Thermodynamic Entropy eq shannon Entropy
CanonicalEnsemble.thermodynamicEntropy_eq_shannonEntropy
Plain-language statement
In the finite, nonempty case the thermodynamic and Shannon entropies coincide. All semi-classical correction factors vanish (dof = 0, phaseSpaceUnit = 1), so the absolute thermodynamic entropy reduces to the discrete Shannon form.
Exact Lean statement
theorem thermodynamicEntropy_eq_shannonEntropy [MeasurableSingletonClass ฮน] [IsFinite ๐]
(T : Temperature) :-- (hT : 0 < T.val) :
๐.thermodynamicEntropy T = ๐.shannonEntropy TFormal artifact
Lean source
theorem thermodynamicEntropy_eq_shannonEntropy [MeasurableSingletonClass ฮน] [IsFinite ๐] (T : Temperature) :-- (hT : 0 < T.val) : ๐.thermodynamicEntropy T = ๐.shannonEntropy T := by have h_thermo_eq_diff : ๐.thermodynamicEntropy T = ๐.differentialEntropy T := by unfold CanonicalEnsemble.thermodynamicEntropy CanonicalEnsemble.differentialEntropy have h_log : (fun i => Real.log (๐.physicalProbability T i)) = (fun i => Real.log (๐.probability T i)) := by funext i simp [CanonicalEnsemble.physicalProbability, IsFinite.dof_eq_zero (๐:=๐), IsFinite.phase_space_unit_eq_one (๐:=๐)] simp_all only [physicalProbability_def] have h_shannon : ๐.shannonEntropy T = ๐.differentialEntropy T := (shannonEntropy_eq_differentialEntropy (๐:=๐) T) calc ๐.thermodynamicEntropy T = ๐.differentialEntropy T := h_thermo_eq_diff _ = ๐.shannonEntropy T := h_shannon.symm- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/StatisticalMechanics/CanonicalEnsemble/Finite.lean:284-305
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