
|
|
| lemma IsGroundVector.rayleigh_larger {n m : ℕ} {L : Length} {a : ℝ} |
| {u : Jet n L} (hu : IsGroundVector n L a u) (hn : 0 < n) (hnm : n ≤ m) |
| (v : Jet m L) (hv : v∈dirichletDomain m L a) : |
| (energy u/(n:ℝ))*(m:ℝ)*‖v.1‖^2≤energy v := by |
| apply HardSphere.rayleigh_larger hn hnm (c := energy u/(n:ℝ)) ?_ v hv |
| intro w hw |
| rw [div_mul_cancel₀ _ (by exact_mod_cast hn.ne')] |
| exact hu.rayleigh_all w hw |
|
|
| lemma IsGroundVector.energy_per_particle_mono {n m : ℕ} {L : Length} {a : ℝ} |
| {u : Jet n L} {v : Jet m L} (hu : IsGroundVector n L a u) |
| (hv : IsGroundVector m L a v) (hn : 0 < n) (hnm : n ≤ m) : |
| energy u/(n:ℝ)≤energy v/(m:ℝ) := by |
| have h := hu.rayleigh_larger hn hnm v hv.1 |
| rw [hv.2.2.1,pow_two,mul_one,mul_one] at h |
| exact (le_div_iff₀ (by exact_mod_cast hn.trans_le hnm)).mpr h |
|
|
| end HardSphere |
|
|
| end |
|
|
| end OAI |
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|
