Thermodynamics is the paradigm example in physics of a time-asymmetric theory, but the origin of the asymmetry lies deeper than the second law. A primordial arrow can be defined by the way of the equilibration principle (“minus first law”). By appealing to this arrow, the nature of the well-known ambiguity in Carathéodory's 1909 version of the second law becomes clear. Following Carathéodory's seminal work, formulations of thermodynamics have gained ground that highlight the role of the binary relation of adiabatic accessibility between equilibrium states, the most prominent recent example being the important 1999 axiomatization due to Lieb and Yngvason. This formulation can be shown to contain an ambiguity strictly analogous to that in Carathéodory's treatment.

## References

See Ref. 4.

See Ref. 7, Sec. 9.

This is the way the postulate is commonly stated in the literature. Carathéodory's actual axioms was (p. 236) “In every arbitrarily close neighborhood of a given initial state there exist states that cannot be approached arbitrarily closely by adiabatic processes.”^{12}

More specifically, quasi-static adiabatic processes are defined as those in which the difference between the work done externally and the limit of the energy change defined when the derivatives of the thermodynamic parameters converge uniformly to zero is less than experimental uncertainty. It is noteworthy that Carathéodory does not define quasi-static in the general case.

Reference 12, Sec. 9. The recourse to experiment is further justified in Constantin Carathédory, “Über die Bestimmung der Energie und der absoluten Temperatur mit Hilfe von reversiblen Prozessen,” Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse 1 (1925), pp. 39–47; a discussion of historical responses to this feature of his theory is found in Ref. 7.

This is related to the local nature of Frobenius' Theorem. See Ref. 21, pp. 183, 184. Note that the energy ambiguity is again resolved by appeal to experience on p. 186.

Uffink has argued that the definition of adiabatic accessibility therein is subtly different from most definitions in the literature—and in particular that of Carathéodory; see Ref. 7, p. 381 *ff*. We need not dwell on this point.

The first incentive strikes us as unnecessary, given that Carathéodory had shown that heat is not fundamental, and that it is perfectly well-defined given his first law and the conservation of energy, both of which feature in the L-Y approach. Their claim (p. 6) that no one “has ever seen heat, nor will it ever be seen, smelled or touched”^{11} can also be said to apply to entropy! The second incentive also strikes us as curious, particularly because the authors themselves stress (Sec. G) that if irreversible processes did not exist, “it would mean that nothing is forbidden, and hence there would be no second law.”^{11}

Here we part company with Ref. 11, p. 24, where it is claimed that if all states are adiabatically equivalent, entropy is constant.

Reference 11, p. 44. There are two notions of universality involved here, and Lieb and Yngvason are anticipating two theorems. The first is Theorem 3.3, which states that the forward sectors *of all states* in the state space of a simple system (i.e., the set of states adiabatically accessible from the given state) point the same way energy-wise. The second is Theorem 4.2, which states that the forward sectors *of all simple systems* point the same way. It is worth emphasizing that the proofs of these theorems depend on several axioms additional to the set A1-A6 needed for the entropy principle. The content of one of these additional axioms, A7, is discussed in Sec. V D below.

This is Theorem 3.4, Ref. 11, p. 45.

See, for example, both Papers in Ref. 15.

For many systems, the axiom is a direct consequence of the equilibration principle, which implies that for any two samples of a gas in states *X*_{1} and *X*_{2}, when you partition the samples as just described and then connect the two partitioned regions, the resulting system will eventually reach an equilibrium state *Y*. This process can be done adiabatically, and by conservation of energy state *Y* will have energy $U=tU1+(1\u2212t)U2$. By assumption, the state also has volume $V=tV1+(1\u2212t)V2$. Compare with p. 31 in Ref. 11.

From Sec. III onwards in Ref. 11, the state space for a simple system is taken to be a convex subset of $R(n+1)$, n being the number of deformation coordinates.

Orthogonality is defined with respect to the canonical scalar product on $R(n+1)$.

The role of A7 in determining the directionality of processes in the L-Y approach is also pointed out and explained by Henderson in Ref. 13, though in a different way to our own.

See Theorem 5.4, Ref. 11, pp. 66–67.

See Ref. 11, pp. 44 and 62.

Compare the discussion of the axiom *T*_{1} of thermal contact in Ref. 11, p. 52.

We are grateful to Jakob Yngvason for clarification of this point.

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