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67aee1d | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | 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
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