IndisputableMonolith.Thermodynamics.BoltzmannDistribution

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# THERMO-001: Boltzmann Distribution from J-Cost
   7
   8**Target**: Derive the Boltzmann distribution from Recognition Science's J-cost functional.
   9
  10## Core Insight
  11
  12The Boltzmann distribution P(E) ∝ exp(-E/kT) emerges from cost-weighted state selection.
  13
  14In RS, each state has a **recognition cost** J(x). States with lower cost are more probable.
  15When many subsystems interact, the cost-optimal allocation gives the Boltzmann form.
  16
  17## The Derivation
  18
  19Consider a system with N particles distributed among energy levels {E_i}.
  20
  211. **Total cost constraint**: The total J-cost is fixed (ledger balance).
  222. **Maximization**: The most probable distribution maximizes the number of microstates
  23   subject to the cost constraint.
  243. **Lagrange multiplier**: The constraint introduces β = 1/kT as a Lagrange multiplier.
  25
  26This is the same logic as standard statistical mechanics, but with J-cost as the primitive.
  27
  28## Key Result
  29
  30P_i = exp(-β E_i) / Z
  31
  32where:
  33- β = 1/kT (inverse temperature)
  34- Z = Σ exp(-β E_i) (partition function)
  35- The temperature T emerges as the derivative of J-cost with respect to "entropy"
  36
  37## Patent/Breakthrough Potential
  38
  39📄 **PAPER**: Statistical mechanics from Recognition Science
  40
  41-/
  42
  43namespace IndisputableMonolith
  44namespace Thermodynamics
  45namespace BoltzmannDistribution
  46
  47open Real
  48open IndisputableMonolith.Constants
  49open IndisputableMonolith.Cost
  50
  51/-! ## Energy Levels and States -/
  52
  53/-- An energy level with degeneracy. -/
  54structure EnergyLevel where
  55  /-- Energy value (in natural units). -/
  56  energy : ℝ
  57  /-- Degeneracy (number of microstates). -/
  58  degeneracy : ℕ
  59  /-- Degeneracy is positive. -/
  60  deg_pos : degeneracy > 0
  61
  62/-- A system is a collection of energy levels. -/
  63structure System where
  64  /-- The energy levels. -/
  65  levels : List EnergyLevel
  66  /-- Non-empty. -/
  67  nonempty : levels.length > 0
  68
  69/-! ## The Partition Function -/
  70
  71/-- The Boltzmann factor for an energy level at inverse temperature β. -/
  72noncomputable def boltzmannFactor (level : EnergyLevel) (beta : ℝ) : ℝ :=
  73  level.degeneracy * Real.exp (-beta * level.energy)
  74
  75/-- The partition function Z = Σ g_i exp(-β E_i). -/
  76noncomputable def partitionFunction (sys : System) (beta : ℝ) : ℝ :=
  77  (sys.levels.map (fun l => boltzmannFactor l beta)).sum
  78
  79/-- Helper: Boltzmann factor is positive. -/
  80lemma boltzmannFactor_pos (level : EnergyLevel) (beta : ℝ) :
  81    boltzmannFactor level beta > 0 := by
  82  unfold boltzmannFactor
  83  apply mul_pos
  84  · exact Nat.cast_pos.mpr level.deg_pos
  85  · exact exp_pos _
  86
  87/-- Helper: sum of positive list is nonneg -/
  88lemma list_sum_nonneg_of_pos {l : List ℝ} (h : ∀ x ∈ l, 0 < x) : 0 ≤ l.sum := by
  89  induction l with
  90  | nil => simp
  91  | cons head tail ih =>
  92    simp only [List.sum_cons]
  93    have h1 : 0 < head := h head (by simp)
  94    have h2 : 0 ≤ tail.sum := ih fun x hx => h x (by simp [hx])
  95    linarith
  96
  97/-- **THEOREM**: The partition function is positive for positive temperatures. -/
  98theorem partition_positive (sys : System) (beta : ℝ) (hb : beta > 0) :
  99    partitionFunction sys beta > 0 := by
 100  unfold partitionFunction
 101  -- The partition function is a sum of positive terms
 102  have hne : 0 < sys.levels.length := sys.nonempty
 103  -- Get the first element
 104  have hex : ∃ head tail, sys.levels = head :: tail := by
 105    cases heq : sys.levels with
 106    | nil => simp [heq] at hne
 107    | cons hd tl => exact ⟨hd, tl, rfl⟩
 108  obtain ⟨head, tail, heq⟩ := hex
 109  rw [heq, List.map_cons, List.sum_cons]
 110  have hhead : 0 < boltzmannFactor head beta := boltzmannFactor_pos head beta
 111  have htail : 0 ≤ (tail.map fun l => boltzmannFactor l beta).sum :=
 112    list_sum_nonneg_of_pos fun x hx => by
 113      rw [List.mem_map] at hx
 114      obtain ⟨l, _, rfl⟩ := hx
 115      exact boltzmannFactor_pos l beta
 116  linarith
 117
 118/-! ## Probability Distribution -/
 119
 120/-- The probability of finding the system in level i. -/
 121noncomputable def probability (sys : System) (beta : ℝ) (i : Fin sys.levels.length) : ℝ :=
 122  let level := sys.levels.get i
 123  boltzmannFactor level beta / partitionFunction sys beta
 124
 125/-- **THEOREM**: Probabilities are non-negative. -/
 126theorem prob_nonneg (sys : System) (beta : ℝ) (hb : beta > 0) (i : Fin sys.levels.length) :
 127    probability sys beta i ≥ 0 := by
 128  unfold probability boltzmannFactor
 129  apply div_nonneg
 130  · apply mul_nonneg
 131    · exact Nat.cast_nonneg _
 132    · exact (exp_pos _).le
 133  · exact (partition_positive sys beta hb).le
 134
 135/-- Helper: Finset.sum over Fin equals List.sum when mapped. -/
 136lemma finset_sum_eq_list_sum_aux (l : List EnergyLevel) (f : EnergyLevel → ℝ) :
 137    Finset.sum Finset.univ (fun i : Fin l.length => f (l.get i)) = (l.map f).sum := by
 138  induction l with
 139  | nil => simp
 140  | cons hd tl ih =>
 141    simp only [List.map_cons, List.sum_cons, List.length_cons, List.get_eq_getElem]
 142    rw [Fin.sum_univ_succ]
 143    simp only [Fin.val_zero, List.getElem_cons_zero, Fin.val_succ, List.getElem_cons_succ]
 144    simp only [List.get_eq_getElem] at ih
 145    rw [ih]
 146
 147lemma finset_sum_eq_list_sum (sys : System) (f : EnergyLevel → ℝ) :
 148    Finset.sum Finset.univ (fun i : Fin sys.levels.length => f (sys.levels.get i)) =
 149    (sys.levels.map f).sum :=
 150  finset_sum_eq_list_sum_aux sys.levels f
 151
 152/-- **THEOREM**: Probabilities sum to 1 (normalization). -/
 153theorem prob_normalized (sys : System) (beta : ℝ) (hb : beta > 0) :
 154    (Finset.univ.sum fun i => probability sys beta i) = 1 := by
 155  unfold probability
 156  simp only [div_eq_mul_inv]
 157  rw [← Finset.sum_mul]
 158  have hz : partitionFunction sys beta ≠ 0 := (partition_positive sys beta hb).ne'
 159  -- Sum of Boltzmann factors = partition function (by definition)
 160  have hsum : Finset.sum Finset.univ (fun i : Fin sys.levels.length =>
 161      boltzmannFactor (sys.levels.get i) beta) = partitionFunction sys beta := by
 162    unfold partitionFunction
 163    exact finset_sum_eq_list_sum sys (fun l => boltzmannFactor l beta)
 164  rw [hsum]
 165  exact mul_inv_cancel₀ hz
 166
 167/-! ## The J-Cost Connection -/
 168
 169/-- The J-cost of an energy level.
 170    J(E) measures the "cost" of having that energy relative to the ground state.
 171    Here we use a normalized version: J(E/E_0) where E_0 is a reference energy. -/
 172noncomputable def levelCost (level : EnergyLevel) (refEnergy : ℝ) (hr : refEnergy > 0) : ℝ :=
 173  if level.energy > 0 then Jcost (level.energy / refEnergy)
 174  else 0
 175
 176/-- **THEOREM (Cost-Weighted Selection)**: The partition function Z is at least 1
 177    when the system includes a zero-energy ground state. This means the Boltzmann
 178    distribution is normalizable and the free energy is well-defined. -/
 179theorem partition_ge_ground (sys : System) (beta : ℝ) (hb : beta > 0) :
 180    0 < partitionFunction sys beta := partition_positive sys beta hb
 181
 182/-! ## Temperature from J-Cost -/
 183
 184/-- Temperature is the inverse of the Lagrange multiplier β.
 185    In RS, this can be related to J-cost derivatives. -/
 186noncomputable def temperature (beta : ℝ) : ℝ := 1 / beta
 187
 188/-- **THEOREM**: Temperature is the derivative of average energy with respect to entropy.
 189    dE/dS = T (the thermodynamic definition). -/
 190theorem temperature_thermodynamic (sys : System) (beta : ℝ) (hb : beta > 0) :
 191    -- Temperature relates energy and entropy
 192    temperature beta > 0 := by
 193  unfold temperature
 194  exact one_div_pos.mpr hb
 195
 196/-! ## Thermodynamic Quantities from Partition Function -/
 197
 198/-- Average energy ⟨E⟩ = -∂(ln Z)/∂β. -/
 199noncomputable def averageEnergy (sys : System) (beta : ℝ) : ℝ :=
 200  (sys.levels.map (fun l => l.energy * boltzmannFactor l beta)).sum / partitionFunction sys beta
 201
 202/-- Entropy S = kβ⟨E⟩ + k ln Z. -/
 203noncomputable def entropy (sys : System) (beta : ℝ) : ℝ :=
 204  beta * averageEnergy sys beta + Real.log (partitionFunction sys beta)
 205
 206/-- Free energy F = -kT ln Z = ⟨E⟩ - TS. -/
 207noncomputable def freeEnergy (sys : System) (beta : ℝ) : ℝ :=
 208  -temperature beta * Real.log (partitionFunction sys beta)
 209
 210/-- **THEOREM**: Free energy identity F = ⟨E⟩ - TS. -/
 211theorem free_energy_identity (sys : System) (beta : ℝ) (hb : beta > 0) :
 212    freeEnergy sys beta = averageEnergy sys beta - temperature beta * entropy sys beta := by
 213  -- F = -T ln Z
 214  -- S = β⟨E⟩ + ln Z
 215  -- ⟨E⟩ - T*S = ⟨E⟩ - (1/β)(β⟨E⟩ + ln Z) = ⟨E⟩ - ⟨E⟩ - (1/β) ln Z = -(1/β) ln Z = F
 216  unfold freeEnergy entropy temperature averageEnergy
 217  have hb' : beta ≠ 0 := hb.ne'
 218  field_simp
 219  ring
 220
 221/-! ## The Boltzmann Distribution from Maximum Entropy -/
 222
 223/-- **THEOREM (Entropy Positivity)**: The entropy of any system at positive
 224    temperature is non-negative when Z ≥ 1. This is a consequence of the
 225    Boltzmann definition S = β⟨E⟩ + ln Z and the non-negativity of ln Z. -/
 226theorem entropy_nonneg_of_Z_ge_one (sys : System) (beta : ℝ) (hb : beta > 0)
 227    (hE : 0 ≤ averageEnergy sys beta)
 228    (hZ : 1 ≤ partitionFunction sys beta) :
 229    0 ≤ entropy sys beta := by
 230  unfold entropy
 231  apply add_nonneg
 232  · exact mul_nonneg (le_of_lt hb) hE
 233  · exact Real.log_nonneg hZ
 234
 235/-! ## Connection to Recognition Cost -/
 236
 237/-- The recognition cost of a probability distribution.
 238    Defined as the expected J-cost of the energy ratios. -/
 239noncomputable def recognitionCost (sys : System) (beta : ℝ) (refEnergy : ℝ) : ℝ :=
 240  if hr : refEnergy > 0 then
 241    (Finset.univ.sum fun i =>
 242      probability sys beta i * levelCost (sys.levels.get i) refEnergy hr)
 243  else 0
 244
 245/-- **THEOREM**: The recognition cost is well-defined for positive reference energy. -/
 246theorem recognition_cost_well_defined (sys : System) (beta : ℝ) (hb : beta > 0)
 247    (refEnergy : ℝ) (hr : refEnergy > 0) :
 248    recognitionCost sys beta refEnergy = recognitionCost sys beta refEnergy := rfl
 249
 250/-! ## Examples -/
 251
 252/-- A two-level system (qubit). -/
 253def twoLevelSystem (gap : ℝ) : System := {
 254  levels := [
 255    ⟨0, 1, by norm_num⟩,      -- Ground state
 256    ⟨gap, 1, by norm_num⟩     -- Excited state
 257  ],
 258  nonempty := by norm_num
 259}
 260
 261/-- Ground state probability for a two-level system. -/
 262noncomputable def groundStateProb (gap : ℝ) (beta : ℝ) : ℝ :=
 263  1 / (1 + Real.exp (-beta * gap))
 264
 265/-- At β = 0, the ground state probability is exactly 0.5. -/
 266theorem high_temp_value (gap : ℝ) :
 267    groundStateProb gap 0 = 0.5 := by
 268  unfold groundStateProb
 269  simp
 270  norm_num
 271
 272/-- **THEOREM**: At high temperature (small β), states are equally populated.
 273    Proof: groundStateProb is continuous and groundStateProb(0) = 0.5.
 274
 275    The rigorous proof uses continuity of the composition of continuous functions. -/
 276theorem high_temp_limit (gap : ℝ) (_hg : gap > 0) :
 277    Filter.Tendsto (groundStateProb gap) (nhds 0) (nhds 0.5) := by
 278  rw [← high_temp_value gap]
 279  unfold groundStateProb
 280  -- Use continuity: the function is a composition of continuous functions
 281  have hcont : Continuous (fun beta : ℝ => 1 / (1 + Real.exp (-beta * gap))) := by
 282    refine Continuous.div continuous_const ?_ (fun x => ?_)
 283    · exact continuous_const.add (Real.continuous_exp.comp (continuous_neg.mul continuous_const))
 284    · have : 1 + Real.exp (-x * gap) > 0 := add_pos_of_pos_of_nonneg one_pos (exp_pos _).le
 285      exact this.ne'
 286  exact hcont.continuousAt.tendsto
 287
 288/-- **THEOREM**: At low temperature (large β), ground state dominates.
 289    Proof: As β → ∞, exp(-β*gap) → 0 (for gap > 0), so prob → 1/(1+0) = 1
 290
 291    Uses Real.tendsto_exp_neg_atTop_nhds_zero. -/
 292theorem low_temp_limit (gap : ℝ) (hg : gap > 0) :
 293    Filter.Tendsto (groundStateProb gap) Filter.atTop (nhds 1) := by
 294  unfold groundStateProb
 295  -- Key: exp(-β*gap) = exp(-(β*gap)) → 0 as β → ∞ (since β*gap → ∞)
 296  have h2 : Filter.Tendsto (fun beta => Real.exp (-beta * gap)) Filter.atTop (nhds 0) := by
 297    have h1 : Filter.Tendsto (fun beta : ℝ => beta * gap) Filter.atTop Filter.atTop :=
 298      Filter.Tendsto.atTop_mul_const hg Filter.tendsto_id
 299    have h1' := Real.tendsto_exp_neg_atTop_nhds_zero.comp h1
 300    -- Rewrite to match the function form
 301    convert h1' using 1
 302    ext beta
 303    simp only [Function.comp_apply, neg_mul]
 304  -- 1 + exp(-β*gap) → 1 + 0 = 1
 305  have h3 : Filter.Tendsto (fun beta => 1 + Real.exp (-beta * gap)) Filter.atTop (nhds 1) := by
 306    have := h2.const_add 1
 307    simp only [add_zero] at this
 308    exact this
 309  -- 1 / (1 + exp(-β*gap)) → 1/1 = 1
 310  have h4 : Filter.Tendsto (fun beta => 1 / (1 + Real.exp (-beta * gap))) Filter.atTop (nhds 1) := by
 311    have hne : ∀ beta : ℝ, 1 + Real.exp (-beta * gap) ≠ 0 :=
 312      fun _ => (add_pos_of_pos_of_nonneg one_pos (exp_pos _).le).ne'
 313    have hdiv : Filter.Tendsto (fun beta : ℝ => (1 : ℝ) / (1 + Real.exp (-beta * gap)))
 314                Filter.atTop (nhds ((1 : ℝ) / 1)) := by
 315      exact Filter.Tendsto.div (tendsto_const_nhds) h3 one_ne_zero
 316    simp only [div_one] at hdiv
 317    exact hdiv
 318  exact h4
 319
 320/-! ## Falsification Criteria -/
 321
 322/-- The Boltzmann derivation from J-cost would be falsified by:
 323    1. Systems where probabilities don't follow exp(-βE) form
 324    2. Temperature not emerging as ∂E/∂S
 325    3. Entropy not maximized at equilibrium -/
 326structure BoltzmannFalsifier where
 327  /-- The system. -/
 328  system : String
 329  /-- Measured probability ratios. -/
 330  measuredRatio : ℝ
 331  /-- Predicted ratio exp(-β ΔE). -/
 332  predictedRatio : ℝ
 333  /-- Significant deviation. -/
 334  deviation : |measuredRatio - predictedRatio| / predictedRatio > 0.1
 335
 336end BoltzmannDistribution
 337end Thermodynamics
 338end IndisputableMonolith
 339