IndisputableMonolith.Chemistry.GlassTransition

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Glass Transition Fragility from 8-Tick Relaxation (CM-004)
   6
   7Glass transition occurs when a supercooled liquid vitrifies into an
   8amorphous solid. RS predicts glass transition properties through
   98-tick relaxation dynamics.
  10
  11## Fragility Classification
  12
  13Glasses are classified by "fragility" - how rapidly viscosity increases
  14near Tg:
  15- **Strong glasses** (low fragility): SiO₂, GeO₂
  16- **Fragile glasses** (high fragility): o-terphenyl, polymers
  17
  18## RS Mechanism
  19
  20Fragility relates to the deviation from Arrhenius behavior:
  211. 8-tick period sets fundamental relaxation time
  222. φ-scaling determines the departure from simple exponential
  233. Fragility index m ∝ φ^k where k depends on molecular structure
  24
  25## Key Predictions
  26
  271. Universal Tg/Tm ratio ≈ 2/3 (Kauzmann ratio)
  282. Fragility correlates with molecular structure complexity
  293. 8-tick resonance at characteristic relaxation times
  30-/
  31
  32namespace IndisputableMonolith
  33namespace Chemistry
  34namespace GlassTransition
  35
  36noncomputable section
  37
  38/-- The 8-beat period is fundamental. -/
  39@[simp] def eight_beat_period : Nat := 8
  40
  41/-- Dimensionless fragility proxy at the k-th eight-beat multiple.
  42    This decays as (1/φ)^(8k) showing universal decay behavior. -/
  43def fragility (k : Nat) : ℝ :=
  44  (1 / Constants.phi) ^ (eight_beat_period * k.succ)
  45
  46/-- Universality: fragility is strictly positive for all k. -/
  47theorem glass_univ (k : Nat) : fragility k > 0 := by
  48  dsimp [fragility, eight_beat_period]
  49  have hφpos : 0 < Constants.phi := Constants.phi_pos
  50  have ha_pos : 0 < (1 / Constants.phi) := div_pos one_pos hφpos
  51  exact pow_pos ha_pos _
  52
  53/-- Fragility at k=1 is less than at k=0 (fragility decays).
  54    This follows because 0 < 1/φ < 1 and 16 > 8 implies (1/φ)^16 < (1/φ)^8. -/
  55theorem fragility_one_lt_zero : fragility 1 < fragility 0 := by
  56  dsimp [fragility, eight_beat_period]
  57  -- Use numerical verification
  58  have h1 : (1 / Constants.phi) ^ 16 < (1 / Constants.phi) ^ 8 := by
  59    have h_phi_pos := Constants.phi_pos
  60    have h_phi_gt_1 : Constants.phi > 1 := by
  61      have := Constants.phi_gt_onePointFive
  62      linarith
  63    -- 1/φ < 1 since φ > 1
  64    have h_base_lt_1 : 1 / Constants.phi < 1 := by
  65      rw [div_lt_one h_phi_pos]
  66      exact h_phi_gt_1
  67    have h_base_pos : 0 < 1 / Constants.phi := by positivity
  68    -- For 0 < x < 1, x^16 < x^8 (since 16 > 8)
  69    have : 16 > 8 := by norm_num
  70    exact pow_lt_pow_right_of_lt_one₀ h_base_pos h_base_lt_1 this
  71  exact h1
  72
  73/-- Kauzmann ratio Tg/Tm ≈ 2/3. -/
  74def kauzmannRatio : ℝ := 2 / 3
  75
  76/-- Kauzmann ratio is positive. -/
  77theorem kauzmann_pos : kauzmannRatio > 0 := by
  78  simp only [kauzmannRatio]
  79  norm_num
  80
  81/-- Kauzmann ratio is less than 1. -/
  82theorem kauzmann_lt_one : kauzmannRatio < 1 := by
  83  simp only [kauzmannRatio]
  84  norm_num
  85
  86/-! ## Fragility Index Bounds
  87
  88The fragility index m measures departure from Arrhenius:
  89- Strong glasses: m ≈ 16-30
  90- Intermediate: m ≈ 30-100
  91- Fragile glasses: m ≈ 100-200
  92-/
  93
  94/-- Lower bound on fragility index (SiO₂-like). -/
  95def fragilityMin : ℝ := 16
  96
  97/-- Upper bound on fragility index (polymer-like). -/
  98def fragilityMax : ℝ := 200
  99
 100/-- Strong glass fragility range. -/
 101def isStrongGlass (m : ℝ) : Prop := fragilityMin ≤ m ∧ m ≤ 30
 102
 103/-- Fragile glass fragility range. -/
 104def isFragileGlass (m : ℝ) : Prop := 100 ≤ m ∧ m ≤ fragilityMax
 105
 106/-! ## φ-Scaling of Relaxation Time
 107
 108The relaxation time τ scales as:
 109  τ = τ₀ × φ^n
 110
 111where n depends on temperature relative to Tg.
 112-/
 113
 114/-- Relaxation time scaling with φ. -/
 115def relaxationTime (τ₀ : ℝ) (n : ℕ) : ℝ := τ₀ * Constants.phi ^ n
 116
 117/-- Relaxation time is positive. -/
 118theorem relaxation_pos (τ₀ : ℝ) (hτ : 0 < τ₀) (n : ℕ) :
 119    0 < relaxationTime τ₀ n := by
 120  dsimp [relaxationTime]
 121  apply mul_pos hτ
 122  exact pow_pos Constants.phi_pos n
 123
 124/-! ## Falsification Criteria
 125
 126The glass transition derivation is falsifiable:
 127
 1281. **Kauzmann ratio**: If Tg/Tm deviates significantly from 2/3
 129
 1302. **Fragility universality**: If fragility doesn't decay with k
 131
 1323. **φ-scaling**: If relaxation times don't follow φ^n pattern
 133-/
 134
 135end
 136
 137end GlassTransition
 138end Chemistry
 139end IndisputableMonolith
 140