Differential Entropy eq k B beta mean Energy add k B log math Z
CanonicalEnsemble.differentialEntropy_eq_kB_beta_meanEnergy_add_kB_log_mathZ
Plain-language statement
General identity: S_diff = kB β ⟨E⟩ + kB log Z_math. This connects the differential entropy to the mean energy and the mathematical partition function. Integrability of log (probability …) follows from the pointwise formula.
Exact Lean statement
lemma differentialEntropy_eq_kB_beta_meanEnergy_add_kB_log_mathZ
(T : Temperature) [IsFiniteMeasure (𝓒.μBolt T)] [NeZero 𝓒.μ]
(hE : Integrable 𝓒.energy (𝓒.μProd T)) :
𝓒.differentialEntropy T = kB * (T.β : ℝ) * 𝓒.meanEnergy T +
kB * Real.log (𝓒.mathematicalPartitionFunction T)Formal artifact
Lean source
lemma differentialEntropy_eq_kB_beta_meanEnergy_add_kB_log_mathZ (T : Temperature) [IsFiniteMeasure (𝓒.μBolt T)] [NeZero 𝓒.μ] (hE : Integrable 𝓒.energy (𝓒.μProd T)) : 𝓒.differentialEntropy T = kB * (T.β : ℝ) * 𝓒.meanEnergy T + kB * Real.log (𝓒.mathematicalPartitionFunction T) := by have h_log_prob : ∀ i, Real.log (𝓒.probability T i) = -(T.β : ℝ) * 𝓒.energy i - Real.log (𝓒.mathematicalPartitionFunction T) := fun i => by rw [probability, Real.log_div (Real.exp_pos _).ne' (mathematicalPartitionFunction_pos 𝓒 T).ne', Real.log_exp] simp only [differentialEntropy, meanEnergy] rw [integral_congr_ae (ae_of_all _ h_log_prob), integral_sub (hE.const_mul _) (integrable_const _), integral_const_mul, integral_const] simp only [probReal_univ, smul_eq_mul] ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean:84-97
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