Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-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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Ideal-bath heat-engine efficiency bound

Physical scope

Conditional efficiency bound for an admitted closed engine cycle between positive-temperature ideal baths inside an isolated compound with an ideal work store; finite reservoirs, unlisted transfers, and empirical engine performance are excluded.

Statement

Let an admitted heat-engine cycle in the sense of Ideal-reservoir thermodynamic cycles operate between fixed ideal-bath temperatures $0 < T_{c} < T_{h}$ , absorb $Q_{h} > 0$ , reject $Q_{c} > 0$ , and deliver $W_{o u t} > 0$ . Define its efficiency by $η = W_{o u t} / Q_{h}$ . Then $η \leq 1 - \frac{T_{c}}{T_{h}} .$ Equality holds exactly when the total entropy change of the isolated compound is zero. In particular, a reversible cycle in the stated boundary model saturates the bound; zero entropy change is not here asserted to imply full reversibility.

Facts & Assumptions

Given: A cyclic working system with $0 < T_{c} < T_{h}$ , $Q_{h} > 0$ , $Q_{c} > 0$ , no unlisted transfers, and the full working system, baths, and work store inside the declared isolated compound.

[F1]

Ideal-reservoir thermodynamic cycles: In engine mode $Q [P] = Q_{h} - Q_{c}$ , $W_{o n} [P] = - W_{o u t}$ , the hot and cold bath energy changes are $- Q_{h}$ and $+ Q_{c}$ , their endpoint entropy changes are $- Q_{h} / T_{h}$ and $+ Q_{c} / T_{c}$ , the working system returns to its initial state, and the work store gains $Δ E_{w} = W_{o u t}$ with zero entropy change. The admitted total-compound endpoint tuples contain those same working-system endpoints and bath/store energy coordinates, lie in one feasible set for the second-law postulate, and use the declared additive entropy split.

[F2]

First-law energy balance for closed process records: For an admitted closed fixed-constitution process with zero endpoint changes in separately tracked center-of-mass kinetic and external-potential energies, $Δ U = Q [P] + W_{o n} [P]$ .

[F3]

Macroscopic second law for isolated compounds: For an admitted isolated compound process with equilibrium endpoints and the declared additive total entropy, $Δ S_{t o t} \geq 0$ ; an admitted reversible operation has $Δ S_{t o t} = 0$ .

Proof

Given: The fixed-bath cycle hypotheses in the Statement and the heat/work signs in [F1].

1.1givenF1F2algebra

The working system returns to its initial state, so $Δ U = 0$ . Applying [F2] with [F1] gives $0 = Q_{h} - Q_{c} - W_{o u t}$ , hence $W_{o u t} = Q_{h} - Q_{c}$ and $η = 1 - Q_{c} / Q_{h}$ .

1.2givenF1F3algebra

The hot bath loses energy $Q_{h}$ , the cold bath gains $Q_{c}$ , and their entropy slopes are constant. Thus their entropy changes are $- Q_{h} / T_{h}$ and $Q_{c} / T_{c}$ . The working system and work store have zero entropy change, so [F3] gives $- \frac{Q_{h}}{T_{h}} + \frac{Q_{c}}{T_{c}} = Δ S_{t o t} \geq 0.$

2.1givenstep 1.1step 1.2algebra

Since $Q_{h}, T_{c}, T_{h} > 0$ , step 1.2 implies $Q_{c} / Q_{h} \geq T_{c} / T_{h}$ . Substituting into step 1.1 yields $η \leq 1 - T_{c} / T_{h}$ . Equality in this algebraic bound holds exactly when $Δ S_{t o t} = 0$ .

3.1F3step 2.1∎

If the full admitted cycle is reversible, [F3] gives $Δ S_{t o t} = 0$ , so step 2.1 is saturated. No converse about reversibility follows from the entropy calculation.

Ideal-bath heat-engine efficiency bound

Statement

Facts & Assumptions

Proof

Depends on

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