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EVMPR theorem twoState_helmholtzFreeEnergy_eq #1474

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jstoobysmith merged 3 commits into from
Aug 1, 2026
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EVMPR theorem twoState_helmholtzFreeEnergy_eq #1474

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EVMPR twoState_helmholtzFreeEnergy_eq for T \geq 0. This can be simpl…
korolrom Jul 31, 2026
0dcf58c
Condensed the proof into fewer lines based on PR feedback
korolrom Jul 31, 2026
75d4153
Line 119 too long. Split it into 2
korolrom Jul 31, 2026
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26 changes: 23 additions & 3 deletions Physlib/StatisticalMechanics/CanonicalEnsemble/TwoState.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -104,8 +104,28 @@ informal_lemma twoState_entropy_eq where
deps := [``twoState, ``thermodynamicEntropy]

/-- A simplification of the `helmholtzFreeEnergy` of the two-state canonical ensemble. -/
informal_lemma twoState_helmholtzFreeEnergy_eq where
tag := "EVMPR"
deps := [``twoState]
lemma twoState_helmholtzFreeEnergy_eq (E₀ E₁ : ℝ) (T : Temperature) :
(twoState E₀ E₁).helmholtzFreeEnergy T =
(β T * (E₀ + E₁) / 2 - Real.log
(2 * Real.cosh (β T * (E₁ - E₀) / 2))) / β T := by
set x := β T * (E₁ - E₀) / 2
set C := β T * (E₀ + E₁) / 2
have hE0 : -β T * E₀ = x +(- C) := by
simp [x, C]
ring
have hE1 : -β T * E₁ = -x + (- C) := by
simp [x, C]
ring
rw [helmholtzFreeEnergy, twoState_partitionFunction_apply,
show (T.val : ℝ) = T.toReal by rfl,hE0, hE1]
have hfactor :
Real.exp (x + (- C)) + Real.exp (-x + (- C)) = Real.exp (-C) * (2 * Real.cosh x) := by
rw [Real.exp_add, Real.exp_add, Real.cosh_eq]
ring
rw [hfactor, Real.log_mul
(Real.exp_pos _).ne'
(by positivity : 2 * Real.cosh x ≠ 0)]
simp only [Real.log_exp,Temperature.β_toReal,div_eq_mul_inv, one_mul, inv_inv]
ring

end CanonicalEnsemble
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