`evolve_zero`

proved

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module: IndisputableMonolith.Thermodynamics.SecondLaw
domain: Thermodynamics
line: 152 · github
papers citing: none yet

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IndisputableMonolith.Thermodynamics.SecondLaw on GitHub at line 152.

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formal source

 149  | n + 1, q => R.step (R.evolve n q)
 150
 151omit [Nonempty Ω] in
 152@[simp] lemma evolve_zero {peq : ProbabilityDistribution Ω}
 153    (R : JDescentOperator peq) (q : ProbabilityDistribution Ω) :
 154    R.evolve 0 q = q := rfl
 155
 156omit [Nonempty Ω] in
 157@[simp] lemma evolve_succ {peq : ProbabilityDistribution Ω}
 158    (R : JDescentOperator peq) (n : ℕ) (q : ProbabilityDistribution Ω) :
 159    R.evolve (n + 1) q = R.step (R.evolve n q) := rfl
 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 _

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