Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Intensive equilibrium conditions for coexisting phases

Physical scope

Finite coexisting phase branches in positive-temperature C2 charts at an attained global entropy maximum, and only the energy, volume, or species transfers declared pairwise feasible in both directions.

Statement

Let $P \geq 1$ named phase branches of the same finite species list occupy a separated compound at an attained global entropy-maximizing endpoint with the fixed totals and exact feasible set declared by Phase branches and coexistence domains and Isolated compound and feasible equilibrium set. Assume each phase lies in an open C2 single-phase chart with finite positive temperature. For every pair of phases, assume the following about the feasible set at this endpoint: each named exchange channel permits a sufficiently small nonzero amount of transfer in both directions, while all other state coordinates may be held fixed.

Then, for every pair $α, β$ :

if energy exchange is permitted, $T_{α} = T_{β}$ ;
if volume exchange is permitted, $p_{α} / T_{α} = p_{β} / T_{β}$ , and if energy exchange is also permitted then $p_{α} = p_{β}$ ;
if species $j$ is transferable, $μ_{α, j} / T_{α} = μ_{β, j} / T_{β}$ , and if energy exchange is also permitted then $μ_{α, j} = μ_{β, j}$ .

The pressure and chemical potentials are the derivative-defined state functions of Entropic temperature, pressure, and molar chemical potentials. The theorem makes no equality claim for a retained coordinate, a one-sided boundary exchange, or a channel absent from the declared feasible set. It is a model deduction, not evidence that these phases or exchange channels occur in nature.

Facts & Assumptions

Given: A finite number of named phase branches, a positive-mole composition chart, an attained global entropy maximum on the exact feasible set, C2 entropy on each phase chart, finite positive temperatures, additive separated-system coordinates and entropy, and the bilateral local feasibility of each exchange direction used below. Use SI units.

[A1]

Stable equilibrium requires an attained global entropy maximum: A stable endpoint attains a global maximum of total entropy on its exact feasible endpoint set; local maxima alone are not called stable equilibrium.

[F1]

Phase branches and coexistence domains and Isolated compound and feasible equilibrium set: Phase amounts and states lie in the declared branch charts and exact feasible set; only the listed totals and exchanges may vary.

[F2]

Additive separated systems and tagged bulk scaling and Concave macroscopic fundamental entropy: For the specified separated compound, energy, volume, aligned species amounts, and entropy add; on each selected single-phase chart the entropy is finite and C2 with $S_{U} > 0$ .

[F3]

Entropic temperature, pressure, and molar chemical potentials: $S_{U} = 1 / T$ , $S_{V} = p / T$ , and $S_{n_{j}} = - μ_{j} / T$ , each partial holding the other physical coordinates fixed.

[L1]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : The derivative of a differentiable composition is the composition of its derivatives.

Proof

1.1givenA1F1F2L1construct

Fix any pair $α, β$ and an allowed energy transfer. For any fixed nonzero increment $h$ joules, bilateral local feasibility supplies the curve $U_{α} (t) = U_{α}^{*} + t h$ , $U_{β} (t) = U_{β}^{*} - t h$ for all sufficiently small $t$ of either sign, with all other coordinates fixed. The total entropy along this curve has a local maximum at $t = 0$ by [A1]. Its derivative exists by [F2]; the right difference quotients are nonpositive and the left difference quotients are nonnegative, so the derivative is zero. By additivity and [L1], this derivative is $h (S_{α, U} - S_{β, U})$ . Since $h \neq 0$ , $S_{α, U} = S_{β, U}$ .

2.1givenF3step 1.1algebra

By [F3], $S_{α, U} = 1 / T_{α}$ and $S_{β, U} = 1 / T_{β}$ . Both temperatures are finite and positive, so equality of the reciprocals gives $T_{α} = T_{β}$ .

3.1givenA1F1F2F3L1step 2.1construct

If volume transfer is permitted, take a nonzero increment $k$ in $m^{3}$ and vary $V_{α} (t) = V_{α}^{*} + t k$ , $V_{β} (t) = V_{β}^{*} - t k$ with both signs feasible. The same local-maximum quotient argument gives $k (S_{α, V} - S_{β, V}) = 0$ , hence $S_{α, V} = S_{β, V}$ . By [F3], $p_{α} / T_{α} = p_{β} / T_{β}$ . When energy transfer is also permitted, step 2.1 gives equal positive temperatures, so $p_{α} = p_{β}$ . These are derivative pressures; a mechanical piston interpretation requires its separately declared coupling.

3.2givenA1F1F2F3L1step 2.1construct

If species $j$ is transferable, vary its mole amounts by $n_{α, j} (t) = n_{α, j}^{*} + t q$ and $n_{β, j} (t) = n_{β, j}^{*} - t q$ for a nonzero increment $q$ in moles. Both signs are feasible. The same quotient argument gives $q (S_{α, n_{j}} - S_{β, n_{j}}) = 0$ , hence equal entropy partials. By [F3], $μ_{α, j} / T_{α} = μ_{β, j} / T_{β}$ . If energy transfer is also permitted, step 2.1 gives equal positive temperatures and therefore equal chemical potentials.

4.1givenstep 1.1step 2.1step 3.1step 3.2∎

Every pair was arbitrary, so the asserted pairwise equalities hold for all named phases and only for the channels whose bilateral feasibility was assumed. If $P = 1$ , there are no pairs and the conditions are vacuous. No derivative equality follows for a retained or boundary coordinate because its required two-sided feasible curve is absent.

Intensive equilibrium conditions for coexisting phases

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources