Heat Capacity eq deriv mean Energy Beta
CanonicalEnsemble.heatCapacity_eq_deriv_meanEnergyBeta
Plain-language statement
Relates C_V = dU/dT to dU/dβ. C_V = dU/dβ * (-1/(kB T²)).
Exact Lean statement
lemma heatCapacity_eq_deriv_meanEnergyBeta
(𝓒 : CanonicalEnsemble ι) (T : Temperature) (hT_pos : 0 < T.val)
(hU_deriv :
HasDerivWithinAt (𝓒.meanEnergyBeta)
(derivWithin (𝓒.meanEnergyBeta) (Set.Ioi 0) (T.β : ℝ))
(Set.Ioi 0) (T.β : ℝ)) :
𝓒.heatCapacity T
= (derivWithin (𝓒.meanEnergyBeta) (Set.Ioi 0) (T.β : ℝ))
* (-1 / (kB * (T.val : ℝ)^2))Formal artifact
Lean source
lemma heatCapacity_eq_deriv_meanEnergyBeta (𝓒 : CanonicalEnsemble ι) (T : Temperature) (hT_pos : 0 < T.val) (hU_deriv : HasDerivWithinAt (𝓒.meanEnergyBeta) (derivWithin (𝓒.meanEnergyBeta) (Set.Ioi 0) (T.β : ℝ)) (Set.Ioi 0) (T.β : ℝ)) : 𝓒.heatCapacity T = (derivWithin (𝓒.meanEnergyBeta) (Set.Ioi 0) (T.β : ℝ)) * (-1 / (kB * (T.val : ℝ)^2)) := by have h_U_eq_comp : (𝓒.meanEnergy_T) = fun t : ℝ => (𝓒.meanEnergyBeta) (betaFromReal t) := by funext t simp [meanEnergy_T, meanEnergyBeta, betaFromReal] have h_UD : UniqueDiffWithinAt ℝ (Set.Ioi (0 : ℝ)) (T.val : ℝ) := isOpen_Ioi.uniqueDiffWithinAt hT_pos unfold heatCapacity rw [h_U_eq_comp] exact (chain_rule_T_beta (F := 𝓒.meanEnergyBeta) T hT_pos hU_deriv).derivWithin h_UD- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean:421-437
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