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Project-declaredLean 4.32.0 ยท mathlib@81a5d257c8e4

Helmholtz Free Energy eq mean Energy sub temp mul thermodynamic Entropy

CanonicalEnsemble.helmholtzFreeEnergy_eq_meanEnergy_sub_temp_mul_thermodynamicEntropy

physicsquantum field theoryrelativitylemmas

Project documentation

The Helmholtz free energy F is related to the mean energy U and the absolute thermodynamic entropy S by the identity F = U - TS. This theorem shows that the statistically-defined quantities in this framework correctly satisfy this principle of thermodynamics.

Exact Lean statement

theorem helmholtzFreeEnergy_eq_meanEnergy_sub_temp_mul_thermodynamicEntropy
    (T : Temperature) (hT : 0 < T.val)
    [IsFiniteMeasure (๐“’.ฮผBolt T)] [NeZero ๐“’.ฮผ]
    (hE : Integrable ๐“’.energy (๐“’.ฮผProd T)) :
    ๐“’.helmholtzFreeEnergy T
      = ๐“’.meanEnergy T - T.val * ๐“’.thermodynamicEntropy T

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem helmholtzFreeEnergy_eq_meanEnergy_sub_temp_mul_thermodynamicEntropy    (T : Temperature) (hT : 0 < T.val)    [IsFiniteMeasure (๐“’.ฮผBolt T)] [NeZero ๐“’.ฮผ]    (hE : Integrable ๐“’.energy (๐“’.ฮผProd T)) :    ๐“’.helmholtzFreeEnergy T      = ๐“’.meanEnergy T - T.val * ๐“’.thermodynamicEntropy T := by  have hTne : (T.val : โ„) โ‰  0 := by exact_mod_cast hT.ne'  have hkฮฒT : T.val * (kB * (T.ฮฒ : โ„)) = 1 := by    rw [kB_mul_beta T hT, mul_one_div, div_self hTne]  rw [helmholtzFreeEnergy_def, log_partitionFunction,    ๐“’.thermodynamicEntropy_eq_differentialEntropy_sub_correction (T := T) hE,    ๐“’.differentialEntropy_eq_kB_beta_meanEnergy_add_kB_log_mathZ (T := T) hE]  linear_combination ๐“’.meanEnergy T * hkฮฒT
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean:173-185

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Plain-language statement

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physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

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Project-declaredLean 4.32.0

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adiabatic_relation_UaUbVaVb

Plain-language statement

Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record