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.

1.
Accelerating clocks differ from inertial ones in the sense that their behavior generally depends on their constitution; see
Harvey R.
Brown
, “
The role of rods and clocks in general relativity and the meaning of the metric field
,” e-print arXiv:0911.4440v1 [gr-qc] (
2009
).
2.
Stephen
Hawking
, “
The No Boundary Condition And The Arrow Of Time
,” in
Physical Origins of Time Asymmetry
, edited by
J. J.
Halliwell
,
J.
Pérez-Mercador
, and
W. H.
Zurek
(
Cambridge U.P.
,
Cambridge
1994
), pp.
346
357
; see p. 348.
3.
This point was stressed by Hans Reichenbach in Chapter II, Sec. 5 of his 1956 book
The Direction of Time
, edited by
Maria
Reichenbach
(
Dover
,
Mineola, New York
,
1999
).
4.
Truesdell claims that all pioneers of thermodynamics regarded time as primitive, in
Clifford A.
Truesdell
and
Subramanyam
Bharatha
,
The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines: Rigorously Constructed Upon the Foundation Laid by S. Carnot and F. Reech
(
Springer-Verlag
,
New York
,
1977
), p.
viii
.
5.
Owen J. E.
Maroney
, “
Does a computer have an arrow of time?
,”
Found. Phys.
40
,
205
238
(
2010
).
6.
Ryan
Smith
, “
Do brains have an arrow of time?
,”
Philos. Sci.
81
,
265
275
(
2014
). An earlier identification of the psychological arrow of time with the thermodynamic arrow, based on the brain as (inter alia) a “registering instrument,” is found in Hans Reichenbach, “Les Fondements logiques de la mechanic de quanta,” Annales de l'Institut Henri Poincaré, Tome XIII. Fasc. II (Paris, 1953), pp. 156–157. English translations of relevant paragraphs are published in the Appendix to Reichenbach's 1956 book, Ref. 3. Although Reichenbach associated the thermodynamic arrow with non-decreasing entropy, it seems to us that the argument has more to do with the equilibration principle, which we are about to discuss.
7.
Jos
Uffink
, “
Bluff your way in the second law of thermodynamics
,”
Stud. Hist. Philos. Mod. Phys.
32
(
3
),
305
394
(
2001
).
8.
For the origin of both these titles, see
Harvey R.
Brown
and
Jos
Uffink
, “
The origins of time asymmetry in thermodynamics: The minus first law
,”
Stud. His. Philos. Mod. Phys.
32
(
4
),
525
538
(
2001
). This Paper contains a detailed account of the equilibration principle, the emphasis it has received in the literature, and its role in the foundations of thermodynamics.
9.
Meir
Hemmo
and
Orly R.
Shenker
,
The Road to Maxwell's Demon: Conceptual Foundations of Statistical Mechanics
(
Cambridge U.P.
,
Cambridge
,
2012
), Chap. 2.
10.

See Ref. 4.

11.
Elliott H.
Lieb
and
Jakob
Yngvason
, “
The physics and mathematics of the second law of thermodynamics
,”
Phys. Rep.
310
,
1
96
(
1999
).
The 1999 online version of this Paper, to which our page numbers refer, is at arXiv:cond-mat/9708200v2. A shorter, more informal version of this Paper is found in
Elliott H.
Lieb
and
Jakob
Yngvason
, “
A fresh look at entropy and the second law of thermodynamics
,”
Phys. Today
53
(
4
),
32
37
(
2000
).
12.
Constantin
Carathédory
, “
Untersuchungen über die Grundlagen der Thermodynamik
,”
Math. Ann.
67
,
355
386
(
1909
).
English translation by
Joseph
Kestin
: “
Investigation into the Foundations of Thermodynamics
,” in
the Second Law of Thermodynamics: Benchmark Papers on Energy
, edited by
Joseph
Kestin
(
Dowden, Hutchinson and Ross
,
Stroudsberg, Pennsylvania
,
1976
), Vol. 5, pp.
229
256
. Page numbers refer to the translation.
13.
See Ref. 7, Sec. 11. A critique of this analysis and a related one in Ref. 8 is found in
Leah
Henderson
, “
Can the second law be compatible with time reversal invariant dynamics?
,”
Stud. His. Philos. Mod. Phys.
47
,
90
98
(
2014
).
14.
See, for example,
J. B.
Boyling
, “
An axiomatic approach to classical thermodynamics
,”
Proc. R. Soc. London, Sec A
329
,
35
70
(
1972
), pp. 42–43.
15.
See, for example,
Louis A.
Turner
, “
Simplification of Carathéodory's treatment of thermodynamics. II
,”
Am. J. Phys.
30
,
506
508
(
1962
),
and
Francis W.
Sears
, “
Modified form of Carathéodory's second axiom
,”
Am. J. Phys.
34
,
665
666
(
1966
).
16.

See Ref. 7, Sec. 9.

17.
See Ref. 7. A stronger analogy is with the geometric treatment of the conformal structure of special relativistic space-time found in
A. A.
Robb
,
The Absolute Relations of Time and Space
(
Cambridge U.P.
,
Cambridge
,
1921
);
see p. 144 of
Jos
Uffink
, “
Irreversibility and the second law of thermodynamics
,” in
Entropy
, edited by
Andreas
Greven
,
Gerhard
Keller
, and
Gerald
Warnecke
(
Princeton U.P.
,
New Jersey
,
2003
), pp.
121
146
.
18.

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 

19.

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.

20.
Turner has argued that the zeroth law is not strictly necessary in Carathéodory's treatment. See
Louis A.
Turner
, “
Temperature and Carathéodory's treatment of thermodynamics
,”
J. Chem. Phys.
38
,
1163
1167
(
1963
).
21.
A modern geometric treatment of Carathéodory's Theorem in the context of the Frobenius Theorem and nonholonomic constraints is found in
Theodore
Frankel
,
The Geometry of Physics
, 2nd ed. (
Cambridge U.P.
,
New York
,
2004
); see Secs. 6.3d–6.3f.
22.
See, for example,
Peter T.
Landsberg
,
Thermodynamics and Statistical Mechanics
(
Dover Publications, Inc.
,
New York
,
1990
), Sec. 5.2;
Louis A.
Turner
, “
Simplification of Carathéodory's treatment of thermodynamics
,”
Am. J. Phys.
28
,
781
786
(
1960
);
and
Francis W.
Sears
, “
A simplified simplification of Carathéodory's treatment of thermodynamics
,”
Am. J. Phys.
31
(
10
),
747
752
(
1963
).
23.
For a discussion of these assumptions, see
Peter T.
Landsberg
, “
On suggested simplifications of Carathéodory's thermodynamics
,”
Phys. Status Solidi B
1
(
2
),
120
126
(
1961
).
24.

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.

25.
See
Peter T.
Landsberg
, “
Foundations of thermodynamics
,”
Rev. Mod. Phys.
28
,
363
392
(
1956
)
and
J.
Dunning-Davies
, “
Carathéodory's principle and the Kelvin statement of the second law
,”
Nature
208
,
576
577
(
1965
).
26.
See
Arthur E.
Ruark
, “
LXIII. The proof of the corollary of Carnot's Theorem
,”
Philos. Mag. Series 6
49
(
291
),
584
585
(
1925
).
27.
It is remarkable that the connection between the Kelvin-Planck formulation of the second law and Carathédory's inaccessibility principle was clarified only in the 1960s (independently) by
B.
Crawford
, Jr.
and
I.
Oppenheim
, “
The second law of thermodynamics
,”
J. Chem. Phys.
34
(
5
),
1621
1623
(
1961
),
and
Peter T.
Landsberg
, “
A deduction of Carathéodory's principle from Kelvin's principle
,”
Nature
201
,
485
486
(
1964
); see also Dunning-Davies in Ref. 25.
28.
B.
Bernstein
, “
Proof of Carathéodory's local Theorem and its global application to thermostatics
,”
J. Math. Phys.
1
,
222
224
(
1960
). See also Ref. 7, pp. 367–368.
29.

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.

30.

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.

31.

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 

32.

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

33.

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.

34.

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

35.

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

36.

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 X1 and X2, 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+(1t)U2. By assumption, the state also has volume V=tV1+(1t)V2. Compare with p. 31 in Ref. 11.

37.

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.

38.

For a discussion of the continuity of such an adiabat, see Ref. 11, p. 42. Figure 1 is a special case of Fig. 3 in Ref. 11, pp. 32 and 96.

39.

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

40.

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.

41.

See Theorem 5.4, Ref. 11, pp. 66–67.

42.

See Ref. 11, pp. 44 and 62.

43.

Compare the discussion of the axiom T1 of thermal contact in Ref. 11, p. 52.

44.
This equation is a consequence of the definition of pressure and a number of continuity assumptions in the L-Y formulation; see pp. 40–41 in Ref. 11. Note that the minus sign is omitted in Eq. (3.6) therein; this is not the case in
Elliott H.
Lieb
and
Jakob
Yngvason
, “
A guide to entropy and the second law of thermodynamics
,” e-print arXiv:math-ph/9805005v1 (
1998
): see Eq. (14).
45.
For an introductory explanation of thermodynamic stability conditions, see
Frederick
Reif
,
Fundamentals of Statistical and Thermal Physics
(
McGraw-Hill
,
Boston
,
1965
).
46.

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

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