Fluctuation dissipation theorem finite
CanonicalEnsemble.fluctuation_dissipation_theorem_finite
Plain-language statement
FDT for finite canonical ensembles: C_V = Var(E) / (k_B Tยฒ).
Exact Lean statement
theorem fluctuation_dissipation_theorem_finite
[MeasurableSingletonClass ฮน] [๐.IsFinite] (T : Temperature) (hT_pos : 0 < T.val) :
๐.heatCapacity T = ๐.energyVariance T / (kB * (T.val : โ)^2)Formal artifact
Lean source
theorem fluctuation_dissipation_theorem_finite [MeasurableSingletonClass ฮน] [๐.IsFinite] (T : Temperature) (hT_pos : 0 < T.val) : ๐.heatCapacity T = ๐.energyVariance T / (kB * (T.val : โ)^2) := by have hฮฒโ_pos : 0 < (T.ฮฒ : โ) := beta_pos T hT_pos let ฮฒโ := (T.ฮฒ : โ) have h_diff_U_beta : DifferentiableWithinAt โ ๐.meanEnergyBeta (Set.Ioi 0) ฮฒโ := by have h_eq_on : Set.EqOn ๐.meanEnergyBeta ๐.meanEnergyBetaReal (Set.Ioi 0) := by intro b hb; exact meanEnergy_Beta_eq_finite ๐ b hb have h_diff' := (differentiable_meanEnergyBetaReal ๐) (T.ฮฒ : โ) exact DifferentiableWithinAt.congr_of_eventuallyEq h_diff'.differentiableWithinAt (eventuallyEq_nhdsWithin_of_eqOn h_eq_on) (h_eq_on hฮฒโ_pos) have h_Var_eq_neg_dUdฮฒ := derivWithin_meanEnergy_Beta_eq_neg_variance ๐ T hT_pos exact CanonicalEnsemble.fluctuation_dissipation_energy_parametric ๐ T hT_pos (by simp_all only [NNReal.coe_pos, neg_neg, ฮฒโ]) h_diff_U_beta- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/StatisticalMechanics/CanonicalEnsemble/Finite.lean:474-487
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Person-level attribution pending.