| import Mathlib |
| import OAI.Analysis.HardSphere.Model |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| noncomputable section |
| open scoped BigOperators ComplexConjugate |
| namespace HardSphere.Variational |
| variable {H J : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [Fintype J] |
| |
| def energy (u : H × (J → H)) : ℝ := ∑ i, ‖u.2 i‖ ^ 2 |
|
|
| def gradientPairing (u v : H × (J → H)) : ℂ := ∑ i, inner ℂ (u.2 i) (v.2 i) |
|
|
| omit [InnerProductSpace ℂ H] in |
| lemma energy_nonneg (u : H × (J → H)) : 0 ≤ energy u := |
| Finset.sum_nonneg fun _ _ ↦ sq_nonneg _ |
|
|
| lemma energy_smul (c : ℂ) (u : H × (J → H)) : |
| energy (c • u) = ‖c‖ ^ 2 * energy u := by |
| simp [energy, norm_smul, mul_pow, Finset.mul_sum] |
| |
| theorem rayleigh (M : Submodule ℂ (H × (J → H))) (u : H × (J → H)) |
| (hmin : ∀ v ∈ M, ‖v.1‖ = 1 → energy u ≤ energy v) |
| (v : H × (J → H)) (hv : v ∈ M) : |
| energy u * ‖v.1‖ ^ 2 ≤ energy v := by |
| by_cases hz : ‖v.1‖ = 0 |
| · simpa [hz] using energy_nonneg v |
| · let c : ℂ := ((‖v.1‖ : ℂ))⁻¹ |
| have hc : ‖c‖ = ‖v.1‖⁻¹ := by simp [c, Complex.norm_real] |
| have hv' : ‖(c • v).1‖ = 1 := by |
| simp only [Prod.smul_fst, norm_smul, hc] |
| exact inv_mul_cancel₀ hz |
| have h := hmin (c • v) (M.smul_mem c hv) hv' |
| rw [energy_smul, hc] at h |
| calc |
| energy u * ‖v.1‖ ^ 2 ≤ (‖v.1‖⁻¹ ^ 2 * energy v) * ‖v.1‖ ^ 2 := |
| mul_le_mul_of_nonneg_right h (sq_nonneg _) |
| _ = energy v := by field_simp |
|
|
| lemma norm_add_real_smul_sq (x y : H) (t : ℝ) : |
| ‖x + (t : ℂ) • y‖ ^ 2 = |
| ‖x‖ ^ 2 + 2 * t * (inner ℂ x y).re + t ^ 2 * ‖y‖ ^ 2 := by |
| rw [norm_add_sq (𝕜 := ℂ), inner_smul_right, norm_smul, mul_pow] |
| simp [Complex.norm_real, sq_abs, mul_assoc] |
| |
| lemma energy_add_real_smul (u v : H × (J → H)) (t : ℝ) : |
| energy (u + (t : ℂ) • v) = |
| energy u + 2 * t * (gradientPairing u v).re + t ^ 2 * energy v := by |
| change (∑ i, ‖u.2 i + (t : ℂ) • v.2 i‖ ^ 2) = _ |
| simp_rw [norm_add_real_smul_sq] |
| simp only [energy, gradientPairing, Finset.sum_add_distrib, Finset.mul_sum, |
| Complex.re_sum] |
|
|
| lemma linear_term_zero {a b : ℝ} (ha : 0 ≤ a) |
| (h : ∀ t : ℝ, 0 ≤ a * t ^ 2 + 2 * b * t) : b = 0 := by |
| have hd : 0 < a + 1 := by linarith |
| have hh := mul_nonneg (h (-b / (a + 1))) (sq_nonneg (a + 1)) |
| have heq : (a * (-b / (a + 1)) ^ 2 + 2 * b * (-b / (a + 1))) * |
| (a + 1) ^ 2 = -(a + 2) * b ^ 2 := by |
| field_simp |
| ring |
| rw [heq] at hh |
| have hb := sq_nonneg b |
| have hprod := mul_nonneg ha hb |
| nlinarith |
|
|
| theorem ground_pairing (M : Submodule ℂ (H × (J → H))) |
| (u : H × (J → H)) (hu : u ∈ M) (hnorm : ‖u.1‖ = 1) |
| (hmin : ∀ v ∈ M, ‖v.1‖ = 1 → energy u ≤ energy v) |
| (v : H × (J → H)) (hv : v ∈ M) : |
| gradientPairing u v = (energy u : ℂ) * inner ℂ u.1 v.1 := by |
| have hre (w : H × (J → H)) (hw : w ∈ M) : |
| (gradientPairing u w).re = energy u * (inner ℂ u.1 w.1).re := by |
| have ha : 0 ≤ energy w - energy u * ‖w.1‖ ^ 2 := |
| sub_nonneg.mpr (rayleigh M u hmin w hw) |
| have hpoly (t : ℝ) : 0 ≤ |
| (energy w - energy u * ‖w.1‖ ^ 2) * t ^ 2 + |
| 2 * ((gradientPairing u w).re - energy u * (inner ℂ u.1 w.1).re) * t := by |
| have h := rayleigh M u hmin (u + (t : ℂ) • w) (M.add_mem hu (M.smul_mem _ hw)) |
| rw [energy_add_real_smul] at h |
| simp only [Prod.fst_add, Prod.smul_fst, norm_add_real_smul_sq, hnorm, one_pow] at h |
| nlinarith |
| exact sub_eq_zero.mp (linear_term_zero ha hpoly) |
| have hI := hre (Complex.I • v) (M.smul_mem _ hv) |
| have hr := hre v hv |
| apply Complex.ext |
| · simpa using hr |
| · simpa [gradientPairing, inner_smul_right, ← Finset.mul_sum, Complex.mul_re, |
| Complex.I_re, Complex.I_im, Complex.mul_im] using hI |
|
|
| end HardSphere.Variational |
|
|
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
