`evolve_equilibrium_eq`

proved

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module: IndisputableMonolith.Thermodynamics.SecondLaw
domain: Thermodynamics
line: 163 · github
papers citing: none yet

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IndisputableMonolith.Thermodynamics.SecondLaw on GitHub at line 163.

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formal source

 160
 161omit [Nonempty Ω] in
 162/-- Equilibrium is a fixed point of the iterated evolution. -/
 163theorem evolve_equilibrium_eq {peq : ProbabilityDistribution Ω}
 164    (R : JDescentOperator peq) (n : ℕ) :
 165    R.evolve n peq = peq := by
 166  induction n with
 167  | zero => rfl
 168  | succ n ih =>
 169      simp [evolve_succ, ih, R.fixes_equilibrium]
 170
 171end JDescentOperator
 172
 173/-! ## §3. KL is monotone non-increasing along the evolution -/
 174
 175omit [Nonempty Ω] in
 176/-- Single-step KL contraction: `D_KL(qₙ₊₁ ‖ peq) ≤ D_KL(qₙ ‖ peq)`. -/
 177theorem kl_step_le {peq : ProbabilityDistribution Ω}
 178    (R : JDescentOperator peq) (q₀ : ProbabilityDistribution Ω) (n : ℕ) :
 179    kl_divergence (R.evolve (n + 1) q₀).p peq.p ≤
 180      kl_divergence (R.evolve n q₀).p peq.p := by
 181  rw [JDescentOperator.evolve_succ]
 182  exact R.kl_non_increasing _
 183
 184omit [Nonempty Ω] in
 185/-- Pointwise KL contraction across a `n ≤ m` interval. -/
 186theorem kl_le_of_le {peq : ProbabilityDistribution Ω}
 187    (R : JDescentOperator peq) (q₀ : ProbabilityDistribution Ω)
 188    {n m : ℕ} (hnm : n ≤ m) :
 189    kl_divergence (R.evolve m q₀).p peq.p ≤
 190      kl_divergence (R.evolve n q₀).p peq.p := by
 191  induction hnm with
 192  | refl => exact le_refl _
 193  | step _ ih => exact le_trans (kl_step_le R q₀ _) ih

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