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EVJJI theorem twoState_entropy_eq - #1605

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Sep 2, 2026
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EVJJI theorem twoState_entropy_eq#1605
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@oe-parks oe-parks commented Sep 1, 2026

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Formalizes the informal lemma tagged EVJJI: a closed form for the
thermodynamic entropy of the two-state canonical ensemble.

Added, both in Physlib/StatisticalMechanics/CanonicalEnsemble/TwoState.lean:

  • twoState_entropy_eq with x = β T * (E₁ - E₀) / 2, gives
    S = kB * (log (2 * cosh x) - x * tanh x). Proved through
    thermodynamicEntropy_eq_shannonEntropy, which carries no side conditions,
    rather than through the Helmholtz free energy. That route needs
    0 < T.val, IsFiniteMeasure, and integrability.

  • twoState_entropy_eq_T_neq_zero is the same statement carrying T ≠ 0, so
    downstream results on positive temperatures get a matching signature without
    having to know the unconditional version is safe at zero.

Removed the Physlib.Meta.Informal.Basic import, now unused in the file.

I left the primary result unconditional, matching the choice made for
twoState_helmholtzFreeEnergy_eq in #1474. Since β 0 = 0, the RHS is
kB * log 2 at T = 0; the docstring notes this.

Notes to reviewer: look first at the statement, since that's the part that has to be right.
Then hp0 and hp1, the two occupation probabilities in closed form. The rest
is algebra. The corollary and the import removal are a line each.

Second contribution to Physlib. Happy to adjust anything. I'm not particularly attached to the T ≠ 0 restatement if you'd rather it weren't there.

Formalize the informal lemma tagged EVJJI: a closed form for the
thermodynamic entropy of the two-state canonical ensemble,

  S = kB * (log (2 * cosh x) - x * tanh x),  where x = beta T * (E1 - E0) / 2

proved via `thermodynamicEntropy_eq_shannonEntropy`, which carries no side
conditions, rather than through the Helmholtz free energy.

Also adds `twoState_entropy_eq_T_neq_zero`, a restatement carrying `T != 0`
so downstream results on positive temperatures get a matching signature, and
drops the `Physlib.Meta.Informal.Basic` import, now unused in this file.

Co-authored-by: Claude Opus 5 <no-reply+claude-opus-5@anthropic.com>
@github-actions github-actions Bot added the small label Sep 1, 2026
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github-actions Bot commented Sep 1, 2026

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@github-actions github-actions Bot added the t-statistical-mechanics Statistical mechanics label Sep 1, 2026

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Approved - looks good to me. Thanks for this PR

@jstoobysmith jstoobysmith added the ready-to-merge This PR is approved and will be merged shortly label Sep 2, 2026
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jstoobysmith merged commit 170a753 into leanprover-community:master Sep 2, 2026
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